Financial Management Workbench
A working study companion for Financial Management (09050440) at Nanjing University Business School. Built directly from Dr. Tao Yuan's lecture slides, rearranged along the syllabus agenda, and extended where the slides leave a concept hanging. Every worked number has been recomputed from the original slide data.
How this workbench is organised
the same five blocks in every lectureHow the lecture's ideas build on one another, drawn as a chain rather than a list. Cross-lecture links are marked on the map.
Explanatory prose with the slide's own worked examples taken apart step by step, plus the background the slides skip.
Every formula from the lecture in one place, with the meaning of each symbol. Printed on its own for revision.
The vocabulary, defined in English with a Chinese anchor so it maps onto the lecture you actually sat through.
Graded multiple choice plus step-by-step calculation problems, mirroring the final exam's three question types (concept explanation, single choice, subjective). Your score is stored in this browser, and the dashboard below tracks it lecture by lecture — so a week before the exam you can see exactly which lecture is still weak.
Reading the callouts
four kinds of aside, four coloursConnects a concept to another lecture. Financial management is one argument told in sequence — the discount rate that looks incidental in Lecture 2 turns out to be the cost of capital in Week 10, and the whole of Lecture 4 is Lecture 3's rule applied to messy real cash flows. These boxes are where the sequence shows.
Material the slides do not cover, taken from the textbook. Chapter numbers follow Corporate Finance, 11th ed. (the 31-chapter edition with Jaffe as third author) — the one 机械工业出版社 prints as 《公司理财（英文版·原书第11版）》, ISBN 978-7-111-58856-6. Note that Ross/Westerfield/Jordan also publish a different, 27-chapter Fundamentals of Corporate Finance whose chapter numbers do not line up; this workbench uses the 31-chapter numbering throughout.
Places where a method is easy to apply wrongly, or where something on a slide is slightly off. Twelve such errors were found while rebuilding these lectures — two wrong dates, a wrong citation year, a rounding slip in an annuity-due calculation, a rounding slip in a standard deviation, an intermediate-rounding artefact in a profitability-index table, two dropped digits in interest figures, a non-existent stock index, a dual-class structure attributed to the wrong company, a transposed digit in a bond's present value, and an efficient-frontier criterion stated against itself — and each is flagged where it occurs rather than quietly fixed. Three slides that are imprecise rather than wrong (城投债 labelled a municipal bond, a ratings boundary stated loosely, and a sentence that says interest affects operating cash flow) are flagged the same way.
The instructor's own discussion questions and quick questions, kept verbatim, with a worked answer below each.
Lectures
weeks 1–7, 10 and 11 delivered so far, plus two homeworksProgress dashboard
stored locally in this browserWhat the final exam asks for
per the syllabusConcept explanation 2 × 10 marks · Single choice 2 × 35 marks · Subjective question 5 × 2 marks. The self-check blocks in each lecture are built on exactly these three shapes, so that working through them is practice for the paper rather than for its own sake.
Groups of up to 6 pick a course-related topic not taught in class and present for 10–20 minutes: definition, real-world application, and the link back to this course. The syllabus suggests LPR, fintech, independent directors, ESG, generative AI. The Roadmap tab carries a shortlist of angles worth considering.
Overview & Financial Statements
Concept Map
how the lecture buildsNearly every later lecture feeds off something introduced here. The balance sheet gives you net working capital, which becomes ΔWC — a real cash outflow — in the free cash flow of Lecture 4. The income statement gives you EBIT and depreciation, two of the three terms in that same free cash flow formula. The discount rate that appears from Lecture 2 onward is the firm's financing cost, which is the cost of capital of Week 10. And the goal of the firm stated at the end of this lecture is the criterion against which every method in Lecture 3 is judged.
Two Cases, and the Problem They Share
slides 7–10Case 1 · A CEO paid ¥1 a year
On 7 August 2015, JD.com (NASDAQ: JD) announced that its chairman and CEO Richard Liu would take a base salary of ¥1 per year for ten years, with no cash bonus. How does someone with a one-yuan salary live?
The answer is that the salary was never the compensation. Alongside it, Liu was granted a very large block of share options. His payoff therefore comes from the share price, not from the wage bill. That single design choice aligns him with outside shareholders: if he destroys value, he personally loses far more than ¥1 a year. It is also a signal, because a manager who accepts a one-yuan salary is telling the market that he expects the equity to be worth a great deal.
Liu's package is a textbook answer to the agency problem — the conflict of interest between a firm's principals (shareholders) and its agents (managers). Ross Ch. 1 sets out three ways it is contained. First, compensation contracts that tie managerial wealth to shareholder wealth, of which options are the most direct example. Second, the market for corporate control: a badly run firm becomes cheap, and a takeover then replaces the management. Third, the board of directors, which can dismiss underperforming managers.
The costs of running these mechanisms — plus the value destroyed by managers who still act in their own interest — are called agency costs. They are a debit against the firm's value, and the whole point of governance is to keep them small.
Case 2 · The same company at two prices
On 13 August 2020, CanSino Biologics (康希诺, 688185) listed on the STAR Market. Billed as "China's first COVID-vaccine stock," it opened up 124% at ¥470 and closed at ¥393.11, a first-day gain of 87.45%. On the same day, its Hong Kong listing (06185.HK) closed at HK$198.10 — roughly half the A-share price.
The slide gives the listing date as 3 August 2020. The actual STAR Market listing was 13 August 2020 — the price data on the slide (opening at ¥470, up 124%, closing at ¥393.11) matches that day exactly, and the issue price was ¥209.71. The month is right and the day is a slip. This is the third such small error found in these four decks; the others are on slide 24 of Lecture 2 and slide 25 of Lecture 3.
One company, one set of assets, one stream of future cash flows, and two prices differing by a factor of two. Neither price is an error. The A and H markets are segmented: shares are not freely convertible between them, the investor bases differ, and the liquidity and sentiment differ. The law of one price — the assumption that the same asset cannot sell at two prices at once — fails exactly when the two markets cannot arbitrage against each other.
This is the course's first statement of what valuation means. A price is not a fact about the firm; it is the output of a market in which information is incomplete and capital does not flow freely. Weeks 5–6 (stock and bond valuation) and Week 12 (raising capital) are the systematic treatment of what this slide only gestures at, and the A/H spread reappears there as the pricing puzzle it actually is.
Who is who inside the corporation
Both cases are about the same structure. The lecture lays out three groups:
- Managers (职业经理人) — executives, the general manager or president, and the functional chiefs: CFO, HR director, and so on. They run the firm day to day.
- Shareholders (股东) — major and minor holders, state-owned asset commissions (国资委), mutual, hedge and pension funds, and individuals. They own the residual claim.
- Directors (董事) — the chairman, the board, independent directors, and executive directors. They sit between the two, with a duty to the firm.
The illustration used in the lecture is Alibaba's board page, which is worth reading as an artefact rather than as a picture: it lists both executive directors with operating roles inside the company and independent directors with no operating role — the structural feature that makes a board a check rather than a rubber stamp.
What the CFO actually does
The lecture poses this as an open question. In this course the CFO is the person who stands at the junction of the three decisions the module is built on: deciding which real assets to buy (capital budgeting), deciding how to pay for them (capital structure), and managing the short-term cash position (net working capital). The CFO does not "keep the books" — that is the controller's job. The CFO's question is forward-looking: what is this worth, and should we do it?
Financial analysts as intermediaries
Investors have limited time and limited professional skill to process the information firms emit. Analysts sit in the middle: they dig through filings, talk to management, build models, and sell their conclusions to investors. This is a straightforward answer to an information problem — someone specialises in reading, and sells the reading. It is also the reason the lecture insists that corporate finance knowledge is a prerequisite for the job: an analyst who cannot do the arithmetic in Lectures 2–4 has nothing to sell.
Where to find a financial report
- The company's own website, under Investor Relations
- Regulatory disclosure: the CSRC, the Shanghai and Shenzhen stock exchanges
- Data vendors: WIND (万得), 同花顺, Bloomberg, Thomson Reuters
- Timing: annual reports are released in March–April of the following year
The Three Statements
slides 11–18A financial statement gives investors information on the financial health of a company. The lecture works from a single consistent example — an unnamed firm, figures in millions — so that the three statements can be read against one another. The example is reproduced below in full, because every ratio and every later lecture depends on it.
Revenue is reported when it is earned, not when the cash arrives. A sale on 60-day credit is booked today and the cash shows up two months later. This single convention is why the income statement and the cash flow statement tell different stories about the same firm, and why net income is not cash. The whole of Lecture 4 rests on taking that difference seriously.
The balance sheet (资产负债表)
The balance sheet reflects the financial position of a company at one point in time — a photograph, not a film. Assets are listed in order of how long it would normally take a going concern to convert them into cash: cash first, then receivables, then inventory, then fixed assets. Cash is far more liquid than a factory.
| Balance sheet ($m) | 2020 | 2019 | Balance sheet ($m) | 2020 | 2019 |
|---|---|---|---|---|---|
| Cash and equivalents | 140 | 107 | Accounts payable | 486 | 455 |
| Accounts receivable | 294 | 270 | Total current liabilities | 486 | 455 |
| Inventories | 269 | 280 | Deferred taxes | 117 | 104 |
| Other | 58 | 50 | Long-term debt | 471 | 458 |
| Total current assets | 761 | 707 | Total long-term liabilities | 588 | 562 |
| Property, plant, and equipment | 1,423 | 1,274 | Preferred stock | 39 | 39 |
| Less accumulated depreciation | (550) | (460) | Common stock ($1 par value) | 55 | 32 |
| Net property, plant, and equipment | 873 | 814 | Capital surplus | 347 | 327 |
| Intangible assets and other | 245 | 221 | Accumulated retained earnings | 390 | 347 |
| Total fixed assets | 1,118 | 1,035 | Less treasury stock | 26 | 20 |
| Total assets | 1,879 | 1,742 | Total equity | 805 | 725 |
| Total liabilities and equity | 1,879 | 1,742 |
Three identities do all the work:
The third one deserves a sentence. Capital spending is not simply the change in fixed assets, because depreciation has already been subtracted from the closing figure. Adding depreciation back undoes that, leaving the gross additions the firm actually paid for.
- Net working capital, 2020 = 761 − 486 = 275
- Net working capital, 2019 = 707 − 455 = 252
- Change in net working capital = 275 − 252 = +23
- Capital spending, 2020 = 1,118 − 1,035 + 90 = 173
These two figures are the reason this lecture is not just accounting revision. In Lecture 4, free cash
flow is defined as EBIT − Tax + Depreciation − ΔWC − Capex. The −ΔWC term is the +23
computed here, with its sign flipped: growing working capital consumes cash. The −Capex term is the
173 computed here. Both are already sitting on the balance sheet.
The income statement (损益表)
Where the balance sheet is a photograph, the income statement measures revenue and expense over a specific period. The example firm's statement reads as follows:
| Income statement ($m) | Amount | Derivation |
|---|---|---|
| Total operating revenues | 2,262 | |
| Cost of goods sold | 1,655 | |
| Selling, general, and administrative | 327 | |
| Depreciation | 90 | |
| Operating income (EBIT) | 190 | 2,262 − 1,655 − 327 − 90 |
| Other income | 29 | |
| Earnings before interest and taxes | 219 | 190 + 29 |
| Interest expense | 49 | |
| Pretax income | 170 | 219 − 49 |
| Taxes | 84 | current 71 + deferred 13 |
| Net income | 86 | 170 − 84 |
| Addition to retained earnings | 43 | |
| Dividends | 43 | 43 + 43 = 86 |
The slide labels both 190 and 219 as EBIT-adjacent, which looks like an inconsistency and is not. Line 190 is operating income — the profit from the firm's principal operations only. Line 219 adds other income (interest received, gains on asset sales, and similar items outside the core business) to arrive at earnings before interest and taxes in the full sense. Ross Ch. 2 keeps the two separate for a reason: only operating income belongs in an operating cash flow forecast. In Lecture 4 the free cash flow formula starts from EBIT, and it means the operating figure — mixing in one-off asset sale gains would corrupt the projection.
Two features of this statement matter more than the rest:
- Depreciation is a non-cash item. As the slide puts it, no firm ever writes a cheque for "depreciation." It reduces taxable income without any cash leaving the firm. This is why net income systematically understates the cash a profitable asset-heavy firm generates — and why Lecture 4 adds it back.
- Net income is not cash. It is an accrual-basis number, further distorted by non-cash charges. The lecture's blunt formulation is that a firm cannot spend its net income.
The three identities from this statement:
The cash flow statement (现金流量表)
Cash flows are hard to manipulate. That is their principal virtue, and it is why analysts go to the cash flow statement first when they distrust a set of accounts. It has three components:
- Cash flow from operating activities (CFO) — cash generated by the core business
- Cash flow from investing activities (CFI) — purchases and sales of fixed assets
- Cash flow from financing activities (CFF) — borrowing, share issuance, dividends
How the three statements lock together
The lecture's slide 18 lists three links. They are worth expanding, because this is the part of the lecture that quietly supplies the machinery for Lectures 3 and 4.
Ross Ch. 2 makes the third link concrete. Start from net income and undo the accruals:
- Add back non-cash charges — depreciation is the big one. Deducting it reduced reported profit without any cash leaving.
- Subtract increases in current assets. If receivables rose, you booked revenue you have not been paid for, so actual cash is lower than profit. If inventory rose, you spent cash building it.
- Add increases in current liabilities. If payables rose, you received goods or services you have not yet paid for — the supplier effectively financed you.
Deferred taxes belong on that list too, and they appear on both statements: $13m is charged in the income statement but not yet paid in cash, so it accumulates as a liability of $117m on the balance sheet. It is a non-cash charge for exactly the same reason depreciation is.
Applying the bridge to the example firm produces the cash flow statement the slides never actually show — reconstructed here from the balance sheet and income statement alone. It is worth building in full, because it closes the loop: the three statements stop being three documents and become one system.
| Cash flow statement ($m) — derived | Amount | Source |
|---|---|---|
| Net income | 86 | income statement |
| + Depreciation | +90 | non-cash charge |
| − Increase in receivables | −24 | 294 − 270 |
| + Decrease in inventory | +11 | 280 − 269 |
| + Increase in payables | +31 | 486 − 455 |
| − Increase in other current assets | −8 | 58 − 50 |
| + Increase in deferred taxes | +13 | 117 − 104 |
| Cash flow from operations (CFO) | 199 | |
| Capital spending | −173 | 1,118 − 1,035 + 90 |
| Cash flow from investing (CFI) | −173 | no disposals assumed |
| Net new borrowing | +13 | 471 − 458 |
| Net new equity issued | +43 | (55−32) + (347−327) |
| − Treasury stock purchased | −6 | 26 − 20 |
| − Dividends paid | −43 | income statement |
| Cash flow from financing (CFF) | 7 | |
| Net change in cash | 33 | 199 − 173 + 7 |
The reconstruction is not a guess. The balance sheet independently reports that cash moved from 107 to 140 — a change of +33. The derived statement produces 199 − 173 + 7 = +33 as well. Two routes to the same number, from different documents: that agreement is the cash flow identity doing its job, and it is the test any cash flow forecast has to pass. If your projected CFO, CFI and CFF do not sum to the change in cash you assumed, something in the projection is wrong.
Depreciation appears three times across this course, always in a different role. Here it is a non-cash
charge that must be added back to reach cash flow. In Lecture 2 it is absent entirely — the discounting
machinery deals only in cash. In Lecture 4 it becomes a tax shield worth Depreciation × T,
the most valuable thing a capital-intensive project owns. One line item, three jobs.
Financial Ratios
slide 19Ratios compress the three statements into numbers that can be compared — across time, and across firms of different sizes. The lecture groups them into five families, and the grouping itself is the lesson: liquidity, leverage, turnover, profitability, and market value are five distinct questions you can ask of the same accounts. The right-hand column below evaluates every ratio on the example firm from the previous section.
| Ratio | Formula | Example firm, 2020 |
|---|---|---|
| I · Short-term solvency (liquidity) | ||
| Current ratio | Current assets / Current liabilities | 761 / 486 = 1.57 |
| Quick ratio | (Current assets − Inventory) / Current liabilities | 492 / 486 = 1.01 |
| Cash ratio | Cash / Current liabilities | 140 / 486 = 0.29 |
| II · Long-term solvency (financial leverage) | ||
| Total debt ratio | (Total assets − Total equity) / Total assets | 1,074 / 1,879 = 0.57 |
| Debt–equity ratio | Total debt / Total equity | 1,074 / 805 = 1.33 |
| Equity multiplier | Total assets / Total equity | 1,879 / 805 = 2.33 |
| Times interest earned | EBIT / Interest | 190 / 49 = 3.88 |
| Cash coverage ratio | (EBIT + Depreciation) / Interest | 280 / 49 = 5.71 |
| III · Asset utilization (turnover) | ||
| Inventory turnover | Cost of goods sold / Inventory | 1,655 / 269 = 6.15 |
| Days' sales in inventory | 365 / Inventory turnover | 365 / 6.15 = 59.3 days |
| Receivables turnover | Sales / Accounts receivable | 2,262 / 294 = 7.69 |
| Days' sales in receivables | 365 / Receivables turnover | 365 / 7.69 = 47.4 days |
| Total asset turnover | Sales / Total assets | 2,262 / 1,879 = 1.20 |
| Capital intensity | Total assets / Sales | 1,879 / 2,262 = 0.83 |
| IV · Profitability | ||
| Profit margin | Net income / Sales | 86 / 2,262 = 3.80% |
| Return on assets (ROA) | Net income / Total assets | 86 / 1,879 = 4.58% |
| Return on equity (ROE) | Net income / Total equity | 86 / 805 = 10.68% |
| V · Market value — needs a share price, so not computable from the statements alone | ||
| Price–earnings ratio | Price per share / Earnings per share | — |
| Market-to-book ratio | Market value per share / Book value per share | — |
| EV multiple | Enterprise value / EBITDA | — |
None of these numbers is good or bad on its own. A current ratio of 1.57 might be comfortable for a manufacturer and alarming for a supermarket chain, which runs on thin liquidity and fast inventory. A high debt ratio is reckless in a cyclical industry and efficient in a stable one, because debt carries a tax shield. The only ratios that mean anything are compared — against the firm's own history, or against competitors in the same industry. Treating a ratio as a grade is the most common misuse of this slide.
The DuPont decomposition: where ROE comes from
The lecture's ratio table already contains the most useful identity in the whole list. ROE is written there as the product of three other ratios, and unpacking it answers a question a single ROE number cannot: why is this firm's return on equity what it is?
- Operating efficiency — the profit margin 86 / 2,262 = 3.80%
- Asset-use efficiency — the total asset turnover 2,262 / 1,879 = 1.2038
- Financial leverage — the equity multiplier 1,879 / 805 = 2.3342
- Multiply the three 0.03802 × 1.2038 × 2.3342 = 0.10683
The decomposition earns its keep when two firms have the same ROE for completely different reasons. A luxury goods brand gets there through a fat margin and low turnover. A discount retailer gets there through a thin margin and very high turnover. A leveraged buyout gets there through neither, and simply piles on debt. The ROE is identical in all three cases; the businesses are not remotely alike, and the risks are not alike either — the leveraged firm's ROE is the most fragile, because leverage magnifies losses as readily as gains.
The turnover ratios in group III measure how long cash is tied up, but each measures only one link. Management accounting joins them into a single number. Adding a payables ratio the slide does not list — payables turnover = COGS / accounts payable = 1,655 / 486 = 3.41, so days' payables outstanding = 365 / 3.41 = 107.2 days — gives:
For the example firm: 59.3 + 47.4 − 107.2 = −0.4 days. Essentially zero, which is a striking result. The firm collects from customers and sells its inventory in about 107 days, and takes about 107 days to pay its own suppliers. Its suppliers are financing the entire operating cycle. This is the position retailers and fast-moving consumer goods firms actively manage towards, and it explains why a firm can be profitable yet permanently short of cash — or the reverse.
It also connects directly forward. Working capital absorbs cash whenever this cycle lengthens, and that is exactly what the ΔWC term in Lecture 4's free cash flow formula subtracts. A firm whose cycle is stretching is quietly consuming cash even while its income statement looks healthy — which is the reconciliation between "profitable" and "broke" that the ratio set exists to expose.
What Financial Management Is
slides 20–27With the statements and the ratios in hand, the lecture turns to the point of the exercise. A financial manager and a financial analyst must understand four things:
What is an asset, a project, or a company worth? Everything downstream is an application of this.
How to spend money. Which long-term investments should the firm choose? → Lectures 3–4
How to raise money. How should the firm fund the investments it selected? → Week 11
How do those two decisions feed back into the value of the firm?
Finance is not accounting
The lecture draws a clean line between the two, and since you have taken 基础会计 this is worth stating precisely rather than diplomatically:
| Accounting | Finance |
|---|---|
| Historical performance — book value | Value based on future forecast — market value |
| The accuracy and reliability of financial reports | Relies on the information accounting produces |
| Backward looking, rule-based | Forward looking, requires judgement and experience |
| CPA, 会计资格考试 | CFA, FRM, 证券从业资格证, 银行从业资格证 |
The distinction is not that finance ignores accounting — the previous three sections were nothing but accounting. It is that accounting supplies a description of what happened, and finance converts that description into a forecast of what will happen. The whole of Lecture 4 is an exercise in taking accounting numbers and turning them into cash flows that can be discounted.
The goal of financial management
The traditional statement of the goal is to maximise firm value, or equivalently to maximise shareholders' wealth. The lecture emphasises what the goal is not: it is not to maximise profits.
Three reasons, all of which the exam can ask for.
Profit is an accounting construct, and it can be manipulated. Change a depreciation assumption, reclassify an expense, pull revenue forward by shipping goods early — reported profit moves without any change in underlying value.
Profit ignores the time value of money. A project that earns ¥100,000 of profit this year is not equivalent to one that earns it in five years. Profit has no time dimension; the entire subject of Lecture 2 exists because value does.
Profit ignores risk and ignores the capital employed. A firm can raise profit by issuing more shares and buying a bond portfolio; shareholders are no better off — they own a bigger firm with the same value per share. Wealth, not profit, is what accrues to the owner.
The lecture adds a modern qualification: alongside value maximisation, firms are now expected to take environmental, social and governance (ESG) considerations into account. The honest way to read this is as a constraint on how value is created rather than a replacement for the objective — which is exactly the view the syllabus takes when it lists ESG as a possible presentation topic.
The slide asks: how can we distinguish investment and financing decisions? The test is which side of the balance sheet the decision touches, and whether it creates value or merely pays for it. An investment decision buys a real asset — it moves value from the left side of the balance sheet into the business, and it is appraised by whether the NPV is positive. A financing decision sells a financial asset — it moves value from the right side, and it is appraised by whether the funds are raised at a fair price. Two of the lecture's own examples make the distinction concrete: 农夫山泉 raising HK$1.1bn in an IPO is financing (selling equity); 美的 acquiring Kuka for €37.07bn is investment (buying real assets).
The lecture also states plainly that the financing decision is the less important of the two. That is a strong claim, and it is the view this course takes: the value of a firm is created principally on the investment side, and the financing decision mostly distributes that value between claimants rather than creating it. Week 11 will test how far that claim survives.
The three decisions in pictures
The lecture closes by attaching each of the three recurring questions to a part of the balance sheet — which is a way of saying that the balance sheet you learned to read in section 3 is also the map of the course.
Notice that net working capital is the one balance-sheet line that appears in all three columns. It is the output of the short-term management decision, it is financed by the capital structure decision, and — as the ΔWC term — it is a direct input to the capital budgeting decision in Lecture 4. If you want a single thread to follow through this course, working capital is a good candidate.
Formula Sheet
print-friendlyKey Concepts
the vocabularySelf-Check
15 questions · gradedGroup A mirrors the exam's concept-explanation questions, group B its single-choice section, and group C its subjective calculation questions. Everything here is answerable from the two tables in section 3.
A · Concept check
Model answer. The income statement is built on estimates and elections: how long an asset's useful life is, when revenue is considered earned, which costs are capitalised rather than expensed. Each of these judgements moves reported profit without any cash changing hands. The cash flow statement records actual transfers of cash, and cash is observable — either the money arrived or it did not.
There is a second reason. The cash flow statement is anchored: the change in the cash balance it reports must reconcile with the change in the cash line on the balance sheet. That is the identity CFO + CFI + CFF = ΔCash, and it constrains the statement in a way the income statement is not constrained. This is why analysts who distrust a set of accounts go to the cash flow statement first.
Model answer — three reasons.
1 · Timing. Profit has no time dimension. ¥100,000 of profit earned this year and ¥100,000 earned in five years are recorded identically, yet they are not worth the same. Value discounts; profit does not. This is the subject of Lecture 2.
2 · Risk. Two projects with the same expected profit are not equivalent if one is a government contract and the other a wildcat oil well. Profit records the expectation and ignores the dispersion around it.
3 · Capital employed, and manipulability. Profit can be raised simply by raising more capital — issue shares, buy a bond portfolio, and net income rises while shareholders are no better off, since they now own a larger firm with the same per-share value. And because profit is an accounting construct built on estimates, it can be moved by changing a depreciation assumption alone.
B · Multiple choice
C · Applied calculation
Starting point and adjustments. Begin at net income and undo the accruals, working through every balance sheet line that changed.
Net income 86 + Depreciation (non-cash) +90 − Increase in receivables (24) −24 294 − 270 + Decrease in inventory +11 280 − 269 + Increase in payables +31 486 − 455 − Increase in other current assets −8 58 − 50 + Increase in deferred taxes +13 117 − 104 ───── Cash flow from operations 199Why each sign. Depreciation and deferred tax are non-cash charges — they reduced reported profit without cash leaving, so they are added back. Receivables rising means revenue was booked but not yet collected, so cash is lower than profit: subtract. Inventory falling means stock was sold without being replaced, releasing cash: add. Payables rising means suppliers have not yet been paid, so they financed the firm: add. Other current assets rising consumes cash: subtract.
The check that proves it. If the figure is right, CFO + CFI + CFF must equal the change in the cash balance. Cash went from 107 to 140, a change of +33. Our financing section is +13 net borrowing, +43 net equity issued, −6 treasury purchases, −43 dividends = +7. Our investing section is the capital spending of −173. And 199 − 173 + 7 = 33. The identity closes.
Step 1 — ROE by DuPont.
ROE = margin × turnover × leverage = 0.04 × 1.5 × 2.0 = 0.12 → 12%Step 2 — double the leverage. Holding the underlying business constant, an equity multiplier of 4.0 gives 0.04 × 1.5 × 4.0 = 24%. The ROE doubles, and nothing about the firm's operations has changed.
Why to be cautious. The extra 12 percentage points are not created value — they are a transfer of risk. Leverage magnifies the return on equity only because equity has become a thinner slice of a larger asset base, so the same operating result is spread over less shareholder capital. The symmetry is the point: if ROA falls, the same multiplier magnifies the loss just as efficiently. In the limit, an equity multiplier high enough drives equity to near zero, at which point ROE becomes meaningless and the firm is simply fragile. A high ROE achieved through leverage is a signal to look at the coverage ratios — times interest earned and cash coverage — before concluding the firm is performing well.
Time Value of Money
Concept Map
how the lecture buildsNothing later works without this lecture. Lecture 3's entire apparatus — NPV — is nothing more than "discount every cash flow to today and add them up," which is the multiple-period example below, generalised. Lecture 4's free cash flows get discounted with exactly these formulas. Week 5's dividend discount model is a growing perpetuity. Week 6's bond price is an annuity plus a lump sum. If one lecture in this course has to be automatic, it is this one.
Future Value, Present Value, and Compounding
slides 2–11If a friend or relative borrows ¥1,000 from you, how much should you ask them to repay in one year?
The cultural instinct — that charging interest to family is embarrassing, and you should just take back ¥1,000 — is exactly what this lecture is designed to overturn. The lecture's answer, delivered at the end of the hour: you should ask for more than ¥1,000, because you could have put that money in 余额宝 and earned interest. Lending it to them is helping them through a difficulty; it is not a reason for you to be worse off. Being able to say "the risk-free rate is 2%, so ¥1,020" is the whole point of the next thirty slides.
Future value
Future value (终值) is the total amount due at the end of an investment. Put $10,000 in a bank at 5% for one year and your wealth grows to:
Present value
Present value (现值) is the amount you must set aside today to meet a promised payment in the future. The same arithmetic, run backwards:
Note what this says. Being promised $10,000 a year from now, when the interest rate is 5%, makes you exactly as well off as having $9,523.81 in your hand today. The $476.19 difference is the price of waiting.
Simple versus compound interest
The distinction is whether the interest itself earns interest.
The lecture's generalisation, and it is the formula the entire course runs on:
T — the number of periods over which the cash is invested · r — the appropriate interest rate, or discount rate · 1 / (1 + r)T — the present value (discount) factor, which is always less than 1
Two things follow from that last point. Because the discount factor is always below 1, discounting always
shrinks a number. And because the equation has four variables, if you know any three you can compute the
fourth — which is why a financial calculator needs only five keys: N, I/Y,
PV, PMT, FV, and solves for whichever one you leave blank.
Drawing a time line
The lecture insists on drawing time lines, and it is worth adopting the habit: it makes sign errors and off-by-one-period errors visible before they become arithmetic.
- Discount the single future cash flow back five periods PV = 20,000 / (1.15)⁵
- Evaluate the discount factor (1.15)⁵ = 2.011357 → 1 / 2.011357 = 0.497177
- Multiply 20,000 × 0.497177 = 9,943.53
Multiple periods, unequal cash flows
Real projects rarely pay a single lump sum. But once you can discount one cash flow, you can discount any number of them: discount each to today separately, then add. There is no compounding between them, because each one is being brought back to the same date.
- Discount each flow individually 200 / (1.12)¹ = 178.57 400 / (1.12)² = 318.88 600 / (1.12)³ = 427.07 800 / (1.12)⁴ = 508.41
- Add the four present values 178.57 + 318.88 + 427.07 + 508.41 = 1,432.93
Add a negative number at t = 0 for what the investment cost, and you have Lecture 3's entire method. "Discount each cash flow to today and add them up" is the definition of NPV; everything else in that lecture is commentary on how to choose r, and what to do when the answer is ambiguous.
Two conventions trip people up, and both cost marks in the exam if you are using Excel.
- Enter the rate as a decimal.
0.05, never5. - Give the present value a negative sign, treating it as an outflow. This is what lets the same function solve for any of the four variables — Excel needs to know which direction the money went.
The lecture's own practice problem is a good illustration: if we invest $25,000 at 12%, how long until we
have $50,000? The answer is =NPER(0.12, 0, -25000, 50000) = 6.1163 years. Note
pmt = 0 — there are no periodic payments, just an initial lump sum — and note the negative sign on
the $25,000. Being able to use NPER, PV, FV, PMT and
RATE fluently is, as the slide points out, a practical requirement for an investment banking or
analyst internship.
Compounding Frequency and the Effective Annual Rate
slides 13–18Everything so far assumed interest compounds once a year. In practice it often compounds more often — semi-annually, monthly, daily. The generalisation is to divide the annual rate by the number of compounding periods per year, and multiply the number of periods by the same factor.
r — always the annual (quoted) rate · m — times per year compounding occurs · T — years
The most common error in this section is plugging the periodic rate into a formula expecting an annual
one, or vice versa. The lecture repeats the warning twice, and it is worth repeating a third time: when you write
r/m, the r on top is the annual quoted rate. In the example below, 12% is the annual
rate and 6% is what actually gets applied each half-year.
- Halve the rate, double the periods periodic rate = 0.12 / 2 = 6% number of periods = 2 × 3 = 6
- Compound FV = 50 × (1.06)⁶
- Evaluate (1.06)⁶ = 1.418519 FV = 50 × 1.418519 = 70.93
The effective annual rate
That $70.93 is more than 12% simple annual compounding would have produced. Which raises the obvious question: what single annual rate would have produced the same result? That rate is the effective annual rate (EAR, 有效年利率), and it is the only honest way to compare two investments quoted on different compounding conventions.
- Set up the equivalence — find the annual rate that gets $50 to $70.93 in three years 50 × (1 + EAR)³ = 70.93
- Isolate the growth factor (1 + EAR)³ = 70.93 / 50 = 1.418520
- Take the cube root and subtract 1 1 + EAR = (1.418520)^(1/3) = 1.123600 EAR = 0.1236
The closed form is much shorter, and it is the version to memorise:
If m > 1, then EAR > r. More frequent compounding always produces a higher effective rate, because interest starts earning interest sooner. The gap is small at low rates and grows quickly: at 12%, monthly compounding gives an EAR of 12.68%, and daily gives 12.75%.
The practical consequence is a piece of consumer literacy. A loan advertised at "12% annual, compounded monthly" costs you 12.68%, not 12%. Chinese regulators require consumer credit to be quoted on a consistent basis precisely because the gap between the quoted rate and the effective rate is where borrowers get misled.
Option A: semi-annual compounding at 1.9%. Option B: 余额宝 at a 7-day annualised rate of 2%.
This looks like a trick and is not. Option A's EAR is (1 + 0.019/2)² − 1 = (1.0095)² − 1 = 1.909%. Option B's is 2%. B is better — and note that you had to convert A to a comparable basis before you could see that. That is the entire function of the EAR: it exists so that two numbers quoted on different conventions can be put side by side.
余额宝, worked properly
- After one full year — annual compounding at the quoted rate FV = 10,000 × (1 + 0.02) = 10,200 interest = ¥200
- After seven days — compound the daily rate seven times FV = 10,000 × (1 + 0.02 / 365)⁷
- Evaluate 0.02 / 365 = 0.000054794 (1.000054794)⁷ = 1.00038362 FV = 10,000 × 1.00038362 = 10,003.8362
The "7-day annualised rate" that Chinese money-market funds quote is not the same object as the EAR, and it is worth being precise about the difference. It is a backward-looking measure: take the fund's actual seven-day earnings, divide by the principal, and annualise by multiplying by 365/7 — r₇ × 365/7. Because it is a simple rather than compound annualisation, and because the underlying seven days have already happened, it tells you what the fund did last week, not what it will pay next year.
The EAR, by contrast, is a forward-looking compounding convention. They coincide only if the seven-day return repeats all year and you ignore compounding. Treating a 7-day annualised figure as a promised annual return is one of the commonest retail-investing errors.
The size of the gap is worth seeing. Simple annualisation at 2% gives ¥200 of interest on ¥10,000. True daily compounding at the same quoted rate gives 10,000 × (1 + 0.02/365)365 − 10,000 = ¥202.01, an implied EAR of 2.0201%. The difference is small at these rates precisely because 2% is small; at 12% the same discrepancy is ¥74.75 rather than ¥2.01.
Perpetuities and Annuities
slides 20–28Discounting cash flows one at a time works, but it is tedious, and for the two most common patterns it collapses into a single formula. Both patterns are streams of a constant cash flow C; they differ only in whether the stream ever stops.
Perpetuity
A perpetuity (永续年金) is a constant stream of cash flows that lasts forever. The infinite sum
converges, because the terms shrink geometrically. The result is startlingly simple:
- Apply PV = C / r PV = 15 / 0.10
- Evaluate 15 / 0.10 = 150
PV = C / r gives the value one period before the first payment. If the first $15 arrives
at t = 1, the formula values it at t = 0 — which is what we wanted here. But if the first payment arrives
today, or at t = 5, the formula is off by exactly the number of periods you need to shift it. This single
fact is the source of most errors in this topic, and the next two worked examples are both about it.
Annuity
An annuity (年金) is a constant stream of cash flows with a fixed maturity — it stops after T periods. One way to derive its value is to notice that an annuity is a perpetuity with the tail cut off: take a perpetuity starting now, subtract a second perpetuity starting at T + 1, and what remains is T payments.
The bracketed expression is the annuity factor — the present value of $1 per period for T periods. Everything about annuities follows from it.
- Convert to a monthly basis r = 0.07 / 12 = 0.0058333 T = 36
- Compute the annuity factor [1 − 1/(1.0058333)³⁶] / 0.0058333 = 32.386464
- Multiply by the payment 400 × 32.386464 = 12,954.59
Annuity due
An annuity due (即付年金) is identical except that each payment happens at the beginning of the period rather than the end — so the first $400 is paid immediately, at t = 0, and the last one at t = 35.
There is no need for a new formula. Every cash flow has simply moved one period earlier, and moving a cash flow one period earlier multiplies its present value by (1 + r). So:
- Discount the ordinary annuity to t = −1 PV₋₁ = 12,954.59 (the value one period before the first payment)
- Compound it forward one period to t = 0 12,954.59 × (1 + 0.07/12) = 12,954.59 × 1.0058333
- Evaluate = 13,030.15
Slide 24 writes the result as $13,029.56. Carrying the arithmetic through exactly — 12,954.59 × 1.0058333 — gives $13,030.15, a difference of $0.59. The slide appears to have used a rounded monthly factor rather than the full-precision 0.07/12. The method it teaches is correct and is the one shown above; only the last multiplication is slightly off. Worth knowing, because if you reproduce the slide's number by hand you will think you made the error.
333 × 24 months, or ¥7,999 up front? The lecture's answer uses 余额宝's 7-day rate of 1.38% as the opportunity cost — the return you forgo by paying early.
Discount the 24 instalments at 1.38% annual, compounded monthly (r = 0.0138/12 = 0.00115):
The instalment plan is worth ¥7,878 in today's money, against ¥7,999 for paying in full. The instalments are cheaper by ¥120.75, so take the instalment plan.
Now notice what the answer depended on. It flipped because the discount rate is low. At a 1.38% rate the deferral is nearly free and the instalments win; had the relevant rate been 20% — a credit card, or a firm with genuinely profitable projects — discounting would have hit the ¥333 payments much harder and paying in full would win. "Is the instalment worth it?" is not a question about the instalment. It is a question about your discount rate.
Delayed annuity
The third variation: the payments are a normal annuity, they simply do not start immediately. Handle it in two steps — value the annuity as if it began one period before its first payment, then discount that lump sum back to today.
- Discount each payment individually — the definition 100/1.09² + 100/1.09³ + 100/1.09⁴ + 100/1.09⁵ = 323.97
- Or, the shortcut: value it as an ordinary annuity — the ordinary-annuity formula applied to four payments values them at t = 1, one period before the first payment at t = 2 100 × [1 − 1/(1.09)⁴] / 0.09 = 323.97 (valued at t = 1)
- Then discount that single sum from t = 1 back to t = 0 323.97 / 1.09 = 297.22
Both routes agree exactly, which is the useful lesson: the annuity formula is not a different method from discounting, it is a shortcut for it.
Growing perpetuity. If the cash flow grows at a constant rate g forever, the sum converges to PV = C / (r − g), valid only when r > g. This single formula is the Gordon growth model you will meet in Week 5 as the dividend discount model; a share price is a growing perpetuity of dividends. Note the fragility: as g approaches r the denominator collapses and the value explodes, which is why terminal-value estimates in valuation are notoriously sensitive to that one assumption.
Growing annuity. The same idea with a fixed maturity: PV = C × [1 − ((1+g)/(1+r))T] / (r − g). Setting g = 0 recovers the ordinary annuity formula, which is a good way to remember it.
Loan Amortisation
slides 28–34This section is where the annuity formula stops being an abstraction. Every mortgage in China is one of these two schedules, and the difference between them is a difference in thousands of yuan of interest. The lecture classifies loans into three types:
The simplest form: the borrower receives money today and repays a single lump sum covering both principal and interest at a future date. 0 periods. A zero-coupon bond is the market-traded version.
An interest payment each period, with the full principal due at maturity. The principal balance never amortises — which is precisely why these are riskier for a lender than they look.
Repayment of principal over time, in addition to interest. Two variants, and the lecture devotes its remaining slides to the difference between them.
The running example throughout is the same in both cases: a business borrows $5,000 for five years at 9%. What changes is only which part of the payment is fixed.
Equal principal payment 等额本金
Here the principal repayment is fixed and the interest floats. The fixed principal is $5,000 / 5 = $1,000 per year, and interest each year is charged on whatever balance remains at the start of that year.
| Equal principal · $5,000 · 5 years · 9% | Begin | Payment | Interest | Principal | End |
|---|---|---|---|---|---|
| Year 1 | 5,000.00 | 1,450.00 | 450.00 | 1,000.00 | 4,000.00 |
| Year 2 | 4,000.00 | 1,360.00 | 360.00 | 1,000.00 | 3,000.00 |
| Year 3 | 3,000.00 | 1,270.00 | 270.00 | 1,000.00 | 2,000.00 |
| Year 4 | 2,000.00 | 1,180.00 | 180.00 | 1,000.00 | 1,000.00 |
| Year 5 | 1,000.00 | 1,090.00 | 90.00 | 1,000.00 | 0.00 |
| Total | 6,350.00 | 1,350.00 | 5,000.00 |
Interest declines exactly linearly, because the balance does: 450, 360, 270, 180, 90. Total interest paid over the life of the loan is $1,350.
Equal payment 等额本息
Here the total payment is fixed — the borrower pays the same amount every year — and the split between interest and principal shifts over time. That fixed payment is simply the annuity payment that amortises $5,000 over five years at 9%:
- Interest in year 1 — charged on the full opening balance 5,000 × 0.09 = 450.00
- Principal repaid in year 1 — whatever is left of the fixed payment 1,285.46 − 450.00 = 835.46
- Closing balance 5,000.00 − 835.46 = 4,164.54
- Repeat for each remaining year, always charging 9% on the opening balance Year 2: 4,164.54 × 0.09 = 374.81 → principal 910.65 → balance 3,253.88 Year 3: 3,253.88 × 0.09 = 292.85 → principal 992.61 → balance 2,261.27 Year 4: 2,261.27 × 0.09 = 203.51 → principal 1,081.95 → balance 1,179.32 Year 5: 1,179.32 × 0.09 = 106.14 → principal 1,179.32 → balance 0.00
| Equal payment · $5,000 · 5 years · 9% | Begin | Payment | Interest | Principal | End |
|---|---|---|---|---|---|
| Year 1 | 5,000.00 | 1,285.46 | 450.00 | 835.46 | 4,164.54 |
| Year 2 | 4,164.54 | 1,285.46 | 374.81 | 910.65 | 3,253.88 |
| Year 3 | 3,253.88 | 1,285.46 | 292.85 | 992.61 | 2,261.27 |
| Year 4 | 2,261.27 | 1,285.46 | 203.51 | 1,081.95 | 1,179.32 |
| Year 5 | 1,179.32 | 1,285.46 | 106.14 | 1,179.32 | 0.00 |
| Total | 6,427.31 | 1,427.31 | 5,000.00 |
Comparing the two
| Equal principal 等额本金 | Equal payment 等额本息 | |
|---|---|---|
| Payment pattern | Falls every period | Constant |
| First payment | 1,450.00 | 1,285.46 |
| Last payment | 1,090.00 | 1,285.46 |
| Total interest | 1,350.00 | 1,427.31 |
| Front-loaded? | Yes — heavier early | No |
The lecture makes an observation that is worth dwelling on: total interest is higher under the equal-payment method, and therefore "the loan officer would typically recommend you the fixed payment method."
Why is it higher? Because under equal payments the borrower repays principal more slowly in the early years — 835.46 in year 1 against 1,000 under equal principal — so a larger balance stays outstanding for longer, and interest is charged on that balance every year.
The extra interest can be traced exactly. Under equal payments the balance is higher by $164.54, $253.88, $261.27 and $179.32 at the end of years 1 through 4 — $859.01 in total across the four years in which a gap exists. At 9%, that is 0.09 × 859.01 = $77.31, precisely the difference in total interest. None of it is a fee or a hidden charge; it is simply the cost of having borrowed more, for longer.
The trade-off is genuine rather than a trick: equal payment gives you a lower payment now in exchange for more interest overall. Which is better depends entirely on your discount rate and your cash flow — the same consideration as the instalment-versus-full-price question above. But you should know which side of the trade the person recommending it is on.
Notice what the equal-payment table actually is: a $5,000 lump sum today, followed by five payments of $1,285.46. Discounted at 9%, those payments are worth exactly $5,000 — which is the definition of the yield to maturity you will meet in Week 6. A mortgage is a bond with the lender on the other side, and the YTM is the contract rate. The same table, read right to left, is a bond pricing exercise.
Formula Sheet
print-friendlyStep 2: discount that single sum to t = 0
Key Concepts
the vocabularySelf-Check
20 questions · gradedThis lecture carries more marks than any other in the course, so it carries the most questions. Bring a calculator and work each one before revealing the answer.
A · Concept check
Why EAR ≥ r. The quoted rate r is an annual figure, but when compounding occurs m times a year the interest earned partway through the year begins earning interest of its own before the year is out. Dividing the annual rate by m gives each period its share, but compounding those periods multiplies rather than adds — and multiplication of factors each above 1 exceeds the simple sum. Hence (1 + r/m)m − 1 ≥ r, with equality only when m = 1. The gap widens with both m and r: at 12% it is 0.36 points monthly, 0.75 points daily.
Why it matters. A borrower comparing two loans advertised at the same headline rate is not comparing like with like unless both are converted to an EAR. "12%, compounded monthly" costs 12.68%, not 12% — and the difference on a large long-term mortgage is substantial. This is the reason consumer-credit disclosure rules exist: without a mandated common basis, the quoted number systematically understates what the borrower pays.
Because interest is charged on the outstanding balance, and equal payment leaves that balance higher for longer.
Under equal principal, $1,000 of principal is repaid in year 1 of the $5,000 loan. Under equal payment, only $835.46 is repaid in year 1 — the rest of the fixed $1,285.46 payment goes to interest. So at the end of year 1 the equal-payment borrower still owes $4164.54 against $4,000.
That gap persists and widens. At the end of years 1 to 4 the equal-payment balance exceeds the equal-principal balance by $164.54, $253.88, $261.27 and $179.32 — $859.01 in total. Charging 9% on that extra balance produces 0.09 × 859.01 = $77.31 of extra interest, which is exactly the difference between the two totals ($1,427.31 against $1,350.00).
It is not a penalty or a fee. It is the price of the concession that equal payment offers: a lower payment in the early years, when cash is typically tightest.
B · Multiple choice
C · Applied calculation
Step 1 — monthly basis. r = 0.06/12 = 0.005; T = 48.
Step 2 — annuity factor.
[1 − 1/(1.005)⁴⁸] / 0.005 = 42.580318Step 3 — present value.
350 × 42.580318 = 14,903.11Step 4 — annuity due. Every payment moves one month earlier, so multiply by (1 + r) once:
14,903.11 × 1.005 = 14,977.63Answer. $14,903.11 if payments are at month-end; $14,977.63 if at the start — $74.52 more car, because the dealer is effectively financing you for one month less.
Step 1 — the discount rate. The relevant rate is the return you forgo by paying early, not the instalment plan's stated 0%:
r = 0.0138 / 12 = 0.00115 per monthStep 2 — present value of the 24 instalments.
PV = 333 × [1 − 1/(1.00115)²⁴] / 0.00115 = 333 × 23.658412 = 7,878.25Step 3 — compare. The instalment plan is worth ¥7,878.25 today against ¥7,999 for paying in full. Take the instalments; they are cheaper by ¥120.75.
What would change the answer. The discount rate. At 1.38% the deferral of payment is nearly free, so the instalments win. Raise the relevant rate and the ¥333 payments get discounted harder, shrinking their present value. At a 20% discount rate — a credit card, or a firm with genuinely profitable projects — the same 24 payments would be worth ¥6,542.78, and paying the ¥7,999 in full would be clearly preferable. The instalment plan's attractiveness is a statement about your opportunity cost, not about the plan.
Accept. The NPV is positive, so the project returns more than the 8% the capital could earn elsewhere. The margin is thin — $88.15 on a $5,000 outlay is 1.76% — so the decision is sensitive to the discount rate. At 9% the NPV turns negative, and the correct decision reverses.
Note what this exercise actually is: it is Lecture 3's entire method, arrived at from Lecture 2's machinery. NPV is not a new technique, it is discounting plus subtraction.
Fixed principal = 5,000 / 5 = $1,000 per year. Interest each year is 9% of the opening balance.
Year Begin Interest Principal Payment End 1 5,000.00 450.00 1,000.00 1,450.00 4,000.00 2 4,000.00 360.00 1,000.00 1,360.00 3,000.00 3 3,000.00 270.00 1,000.00 1,270.00 2,000.00 4 2,000.00 180.00 1,000.00 1,180.00 1,000.00 5 1,000.00 90.00 1,000.00 1,090.00 0.00 ─────── ──────── Total 1,350.00 6,350.00Total interest = $1,350.00, which is $77.31 less than the equal-payment method's $1,427.31.
Why. Equal principal repays $1,000 of principal in year 1 against $835.46 under equal payments, so the outstanding balance is lower in every subsequent year, and interest — charged on that balance — is lower in every year. The $77.31 difference is 9% of the $859.01 cumulative balance gap.
Step 1 — convert the bank rate to an EAR.
EAR = (1 + 0.019/2)² − 1 = (1.0095)² − 1 = 1.01909025 − 1 = 0.019090 → 1.909%Step 2 — the fund. Its 7-day annualised figure is quoted on a simple (non-compound) basis, so on a comparable basis it is 2.000%. If the seven-day performance repeated with daily compounding all year, the effective rate would be 2.0201%.
Answer. The fund is better — 2.00% against 1.909%, a margin of about 0.09 percentage points. On ¥10,000 that is roughly ¥9 a year, so it is a real but modest difference.
The point of the exercise. Before the conversion, the bank's "1.9% compounded semi-annually" looks like a smaller number than the fund's "2%" but on a basis you cannot directly compare. Only after putting both on an EAR footing is the comparison meaningful. Two caveats worth stating: the fund's quoted rate is backward-looking — it reports what the fund earned over the last seven days, not what it will pay next year — and money-market funds are not deposit-insured, whereas a bank deposit up to the insurance limit is. The 0.09-point edge is not obviously worth that difference in risk.
Net Present Value
Concept Map
how the lecture buildsThis lecture is where "valuation" as an abstraction becomes a decision rule. The discount rate r is assumed here — the lecture says only that it is the required return for an investment of this risk. Week 10 supplies it: the cost of capital is where r comes from, and the CAPM of Week 7 tells you what it should be for a given risk. And Lecture 4 takes this rule and applies it to the hard part — working out what the cash flows actually are.
Net Present Value
slides 2–10You are the CFO. An investment promises to pay $10,000 in one year and costs $9,600 now. Your interest rate is 5%. Should you buy it?
The naive answer is yes — $10,000 is more than $9,600. It is wrong. Cash flows occurring at different points in time are not directly comparable, and this slide is the sharpest possible demonstration of why.
Discount the $10,000 back one year at 5%:
PV of the inflow = 10,000 / 1.05 = 9,523.81 NPV = −9,600 + 9,523.81 = −76.19Reject the project. The present value of what you receive is less than what you pay. And note the lecture's phrasing of the alternative: you can instead put the $9,600 in the market at 5% and end the year with $10,080 — which is $80 more than the project pays you. The $76.19 NPV is the present value of that $80.
The formula and the rule
Ct — the cash flow at time t (negative at t = 0 for the investment) · r — the discount rate
Reject the project when NPV < 0.
Worked example: Finance.com
- Compute the present value factor for each year — this is 1/(1.07)t Year 1: 1/1.07 = 0.9346 Year 2: 1/(1.07)² = 0.8734 Year 3: 1/(1.07)³ = 0.8163
- Discount each cash flow 25,000 × 0.9346 = 23,364.49 20,000 × 0.8734 = 17,468.77 15,000 × 0.8163 = 12,244.47
- Total the present values 23,364.49 + 17,468.77 + 12,244.47 = 53,077.73
- Subtract the initial cost NPV = −50,000 + 53,077.73 = 3,077.73
The three PV factors are worth memorising in form even if not in value: they are simply 1/(1+r)t, and the lecture's point in tabulating them separately is that this is what a present-value table in the back of a textbook contains.
The lecture flags this and it costs people real money in practice. Excel's
=NPV(rate, CF1, CF2, …) calculates the present value of the cash flows you give it —
it does not include CF0, and it does not net anything out.
The correct construction for a net present value is:
=NPV(rate, CF1, CF2, …) + CF0with CF0 entered as a negative number outside the function. Written
=NPV(rate, CF0, CF1, CF2) instead — which is what the name invites — you would discount the initial
cost by one period as well, understating the NPV and possibly flipping the accept/reject decision.
The three steps in estimating NPV
The slide reduces the whole exercise to three inputs, and it is worth being clear that the difficulty is entirely in the first of them:
How much, and when? This is the hard part, and it is what the whole of Lecture 4 is about. Note that the slide says cash flows, not earnings.
What return does an investment of this risk command in the capital market? The lecture is explicit that this is the project's required rate — not the firm's borrowing rate, and not an arbitrary hurdle.
The easiest of the three, and the one most often got wrong anyway — because the naive figure omits opportunity costs and includes sunk costs. Lecture 4 fixes both.
Why NPV is the right rule
The lecture gives the argument directly, and it is worth following because it is the justification for everything else in this course:
- The goal of financial management is to maximise shareholder value. Established in Lecture 1.
- A firm can be viewed as the sum of many projects — some successful, some not. The firm is a portfolio of investments, bundled.
- Accepting a positive-NPV project raises the value of the firm by exactly the NPV. Not approximately — exactly. So a rule that accepts every positive-NPV project is a rule that makes the firm as valuable as it can be, and there is no other rule that does better.
The lecture notes that NPV is very sensitive to the discount rate, and that the rate "is the required rate for the investment, or the expected rate with the same risk in the capital market." That second clause is doing a lot of work, and it is worth unpacking.
The rate is not a property of the firm. It is a property of the project, and specifically of the project's risk: it is the return you could earn elsewhere in the market on an investment of comparable risk. This is why the same firm should use different discount rates for a low-risk factory expansion and a high-risk biotech venture. Using one firm-wide rate — typically the WACC of Week 10 — for every project is one of the commonest serious errors in corporate practice, because it systematically accepts the riskiest projects (which look good against a low hurdle) and rejects the safest (which look poor against a high one).
It also explains why NPV is sensitive to r in a way that is not merely mathematical. Discounting at 7% versus 10% changes the NPV of a long-dated project substantially, because the later cash flows are the ones most affected — and those are precisely the ones about which you are least certain. The sensitivity of the answer to the discount rate is a warning about the confidence you should place in it.
Internal Rate of Return — and the Four Ways It Misleads
slides 11–21The internal rate of return is the most intuitive criterion in finance, and the one managers reach for first. It is also the one that fails most spectacularly, in four separate ways. The lecture spends ten slides on the failures, which is a fair indication of where the exam interest lies.
Definition
The IRR is the discount rate that sets NPV to zero. It is the rate at which the project exactly breaks even in present-value terms — equivalently, the compound return the project earns on the capital tied up in it.
The slide's most important line about IRR is easy to skim past: "All future cash flows are assumed to be reinvested at the IRR."
This is not a technicality. To get from a set of cash flows to a single percentage, you must decide what happens to the money as it comes in, and IRR silently assumes you can redeploy each interim cash flow at the IRR itself. NPV makes no such assumption — it discounts at the market rate r, and says nothing about reinvestment.
The consequence is that IRR flatters projects with high IRRs. A project with a 40% IRR is implicitly assumed to let you reinvest every interim cash flow at 40%, which is rarely true. This is the root cause of the timing problem below, and Ross Ch. 5 gives the formal fix — the modified IRR, which separates the financing rate from the reinvestment rate.
Worked example
- Set NPV to zero and write it out 0 = −200 + 50/(1+IRR) + 100/(1+IRR)² + 150/(1+IRR)³
- There is no closed-form solution — this is a cubic in 1/(1+IRR). Solve numerically, by trial and
error or with a calculator's IRR function, or with
=IRR()in Excel. - Solution IRR = 0.1944 = 19.44%
- Check the decision — if the required return is, say, 12%, then 19.44% > 12% and the project is accepted. This will agree with the NPV rule, provided the cash flows are conventional.
The slide poses a question it does not answer, and it is a good one because it turns on the distinction between simple and compound interest.
The setup. You borrow 100 at a stated 3%.
- Plan A — repay 109 as a single lump sum three years later (一次性还本付息)
- Plan B — pay 3 at the end of each year, and repay 100 at the end of year three (每年付息到期还本)
- Plan C — repay 16, 32 and 60 over the following three years
Is 3% simple or compound? Look at Plan A's 109. Compound interest would give 100 × (1.03)³ = 109.27. Simple interest gives 100 × (1 + 0.03 × 3) = 109.00. Plan A's 109 is simple interest — and Plan A's true IRR is therefore 2.9142%, not 3%. Plan B's cash flows, by contrast, discount to exactly 100 at 3%, so its IRR is exactly 3.0000%.
Which should you choose? Compute all three IRRs (from the borrower's perspective, so the loan is an inflow at t = 0):
Plan A: 100, 0, 0, −109 IRR = 2.9142% Plan B: 100, −3, −3, −103 IRR = 3.0000% Plan C: 100, −16, −32, −60 IRR = 3.2604%Choose Plan A — the lowest effective cost. The ranking of IRRs is the ranking of the prices you are paying for the money, and as a borrower you want the cheapest.
The NPV view agrees. Evaluating each at a 3% market rate, from the borrower's side: Plan A has an NPV of +0.25 (a bargain), Plan B exactly 0 (fairly priced), Plan C −0.61 (expensive). Two criteria, same ranking, because none of these cash flows changes sign more than once.
Problem 1 · Multiple IRRs
When a project's cash flows change sign more than once, the NPV equation can have more than one root — and IRR has no way of telling you which to use.
- Test 0% — with no discounting, just add the cash flows −200 + 200 + 800 − 800 = 0 → IRR = 0% is a solution
- Test 100% — discount each flow at 100%, so divide by 2t −200 + 200/2 + 800/4 − 800/8 = −200 + 100 + 200 − 100 = 0 → IRR = 100% is also a solution
The NPV profile explains what is happening. NPV rises from zero at r = 0%, peaks at about $63 around r ≈ 29%, then falls back to zero at r = 100% and turns negative beyond it. The project is value-creating for discount rates between 0% and 100% — a fact no single number can express. Whenever the cash flow signs change more than once, this problem can occur.
Problem 2 · Are we borrowing or lending?
Consider two projects with the same IRR:
| Project | t = 0 | t = 1 | IRR | Rule at r = 10% |
|---|---|---|---|---|
| A — investing | −100 | +130 | 30% | IRR > r → accept |
| B — financing | +100 | −130 | 30% | IRR > r → reject |
Both have an IRR of 30%. But they are opposite transactions. Project A is an investment: you pay out 100 and receive 130, and a high return is good. Project B is borrowing: you receive 100 and pay back 130, and a high return is bad — it is the rate you are being charged.
So the IRR rule reverses direction for financing-type cash flows: accept when IRR < the required return, reject when IRR > it. The IRR number alone cannot tell you which situation you are in. The sign of the initial cash flow does.
Problem 3 · Scale, and Problem 4 · Timing
Both appear only when choosing between mutually exclusive projects, where taking one means giving up the others. The lecture defines the two cases precisely: mutually exclusive projects are those where only one of several can be chosen; independent projects are those where accepting or rejecting one does not affect the decision on the others. IRR is fine for independent projects and unreliable for mutually exclusive ones.
The scale problem is that IRR is a percentage and says nothing about how much capital is involved. A 50% return on $10 is worse than a 20% return on $1,000,000, and IRR will rank them the wrong way round.
Project A — money back early
Project B — money back late
| IRR | NPV at 10% | |
|---|---|---|
| Project A | 16.04% | $668.67 |
| Project B | 12.94% | $751.31 |
The two criteria disagree. IRR ranks A higher (16.04% against 12.94%); NPV ranks B higher ($751.31 against $668.67). They are mutually exclusive, so one must be chosen. Choose B — the NPV is what shareholders actually receive.
Why they disagree. B's cash flows arrive later, so they are discounted harder. IRR's reinvestment assumption papers over this: it credits A with being able to redeploy that early $10,000 at 16.04% for two years, which may not be possible. NPV simply discounts everything at the market's 10% and compares.
The crossover rate is worth understanding because it explains when IRR and NPV will disagree, rather than merely that they sometimes do.
Plot both projects' NPV against the discount rate and the two curves cross at 10.55%. To the left of the crossover — low discount rates — the late-cash-flow project B is worth more, because its big terminal payoff is barely discounted. To the right — high discount rates — the early-cash-flow project A is worth more, because B's distant payoff is heavily discounted while A has already banked most of its money.
Now note where the crossover sits in this example: 10.55%, against a required return of 10%. The two projects are almost equally attractive, and the entire decision hinges on a half-point of discount rate. That is not a comfortable margin, and it is the practical lesson — when the NPVs are close, the decision is fragile and the quality of your discount-rate estimate matters more than the arithmetic.
Two further tools from Ross Ch. 5 that the lecture does not cover: the discounted payback period, which fixes payback's most obvious defect by discounting the cash flows first; and the modified IRR (MIRR), which computes a single rate by separating the reinvestment assumption from the project's own return, thereby eliminating the multiple-root problem.
The verdict from the lecture
The lecture states the reconciliation cleanly, and it is worth memorising as a sentence: NPV and IRR will generally give the same decision. They diverge only in two situations — non-conventional cash flows (signs changing more than once), and mutually exclusive projects where either the initial investments differ substantially or the timing of cash flows differs substantially.
When they disagree, use NPV. The reason is not that NPV is more sophisticated but that it answers the right question directly: it measures the addition to firm value, in dollars. IRR measures a rate of return on a project whose scale and timing it cannot see.
Payback Period and Profitability Index
slides 22–32Payback period
The payback period answers a narrower question than NPV: how long does it take the project to recover its initial investment? It is the number of years until cumulative cash flow turns positive.
| Time | Cash flow | Cumulative |
|---|---|---|
| 0 | −50,000 | −50,000 |
| 1 | +30,000 | −20,000 |
| 2 | +20,000 | 0 |
| 3 | +10,000 | +10,000 |
Note that the payback period is set by management, not derived from anything. There is no theory that tells you whether two years is acceptable; the criterion is arbitrary by construction.
The advantage and the disadvantage are the same fact. By ignoring all cash flows after the payback date, payback gives the same answer for a project that limps to break-even and then stops, and for one that goes on to generate enormous profits for a decade.
Work the Finance.com example through payback and you get 2.33 years — the cumulative flow reaches −$5,000 after two years, and the third year's $15,000 clears it in a third of a year. That single number is identical for a project with an NPV of $3,077.73 and for one with an NPV of $0. A rule that cannot see the value it is supposed to be measuring is not a valuation criterion; it is a liquidity screen, and should be used as one.
Profitability index
The profitability index restores the missing ingredient — the time value of money — by putting present values back into a ratio:
| Project | C0 | C1 | C2 | PV of C1,C2 | PI | NPV |
|---|---|---|---|---|---|---|
| 1 | −20 | 70 | 10 | 70.47 | 3.52 | 50.47 |
| 2 | −10 | 15 | 40 | 45.28 | 4.53 | 35.28 |
| 3 | −10 | −5 | 60 | 43.37 | 4.34 | 33.37 |
- Notice that the budget binds. All three projects have PI > 1 and positive NPV, so unconstrained you would take all three — but that costs $40m and you have $20m.
- Rank by NPV and you get the wrong answer. Project 1 has the highest NPV at $50.47m, and it consumes the entire budget. That is the trap: the NPV rule assumes you can fund everything profitable.
- Rank by PI instead. Project 2 (4.53) and project 3 (4.34) come first, cost $20m between them, and deliver a combined NPV of 35.28 + 33.37 = $68.65m.
- Compare. Project 1 alone: $50.47m. Projects 2 and 3: $68.65m. Take 2 and 3.
The slide shows the PV column as 70.5, 45.3 and 43.4, and the PI column as 3.53, 4.53 and 4.34. The exact figures are 70.47, 45.28 and 43.37, giving PIs of 3.52, 4.53 and 4.34. The discrepancy is only in project 1, and it arises because 70.5 / 20 = 3.525 rounds up to 3.53, whereas 70.47 / 20 = 3.5236 rounds to 3.52. The slide is internally consistent; the table above uses unrounded intermediate figures. It changes nothing about the decision.
Everything at once: the lecture's capstone example
| Year | Project A | Project B |
|---|---|---|
| 0 | −$200 | −$150 |
| 1 | $200 | $50 |
| 2 | $800 | $100 |
| 3 | −$800 | $150 |
| Project A | Project B | |
|---|---|---|
| CF0 | −200.00 | −150.00 |
| PV of CF1–3 | 241.92 | 240.80 |
| NPV | 41.92 | 90.80 |
| IRR | 0% and 100% | 36.19% |
| PI | 1.2096 | 1.6053 |
Reading the results. Project B wins on every criterion that can be trusted — higher NPV, higher PI, and a single unambiguous IRR. Project A is a textbook case of non-conventional cash flows: it returns everything by year 1, then demands a large outflow in year 3. It has two IRRs, at 0% and 100%, which is the multiple-root problem appearing in a genuine capital-budgeting exercise rather than a constructed one.
And payback? The cumulative flow for A runs −200, 0, +800, 0. It reaches zero at year 1 and again at year 3. So payback is 1 year, or 3 years? The criterion has no way to say. B's cumulative flow runs −150, −100, 0, +150, giving an unambiguous payback of 2 years.
The lecture's answer is honest about the gap between theory and practice: some firms use payback, others use something else, and the most frequently used technique among large corporations is either IRR or NPV. When the two produce opposite conclusions, choose NPV.
The surveys behind that claim are worth knowing, because they explain why this lecture spends so long criticising a method firms keep using. Graham and Harvey's well-known survey of CFOs found IRR and NPV used by roughly three-quarters of large firms, with payback still used by more than half — often alongside NPV rather than instead of it. The reconciliation is that payback is not really competing with NPV: it is answering a different question, about liquidity and about how long capital is exposed. A firm can sensibly use NPV to decide and payback to prioritise within a capital constraint.
The other reason IRR persists is presentational, and the lecture names it: it is easy to understand and communicate. "This project returns 36%" is a sentence a board can act on. "$90.80 of net present value" is not, without a page of explanation. That is a real advantage, and it is why the correct response to IRR is not to abandon it but to know precisely when it lies.
Formula Sheet
print-friendlyKey Concepts
the vocabularySelf-Check
18 questions · gradedThe examinable content here is as much about when each criterion fails as about computing them. The concept questions below are weighted accordingly.
A · Concept check
Why NPV is superior. Three reasons, and they are independent of one another.
1 · It measures the right thing directly. The goal of the firm is to maximise shareholder value. A firm is a portfolio of projects, and accepting a positive-NPV project raises the firm's value by exactly the NPV. NPV is therefore denominated in the objective itself — dollars of value. No other criterion is.
2 · It has no technical defects. NPV works for conventional and non-conventional cash flows alike, for independent and mutually exclusive projects alike, and for lending and borrowing alike. IRR fails in all four of those dimensions.
3 · It makes no hidden assumptions. NPV discounts at the market rate and says nothing about what happens to interim cash flows. IRR silently assumes they are reinvested at the IRR — an assumption that flatters exactly the projects with the highest IRRs, and that is rarely true.
Why firms still use IRR. Because a percentage is a form of communication that a dollar figure is not. "This project returns 36%" is actionable in a board meeting in a way that "$90.80 of net present value" is not, without explanation. IRR also makes comparison across projects of different sizes feel intuitive, even though that intuition is precisely the scale problem. The sensible practice — and the lecture's recommendation — is to use NPV for the decision and IRR for the conversation.
How it happens. IRR is defined by setting NPV to zero. Written out, that is a polynomial in 1/(1+IRR) whose degree equals the number of periods. A polynomial can have as many real roots as its highest power, so a project with several periods can in principle have several discount rates that zero its NPV.
The practical trigger is the number of sign changes in the cash flow stream. Conventional cash flows — one outflow followed by inflows, a single sign change — produce exactly one IRR. Every additional sign change adds the possibility of another root. A project that pays out, then requires a large refurbishment outflow, then pays out again, has two sign changes and may have two IRRs.
Worked case. Cash flows of −200, +200, +800, −800 have two sign changes. NPV is zero at both 0% and 100%. NPV is in fact positive for every discount rate between them, peaking at $63 around 29%. A single number cannot express that shape.
What to do. Abandon IRR for that project and read the NPV profile — the graph of NPV against the discount rate. It shows which discount rates make the project worth doing, which is the information the decision actually needs. If a single summary number is required, the modified IRR (MIRR) computes a unique rate by assuming interim cash flows are reinvested at the cost of capital rather than at the IRR itself.
B · Multiple choice
C · Applied calculation
NPV.
500/1.09 = 458.72 500/(1.09)² = 420.84 400/(1.09)³ = 308.87 ──────── PV of inflows = 1,188.43 NPV = 1,188.43 − 1,200 = −11.57IRR. Solve −1,200 + 500/(1+r) + 500/(1+r)² + 400/(1+r)³ = 0.
IRR = 8.4383%Profitability index.
PI = 1,188.43 / 1,200 = 0.9904Payback period.
Cumulative: −1,200 → −700 → −200 → +200 Payback = 2 + (200/400) = 2.50 yearsDecision: reject. All three value criteria agree, which is worth noting — they usually will for a conventional cash flow stream. NPV is negative (−$11.57); IRR of 8.44% is below the required 9%; PI is below 1 at 0.9904.
The instructive part. Payback is 2.5 years, which sounds unambiguously acceptable — most managers would call a two-and-a-half-year payback a good project. But the project destroys value. Payback cannot see this, because it cannot see that the $400 in year 3, discounted at 9%, is worth only $308.87. This is the lecture's criticism of payback demonstrated on a single project rather than asserted.
The conflict. IRR ranks A higher; NPV ranks B higher. Because the projects are mutually exclusive, choosing one means forgoing the other, so the two criteria give genuinely incompatible advice.
The cause. This is the timing problem. B's large payoff is deferred to year 3, while A returns almost everything in year 1. IRR's reinvestment assumption credits A with redeploying that early $10,000 at 16.04% for two further years — an assumption that may not be achievable. NPV makes no such assumption; it discounts every cash flow at the market's 10% and simply compares totals.
Which governs: NPV. The objective of the firm is to maximise shareholder value, and NPV measures the addition to that value in dollars. B adds $751.31 of value against A's $668.67 — $82.64 more. Choosing A would leave $82.64 on the table.
The fragility worth noting. The crossover rate at which the two NPVs are equal is 10.55% — only half a percentage point above the required return. A small error in estimating the discount rate would flip the decision. When NPVs are this close, the quality of the discount-rate estimate matters more than the arithmetic.
No — a raw NPV comparison is not valid here, because the two machines have different lives. B's higher NPV partly reflects the fact that it is delivering benefits for five years rather than two. To compare them you must put them on the same footing, which means converting each NPV into an equivalent annual amount.
NPV of each.
Annuity factor, 2 yrs @10% = [1 − 1/(1.1)²]/0.1 = 1.7355 Annuity factor, 5 yrs @10% = [1 − 1/(1.1)⁵]/0.1 = 3.7908 Machine A: NPV = −200 + 140 × 1.7355 = −200 + 242.98 = 42.98 Machine B: NPV = −400 + 130 × 3.7908 = −400 + 492.80 = 92.80Equivalent annual annuity. Divide each NPV by its own annuity factor:
EAA_A = 42.98 / 1.7355 = 24.76 per year EAA_B = 92.80 / 3.7908 = 24.48 per yearAnswer: choose Machine A. On a per-year basis it delivers $24.76 of value against B's $24.48.
Why the ranking flipped. NPV asks "which project is worth more in total?" and answers B, because B runs for five years. EAA asks "which project creates more value per year of operation?" and answers A. When the machines are alternatives providing the same service, the second question is the right one — you will buy replacements either way.
Sanity check. Over a ten-year horizon, you would buy A five times (2-year life each) or B twice (5-year life each). Discounting the equivalent annual amounts over ten years:
A: 24.76 × 6.1446 = 152.15 B: 24.48 × 6.1446 = 150.43A wins, by a margin of $1.72. The margin is small — this is genuinely close — but the method is what matters: comparing unequal lives on raw NPV would have given the wrong answer with considerably more confidence.
Capital Budgeting
Concept Map
how the lecture buildsLook at the free cash flow formula again: EBIT − Tax + Depreciation − ΔWC − Capex. Every term is
something Lecture 1 introduced. EBIT and depreciation come from the income statement. ΔWC is the
change in net working capital — the 761 − 486 arithmetic from the balance sheet. Capex is that same lecture's
capital spending = ending fixed assets − beginning fixed assets + depreciation.
What Lecture 4 adds is not new arithmetic. It is the judgement about which of those accounting quantities belong in the forecast at all, and the discipline of converting accrual figures into cash.
Which Cash Flows Count
slides 3–8The lecture states its six rules on a single slide, and then spends five slides unpacking them. The first is the principle; the rest are its consequences.
Not accounting earnings. The difference between the cash flows with the project and the cash flows without it.
Do not matter. Already incurred, whatever you decide.
Do matter. Using an asset forgoes whatever else it could have earned.
Do matter — both erosion and synergy.
We want incremental after-tax cash flows. Tax is a cash flow, and it is often the largest single line.
Does matter. Compare real cash flows at real rates, or nominal at nominal — never mix them.
Discount cash flows, not earnings. The lecture's justification is a single sentence worth remembering verbatim: earnings do not represent real money that you can spend.
This is not a technical preference. Earnings are an accrual construct shaped by depreciation schedules, revenue-recognition rules and provisions — all of which are choices. Two identical projects can report different earnings because of a different depreciation method, while having identical cash flows. Cash flows are what the firm actually receives and pays, and they are what the discounting machinery of Lecture 2 can operate on.
The lecture's honest framing of the workload is worth noting too: much of the work in evaluating a project lies in taking accounting numbers and generating cash flows. The finance is easy; the conversion is the job.
Sunk costs 沉没成本
A sunk cost has already occurred, regardless of whether the project goes ahead. Because it is unchanged by the decision, it is not incremental — and therefore irrelevant to it.
The lecture's example is a consulting fee paid before the investment decision was made. In the Baldwin case below, a $250,000 marketing test has already been paid. It is a real cost, it will appear in the accounts, and it must be excluded from the analysis entirely.
Excluding sunk costs is simple in principle and psychologically difficult in practice. Having spent $250,000 on a study, the instinct is that the money should count for something — that abandoning the project now would "waste" it. But the $250,000 is gone either way. The only question that matters is whether the remaining investment is worth making, and the answer to that question cannot depend on a cost that has already been paid.
This is the sunk cost fallacy, and it is the single most expensive error in corporate investment decisions. The rational question is always forward-looking: given where we are now, is the next dollar worth spending?
Opportunity costs 机会成本
An opportunity cost arises when an asset could be used for something else. If the project consumes it, the revenue from that alternative use is lost — and that loss is a genuine cost of the project, even though no invoice is ever issued.
The lecture's example: a warehouse the firm owns. If the project is not taken, the warehouse can be sold. Taking the project means forgoing that sale, so the forgone proceeds are a cash flow attributable to the project. In the Baldwin case the same logic applies to the factory site, valued at $150,000.
The slide asks: if taking the project means you lose the return you could have earned by putting the money in the capital market — a bank deposit, say — is that an opportunity cost?
Economically, yes. It is a real cost, and a substantial one. But it is not treated as a cash flow in the capital budgeting calculation, and understanding why is one of the most important points in the lecture.
The lecture's resolution is on the next slide, and it is a matter of the accounting rather than the economics: financing cost is not included in the calculation of cash flows, but is reflected in the discount rate.
Why? Because double-counting is the alternative. The discount rate r is the required return of the firm's investors — the same forgone return, expressed as a rate. Discounting the project's cash flows at r already charges the project for the capital it uses. If you also subtracted the forgone interest from the cash flows, you would be charging it twice, and you would reject projects that should be accepted.
So the rule to carry into the exam: interest expense never appears in a project's cash flows. It is in the discount rate, where it belongs.
Financing costs 融资成本
The lecture lists the sources of project finance — internal cash flow, debt financing, equity financing — and makes clear that none of them is free. Each carries a cost: opportunity cost, interest expense, dividends, or the dilution implied by issuing shares at a low price.
That identity is the hinge between this lecture and Week 10. The cost the firm bears for its capital is exactly the return its investors demand — the same number observed from the two sides of the transaction. And it is captured in one place only:
Side effects 副作用
A project does not exist in isolation: it acts on the firm's other products. Both directions count, and both must be taken into the incremental cash flow.
Erosion is often treated as a reason to reject a project — "it would cannibalise our own sales." That framing is usually wrong, and the reason is worth stating.
The question is never whether a new product takes sales from an old one. It is what would happen if you did not launch it. If a competitor launches the competing product instead, your customers leave anyway — and you lose the sales without gaining the new product's margin. In that case launching and cannibalising yourself is strictly better than being cannibalised by someone else.
Erosion only counts as a genuine cost to the extent that the lost sales would not have been lost anyway. This is why incremental thinking is the discipline the whole lecture is built on: the relevant comparison is always "the firm with the project" against "the firm without it," not "with the new product" against "a hypothetical world where nothing ever changes."
Free Cash Flow
slides 9–10Free cash flow is the quantity that gets discounted. The lecture defines it as the cash flow left to shareholders and debtholders under the going concern assumption. The word "free" is doing specific work: it means the firm is free to spend the money — nothing further is required to keep the project running.
The lecture notes that the going concern assumption means "certain investment in fixed assets and working capital is required." That is the whole reason Part 2 of the formula exists.
A project that is assumed to continue operating must replace worn-out equipment and must carry the inventory and receivables that its sales require. Neither is optional, and both consume cash. So free cash flow is not simply the profit from operations — it is what is left after funding the investments needed to keep the operation going. A project can be profitable and still have negative free cash flow in a growth year, because growth in working capital absorbs more cash than the profit generates.
The two parts
The formula's treatment of depreciation looks contradictory at first: it is subtracted to reach EBIT, then added straight back. The reason is that depreciation does two separate jobs.
As an expense, it reduces taxable income — so EBIT is lower than it would be, and the tax bill is lower. That is a real cash benefit. As a non-cash charge, it involves no cash leaving the firm, so adding it back restores the cash figure.
Written the other way round, the same fact gives the tax shield form of the formula:
Now it is explicit: the project's operating cash flow is its after-tax margin, plus Depreciation × T
— a cash saving that exists only because depreciation is deductible. For a capital-intensive project, that second
term is often the difference between a positive and a negative NPV. It is also why Lease-versus-Buy decisions
(Ross Ch. 21) turn almost entirely on who can claim the depreciation shield.
The Baldwin Company
slides 11–19 · Ross 6.2This is the lecture's centrepiece and the exam's most likely long question: a complete capital budgeting analysis, built from the facts up. The slide header identifies it as Textbook 6.2, which is Ross Corporate Finance 11e, §6.2 — the same example, with the same numbers.
Every figure in this example is in $1,000s. A revenue of "100.00" means $100,000. The slide notes "Unit (1,000)" at the bottom of the first table, and it is easy to lose a factor of a thousand by the end of a five-table calculation.
The facts
Market value of the proposed factory site, which the firm already owns: $150,000
Cost of the bowling ball machine: $100,000, depreciated under MACRS 5-year
Increase in net working capital: $10,000
Production, in units, over the machine's five-year life: 5,000 · 8,000 · 12,000 · 10,000 · 6,000
Price in year 1: $20, rising 2% per year thereafter
Production cost per unit in year 1: $10, rising 10% per year thereafter
Tax rate: 34% · Discount rate: 10%
Step 0 · Separating the relevant cash flows
Before any arithmetic, the six rules from section 2 dispose of two of the facts above:
Step 1 · Capital spending and working capital — Cash Flow Part 2
The investing side of the project. There are three components: the machine, the factory site, and the working capital commitment. All figures in $1,000s.
| Year | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| (1) Bowling ball machine | −100.00 | 21.76 | ||||
| (2) Accumulated depreciation | 20.00 | 52.00 | 71.20 | 82.72 | 94.24 | |
| (3) Adjusted basis after depreciation | 80.00 | 48.00 | 28.80 | 17.28 | 5.76 | |
| (4) Opportunity cost (factory site) | −150.00 | 150.00 | ||||
| (5) Net working capital (end of year) | 10.00 | 10.00 | 16.32 | 24.97 | 21.22 | 0 |
| (6) Change in net working capital | −10.00 | −6.32 | −8.65 | 3.75 | 21.22 | |
| (7) Cash Flow Part 2 = (1) + (4) + (6) | −260.00 | −6.32 | −8.65 | 3.75 | 192.98 |
This trips people up. Look at row (5): net working capital is 10.00 at the end of year 0 and 10.00 at the end of year 1. The change is therefore zero, and the slide leaves the cell blank rather than printing a 0.
The $10,000 commitment is made at t = 0, which is why row (6) shows −10.00 in the year-0 column. It is not made again in year 1. If you mechanically subtracted successive NWC figures you would double-count it and understate the NPV.
The working capital requirement is driven by the scale of operations. It starts at 10.00 at t = 0, and the slide gives the balances through to zero at the end of year 5, when the project winds down and the working capital is released.
| Year end | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| NWC balance | 10.00 | 10.00 | 16.32 | 24.97 | 21.22 | 0 |
| Change | +10.00 | 0 | +6.32 | +8.65 | −3.75 | −21.22 |
| Cash flow impact (−ΔWC) | −10.00 | 0 | −6.32 | −8.65 | +3.75 | +21.22 |
The timing is why it matters at all. In years 2 and 3 the project is growing and absorbs cash; in year 5 it releases 21.22 back in a single lump. Discounted at 10%, that sub-stream has a present value of −5.98: committing cash early and recovering it five years later costs about $5,980 in today's money. A growing project therefore destroys value through working capital alone, before any question of operating profit arises.
Step 2 · Sales revenue slide 15
Revenue each year is units × price, where the price grows at 2%:
- Year 1 — the base case, no growth applied 5,000 × $20.00 = $100,000 → 100.00
- Year 2 — price has grown once 8,000 × [$20 × (1.02)¹] = 8,000 × $20.40 = $163,200 → 163.20
- Year 3 — price has grown twice 12,000 × [$20 × (1.02)²] = 12,000 × $20.808 = $249,696 → 249.70
- Years 4 and 5 10,000 × $21.22416 = $212,242 → 212.24 6,000 × $21.64864 = $129,892 → 129.89
Step 3 · Operating costs slide 16
The same structure, but the cost per unit grows at 10% — much faster than the 2% price growth. This divergence is deliberate in the example, and it is why the project's margins compress sharply by year 5.
- Year 1 5,000 × $10.00 = $50,000 → 50.00
- Year 2 8,000 × [$10 × (1.10)¹] = 8,000 × $11.00 = $88,000 → 88.00
- Years 3 to 5 12,000 × $12.10 = $145,200 → 145.20 10,000 × $13.31 = $133,100 → 133.10 6,000 × $14.641 = $87,846 → 87.85
Note what is happening to the margin. In year 1 the firm makes $10 per unit. By year 3 the cost is $12.10 against a price of $20.81 — a margin of $8.71. By year 5 the cost is $14.64 against a price of $21.65 — a margin of just $7.01. Costs are growing five times as fast as price.
Step 4 · Depreciation and the MACRS schedule slide 17
The machine costs $100,000 and is depreciated under the Modified Accelerated Cost Recovery System (MACRS) on a five-year schedule. MACRS is an accelerated method: it front-loads the depreciation, which front-loads the tax shield.
| MACRS 5-year | Year 1 | Year 2 | Year 3 | Year 4 | Year 5 | Year 6 | Total |
|---|---|---|---|---|---|---|---|
| Rate | 20.00% | 32.00% | 19.20% | 11.52% | 11.52% | 5.76% | 100.00% |
| Depreciation on $100,000 | 20.00 | 32.00 | 19.20 | 11.52 | 11.52 | 5.76 | 100.00 |
The schedule runs to six years, but the machine is sold at the end of year 5. The final 5.76% — $5,760 — is therefore never taken as a depreciation charge, and that is exactly why the machine's book value at sale is $5,760.
This is not a coincidence and it is worth seeing: accumulated depreciation at the end of year 5 is 20.00 + 32.00 + 19.20 + 11.52 + 11.52 = 94.24, so the adjusted basis is 100.00 − 94.24 = 5.76 — precisely the unclaimed final year. The row (3) figures in the Step 1 table are this running calculation.
Step 5 · Taxable income, tax, and net income slides 17–18
| Year | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| (8) Sales revenue | 100.00 | 163.20 | 249.70 | 212.24 | 129.89 |
| (9) Operating costs | −50.00 | −88.00 | −145.20 | −133.10 | −87.85 |
| (10) Depreciation | −20.00 | −32.00 | −19.20 | −11.52 | −11.52 |
| (11) Income before taxes | 30.00 | 43.20 | 85.30 | 67.62 | 30.53 |
| (12) Tax at 34% | −10.20 | −14.69 | −29.00 | −22.99 | −10.38 |
| (13) Net income | 19.80 | 28.51 | 56.30 | 44.63 | 20.15 |
Net income is computed and then — in a sense — set aside. It never enters the final cash flow calculation.
It appears because the lecture is following the textbook's exposition, and because seeing the accrual figure next to the cash figure is the point. Check the two against each other for year 1: net income is 19.80, but cash flow from operations is 39.80. The difference is exactly the 20.00 of depreciation — the non-cash charge.
So the final table subtracts taxes from revenue and costs, not net income. Tax is a cash outflow; net income is an accounting construct that has already had a non-cash charge taken out of it.
Step 6 · Cash Flow Part 1 slide 19
The operating side of the project: revenue, less the cash costs, less the tax actually paid. Note that depreciation is not subtracted — it is not a cash cost. Its only effect on cash is through the tax bill, which is already accounted for.
| Year | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| (1) Sales revenue | 100.00 | 163.20 | 249.70 | 212.24 | 129.89 |
| (2) Operating costs | −50.00 | −88.00 | −145.20 | −133.10 | −87.85 |
| (3) Taxes | −10.20 | −14.69 | −29.00 | −22.99 | −10.38 |
| (4) Cash Flow Part 1 = (1) − (2) − (3) | 39.80 | 60.51 | 75.50 | 56.15 | 31.67 |
The lecture gives three equivalent approaches to Part 1, and checking that they agree is the best available self-test of whether you have understood the tax treatment.
| Approach | Formula | Year 1 |
|---|---|---|
| Top-down | Sales − Costs − Taxes | 100 − 50 − 10.20 = 39.80 |
| Bottom-up | Net income + Depreciation | 19.80 + 20.00 = 39.80 |
| Tax shield | (Sales − Costs)(1 − T) + Dep × T | 50 × 0.66 + 20 × 0.34 = 39.80 |
All three give 39.80, and they will keep agreeing in every year — the algebra is identical, just rearranged. Use whichever suits the data you have been given. The tax shield form is the most informative when depreciation policy is the thing in question; the bottom-up form is the fastest when you already have a projected income statement.
Step 7 · Combining the two parts, and the NPV slide 19
| Year | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| (4) Cash Flow Part 1 | 39.80 | 60.51 | 75.50 | 56.15 | 31.67 | |
| (5) Cash Flow Part 2 | −260 | −6.32 | −8.65 | 3.75 | 192.98 | |
| (6) Incremental cash flow (4) + (5) | −260 | 39.80 | 54.19 | 66.85 | 59.90 | 224.65 |
| Year | Cash flow | Discount factor | Present value |
|---|---|---|---|
| 0 | −260.00 | 1.0000 | −260.00 |
| 1 | 39.80 | 0.9091 | 36.18 |
| 2 | 54.19 | 0.8264 | 44.79 |
| 3 | 66.85 | 0.7513 | 50.23 |
| 4 | 59.90 | 0.6830 | 40.91 |
| 5 | 224.65 | 0.6209 | 139.49 |
| NPV | 51.59 |
Small print worth knowing. Discounting the table's rounded cash flows gives 51.5948; the rounded present-value column above sums to 51.60, a 0.01 difference from column rounding. Carrying full precision through every intermediate step instead gives 51.5913. All three agree at 51.59, which is the figure the lecture reports — so the discrepancy is presentational, not substantive. Use the slide's own cash flows in the exam and round only at the end.
The salvage value, in full
The year-5 cash flow of 224.65 contains three separate items, and the tax treatment of the machine's sale is worth working through on its own because it is a common exam question.
- Find the book value (adjusted basis) — what depreciation has left on the books 100.00 − 94.24 = 5.76 ($5,760)
- Compute the taxable gain — the market value exceeds the book value, so there is a capital gain 30.00 − 5.76 = 24.24
- Compute the tax due on that gain 0.34 × 24.24 = 8.242 ($8,242)
- Arrive at the after-tax salvage value — the cash you actually keep 30.00 − 8.242 = 21.758 → 21.76
So the year-5 Cash Flow Part 2 of 192.98 is:
The commonest error here is to tax the full $30,000 of proceeds. That is wrong, and the reason is that depreciation has already given the firm a tax benefit on most of the machine's cost.
Over five years the firm deducted $94,240 of depreciation, reducing its taxable income by that amount and saving 0.34 × 94,240 = $32,042 in tax along the way. If the machine could then be sold for $30,000 with no tax consequence, the firm would have deducted nearly the whole cost and recovered a third of it in cash — a double benefit. Taxing the gain of $24.24 claws the excess back.
Note what this means in the loss case. If the machine sold for less than its $5,760 book value, the project would generate a capital loss and produce a tax credit — a positive cash flow. An asset whose market value has collapsed is worth less than nothing on the tax side, and the analysis has to reflect that.
If you remember nothing else from this example, remember that the analysis consumed $260,000 up front — machine, forgone factory sale, and working capital — and produced an NPV of $51.59 thousand. And remember the number that is not there: the $250,000 marketing test, excluded as a sunk cost. In the exam, being able to say why it is excluded is worth more than reproducing the table.
Three Routes to the Same Number, Inflation, and Risk
slides 20–23Three ways to calculate the operating cash flow
All three assume there is no interest expense — which, per the financing-cost rule, there never is.
| Approach | Formula | Use it when |
|---|---|---|
| Top-down 自上而下 | Sales − Costs − Taxes | You have revenue and cost projections. Do not subtract non-cash deductions — depreciation is not here. |
| Bottom-up 自下而上 | Net income + Depreciation | You already have a projected income statement. Fastest, and it makes the non-cash add-back explicit. |
| Tax shield 税盾法 | (Sales − Costs)(1 − T) + Depreciation × T | Depreciation policy is the question — leasing, or a change in tax regime. Isolates the shield. |
The Baldwin machine is depreciated on a five-year MACRS schedule, which is accelerated: 20% in year 1 rather than the 20% a straight-line method would also give, but 32% in year 2 rather than 20%, and so on.
Acceleration does not change the total depreciation — $100,000 either way. It changes the timing, and timing is value. Shifting depreciation earlier shifts the tax shield earlier, and a dollar of tax saved sooner is worth more than the same dollar saved later.
Take the shield in year 2 alone. Under MACRS the year-2 charge is 32.00, worth 0.34 × 32.00 = 10.88 of tax saved. Under straight-line (20% each year) it would be 0.34 × 20.00 = 6.80. The difference of 4.08 is a real cash advantage, arriving one year earlier than it otherwise would. For a capital-intensive project this is not a rounding detail — it is a large part of why accelerated depreciation is politically contested and why firms lobby over it.
It also explains the year-6 tail. MACRS 5-year property is written off over six tax years because the rates sum to exactly 100% only with that final 5.76%. The Baldwin machine is sold before that tail is claimed, which is precisely why it has a book value of 5.76 at sale — and therefore a taxable gain.
Inflation
The lecture states the rule in one line: in capital budgeting, one must compare real cash flows discounted at real rates, or nominal cash flows discounted at nominal rates. Never mix the two — that is the error, and it is a common one.
The lecture also gives the approximation that everyone actually uses in conversation:
- Exact, from the Fisher equation (1 + real) = 1.10 / 1.05 = 1.047619 real = 4.7619%
- The approximation 10% − 5% = 5.0000%
- The difference 5.0000 − 4.7619 = 0.2381 percentage points
It always overstates, and the error grows with the size of the rates: the cross-product term that the approximation drops is real × inflation. At 4.76% and 5% that is 0.238 points; at 20% and 15% it would be 3 points — large enough to change a decision. Use the approximation for intuition, the exact form for arithmetic.
The rule sounds like bookkeeping hygiene. It is not — the two routes give the same answer only if you are consistent, and mixing them gives a wrong answer that looks perfectly reasonable.
Suppose a project generates a real cash flow of 100 a year for three years, the real discount rate is 6%, and inflation is 4%. Route one, discounting real at real:
PV = 100 × [1 − 1/(1.06)³] / 0.06 = 100 × 2.673012 = 267.30Route two, converting everything to nominal first. The nominal cash flows are 104, 108.16 and 112.49; the nominal rate from Fisher is 1.06 × 1.04 − 1 = 10.24%, not 10%:
PV = 104/(1.1024) + 108.16/(1.1024)² + 112.49/(1.1024)³ = 267.30The same answer, to the cent. But now see what happens when the rule is broken in the two ways it usually is.
Using the approximation 10% instead of the exact 10.24% gives 268.45 against 267.30 — an overstatement of about 1.15, or 0.4%. Small at these rates, but it grows with the horizon and with the size of the rates.
Discounting the nominal cash flows at the real rate of 6% — the actual mixing error, and much the more dangerous of the two — gives 288.82 against the correct 267.30. That is an overstatement of 8.1%. A project appraised that way would look comfortably profitable when it is marginal.
The consistency rule is what guarantees the two routes agree; breaking it is how the error enters. The safe habit is to write the rate and the cash flows down together and check that both are labelled real or both nominal before touching the calculator.
Risk analysis
The lecture's closing observation is that the key inputs — future cash flows and the discount rate — are estimates, not facts. The response is not to pretend otherwise but to see how much the answer moves when the inputs do.
Change one input at a time and observe the effect on NPV. How does NPV change if revenues rise 10%? The virtue is identifying which assumptions the decision actually depends on — usually far fewer than the model contains.
Change several inputs together in a coherent story — a recession scenario, a competitive-entry scenario — each with a probability. Captures the fact that bad things tend to arrive together: volumes fall and prices fall and costs rise.
A step beyond both. Rather than choosing a few scenarios by hand, specify a probability distribution for each input and let the computer draw thousands of combinations. The name comes from the casino.
- Specify the basic model — the NPV formula, with its inputs named NPV = f(units, price, cost per unit, tax rate, discount rate, …)
- Specify a distribution for each variable — not a single number but a range and a shape units ~ Normal(8,000, 1,500) price ~ Normal(20, 2) …
- The computer draws one outcome — one value from each distribution units = 7,340 price = 19.12 … → one NPV
- Repeat, e.g. 10,000 times — yielding 10,000 NPVs, not one
- Calculate the distribution of NPV — mean, spread, and the probability of a negative NPV
Ross Ch. 7 is candid about the limitations, and the exam may reward knowing them.
Sensitivity analysis has a known flaw: it changes one variable at a time, but real variables move together. If demand falls, price usually falls too. Sensitivity analysis will therefore systematically understate how bad the bad case is. Its real value is diagnostic — identifying which variable the decision hinges on — rather than predictive.
Monte Carlo's weakness is the opposite: it is only as good as the distributions you feed it, and those distributions are themselves guesses. There is a temptation to treat a simulation's output as objective because a computer produced it. It is not — it is your assumptions, propagated. The technique's genuine contribution is to make the shape of the risk visible, which a point estimate cannot do.
Ross also covers two techniques the lecture does not reach: break-even analysis, which solves for the sales level at which NPV is zero (and for the accounting break-even, at which net income is zero — a different and much less useful number), and real options, the recognition that a manager can expand, defer or abandon a project as information arrives, which is worth something and which plain NPV ignores entirely.
Formula Sheet
print-friendlyKey Concepts
the vocabularySelf-Check
16 questions · gradedThe examinable core of this lecture is the Baldwin analysis and the six relevance rules that drive it. If you can reproduce the first table and explain why the marketing test is missing from it, everything else follows.
A · Concept check
The general reason. Earnings are an accrual construct, shaped by accounting elections that have nothing to do with cash. Two identical projects can report different net income purely because they chose different depreciation methods. Cash flows are what the firm actually receives and pays, and they are the only thing the discounting machinery of Lecture 2 can operate on.
The specific items. Four bridges from net income to cash:
- Depreciation — deducted to reach net income (lowering it and the tax bill) but involving no cash outflow. Added back in full. In Baldwin year 1, net income of 19.80 becomes operating cash flow of 39.80 once the 20.00 of depreciation is restored.
- Changes in working capital — accrual accounting books revenue when earned, not when collected. Rising receivables and inventory consume cash that profit never reflects.
- Capital spending — the purchase of fixed assets does not appear in net income at all, beyond its depreciation. It is a cash outflow nonetheless, and often the largest one.
- Taxes actually paid — deferred tax charges reduce reported income without cash leaving the firm, in the same way depreciation does.
The consequence. A firm can be profitable and still run out of cash. Free cash flow is what is left after funding the investments the going concern requires, and it is that figure — not earnings — which determines what the project is worth.
B · Multiple choice
C · Applied calculation
Step 1 — annual depreciation. Straight-line over three years:
300,000 / 3 = 100,000 per yearStep 2 — annual operating cash flow.
EBIT = 400,000 − 150,000 − 100,000 = 150,000 Tax = 150,000 × 0.30 = 45,000 FCF = EBIT − Tax + Depreciation − ΔWC − Capex = 150,000 − 45,000 + 100,000 − 0 − 0 = 205,000 per yearCross-check with the tax shield form: (400,000 − 150,000) × 0.70 + 100,000 × 0.30 = 175,000 + 30,000 = 205,000. Matches.
Step 3 — assemble the cash flows. The working capital is an outflow at t = 0 and an inflow at t = 3. Note that it is not deducted from the annual operating cash flow — only the change in working capital enters FCF, and there is none in years 1 and 2.
t = 0: −300,000 − 25,000 = −325,000 t = 1: 205,000 t = 2: 205,000 t = 3: 205,000 + 25,000 = 230,000Step 4 — discount at 10%.
−325,000 / 1.000 = −325,000.00 205,000 / 1.100 = 186,363.64 205,000 / 1.210 = 169,421.49 230,000 / 1.331 = 172,802.40 ─────────── NPV = 203,587.53Answer. NPV = $203,587.53. Accept the project. The margin is comfortable — roughly 63% of the initial outlay — so this decision is not sensitive to small errors in the estimate.
Two things to notice. First, the $25,000 of working capital costs less than $25,000 in present value terms, because it comes back: its net present value is −25,000 + 25,000/1.331 = −$6,217. Second, the depreciation tax shield contributes 100,000 × 0.30 = $30,000 a year, or about $74,600 of present value — over a third of the total NPV comes from the tax treatment of the equipment, not from its operation.
Step 1 — book value at the end of year 5. The 5-year MACRS schedule claims 20% + 32% + 19.2% + 11.52% + 11.52% = 94.24% over five years; the final 5.76% would be claimed in year 6, but the machine is sold first.
Book value = 500,000 × (1 − 0.9424) = 500,000 × 0.0576 = 28,800Step 2 — taxable gain.
Gain = 120,000 − 28,800 = 91,200Step 3 — tax due and after-tax salvage.
Tax = 91,200 × 0.34 = 31,008 After-tax salvage = 120,000 − 31,008 = 88,992Answer. Book value $28,800; tax $31,008; after-tax salvage $88,992.
If it sold for $20,000 instead. Market value of $20,000 is now below the book value of $28,800, so the sale generates a capital loss of $8,800 rather than a gain. Assuming the loss is deductible against other taxable income, it produces a tax credit:
Loss = 20,000 − 28,800 = −8,800 Tax credit = 8,800 × 0.34 = +2,992 After-tax salvage = 20,000 + 2,992 = 22,992The after-tax salvage value of $22,992 exceeds the $20,000 sale price. That is not an anomaly — it is the depreciation schedule unwinding. The firm over-depreciated the machine relative to its actual decline in value, and the tax system refunds part of that when the asset is disposed of below book value.
Route 1 — real cash flows at the real rate.
Annuity factor, 3 yrs @ 6% = [1 − 1/(1.06)³] / 0.06 = 2.673012 PV = 100 × 2.673012 = 267.30Route 2 — convert everything to nominal. The nominal cash flows grow with inflation; the nominal rate comes from the Fisher equation, not from adding 6% and 4%.
Nominal rate = (1.06 × 1.04) − 1 = 0.1024 → 10.24% Nominal cash flows: 100 × 1.04¹ = 104.0000 100 × 1.04² = 108.1600 100 × 1.04³ = 112.4864 PV = 104.0000/1.1024 + 108.1600/1.1024² + 112.4864/1.1024³ = 94.2394 + 88.9903 + 84.0706 = 267.30The two routes agree to the cent. That agreement is the whole point of the consistency rule, and it is not a coincidence — it is guaranteed by using the Fisher rate rather than the approximation.
What goes wrong if you mix them. Discounting the nominal cash flows at the real rate of 6% — the actual error, and an easy one to make when the two numbers sit on different lines of your spreadsheet:
PV = 104/1.06 + 108.16/1.06² + 112.4864/1.06³ = 288.82That is 288.82 against the correct 267.30 — an overstatement of 8.1%. A project appraised this way would look comfortably profitable when it is marginal, and the error grows with both the inflation rate and the project's horizon. The safeguard is procedural rather than intellectual: label every rate and every cash flow as real or nominal before starting the arithmetic, and check that the labels match.
Stock Valuation
Concept Map
how the lecture buildsThe constant-growth model is not new arithmetic. Lecture 2 established that a growing perpetuity paying Div1 next period, growing at g, discounted at R, is worth Div1 / (R − g) — and then immediately flagged that the formula only holds when R > g. That formula appears here twice: once as the dividend discount model, and again divided by the share count as the DCF method.
The free cash flow definition is lifted straight from Lecture 4 — EBIT − Tax + Depreciation − ΔWC −
Capex — with one addition, the from-net-income form on slide 25, which is where the interest add-back
question lives. Slides 26 to 31 then work a numerical example that is deliberately a refresher of the Baldwin
case.
What a Share Is, and Where It Trades
slides 2–9The lecture opens not with a formula but with a chart of 宁德时代 (CATL, 300750) running from under ¥200 to ¥467 and back down again, overlaid with moving averages — and a question underneath it: 好的公司一定是一项好的投资吗？ The slide supplies the bait: CATL's 2025 revenue passed ¥400 billion, up 17%.
The slide asks it directly, and answers it at the very end of the lecture (slide 37). Revenue growth of 17% tells you the company is doing well. It tells you nothing about whether the price you pay today is a good price.
The resolution is the distinction this whole lecture is built on: price is what you pay, value is what you get, and they are not the same number. A wonderful company bought at a price above its intrinsic value is a bad investment; a mediocre company bought below its intrinsic value is a good one. The return comes from the gap, not from the quality.
Concretely: if you paid the peak price of ¥467 for CATL and it fell to the ¥195 low, you lost 58% — in a period when the company's revenue was growing. Nothing was wrong with the business. Something was wrong with the price.
The institutional history, in two slides
The lecture spends two slides on context that is easy to skip and worth keeping, because it explains why the rules that follow look the way they do. Slide 3 shows a 1990 Shenzhen Special Zone Daily share-price table and two paper share certificates — 深圳发展银行 and 万科 — the physical objects that a share used to be. The date on the slide is the founding one: 1990年12月1日, the Shenzhen Stock Exchange began centralised trading. Slide 4 then shows a still from 《繁花》, whose subject is the same decade's Shanghai market.
Thirty-five years later a share is an electronic entry, but the legal content has not changed: a firm sells a proportion of the ownership to investors in exchange for funds, and investors become shareholders entitled to voting rights, dividends and capital gains.
What actually changes on the books
When a firm issues new shares, the accounting is mechanical and worth knowing, because it is the one place where the value of equity is recorded at something close to a market price:
The par value is a legal artefact — a nominal amount, historically the floor below which shares could not be issued, and in most jurisdictions now close to meaningless. In China, A-share par value is fixed at ¥1 per share by convention. The capital surplus account carries everything above par.
Why keep both? Because par value is tied to the firm's legal capital — in many jurisdictions it cannot be distributed to shareholders, and it is the figure creditors look to. So a ¥10m issue of 1m shares at ¥10 each does not simply add ¥10m to one line: it adds ¥1m to common stock and ¥9m to capital surplus. The distinction is between capital the firm must keep and capital it may distribute.
This is also why share repurchases and stock dividends are more complicated than they look. Neither raises cash, but both move amounts between these two equity accounts and retained earnings — and the accounting rules constrain which way the amounts may move.
Primary versus secondary market 一级市场 / 二级市场
This is the single most examinable distinction in the lecture's first half, and it is examinable precisely because it is simple enough to be stated in one line: does the money go to the firm, or to another investor?
The new-issue market. Stocks are brought to the market for the first time and sold to investors, and the money flows to the issuing company.
IPO — initial public offering: a firm sells its shares to the public for the
first time.
SEO — seasoned equity offering: a firm that is already public issues new shares
again.
Private placement 定向增发: money raised privately from certain
investors.
Where existing shares are traded among investors. The firm is not a party to the transaction and receives nothing — the money flows among investors.
Everything you see on a stock screen is the secondary market. The price formed there is what the firm's own subsequent share issues will be priced against, which is the channel through which the secondary market still matters to the firm.
The lecture makes the distinction concrete with a pair of October 2024 news stories, and they are worth reading as a matched set:
| Item | Market | Why |
|---|---|---|
| 烽火通信 (600498) 拟向特定对象发行股票，募资不超过 15 亿元 | Primary | A new issue to selected investors. The ¥1.5bn goes into the company, and the slide notes it is earmarked for working capital to support day-to-day operations. |
| A 股成交额连续 4 个交易日超 2 万亿元 | Secondary | Turnover of ¥2.16tn in one day is existing shares changing hands. Not one yuan of it reaches any listed company. |
The slide's own gloss on the second story is the useful one: the market that day contracted (缩量) even while turnover stayed above ¥2tn, the Shanghai Composite rose 1.32%, the Shenzhen Component fell 0.82% and the ChiNext fell 2.95%, with 2,998 stocks up, 2,229 down and 82 at the daily limit.
If the firm gets nothing from a secondary-market trade, the natural question is why a firm would want its shares traded at all. The lecture does not ask this; Ross Ch. 9's own discussion of the stock market is the place to look for it.
Three answers. Liquidity — investors will pay more for an asset they can exit, so a liquid secondary market raises the price at which the firm can sell its next primary issue. Price discovery — the traded price is the market's continuously updated estimate of the firm's value, which is what private placements, SEOs, employee options and takeover offers are all priced against. Governance — a liquid market means a dissatisfied shareholder can exit rather than having to fight, and, in extremis, that a badly run firm becomes cheap enough for someone else to buy and fix. That last channel is the market for corporate control from Lecture 1.
Shareholder Rights, and the Fight for Control
slides 10–17Slide 6 sets out the three-party structure that the rest of this section assumes. Shareholders elect directors; directors appoint and oversee managers; managers run the firm. The slide's arrows point back up from managers to shareholders because the chain is also a chain of accountability.
Directors 董事: 董事长、董事会主席 · 独立董事 · 执行董事
Managers 职业经理人: 高管 · 总经理、总裁 · 财务总监、人力资源总监等
The point of setting it out is that "the firm" is not a single decision-maker. When the lecture says a project should be accepted if its NPV is positive, someone has to want that — and the sections that follow are about who that someone is, and what they are incentivised to want.
Two rights, and they can be separated
A share carries two distinct things, and much of corporate governance is the story of pulling them apart:
Controlling versus minority shareholders
The lecture draws the line where control actually changes hands, and the thresholds are jurisdiction-specific — a detail worth noticing:
A firm may have no controlling shareholder at all if ownership is well diversified. Where there is one, the lecture states the risk plainly: a controlling shareholder may divert firm resources for its own private benefit, jeopardising the wealth of minority shareholders — an agency conflict.
Lecture 1 framed the agency problem as managers versus shareholders — the separation of ownership from control, and the three mechanisms that contain it: compensation contracts, the market for corporate control, and the board.
Slide 10 is a different axis: controlling shareholders versus minority shareholders. The conflict is not between owners and non-owners; both are owners. It is between an owner who can direct the firm and owners who cannot.
The mechanism matters because the remedy is different. Manager-shareholder conflict is contained by aligning incentives. Controlling-shareholder conflict is contained by limiting what a majority vote can do — which is exactly what cumulative voting, proxy access and independent directors are for, and why the lecture spends the next five slides on voting rules rather than treating them as administrative detail.
Voting procedure: cumulative versus straight 累积投票 / 直接投票
The default rule is one share, one vote. What varies is how the votes are cast when several directors are elected at once, and the difference decides whether a minority holder can be represented at all.
Directors are elected all at once, and a shareholder may concentrate all of their votes on a single candidate.
This favours minority shareholders. The lecture gives the threshold: if N directors are up for election, holding 1/(1+N) of the shares is enough to guarantee one seat.
Each director is voted on separately, so the votes cannot be concentrated.
The controlling shareholder dominates and elects every seat. A minority holder can win nothing, however large the holding — short of outright majority.
- Cumulative voting — count the votes. Each share carries one vote per seat: Smith: 20 × 4 = 80 votes Jones: 80 × 4 = 320 votes Total 400 votes
- Can Smith guarantee a seat? Yes — by giving all 80 votes to one candidate. Jones's 320 votes spread over four seats average 80 each, so Smith's concentrated 80 cannot be beaten by any single Jones candidate: Smith concentrates 80 on one nominee Jones's best single nominee: 320 / 4 = 80
- Straight voting — the same numbers, a different answer. Votes are cast seat by seat, and Smith loses every one: Seat 1: Jones 80 vs Smith 20 → Jones Seat 2: Jones 80 vs Smith 20 → Jones Seat 3: Jones 80 vs Smith 20 → Jones Seat 4: Jones 80 vs Smith 20 → Jones
Check the threshold rule against the numbers: with N = 4, a guaranteed seat needs 1/(1+4) = 20% of the shares — which is exactly the 20 shares Smith holds. Smith is precisely on the boundary, and the boundary is exact, not approximate. One share fewer and the 80 votes fall short of Jones's 80-vote average.
The lecture states the rule; it is worth being able to derive it, because the derivation is two lines and the exam may ask for it.
Let S be the shares outstanding and N the number of directors to elect. A minority holder with fraction x of the shares can cast xSN votes for a single candidate. Everyone else has (1−x)S shares and (1−x)SN votes, which they must spread across N seats — so their strongest single candidate gets at most (1−x)SN / N = (1−x)S votes.
The minority holder wins a seat when their concentrated votes exceed that: xSN > (1 − x)S xN > 1 − x x(N+1) > 1 x > 1 / (1 + N)
So the threshold is a strict inequality: you need more than 1/(1+N). Slide 12's Smith holds exactly 1/5 with N = 4, and the rule as the slide states it ("will guarantee a seat") works only because Jones's votes cannot all be concentrated on one candidate — the seat is won on the comparison of single-candidate totals, where 80 ties with 80 and Smith, as the nominee, prevails. Read the formula as a good approximation and the derivation as the exact test; in an exam, show the vote totals.
Proxy voting and proxy fights 代理投票 / 代理权争夺
Proxy voting lets a shareholder transfer their voting right to another party to exercise on their behalf. The lecture's example is the ordinary case: a mutual fund casts votes on behalf of the fund's investors, because those investors cannot practically attend.
A proxy fight is the same mechanism weaponised. Management wants to accumulate as many proxies as possible. An outside group of shareholders can solicit proxies instead, in an attempt to replace management by electing enough directors. Control of the firm can therefore change hands without a single share being bought — which is why proxy rules are regulated as heavily as they are.
The lecture illustrates proxy voting and control with the ownership of 娃哈哈, and asks the question the pie chart is designed to make awkward.
| Shareholder | Stake | What it is |
|---|---|---|
| 杭州上城区文商旅投资控股集团有限公司 | 46% | A Hangzhou district-level state-owned investment vehicle |
| 宗馥莉 | 29% | The founder's family |
| 杭州娃哈哈集团有限公司基层工会联合委员会 | 25% | The employee shareholding committee (职工持股会) |
The answer the chart is fishing for: 46% is the largest single block and is state-owned, but it is below the 50% threshold the previous slide gave as the Chinese control line — and the remaining 54% is split between the family and the employees' committee. So the honest answer is that 娃哈哈 is a mixed-ownership enterprise in which the state is the largest shareholder and the family, through the founder's legacy and the employee block, retains operational control. Neither label in the question is quite right.
The point for this lecture is not the corporate history. It is that control is a function of how the votes combine, not of who owns most — which is exactly what the cumulative-voting example above established, at smaller scale.
Two devices that dilute voting power
Both of the following take votes away from some shareholders, and both are contested — which is why they are in the lecture.
Two separate problems on one slide. Both are worth knowing, because the exam may well be marked against the slide.
1 · Alibaba does not have dual-class shares. The slide says "Jack Ma's A-class shares in Alibaba have 10 times voting rights as B-class shares." Alibaba listed in New York in 2014 with a single class of shares and one vote per share. What it has instead is the 合伙人制度 (Lakeside partnership): the partnership holds the right to nominate a majority of the board. That controls the board, whereas a dual-class structure controls the shareholders' meeting — a genuinely different mechanism, and the reason the HKEX refused Alibaba in 2013 and lost the listing to New York.
Chinese companies that do use supervoting shares, for accurate reference: 小米集团 (first weighted-voting-rights listing in Hong Kong, July 2018 — Class A carries 10 votes), 美团 (second, September 2018), and 京东 (Class B carries 20 votes). If the exam compares mechanisms across companies, that is the correct set.
2 · The blanket claim about China is out of date. The slide says CSRC does not allow this structure for Chinese public companies. That was true until 2019, when the STAR Market listing rules admitted 特别表决权股份; the ChiNext followed in 2020. 优刻得 (UCloud, 688158) listed on the STAR Market on 20 January 2020 as A-share's first dual-class issuer — its founders held about 23.1% of the shares and about 60.1% of the votes, through Class A shares carrying 5 votes each against Class B's one. Its prospectus identified the resulting governance risk explicitly.
The narrower statement that is still true: dual-class structures remain prohibited for ordinary A-share listings on the main boards, and the STAR/ChiNext regime caps the voting ratio and restricts who may hold supervoting shares. If you are asked "is it allowed in China?", the defensible answer is now "on the STAR Market and ChiNext, within limits, since 2019–2020; not on the main boards" — not a flat no.
Preferred stock 优先股
Preferred stock sits between common stock and debt, and the lecture compares it to each in turn.
| Feature | Preferred vs common stock | Preferred vs debt |
|---|---|---|
| Dividends | A preference in receiving dividends — paid before any common dividend | The cash flow pattern is very similar to debt |
| Liquidation | Priority in the distribution of assets in liquidation | But ranked below debt: debt > preferred > common |
| Voting | Typically no voting right — the trade for preference | — |
| Maturity | Typically no maturity date — it is not repaid on a fixed schedule | — |
| Stated value | Has a stated liquidating value | — |
| Accounting | — | Recorded differently from debt — this is the main formal difference that remains |
If preferred stock's cash flows are so close to debt's, and its liquidation rank is only one notch above, why does the lecture single out the accounting treatment as the difference that remains? Because that treatment is what firms actually manage around.
Preferred dividends are generally treated as a distribution of profit, not an expense in the accounts. So a firm that issues preferred stock instead of debt reports higher net income, a lower debt-to-equity ratio and a higher interest-coverage ratio than an otherwise identical firm with debt — without any difference in the cash it must pay out. Leverage ratios and coverage ratios built from the accounts therefore improve.
That is why preferred stock is a favourite instrument of financial institutions, for which regulatory capital ratios are computed from these same numbers, and why the distinction between "debt-like" and "equity-like" is one of the more heavily litigated questions in accounting. The lecture's one-line note — "their accounting recording is different" — is doing more work than it appears to.
The Dividend Discount Model
slides 18–23Start from value, not from price
Slide 18 sets up the whole valuation half of the lecture in four lines.
The price a stock should have, as reflected by the operating strength of the firm. Stock valuation means estimating the intrinsic value.
Price fluctuates around intrinsic value, driven by short-run supply and demand. The strategy that follows is the entire content of the lecture's last slide: buy undervalued stocks, sell overvalued ones.
The slide poses it here and answers it on slide 37. Hold the question: the model below is what turns it from a slogan into a calculation.
Three methods, from three perspectives
Slide 19 lists them in the form the exam is most likely to ask for — by whose cash flows are being discounted, which determines whose claim on the firm is being valued.
| Perspective | Method | Whose cash flow |
|---|---|---|
| Investors' | Discounted dividend method 股利折现法 | The dividends received by a minority shareholder |
| Firm's | Discounted cash flow method 现金流折现法 | The free cash flow of the whole business — hence a controlling claim |
| Industry's | Comparable method 比较法 | Not cash flows at all — the multiples of similar firms |
Slide 20 restates the principle that governs the first two, and it is the same sentence Lecture 3 opened with: the value of any asset is the present value of its expected future cash flows. For a share, those cash flows arrive in exactly two forms — dividends and capital gains.
Case 1 · zero growth
Assume dividends stay at the same level forever: Div1 = Div2 = Div3 = …. The present value of a constant stream is the present value of a plain perpetuity — Lecture 2's formula, unchanged.
Lecture 2 derived that a perpetuity paying C forever, discounted at R, is worth C/R — and used a ¥15 payment at 10% to get ¥150. Nothing has changed here except the symbol. The only condition is R > 0.
The economics is worth naming: a zero-growth stock is a perpetual bond with a fully discretionary coupon. It has no maturity, and the dividend can be cut to zero at the board's discretion — which is why the required return R on it is far above the yield on a government bond of the same duration.
Case 2 · constant growth
Now assume dividends grow at a constant rate g forever, so that Div1 = Div0(1+g), Div2 = Div1(1+g) = Div0(1+g)², and so on. The stream is a growing perpetuity, and Lecture 2 already priced one of those.
1 · Div1, not Div0. The slide flags this itself in red, and it is the single most common error. The formula values next year's dividend. If the question hands you the dividend just paid, you must grow it once before using it. The reason is structural: the perpetuity's first cash flow sits at t = 1, and P0 is the value one period before it.
2 · The formula requires R > g. If g ≥ R the denominator is zero or negative and the formula returns a meaningless (or negative) price. This is not a numerical nuisance; it is the model telling you the assumption is impossible. A firm cannot grow faster than the whole economy forever — it would eventually become the economy. So g must be below the long-run growth rate of nominal GDP, and R is above that, which is why the condition is nearly always satisfied by a sensible g.
3 · Growth is not free. The model treats g as given, but growth requires investment, and investment consumes the cash that would otherwise be paid out. Raising g in the numerator without reducing the dividend is a common and seductive error; the section on NPVGO below is the correction.
- Grow the dividend once — because the slide gives you Div0, not Div1: Div₁ = 0.50 × (1 + 0.02) = 0.5100
- Apply the constant-growth formula: P₀ = 0.5100 / (0.15 − 0.02) = 0.5100 / 0.13 = 3.9231
See the trap: using Div0 directly gives 0.50/0.13 = $3.85, which is $0.08 too low. On a $3.92 stock that is a 2% error — small here, but the error is proportional to g, so at g = 5% the same slip understates the price by 5%.
The lecture hands you g in the question. Ross §9.2 asks the harder question — where would you get it in practice — and the answer is the link between growth and retained earnings.
A firm that pays out a fraction of its earnings and reinvests the rest grows only if those reinvested funds earn a return. Writing b for the retention ratio (the fraction of earnings not paid out) and ROE for the return on equity:
This makes the trade-off explicit and is the single most useful formula beside the model itself. A firm retaining nothing has b = 0 and hence g = 0 — it is the zero-growth case. A firm retaining everything has b = 1 but pays no dividend, so the model has no numerator. And g cannot be raised without either retaining more (which cuts the dividend now) or earning a higher ROE on what is retained (which is the only genuinely value-creating route).
The same algebra runs backwards to give the required return, which is how the model is most often used in practice:
Read it as: your total return is the cash you receive plus the rate at which the asset appreciates — which is what a return is. It also explains why a low dividend yield is not automatically a bad sign: a firm retaining earnings for high-ROE projects should have a low yield and a high g, and the two components sum to the same required return.
Slide 20 lists three cases and marks the third — differential growth — "learn by yourself". It is the case that actually describes most real companies: fast growth for a period, then a transition, then stable growth forever. Ross §9.1 works it as a two- or three-stage model. The technique is to split the dividend stream at the point where growth becomes constant, discount the early dividends individually, and value the rest with the constant-growth formula applied to the first steady dividend.
Two things are worth knowing even if the arithmetic is left for self-study. First, the split is arithmetic only — every piece is either a single discounted cash flow or a growing perpetuity. Second, and more important, in these models the terminal value dominates: typically 70–90% of the answer comes from the constant-growth stage, which is built on an assumption about the distant future. The precision of the early-year forecasts is therefore largely decorative. That is the "healthy sense of skepticism" Ross asks for in §9.2.
Ross also covers NPVGO in this chapter — the firm valued as the present value of its earnings if it never grew, plus the net present value of its growth opportunities. It is the formal version of the caution above: growth adds value only to the extent that the projects it funds have positive NPV, which is Lecture 3's rule applied to the firm's own investment programme.
Discounted Cash Flow — and a Full Worked Example
slides 24–31 · Ross Ch. 4 exhibitsSlide 24 states the big-picture idea in a single sentence, and it is the sentence that justifies the whole method: a firm can be viewed as the sum of many projects, and its value can be calculated by discounting the total expected free cash flows.
Lecture 4 valued one project: forecast the incremental cash flows, discount at the project's required return, accept if the NPV is positive. This slide says the firm is just a bundle of such projects, so the same procedure applies at the top level. The formula is the same formula, the discount rate is the same kind of discount rate, and the only change is the scope of the forecast.
Two consequences follow, and the lecture states both. Because the free cash flow belongs to all the firm's capital providers, discounting it gives the firm value, not the equity value — so equity value = firm value − debt value. And all of these are market values, not book values, with the practical note that the market value of debt is usually very close to its book value.
The slide also unpacks the word "free", and it is the same point Lecture 4 made: free means the cash flows are left for the firm to use freely. Under the going-concern assumption the firm must invest in working capital and fixed assets to keep operating normally — so those outflows come off the top before anything is "free".
Two ways to calculate it — and the interest question
Slide 25 gives both routes, and the second one carries the lecture's only explicit "quick question".
The slide asks it and does not answer it. It is the most examinable question on this slide, so here is the answer in full.
Because net income is what is left after the debtholders have been paid, and free cash flow is what is left before. Free cash flow is the cash available to all capital providers — shareholders and creditors. Net income has already deducted interest, so it belongs to shareholders alone. Adding interest back converts a shareholder figure into a firm figure. This is the same rule as Lecture 4's: financing flows are not part of the project's (or the firm's) cash flows.
But why (1 − Tax rate) rather than the full amount? Because interest is tax-deductible. Every ¥100 of interest actually paid costs the firm only ¥100 × (1 − T), since it reduces taxable income by ¥100 and therefore saves T × 100 in tax. Grossing it up to the full ¥100 would overstate the cash the debtholders received relative to what the firm actually gave up.
And the direction of the mistake matters. If you discount free cash flow at the WACC — a rate that already reflects the tax advantage of debt — and then fail to make this adjustment, you have deducted the interest twice: once in the cash flow and once in the discount rate. The result understates firm value. This is exactly the double-counting trap from Lecture 4, arriving from the other side.
FCF: Example — Cane Distribution, Inc.
Slides 26 to 31 work a complete example. It is the textbook's Cane Distribution case, and the slides' own exhibit numbers (4-1, 4-2, 4-3) identify it as Chapter 4 material — a deliberate refresher of the free cash flow construction before the comparable method arrives.
Initial capital invested in fixed capital of $500,000 and working capital of $60,000
Working capital initially consisted solely of inventory
Fixed capital: $50,000 non-depreciable property + $450,000 depreciable property
Depreciable property has a 10-year life, no salvage value
Task: starting with net income, calculate Cane's FCFF for each year
The opening balance sheet checks out: 224 + 336 = 560 of capital raised, and 500 + 60 = 560 invested. Note that the debt and equity are both stated, so the example is capital-structure-aware from the start — which is why the interest add-back is needed at all.
Exhibit 4-1 · Income statement slide 27, $1,000s
| Years ending 31 December | 2008 | 2009 | 2010 |
|---|---|---|---|
| EBITDA | 200.00 | 220.00 | 242.00 |
| Depreciation expense | −45.00 | −49.50 | −54.45 |
| Operating income | 155.00 | 170.50 | 187.55 |
| Interest expense (at 7 percent) | −15.68 | −17.25 | −18.97 |
| Income before taxes | 139.32 | 153.25 | 168.58 |
| Income taxes (at 30 percent) | −41.80 | −45.97 | −50.58 |
| Net income | 97.52 | 107.28 | 118.00 |
A dropped digit, and three independent checks confirm it. The table above shows the corrected figure; the slide shows 8.97.
Check 1 — the column does not foot. Operating income less interest must equal income before taxes: 187.55 − 8.97 = 178.58 ✗ slide prints 168.58 187.55 − 18.97 = 168.58 ✓
Check 2 — the stated 7% rate. Interest is charged at 7% on long-term debt, and the balance sheet gives long-term debt of $271.04 thousand at the end of 2009: 271.04 × 0.07 = 18.9728 → 18.97 The same check works for the other two years, which is what makes the third one so clearly a typo: 224.00 × 0.07 = 15.68 ✓ and 246.40 × 0.07 = 17.25 ✓.
Check 3 — the solution slide contradicts it. Slide 31's answer table reports "Interest expense × (1 − Tax rate)" of 13.28 for 2010. Working backwards: 13.28 / 0.70 = 18.97, not 8.97. So the instructor's own solution used the correct figure — only the income statement's line is mistyped.
The rest of the 2010 column is self-consistent once the interest is corrected: 168.58 × 0.30 = 50.574 → 50.58, and 168.58 − 50.58 = 118.00, exactly as printed. So this is a transcription slip in one cell, not a modelling error. One further one-cent artefact, for completeness: 2009's tax should round from 153.25 × 0.30 = 45.975 to 45.98, and the slide prints 45.97. The slide is internally consistent after that (153.25 − 45.97 = 107.28), so it is a rounding convention rather than an error — but it is why 45.97 and 45.98 both show up in different printings of this table.
Exhibit 4-2 · Balance sheet slides 28–29, $1,000s
| Years ending 31 December | 2007 | 2008 | 2009 | 2010 |
|---|---|---|---|---|
| Cash | 0.00 | 108.92 | 228.74 | 360.54 |
| Accounts receivable | 0.00 | 100.00 | 110.00 | 121.00 |
| Inventory | 60.00 | 66.00 | 72.60 | 79.86 |
| Current assets | 60.00 | 274.92 | 411.34 | 561.40 |
| Fixed assets | 500.00 | 500.00 | 550.00 | 605.00 |
| Less: accumulated depreciation | −0.00 | −45.00 | −94.50 | −148.95 |
| Total assets | 560.00 | 729.92 | 866.84 | 1,017.45 |
| Accounts payable | 0.00 | 50.00 | 55.00 | 60.50 |
| Current portion of long-term debt | 0.00 | 0.00 | 0.00 | 0.00 |
| Current liabilities | 0.00 | 50.00 | 55.00 | 60.50 |
| Long-term debt | 224.00 | 246.40 | 271.04 | 298.14 |
| Common stock | 336.00 | 336.00 | 336.00 | 336.00 |
| Retained earnings | 0.00 | 97.52 | 204.80 | 322.80 |
| Total liabilities and equity | 560.00 | 729.92 | 866.84 | 1,017.45 |
The tables look like four independent columns of numbers. They are not — every figure is pinned by the others, and checking that is the best way to be sure you have understood the model.
Retained earnings walk forward with no dividends. 0 + 97.52 = 97.52; 97.52 + 107.28 = 204.80; 204.80 + 118.00 = 322.80. The company pays nothing out, so net income accumulates in full.
Long-term debt grows at exactly 10% a year — 224.00 → 246.40 → 271.04 → 298.14 — which is what makes the interest bill grow at 7% on a growing principal.
Cash is the plug, and it reconciles exactly. Start from the Lecture 4 identity, take the cash flow to equity (net income plus depreciation less the investing outflows), add the new borrowing, and you land on the printed cash balance every year: 2008: 0 + (97.52 + 45.00 − 56.00 − 0.00) + (246.40 − 224.00) = 108.92 ✓ 2009: 108.92 + (107.28 + 49.50 − 11.60 − 50.00) + (271.04 − 246.40) = 228.74 ✓ 2010: 228.74 + (118.00 + 54.45 − 12.76 − 55.00) + (298.14 − 271.04) = 360.53 ≈ 360.54 ✓
This is the single most useful thing to take from the example. It confirms that the "investment in fixed capital" line in the solution is net capital spending — the Lecture 1 definition, ending fixed assets minus beginning fixed assets plus depreciation. On the stated gross figures that is not obvious, but on the net figures it drops out: 2008: 455.00 − 500.00 + 45.00 = 0.00 ✓ (slide shows 0.00) 2009: 455.50 − 455.00 + 49.50 = 50.00 ✓ (slide shows 50.00) 2010: 456.05 − 455.50 + 54.45 = 55.00 ✓ (slide shows 55.00) The net fixed asset figures (500.00, 455.00, 455.50, 456.05) are total assets less current assets, since the balance sheet reports fixed assets gross with accumulated depreciation shown separately.
Exhibit 4-3 · Working capital slide 30, $1,000s
Note carefully what is excluded and why: cash is out of current assets, and short-term debt is out of current liabilities. Working capital here means the operating working capital tied up in the business, not the treasury position.
| Years ending 31 December | 2007 | 2008 | 2009 | 2010 |
|---|---|---|---|---|
| Accounts receivable | 0.00 | 100.00 | 110.00 | 121.00 |
| Inventory | 60.00 | 66.00 | 72.60 | 79.86 |
| Total current assets excluding cash | 60.00 | 166.00 | 182.60 | 200.86 |
| Accounts payable | −0.00 | −50.00 | −55.00 | −60.50 |
| Working capital | 60.00 | 116.00 | 127.60 | 140.36 |
| Increase in working capital | 56.00 | 11.60 | 12.76 |
Each year's working capital absorbs cash, so the increase is subtracted in the FCF calculation. And as in Lecture 4, this is a matter of timing, not of value destroyed: the increase in working capital is an outflow while the business grows, and it would all come back if operations ever wound down. Cane Distribution never winds down in this example, so the outflow simply accumulates — 56.00 + 11.60 + 12.76 = 80.36 over the three years, all of it still sitting on the balance sheet as receivables, inventory and the payables offsetting them.
Slide 31 · The solution
| Years ending 31 December | 2008 | 2009 | 2010 |
|---|---|---|---|
| Net income | 97.52 | 107.28 | 118.00 |
| Non-cash charges — depreciation | 45.00 | 49.50 | 54.45 |
| Interest expense × (1 − Tax rate) | 10.98 | 12.08 | 13.28 |
| Investment in fixed capital | −0.00 | −50.00 | −55.00 |
| Investment in working capital | −56.00 | −11.60 | −12.76 |
| Free cash flow to the firm | 97.50 | 107.26 | 117.97 |
- Start from net income — as slide 118.00, confirmed above.
- Add back the non-cash charge: + 54.45 (depreciation, from Exhibit 4-1)
- Add back after-tax interest — the firm-level adjustment from slide 25: 18.97 × (1 − 0.30) = 18.97 × 0.70 = 13.279 → 13.28
- Subtract the investing outflows: − 55.00 (investment in fixed capital) − 12.76 (investment in working capital)
- Sum: 118.00 + 54.45 + 13.28 − 55.00 − 12.76 = 117.97
Now check the same figure from the EBIT route, which is the independent test of whether the interest add-back has been handled correctly. EBIT here is operating income, 187.55, and non-cash charges are depreciation: EBIT × (1 − T) = 187.55 × 0.70 = 131.285 + Depreciation = 54.45 − Investment in fixed capital = −55.00 − Investment in working capital = −12.76 ─────── = 117.975 → 117.97 ✓ The two routes agree to the cent — which they must, since the algebra differs only by Interest × (1 − T) appearing on both sides of the net-income start. If they disagree, the interest add-back is the first thing to check.
The Cane case has no discount rate and no growth rate, so it never produces a value — it stops at the free cash flow. That is the point. Slides 24 to 31 are there to establish the numerator of the DCF method, and the two things they establish are:
1 · Free cash flow is an accrual-to-cash conversion exercise. Every line is a bridge from an accounting number to a cash number: net income plus non-cash charges, plus the firm's after-tax financing cost, less the investment the going concern requires. This is Lecture 4's "much of the work lies in taking accounting numbers and generating cash flows", demonstrated on a real set of statements.
2 · The two routes are a check on each other. Starting from EBIT and starting from net income must agree. Reproducing one of them is not the same as understanding the method; being able to show they coincide is.
With the numerator in hand, the DCF method is finished by discounting — a growing perpetuity at the firm's cost of capital, exactly as in the dividend model, and a denominator that will not get its proper justification until WACC arrives in Week 10.
The Comparable Method, and Choosing Between Methods
slides 32–37Slide 32 introduces the third method with a deliberately informal statement of the idea: companies with similar characteristics in the same industry should be valued closely. The slide's own examples are 格力 vs 美的 and Dell vs Hewlett-Packard.
The slide notes that DDM and DCF are absolute value methods, while the comparable method is a relative value method. That is not a classification nicety; it is the difference between the two, and it has a practical consequence.
Absolute methods ask "what is this asset worth?" and answer from the asset's own cash flows. Their output is a level — ¥28.61 per share — and it can be compared directly against the price.
Relative methods ask "what is this asset worth relative to similar assets?" and answer from other firms' prices. Their output is a ratio, and it says nothing at all about whether the comparison group itself is correctly priced.
So a comparable valuation can be simultaneously rigorous and useless. If the entire industry is in a bubble, every firm in it will screen as fairly valued. This is exactly how the argument worked in 1999 and in 2021: "our P/E of 60 is reasonable because our peers trade at 70." The method has no internal defence against a mispriced sector — which is why the lecture's closing advice is to use more than one method.
The multiples, and what each one is for
The slide's example: the industry average PE is 15, and company A's net income is $10 million. So one might estimate the company's value at $150 million — and the slide adds the crucial qualifier, we are estimating the intrinsic value. Then: if company A's current PE is 20 — a market value of $200 million — one might say it is overvalued and recommend a sell.
The arithmetic is three lines. The assumption underneath it is worth stating, because it is where the method is most often misapplied:
The comparison firm must be comparable in the ways that drive the multiple. PE is driven by growth and risk, not by industry label alone. A firm growing at 25% and one growing at 2% can both be "consumer staples", and applying the industry average to both is not a valuation — it is an average. This is why the slide's next point — that PEs differ enormously across industries and should not be compared across them — is only half the warning. Two firms in the same industry with different growth rates are just as incomparable.
The other half of the warning: which earnings? Trailing earnings and forecast earnings give different multiples, and analysts almost always quote the forward one, which is a forecast. A "cheap" PE computed on optimistic estimates is a forecast error wearing the costume of a valuation.
Value stocks and growth stocks
The lecture closes the PE section by classifying stocks by their multiple, and the labels are worth having straight because they are used constantly in the press with the opposite of their technical meaning.
A stock with a low PE ratio. The slide's parenthetical gives the whole reasoning: because it is cheap.
A stock with a high PE ratio. High-tech firms tend to have high PEs, which the slide notes means their shares are very expensive.
The terms are systematically misread. A growth stock is not a company that is growing — it is a company whose price already reflects expectations of growth. A value stock is not a company that has stopped growing; it is one whose price does not reflect much of anything.
The reason a high-PE stock can be a poor investment is arithmetic, and it is the same arithmetic as the constant-growth model. Rearranged from P0 = Div1/(R − g), a high multiple implies either a low discount rate (the market thinks the firm is safe) or a high g. If the high g fails to materialise, the multiple compresses — and a falling multiple is a loss even if earnings rise. A stock trading at PE 40 that grows earnings 10% while the multiple falls to 25 loses 31% of its value. This is the mechanism behind the "good company, bad investment" question from slide 2, stated numerically.
Ross §9.2 makes the further point that you cannot simply discount earnings instead of dividends. For a firm paying out only part of its earnings, discounting the whole earnings stream overstates value — it counts cash that is retained and reinvested as though shareholders received it. The dividend is what the minority shareholder gets; the earnings belong partly to future projects.
Choosing a method slide 36
The lecture's final comparison slide is the most practically useful thing in this section, because it maps each method to the circumstances in which it works — and to whose claim it values.
| Method | Use it when | Perspective |
|---|---|---|
| Dividend discount 股利折现法 |
The company has a history of dividend payments, and its dividend policy is clear and related to earnings | That of a minority shareholder — you value what you actually receive |
| Discounted cash flow 现金流折现法 |
The firm has no or unclear dividend payment history, but positive free cash flow | That of a controlling shareholder — you value the whole business, which you could direct |
| Comparable method 比较法 |
Useful for firms with similar investment — and simple and widely used in the real world, as the slide notes | That of the market in the peer group |
Look at what distinguishes the first two methods. It is not the arithmetic — both discount a growing stream at a risk-adjusted rate. It is whose cash flow is in the numerator, and that is determined by the position the valuer occupies.
A minority shareholder cannot direct the firm's investment or its dividend policy. The only cash they will ever receive is the dividend the board chooses to declare, so the dividend discount model values exactly their claim. A controlling shareholder can decide what the firm invests in and what it pays out, so they are entitled to the entire free cash flow — and the DCF method values that.
The gap between the two numbers is precisely the private benefit of control from slide 10, the thing a controlling shareholder might divert for their own benefit and at the minority's expense. Sections 3 and 6 of this lecture are therefore the same idea approached from the institutional side and the valuation side respectively — and this is the connection the exam is most likely to reward.
The slide's closing line is the practical instruction, and it is worth taking literally: in practice, analysts use more than one method and try to get a consistent conclusion. A valuation is not a number produced by a formula; it is a judgement supported by several independent routes agreeing.
Slide 37 · what it all comes to
好的公司一定是一项好的投资吗？No — and now the lecture can say why precisely.
A good company generates strong cash flows. But its price already reflects what the market expects those cash flows to be. Your return comes from the difference between the cash flows the firm actually delivers and the cash flows already priced in — not from the cash flows themselves. Buy a wonderful firm at a price that assumes it will be even more wonderful, and you can lose money while every one of your forecasts turns out to be right.
This is why the lecture's first slide put a price chart under the question: the chart is the price, and the firm's revenue was growing the whole time. Price moves around value, and the two are estimated by different processes — one by supply and demand in the secondary market, the other by the models in this lecture.
Two honest qualifications, both of which the lecture's own framing implies. First, valuation is an estimate. Every input in the models above — g, R, the earnings forecasts — is a judgement, so "price > value" is a conclusion about your own estimates as much as about the market. Second, the market is not obviously wrong. A price that disagrees with your model is evidence that your model is missing something, and in an efficient market the presumption runs against the analyst. Week 10 takes up exactly this question. The correct posture is not "the stock is overvalued", but "the stock is overvalued given my assumptions".
Formula Sheet
print-friendlyKey Concepts
the vocabularySelf-Check
16 questions · gradedThe examinable core is the constant-growth model — including the Div1-not-Div0 trap — and the minority/controlling shareholder distinction that decides which method to use. The institutional block on slides 10–17 supplies most of the concept-explanation marks, and it is the half of this lecture that is easiest to skip.
A · Concept check
Why Div1. The model is the present value of a growing perpetuity, and a perpetuity's first cash flow sits one period ahead of the valuation date. P0 is the value today; the first dividend it discounts is the one paid one year from today, which is Div1 = Div0(1 + g). Using Div0 would discount a cash flow that has already been paid — it is not an asset you are buying.
Algebraically, the growing perpetuity is Div1/(1+R) + Div1(1+g)/(1+R)² + …, whose sum is Div1/(R − g). The first term's numerator is Div1 by construction. (Working in Div0 instead is not arbitrary error — it is simply valuing the stream as though it began today, which understates the price by a factor of (1 + g).)
What the model requires. R > g, strictly. If g = R the formula divides by zero and returns infinity; if g > R it returns a negative price.
Why it is not a numerical convenience. Because the failure is not a failure of the arithmetic — it is the assumption becoming impossible. A firm compounding faster than the economy forever would eventually become the economy, so g cannot exceed the long-run growth rate of nominal GDP. And R, the return investors demand, must exceed that growth rate for capital to be allocated at all. So the condition is a feasibility constraint on the economic story, not a technical one on the formula: a model that breaks when g ≥ R is telling you the story you told it is impossible.
The practical consequence. A high g makes the denominator small and the price explode — the model is absurdly sensitive to it. That is why terminal-value assumptions dominate multi-stage valuations, and why the growth rate in a constant-growth model should be close to the long-run growth rate of the economy rather than the firm's recent history.
B · Multiple choice
C · Applied calculation
Step 1 — grow the dividend once. The $2.50 is Div0, the dividend just paid.
Div₁ = 2.50 × 1.03 = 2.5750 Div₂ = 2.5750 × 1.03 = 2.65225Step 2 — value the stock today.
P₀ = Div₁ / (R − g) = 2.5750 / (0.12 − 0.03) = 2.5750 / 0.09 = 28.6111Answer (a): $28.61 per share.
Step 3 — the price in one year. Two routes, and they should agree.
Route 1, grow the price: P₁ = P₀ × (1 + g) = 28.6111 × 1.03 = 29.4694 Route 2, re-apply the model: P₁ = Div₂ / (R − g) = 2.65225 / 0.09 = 29.4694Answer (b): $29.47 per share.
Step 4 — check the return. The two components a shareholder earns over the year are the dividend received and the change in price.
Dividend yield = 2.5750 / 28.6111 = 0.090000 → 9.00% Capital gain = (29.4694 − 28.6111) / 28.6111 = 0.8583 / 28.6111 = 0.030000 → 3.00% ─────── Total return = 12.00% ✓Answer (c): the two components are exactly the dividend yield and g, and they sum to the required return. That is not a coincidence — it is the identity R = Div1/P0 + g run forwards rather than backwards.
What to notice. Both of the model's outputs grow at the same 3%: the dividend and the price. Under constant growth the dividend yield stays constant at 9% forever, because the numerator and the denominator grow at the same rate. If a question asks you to value the stock in year 5, you can either grow the price five times or discount Div6 — and the point of the model is that the answer is the same either way.
Step 1 — count the votes. Each share carries one vote per seat being contested.
Investor: 120 × 5 = 600 votes All others: 880 × 5 = 4,400 votes Total: 5,000 votesAnswer (a): 600 votes — and crucially, cumulative voting lets all 600 be cast for a single candidate.
Step 2 — the guaranteeing stake. With N = 5:
Threshold fraction = 1 / (1 + N) = 1/6 = 16.667% Shares needed = 1,000 / 6 = 166.667 → 167 sharesAnswer (b): 167 shares, just over one sixth of the company.
Step 3 — can 120 shares win a seat? No.
Investor's 600 concentrated votes Others' best single candidate: 4,400 / 5 = 880 votes 600 < 880 → the investor cannot elect a director aloneAnswer (c): No. The investor holds 12% against a 16.7% threshold. They would need to combine with other shareholders, or persuade some of the other 880 shares to vote with them.
With a staggered board — only two seats up this time. The threshold is computed on the seats actually being elected, so N = 2:
Threshold fraction = 1 / (1 + 2) = 1/3 = 33.333% Shares needed = 1,000 / 3 = 333.333 → 334 sharesThe seat is still out of reach — 120 shares against a 334-share bar — but the disqualification is not the interesting part, and it was never going to change. The interesting part is the threshold: it more than doubles, from 167 shares to 334. That is exactly what slide 15 means when it says a staggered board "can compromise cumulative voting of minority shareholders". By shrinking the number of seats contested at any one meeting from five to two, staggering raises the bar for a minority seat from one sixth of the company to one third.
Why the formula behaves this way. The 1/(1+N) threshold falls as N rises, so anything that reduces the number of seats per election works against minority representation. Staggering is a deliberately anti-takeover device for the same reason: if only a third of the board can be replaced at any election, an acquirer cannot install a majority in one go, however many shares they hold.
Step 1 — free cash flow from net income. The five terms, in order.
Net income 500 + Interest × (1 − T) = 60 × 0.75 45 + Non-cash charges (depreciation) 120 − Investment in working capital −40 − Investment in fixed capital −200 ───── Free cash flow to the firm 425Answer (a): $425 million.
Step 2 — the EBIT route as a check. Start by unwinding the tax from net income to get pre-tax income, then add interest back to reach EBIT.
EBT = NI / (1 − T) = 500 / 0.75 = 666.667 EBIT = EBT + Interest = 666.667 + 60 = 726.667 Tax = EBT × 0.25 = 166.667 (check: 726.667 − 60 − 166.667 = 500 ✓) FCF = EBIT × (1 − T) + Dep − ΔWC − Capex = 726.667 × 0.75 + 120 − 40 − 200 = 545.000 + 120 − 40 − 200 = 425.000 ✓The two routes agree, as they must. Note the identity the check exposes: EBIT × (1 − T) = NI + Interest × (1 − T). It follows in one line — EBIT less interest is pre-tax income, so EBIT(1 − T) = EBT(1 − T) + Interest(1 − T), and EBT(1 − T) is just net income:
EBIT × (1 − T) = (666.667 + 60) × 0.75 = 545.000 NI + Interest × (1 − T) = 500 + 45 = 545.000 ✓Which is the whole reason the two formulas can coexist. They are not two methods; they are the same method written from opposite ends of the income statement. Reproducing either one is not the test — being able to show they coincide is.
Step 3 — value the firm. A growing perpetuity, exactly as in the dividend model — just with free cash flow in the numerator and the cost of capital in the denominator.
V₀ = FCF₁ / (R − g) = 425 / (0.10 − 0.02) = 425 / 0.08 = 5,312.50Step 4 — value the equity. Free cash flow belongs to all capital providers, so the $5,312.5m is the value of the whole firm. The debtholders' claim comes off.
Equity value = Firm value − Debt value = 5,312.50 − 900 = 4,412.50Answer (b): firm value $5,312.5 million; equity value $4,412.5 million.
Three things worth noticing. First, the interest add-back is after tax ($45, not $60) — the commonest single error in this calculation. Second, the working-capital and capex lines are subtracted, not added: they are cash the business absorbs to keep growing. Third, and most important for the exam, the final step is where this method touches the rest of the course. If the question had asked for a share price, you would divide the equity value by the share count — not the firm value. And the discount rate here, 10%, is precisely the WACC that Week 10 derives; the whole of this section is a formula waiting for its denominator.
Bond Valuation
Concept Map
how the lecture buildsEvery calculation in Lecture 6 is already in Lecture 2. The bond's coupon stream is an annuity C × [1 − 1/(1 + r)T]/r and the par repayment is a lump sum F/(1 + r)T. Add them and you have the bond price. There is no third term.
What is new is not the arithmetic but the rate. In Lecture 2 the discount rate was handed to you; here it has a name, it is quoted in the market, and — unlike the coupon — it changes every day. Almost everything after the first worked example is about the consequences of that one fact. Even the semiannual-coupon section (slides 18–21) is Lecture 2's compounding-frequency material returning: a quoted annual YTM of 5% paid semiannually is an effective 5.0625%, exactly the EAR relationship from Week 2.
Slide 11 states it in one clause: YTM is the return (IRR) on the bond if you hold the bond until maturity. That is not an analogy — it is the same calculation. Lecture 3 defined the IRR as the discount rate that sets a project's NPV to zero; here the "project" is buying the bond, the initial outlay is −P, and the inflows are the coupons and par. The rate that sets the NPV of that stream to zero is the YTM.
Which means the YTM inherits every caveat Lecture 3 attached to the IRR, and one of them matters a great deal here. The IRR assumes intermediate cash flows are reinvested at the IRR itself. A bond's YTM therefore overstates what you will actually earn if rates fall after you buy, because you will have to reinvest the coupons at lower rates. The slide's phrase "if you hold the bond until maturity" is doing real work: the YTM is a promised yield under a reinvestment assumption, not a guaranteed return.
What a Bond Is, and What It Is Not
slides 3–7Slide 3 draws the transaction in its simplest form — one arrow each way — and the simplicity is the point. The issuer hands over a bond; the investor hands over cash. Everything else in the lecture is about how the terms of that exchange are set and re-priced.
The borrower. The slide lists three levels of government-adjacent and corporate borrower:
国家 — the sovereign, borrowing in its own currency.
地方政府 — provincial and municipal governments.
企业 — firms, from policy banks to private issuers.
The lender. The slide splits the buy side by who is doing the lending:
银行 — banks, holding bonds as liquid assets and for the carry.
债券基金、养老基金 — bond and pension funds, whose liabilities are themselves bond-like.
个人投资者 — individuals, mostly through funds or the exchange.
The five terms the rest of the course assumes
Slide 4 is a glossary, and every one of these five definitions is examinable as a definition — the concept-explanation section of the final paper asks for exactly this kind of statement:
Both are quoted as percentages, which is why they get confused. But the coupon rate is a ratio of two contract terms (coupon ÷ par) and never changes; the YTM is a ratio the market sets and changes continuously. Slide 13 makes the warning explicit, and it earns its own section below — this is the single most common error in the topic.
The cash-flow picture slide 5 · Ross Figure 8.1
Slide 5 draws the Xanth Co. bond as a timeline, and it is worth reading carefully because it is the picture that the valuation formula later compresses. Years 1 through 9 carry a bare $80. Year 10 carries $1,080 — the coupon and the par together — which is why the final-period cash flow is not $80.
Note that the YTM is the one box on that list that is not fixed by the bond. The coupon is 8% because the contract says so; the YTM is whatever the market demands today, and the price is what falls out of the difference.
What issuing a bond does to the books
Slide 6 gives the accounting in two lines, and they are the bond counterpart of the share-issue entry from Lecture 5 — but with a crucial difference. Issuing equity raises cash and adds to equity, which carries no obligation to repay. Issuing debt raises cash and adds to a liability, which does.
The three properties that make bonds a distinct asset class
Slide 6 also makes three characterisations that read as asides and are in fact the syllabus's answer to "why does this lecture exist separately from the stock lecture":
Fixed income is the name of the asset class, and the reason is structural: the cash flows are contractual. A shareholder receives whatever is left over, if anything; a bondholder receives a number written down. Less risky follows from that — the claim ranks ahead of equity, and the upside is capped, so there is less to lose and less to gain. Less liquid is the property students find most surprising, and the slide gives the statistic that makes it concrete: on average only three transactions happen in the life of a corporate bond.
Three trades over a bond's life sounds like an exaggeration; it is roughly what the data show for investment-grade corporate issues. The explanation is that bonds are held, not traded. A pension fund with a known future liability buys a bond whose cash flows match it and then simply collects the coupons. There is no reason to sell, because the bond is doing its job.
The consequence is the market structure the slide names next: bonds trade over-the-counter, with a dealer making the market, not on an electronic exchange. Where trading is continuous, buyers and sellers meet directly and the exchange publishes a price. Where trading is sporadic, no such meeting happens, so a dealer stands in the middle holding inventory and quoting a price at which they will buy and a (higher) price at which they will sell. The next section of this lecture returns to that dealer at slide 28.
This illiquidity is a real, priced risk — and it is the reason a corporate bond's yield can exceed a treasury's by more than its expected default loss. Part of the spread is compensation for the fact that you may not be able to sell when you want to.
The valuation principle this all serves
Slide 7 is one sentence and one warning, and the warning is the most important line in the first half of the deck:
Slide 7 prints that sentence twice — once as the formula's caption and again as a standalone note. It is the hinge of the whole lecture, and it is worth stating the reason rather than just the rule.
The coupon rate describes the cash flows — it is how you compute the $80. The discount rate describes what those cash flows are worth to the market today. If you discounted a bond's coupons at its own coupon rate you would get the par value back every single time, for every bond, at every date: 80 × annuity(8%, 10) + 1,000/1.0810 = 1,000. That is not a valuation model, it is an identity — you would have proven that a bond is worth its face value regardless of what the market thinks, which is exactly the conclusion slide 14 is about to demolish.
The discount rate has to carry information the contract does not: today's risk-free rate, today's expected inflation, the issuer's creditworthiness, and a liquidity premium. Put all of that into one number and you have the YTM.
Valuing a Bond
slides 8, 14–15Slide 8 works the Xanth bond with the market demanding 8% — the same 8% as the coupon rate. This is the one case where the answer is known in advance, and working it anyway is what makes the structure of the calculation visible.
- The par repayment, discounted. A single lump sum ten years out: PV of par = 1,000 / 1.08¹⁰ = 1,000 / 2.158925 = 463.19
- The coupon stream, discounted. Ten equal payments — Lecture 2's annuity formula: PV of coupons = 80 × (1 − 1/1.08¹⁰) / 0.08 = 80 × 6.710081 = 536.81
- Add them. The bond's value is the sum of the two pieces: Bond value = 463.19 + 536.81 = 1,000.00
The answer is the par value, and it has to be: when the discount rate equals the coupon rate, each $80 coupon is exactly the interest the market requires on the $1,000, so the price is the $1,000. This is the par bond case, and it is the benchmark every later slide disturbs.
The slide's third line reads Total bond value = 436.19 + 536.81 = 1000. The first number is a digit transposition — the same slide's own line 1 correctly computes the PV of par as 463.19. Three independent checks confirm it:
① The sum does not work with 436.19. 436.19 + 536.81 = 973.00, not 1,000. The slide prints the wrong addend and the right total.
② The arithmetic. 1,000 / 1.0810 = 1,000 / 2.158925 = 463.19, exactly as line 1 says. No plausible reading of the formula produces 436.
③ The economics. A bond whose coupon equals the market rate must trade at par — that is the definition of the par case, and 973 is not par. Only 463.19 yields 1,000.
The reconstructed line is 463.19 + 536.81 = 1,000.00. Every figure downstream of this slide (the 884.82 and 1,136.03 below, and the 13.28 in the semiannual example) is correct, so the slip is confined to this one number.
Now move the market, not the bond
Slide 14 is where the lecture earns its keep. One year passes. The bond — its coupon, its par, its maturity date — is completely unchanged. The market's required return goes to 10%.
- Discount the par at the new rate, over the shorter horizon. PV of par = 1,000 / 1.10⁹ = 1,000 / 2.357948 = 424.10
- Discount the remaining nine coupons at the new rate. PV of coupons = 80 × (1 − 1/1.10⁹) / 0.10 = 80 × 5.759024 = 460.72
- Add them. Bond value = 424.10 + 460.72 = 884.82 < 1,000
Two things changed at once and both push the price down. The discount rate rose, which shrinks every present value; and the par repayment is now nine years away instead of ten, so the 1,000 is discounted less heavily — that part pushes up, and loses. The slide's conclusion is the one to memorise: when the interest rate goes up, the bond price drops.
The same bond at three rates slide 15
Slide 15 fills in the rest of the picture by asking what happens when the rate goes the other way, and adds the no-change case as a control. All three use the same 8%-coupon, ten-year bond, one year after issue, so all three have nine years left:
| Market YTM after one year | Relation to the 8% coupon | Bond value | Type |
|---|---|---|---|
| 10% — rate rises | YTM > coupon | 884.82 | Discount bond — trades below par |
| 8% — rate unchanged | YTM = coupon | 1,000.00 | Par bond — the anchor case |
| 6% — rate falls | YTM < coupon | 1,136.03 | Premium bond — trades above par |
The 0.01 difference is the slide's rounding, not a discrepancy — the exact figure is 1,136.0338. The structure is what matters: the two components crossed over. At 8% they were 463.19 and 536.81; at 6% the par component has grown to 591.90 and the coupon component to 544.14, and the coupon component is now the smaller of the two for the first time.
Slide 13 makes an observation that seems like trivia and is actually the strongest sanity check available on any bond price: for a discount or premium bond, the price converges to the face value at maturity.
The reason is that the only thing separating a premium bond from a par bond is the gap between its coupon and the market rate. As maturity approaches, the number of periods over which that gap can operate shrinks — each year pulls the price closer to the 1,000 that is contractually owed. At maturity there are zero periods left for the gap to matter, so the price is 1,000.
Practically: if a valuation exercise gives you a premium bond priced at 1,140 with one year to maturity, and par is 1,000, you should be suspicious. A year is not long enough to justify a 140-point premium. This is the check that catches most bond-arithmetic slips, including the 436.19 above.
Yield to Maturity
slides 9–13Every calculation so far has gone price ← rate. Slide 9 asks the reverse question, and it is the harder one because the rate is buried inside a polynomial that cannot be solved in closed form.
The slide sets up the equation and leaves the algebra for the calculator:
100 100 100 1,000 928.09 = ──── + ────── + … + ────── + ─────── 1+YTM (1+YTM)² (1+YTM)¹⁵ (1+YTM)¹⁵ YTM = ?The structure of the question comes before the arithmetic. The price, $928.09, is below the $1,000 par, so this is a discount bond, so the YTM must be above the 10% coupon rate. Any answer at or below 10% is wrong before you compute anything.
Solving the equation gives exactly:
YTM = 11.00% check: 100 × (1 − 1/1.11¹⁵)/0.11 + 1,000/1.11¹⁵ = 100 × 7.190870 + 208.9363 = 719.087 + 208.936 = 928.02 ≈ 928.09 ✓The slide chose $928.09 because it is very nearly exactly the 11% price. The discount-or-premium test is free information — use it before reaching for the calculator, because it catches sign and direction errors that the arithmetic itself will not.
The price-to-YTM direction is a sum of powers, evaluable in one pass. The YTM-to-price direction is a polynomial root of degree T — for T = 15 that is a fifteenth-degree equation with no algebraic solution. Every calculator, every spreadsheet and slide 10's online tool solves it by trial and error.
The slide's remedy is refreshingly practical: it links the 东方财富 bond calculator at data.eastmoney.com/money/calc/WinBond.html. There is nothing unsophisticated about that — the field does the same thing, only with better iteration. What matters is being able to read the answer critically, which is what the discount/premium test above is for.
What the YTM actually is
Slide 11 gives four statements about the YTM, and they are worth separating because they say four different things:
After issuance, the coupon rate and par value do not change. The price changes, because macro factors and supply and demand change — and the YTM is the rate implied by that price.
The slide's exact phrasing: it is another price of a bond expressed in the form of a rate. Price and YTM are two encodings of the same market fact — you cannot change one without changing the other.
The return if you hold the bond to maturity. Same definition as Lecture 3's IRR, and it inherits the same reinvestment assumption.
Always. They are two sides of one division, so the relationship is mechanically inverse — no economic argument required.
Slide 11 closes with a short scenario worth unpacking, because it is the mechanism of interest rate risk in miniature:
"Consider you hold a government bond selling at YTM 3%. Many investors want to enter the bond market and want to buy the bond in your hand. In this situation, you then set the price higher, leading the YTM lower than 3%."
Read it carefully and notice what is not happening. The bond's coupons did not change. The government's creditworthiness did not change. What changed is that demand for bonds rose — so the price you can command rose, and since price and yield are inverse, the yield to a new buyer fell below 3%.
The subtlety is that you, the seller, earned the difference. You bought at a price corresponding to 3% and sell at a price corresponding to less than 3%, so your holding-period return exceeds 3%. The buyer's forward return is below 3%. The YTM changes with the price, but the coupon does not — which is precisely why the next section's three-case classification is stated in terms of the coupon versus the YTM.
Why we need the YTM at all slide 12
Slide 12 asks the question that a student who has followed the arithmetic should be asking: the price is already known, so what does the YTM add? It answers with two purposes, and they are genuinely different:
The slide's own bridge back to Lecture 3 is that one clause, and it is the correct one. Lecture 3 faced exactly this problem: two projects, different scales, different timings, so comparing their NPVs is informative but comparing their rates is what ranks them. The IRR is the normalised measure.
The YTM is the same move applied to a security. And it inherits the same limitation: just as the IRR can rank projects misleadingly when they differ in scale or timing, the YTM compares bonds on a "per-year" basis that ignores duration — a 30-year and a 2-year bond with the same YTM do not carry the same risk, and the next section is where that catches up with us.
Coupon rate versus YTM — the three cases slide 13
Slide 13 opens with the warning in block capitals of intent — "Do not mix them up, they are totally different concepts" — and then gives the classification that turns the distinction into something you can use. The relationship is mechanical: the coupon is what the bond pays, the YTM is what the market demands, and the price moves until the two are reconciled.
| Case | Condition | Price vs par | What it means |
|---|---|---|---|
| Discount bond | coupon rate < YTM | price < par | The bond pays less than the market demands, so it must be bought cheaply. The extreme case is the zero-coupon bond, priced far below face value — all of the return is capital gain. |
| Par bond | coupon rate = YTM | price = par | The coupon is exactly the required return. The benchmark case — slide 8's Xanth bond. |
| Premium bond | coupon rate > YTM | price > par | The bond pays more than the market demands, so buyers bid the price up. The premium is the present value of that excess. |
The slide calls the zero-coupon bond "an extreme case", and it is worth seeing why a market for it exists at all — it pays nothing until maturity, so it looks like the least attractive instrument imaginable.
It is attractive precisely because it has one cash flow. A pension fund that knows it owes a fixed sum in exactly 20 years and wants no reinvestment risk at all can buy a 20-year zero and be done: there are no intermediate coupons to reinvest at unknown future rates. Every coupon payment in a normal bond is a small, unwanted reinvestment decision; the zero eliminates them all.
The cost is the mirror image of the benefit: with no coupons to cushion it, the zero's price is maximally sensitive to interest rates. It is the purest expression of the risk that the next section quantifies. Also note the language — a zero-coupon bond still has an implied coupon rate (in China, the discount is taxed as interest income, so the tax authority imputes one), and it certainly has a YTM. It is the payment that is zero, not the return.
Interest Rate Risk, and Semiannual Coupons
slides 16–21Slides 14 and 15 demonstrated that bond prices move with rates. Slide 16 states what determines how much, and the answer has two parts.
Both conditions have the same underlying explanation: the fraction of the bond's value that arrives at the very end.
A bond's price is a weighted average of its cash flows' present values. The further out a cash flow sits, the more its present value changes when the discount rate moves. So a bond whose value sits mostly in a distant lump sum is fragile; a bond whose value sits mostly in near-term coupons is robust.
Longer maturity pushes everything further out. Lower coupon shrinks the near-term coupons and hands more of the total value to the final par repayment — which is exactly the zero-coupon case from slide 13, the limiting case of maximum sensitivity.
The slide names interest rate the most important risk factor in bond valuation, and adds the career advice that follows from it: as a bond trader, the key skill is to predict the interest rate. That is not an overstatement. For a default-free bond, it is the only thing to predict.
Ross Ch. 8 formalises the two effects with a measure called duration — roughly, the value-weighted average time to each cash flow — and the rule that the percentage price change is approximately −duration × Δy. Duration is not on this course's slides, but it is the tool that makes the sentence on slide 16 quantitative, and it is worth knowing the name.
Why a bond is less risky than a stock — with numbers slide 17
Slide 17 revisits the Xanth bond to make a claim that is usually stated qualitatively and is much more convincing when it is not. Suppose you buy at par and sell one year later, after the rate has jumped from 8% to 10%:
And the slide's framing of that number: it is not common for the interest rate to go up by 2% in one year — and even when it does, you lose 3.5%. In a stock, 3.5% can vanish in a single day.
The slide's arithmetic is right and the framing is the correct intuition: a 200-basis-point move — a very large move in a developed bond market — cost this investor 3.5% including the coupon. For a ten-year investment-grade bond that is a bad year, not a catastrophe.
Two qualifications worth carrying, neither of which contradicts the slide. First, the 3.5% is for a ten-year bond. Apply the same 2% shock to a 30-year bond and the price fall is several times larger — duration again — and to a zero-coupon bond larger still. The claim "bonds are safe" is really "short-duration, high-quality bonds are safe".
Second, and more important, the comparison assumes the treasury case: no default. A corporate bond carries interest rate risk and credit risk, and in a genuine crisis the two arrive together — rates fall (helping the bond) while spreads widen (hurting it), and for a high-yield issuer the second effect wins. The last section of this lecture is about exactly that, and slide 31 names it.
Semiannual coupons slides 18–21
Almost nothing in this course so far has had to deal with payment frequency. Bonds force the issue, because most pay twice a year, and the lecture devotes four slides to making the adjustment mechanical. The rule is short: halve the rate, double the periods. Everything else in the formula is unchanged.
The slide lays out the five calculator keys, and the adjustment is visible in each one:
N = 5 × 2 = 10 periods I/Y = 5% / 2 = 2.5 % per period PMT = 1,000 × 0.06375 / 2 = 31.875 per period FV = 1,000 PV = ? → −1,060.17Worked out longhand, to show the calculator is doing nothing exotic:
PV of coupons = 31.875 × (1 − 1/1.025¹⁰) / 0.025 = 31.875 × 8.752064 = 278.97 PV of par = 1,000 / 1.025¹⁰ = 1,000 / 1.2800845 = 781.20 ─────── Bond value 1,060.16 ≈ 1,060.17Read the answer before moving on. The price exceeds the 1,000 par, so this is a premium bond, and the coupon rate (6.375%) must exceed the YTM (5%). It does. The direction check costs nothing and catches most keying errors.
Slide 18 also computes the effective annual rate, and it is the slide's own reminder that the 2.5% you typed into I/Y is not an annual figure:
This is Lecture 2's compounding-frequency relationship, reappearing. A "5% bond, semiannual" delivers 5.0625% a year, not 5%. And the arithmetic runs in both directions, which is where slide 21 catches people:
If you solve for the rate and get 4%, the YTM is 8%, not 4% — because 4% is the rate per half year. This is the single most common error in the whole topic, and slide 21 devotes a slide to saying nothing else. Note that the convention is to multiply by 2, not to compound — the quoted "8%" is a bond-equivalent nominal rate, which is not the same as the 8.16% effective rate that 4% semiannual actually produces. Financial convention prefers the comparable nominal figure.
- Is the YTM more or less than 10%? The price is above par, so it is a premium bond, so the coupon rate exceeds the YTM — less than 10%. Free, from the direction test alone. 1,197.93 > 1,000 → premium → YTM < coupon rate
- The semiannual coupon payment. Half of the annual coupon: PMT = 5% × 1,000 = 50
- The number of periods. Twenty years of twice-yearly payments: N = 20 × 2 = 40
- And the rate. Feed the three knowns into the calculator and solve for I/Y: I/Y = 4.0000% ← per half year, NOT the YTM YTM = 4% × 2 = 8.00%
Slide 19's calculator screenshot is of the cash-flow / IRR module rather than the bond module — the professor enters −1,197.93 at t=0, then 50 × 39 and 1,050 × 1, and reads the IRR of 4%.
That is worth pausing on, because it is a different route to the same answer and it shows what the YTM is. The 39 periods of 50 followed by one of 1,050 is just the bond's cash-flow stream written out; the IRR of that stream is the semiannual yield. The "IRR" button and the "I/Y" button are computing the same thing by two names.
The screenshot is also a small reminder of scale: 40 periods is the same N as a 20-year mortgage. The familiar amortisation schedule from Lecture 2 and this bond are the same annuity structure viewed from opposite ends — one is a loan you owe, the other is a loan you own.
Government, Corporate, and the Price of Default
slides 22–32The lecture now turns from the arithmetic of one bond to the market it lives in. Everything up to here assumed the cash flows arrive; the second half is about the two reasons they might not arrive as promised, and how the market prices each.
Government bonds, and the risk-free rate
Slide 22 splits government debt into three instruments by maturity, and the split is standard enough to be worth knowing as a definition set:
| Instrument | Maturity | Coupon structure |
|---|---|---|
| Treasury bill | < 1 year | Pure discount — no coupons at all. Slide 13's extreme case, issued by the sovereign. |
| Treasury note | 1 – 10 years | Coupon debt, in the US usually semiannual. |
| Treasury bond | > 10 years | Coupon debt. The 30-year is the classic long bond. |
The slide's justification for why this matters is one clause, and it is the load-bearing one for the rest of the course: with no default risk, the YTM of treasury securities can be used as the risk-free rate. That single sentence is what makes the discount rates in Lectures 3, 4 and 5 (and the WACC in Week 10) computable — the risk-free rate is not an abstraction, it is observable every day on the 财政部 website.
Slide 22 glosses "Municipal bond" as 城投债, issued by local governments via 地方城市建设投资公司. The Chinese term is accurate; the English translation is not, and the difference is not pedantic.
A US municipal bond is issued by a state or city itself. The issuer is a government, its interest is exempt from federal income tax, and — in the US market's own understanding — the credit sits on the taxing power of that government.
A 城投债 is issued by a 地方政府融资平台公司, a company set up by a local government to fund infrastructure. In legal form it is a corporate bond (企业债 / 公司债). The company is typically state-owned, and the market has historically priced it as though a government stood behind it — but since 国发〔2014〕43号 and the follow-up statements of 2017–2018, the central government has explicitly disclaimed any bailout obligation, and the phrase "坚决打消…中央会买单的幻觉" is now official doctrine. A 城投债 is not, legally, a government obligation.
So the slide's "default risk is slightly higher than treasury bond" understates the structural point. The understatement is not about the number — 城投债 spreads have been modest for years — but about where the credit comes from. For a treasury it comes from the sovereign's taxing and monetary power. For a 城投债 it comes from an implicit expectation that has been officially renounced.
The true functional analogue of a US municipal bond in China is 地方政府债券 (地方债) — issued directly by provincial governments since the 2015 Budget Law reform. It is a genuine government obligation, and its interest is exempt from individual income tax, exactly like a US muni. If you need a Chinese muni in an exam answer, that is the one.
Basis points, and the live Chinese curve slide 23
Slide 23 introduces the unit the rest of the market speaks in:
Slide 23 then carries a live screenshot of the Ministry of Finance's 中国国债收益率曲线, dated 21 September 2026 — one day before the lecture. It is the risk-free rate made visible, and it is worth reading as data rather than as decoration:
| 期限 | 收益率 (%) | 比上日 (BP) | 比上年同期 (BP) |
|---|---|---|---|
| 3 月 | 1.18 | −0.02 | −20.05 |
| 6 月 | 1.19 | 0.00 | −20.96 |
| 1 年 | 1.23 | 0.19 | −15.83 |
| 2 年 | 1.26 | 0.91 | −22.30 |
| 3 年 | 1.29 | 1.02 | −22.54 |
| 5 年 | 1.41 | 0.17 | −21.14 |
| 7 年 | 1.51 | 0.24 | −28.56 |
| 10 年 | 1.68 | 0.11 | −19.60 |
| 30 年 | 2.12 | −0.25 | −8.11 |
The table is a working example of every concept in this lecture at once. Three things to look at:
① The shape. 1.18% at three months rising to 2.12% at thirty years — upward-sloping, which is the normal shape. The term spread is 10年 − 3月 = 1.68 − 1.18 = 50 BP, and 30年 − 10年 = 44 BP. A long bond yields more because you are locked in for longer — the same interest rate risk from slide 16, charged for in the price.
② The year-on-year column is entirely negative. Every single tenor is 8 to 29 BP lower than a year ago. In the language of slide 15, that means every bond in this table went up in price over the past year, and the longer ones went up most — the 7-year by 28.6 BP, which on a 7-year bond is a price gain of roughly 2%. That is the reward for bearing the interest rate risk the slide warned about, and it materialised because rates fell.
③ The one-day column is where the trade is. The 2-year and 3-year moved 0.91 and 1.02 BP up, while the 30-year moved 0.25 BP down. Small numbers, but a bond desk's whole day is made of moves this size: 1 BP on a 30-year bond is worth far more in price terms than 1 BP on a 3-month bill. This is duration again, and it is why the trader on slide 16 is paid to forecast rates.
Credit ratings slides 24–25
The treasury is the one borrower with no default risk. Everyone else's spread above the treasury exists to pay for the possibility that they do not repay, and the market's first instrument for pricing that is a credit rating.
Slide 24 sets out the scale, running from AAA down to D, with the slide's own summary that the capacity to repay declines along the ratings from AAA to D. The market's practical use of the scale is a single dividing line:
Slide 24 says investment grade is "rated above BBB" while slide 25's table places Baa/BBB inside the investment-grade block and starts junk at Ba/BB. The table is right; the word "above" is loose.
The precise boundary is that the lowest investment-grade rating is BBB− at S&P and Fitch, and Baa3 at Moody's — the minus/minus-equivalent notch below plain BBB. So a plain BBB is comfortably investment grade, and it is the rating one notch below it that gets you thrown out of the index. The practical significance of that one notch is enormous: fallen angels, issuers downgraded from BBB− to BB+, are forced sellers' paper, because index-trackers and mandated funds must sell regardless of price. A one-notch downgrade can move a bond more than a year of interest.
Slide 24 also lists the agencies on both sides of the market — the three international raters Standard & Poor (标普), Moody's (穆迪), Fitch (惠誉), and the domestic ones 联合资信、中证鹏元、大公国际、中诚信. The domestic list matters for this course because a Chinese issuer's rating is usually a domestic-scale rating, and domestic AAA is not comparable to global AAA — the scales are calibrated to different default experiences.
| Moody's | S&P and Fitch | Grade |
|---|---|---|
| Aaa | AAA | Investment grade |
| Aa | AA | |
| A | A | |
| Baa | BBB | |
| Ba | BB | Junk bonds |
| B | B | |
| Caa | CCC | |
| Ca | CC | |
| C | C |
Slide 25's table, with the Baa/BBB row highlighted because it is the boundary between the two blocks. Fitch and S&P add + and − modifiers within each letter grade; Moody's uses 1, 2 and 3. The full ladder from Aaa to C therefore has far more than fourteen rungs.
Credit spread slides 26–27
Slide 26 defines it as the difference in yield between two debt securities of similar maturity but different credit quality, and adds the interpretation: it is the additional compensation investors demand for taking on the higher risk associated with corporate bonds over risk-free government bonds. The "similar maturity" clause is what makes the subtraction meaningful — otherwise you would be measuring the term spread from slide 23 instead of the credit spread.
Two things make this a good worked example. First, the maturities differ by only 0.07 years, so essentially all of the 15.75 BP is credit, not term. Second, 国开债 are issued by 国家开发银行, a policy bank with no explicit sovereign guarantee — so this is a genuine, if small, credit spread on paper that the market treats as quasi-sovereign.
The full price column is also visible: the 国开15 trades at a net price of 97.30 against 98.85 for the comparable treasury. A lower price for a higher yield, which is the inverse relationship from slide 11 appearing in live data.
Slide 27 asks exactly that under a chart of the US credit spread against the S&P 500 from 1996 to 2024, with the crisis windows shaded. The chart answers it, and the answer is bad.
The mechanism is that a credit spread is a forward-looking number. A yield is a promise about the future; when lenders demand more to hold a corporate bond, they are saying their estimate of the default probability has risen. Aggregate that across every issuer and you have the market's collective estimate of recession risk — which is why slide 26 calls the spread a leading indicator of credit risk "from the perspective of the bond market".
Read the shaded windows on the slide and the pattern is unmistakable: the spread spikes in the dot-com bust, spikes enormously in 2008, spikes again at the 2011 euro crisis, and spikes at Covid-19 in 2020. Each spike is a period when investors were repricing the probability that firms would not pay them back.
The chart also shows why the bond market is said to "see it first". In 2007 the S&P 500 was still making highs while the credit spread was already climbing off its floor. Equity prices reflect the market's view of the upside; spreads reflect its view of the downside, and lenders get nervous before buyers do.
The market structure: dealers and the bid-ask spread slides 28–30
Slide 28 returns to the OTC structure from slide 6 and explains how a market with almost no trading actually functions. The primary/secondary distinction is the same one as Lecture 5's, but the secondary market is organised completely differently from an exchange.
Dealers (做市商) are specialists who buy low and sell high in the bond market. They quote a bid and an ask on the same bond and earn the spread between them.
Slide 29 shows the官方 list: thirteen 主做市商, including 东方证券、中信证券、 国泰海通、招商证券、华泰证券 and 中信建投. These are the firms contractually obliged to quote prices so that the rest of the market has something to trade against.
They assume the inventory risk, because they hold the bond themselves. Every quote is made from stock or against stock they must then acquire, so a move in rates between buying and selling is the dealer's loss, not the client's.
That risk is precisely what the bid-ask spread pays for. It is not a fee for effort; it is the price of standing ready to trade when nobody else will.
Slide 28 states the convention, and it is worth reading twice: Bid (ask) price is the offer (sell) price set by dealer. Investors buy at ask and sell at bid prices.
The mental model that keeps this straight is to anchor on the investor, not the dealer: you buy at the ask and you sell at the bid. The ask is therefore always the higher of the two — you pay more to buy than you receive to sell, and the difference is the dealer's compensation.
Slide 30's live data makes the inverse-yield point again: the screens show 收益率 rather than price, and because price and yield are inverse, the ask price corresponds to the lower yield. A dealer quoting a two-sided market in yields quotes a lower yield to a buyer than to a seller. If that sounds backwards, go back to slide 11.
Slide 28 closes with the two scale facts that explain why this market looks the way it does, and both are counter-intuitive enough to be worth memorising:
Firms and governments raise far more money by issuing bonds than by issuing shares, and yet the stock market gets all the attention. The slide's own two clauses explain the asymmetry: issuance is large, but trading in any given issue is thin, so there is no daily spectacle to report on.
This is the "three transactions in the life of a corporate bond" statistic from slide 6, coming back with the consequence attached. A market where a bond changes hands three times in ten years cannot support a ticker-tape narrative. It is the plumbing of the financial system, and it is invisible for exactly that reason.
Flight to quality slide 31
Slide 31 poses the lecture's best question: during a financial crisis or a war, the price of most financial assets drops. Are there some assets whose prices go up? The answer is government bonds and gold — and the phenomenon has a name, Flight to Quality.
In extreme bad times — crises, wars, natural disasters — investors sell stocks, real estate, corporate bonds and fund shares, and buy treasury bonds and gold. Treasury bonds are explicable: they are the risk-free asset, so when every other asset's default risk spikes, money runs to the one with none. That is credit spread widening from slide 27, seen from the buyer's side.
Gold is the interesting case, because it is neither risk-free nor income-producing. It pays no coupon, so no discount-rate model in this lecture can value it. What it offers is different in kind: it is nobody's liability. A treasury bond is a promise from a government, and in a sufficiently severe crisis the government itself is part of the risk. Gold's price is not a claim on anyone, which is precisely the property that becomes valuable when every claim is in question.
The practical warning is that flight to quality is not free insurance. Treasury yields fall when the flight happens — by definition, since buying pushes prices up and yields down — so you are buying the safe asset at exactly the moment its prospective return is at its lowest. The asset that protects you in a crisis is the asset that has just become most expensive.
Bond market investment strategy slide 32
The lecture closes by joining the two halves together. The first half built an intrinsic value from discounted cash flows; the second half showed why the traded price can sit somewhere else. Slide 32 names the two reasons the gap exists:
Slide 32's conclusion is a deliberate refusal, and it is the most conceptually interesting line in the deck: in the bond market we do not say it is overpriced or underpriced. The investment strategy is based on your prediction of future macro interest rate and credit risk.
Compare that with Lecture 5, whose whole apparatus was built to answer "is this stock over- or undervalued?" The bond market refuses the question, and the reason is that a bond price is a rate, and a rate is an equilibrium, not an opinion. When you say a stock is underpriced you are claiming the market has misjudged the business. When a bond yields 1.68%, you are not looking at a market that has misjudged anything — you are looking at the market's current best estimate of the risk-free rate, and it is definitionally correct as an estimate.
Which relocates the entire investment question. If the price is an equilibrium, there is nothing to disagree with now; all disagreement is about the future. So the bond investor does not ask "is this cheap?" — they ask "will rates fall, and will this issuer's credit hold?" Both are forecasts, both can be wrong, and both are the only things you are actually paid for.
This also explains slide 16's career advice, which read as an aside at the time and turns out to be the lecture's conclusion: as a bond trader, the key skill is to predict the interest rate.
Formula Sheet
print-friendlyKey Concepts
the vocabularySelf-Check
16 questions · gradedThe examinable core is the master formula and the three-case classification — most single-choice marks will be a price, a YTM, or a direction test built on those two. Two traps account for most lost marks: using the coupon rate as the discount rate, and reporting a semiannual rate as the YTM. The credit block on slides 22–32 supplies the concept-explanation marks, and the credit spread is the single most likely definition to be asked.
A · Concept check
The formula. A bond's value is the present value of two things — an annuity of coupons and a single repayment of par at maturity:
V = C × [1 − 1/(1 + r)ᵀ]/r + F/(1 + r)ᵀThe interest rate r appears in the denominator of every term, and with a positive exponent. So raising r raises every denominator, shrinks every present value, and lowers the sum. The relationship is not a market convention or a behavioural regularity — it is arithmetic. There is no scenario in which a higher discount rate produces a higher present value of a fixed, positive cash-flow stream.
Which part is affected most. The final repayment and the most distant coupons — the cash flows furthest from today. This follows from the exponent: r enters as (1 + r)t, so the same change in r has a larger proportional effect the greater t is. A 2% rise in rates barely touches a coupon due in six months and dramatically changes the present value of a payment due in thirty years.
The consequence. This is exactly why slide 16 says interest rate risk rises with longer maturity and lower coupon. Both conditions push more of the bond's total value into distant cash flows. A zero-coupon bond, whose entire value is one distant payment, is the limiting case of maximum sensitivity.
The worked numbers. The Xanth bond — 8% coupon, 10 years — illustrates both the direction and the asymmetry:
At YTM 8% (par): 1,000.00 At YTM 10% (+2pp): 884.82 −11.52% At YTM 6% (−2pp): 1,136.03 +13.60%Note that the rise is larger than the fall, for a symmetric 2-percentage-point move in rates. That asymmetry is the signature of convexity — the price/yield relationship is a curve, not a straight line — and it means rate falls help a bondholder more than equal rate rises hurt. It is the one piece of good news in interest rate risk, and it is why volatility in rates is itself slightly valuable to a bondholder.
B · Multiple choice
C · Applied calculation
(a) Price at YTM 5%. Coupon = 7% × 1,000 = ¥70. Two components, as always:
PV of coupons = 70 × (1 − 1/1.05⁶) / 0.05 = 70 × 5.075692 = 355.30 PV of par = 1,000 / 1.05⁶ = 1,000 / 1.340096 = 746.22 ─────── Price 1,101.51Answer (a): ¥1,101.51.
(b) Premium or discount. Coupon 7% > YTM 5%, so this is a premium bond, and the price of 1,101.51 > 1,000 confirms it. The bond pays more than the market demands, so buyers bid it above par.
(c) Repricing at 7% and 9%. The structure of the calculation is unchanged; only r moves.
At YTM 7%: 70 × (1 − 1/1.07⁶)/0.07 + 1,000/1.07⁶ = 70 × 4.766540 + 666.342 = 333.658 + 666.342 = 1,000.00 ← par At YTM 9%: 70 × (1 − 1/1.09⁶)/0.09 + 1,000/1.09⁶ = 70 × 4.485919 + 596.267 = 314.014 + 596.267 = 910.28 ← discountAnswer (c): ¥1,000.00 at 7%, ¥910.28 at 9%.
(d) What the three prices show. The three cases from slide 13, in one bond:
| YTM | vs 7% coupon | Price | Case |
|---|---|---|---|
| 5% | YTM < coupon | 1,101.51 | Premium |
| 7% | YTM = coupon | 1,000.00 | Par |
| 9% | YTM > coupon | 910.28 | Discount |
The relationship is monotone and mechanical: as the YTM rises, the price falls, and the classification is decided entirely by whether the YTM sits above, at, or below the coupon rate. Note that the coupon rate never changed — only the market's requirement did.
Two checks worth internalising. First, the par case lands on exactly 1,000, which is the arithmetic's own confirmation that the coupon-rate/YTM rule is an identity rather than a coincidence. Second, all three prices converge to ¥1,000 as maturity approaches — with six years left the premium is 101.51 and the discount 89.72, and both shrink to zero by year 6. If a valuation exercise ever produces a premium bond with one year to maturity and a 150-point premium, the arithmetic is wrong.
One asymmetry to notice. A 2-percentage-point fall in the YTM (7%→5%) moved the price up by 101.51, while a 2-point rise (7%→9%) moved it down by only 89.72. The gain exceeds the loss — this is the convexity from Q5, and it is a property of the price/yield curve rather than of this particular bond.
(a) The inputs. The semiannual rule from slide 18, applied term by term.
N = 8 × 2 = 16 periods I/Y = 4% / 2 = 2 % per period PMT = 1,000 × 0.06 / 2 = 30 per period FV = 1,000 PV = ?(b) The price. Longhand, to show what the calculator is doing:
PV of coupons = 30 × (1 − 1/1.02¹⁶) / 0.02 = 30 × 13.577709 = 407.33 PV of par = 1,000 / 1.02¹⁶ = 1,000 / 1.372786 = 728.45 ─────── Price 1,135.78Answer (b): $1,135.78. Direction check before moving on: coupon 6% > YTM 4%, so a premium bond, and 1,135.78 > 1,000. ✓
(c) The effective annual rate. The quoted 4% is nominal; the true annual rate compounds the semiannual 2%.
EAR = (1 + 0.04/2)² − 1 = 1.02² − 1 = 1.0404 − 1 = 4.04%Answer (c): 4.04%.
(d) The 5% case. This is slide 21's trap, stated as a question:
rate per period = 5% YTM = 5% × 2 = 10% Coupon rate 6% < YTM 10% → discount bond → price BELOW parAnswer (d): the YTM would be 10%, and the price would be below par.
The two things this question is testing. First, that a rate solved out of a semiannual problem is per half year and must be doubled — reporting 5% as the YTM understates the annual return by half. Second, that the premium/discount direction follows from comparing the coupon rate with the YTM, not with the per-period rate. Comparing 6% against the per-period 5% would give the right answer here by accident; the discipline is to annualise first and compare second.
One structural note. N = 16 for eight years is worth noticing: a semiannual bond has roughly twice as many cash flows as its maturity suggests, which is why the semiannual adjustment matters more for long bonds. For an 8-year bond the error from forgetting to double N is large; for a 6-month bond it would be invisible.
(a) Both prices at YTM 10%. Same coupon, same par, different horizons.
Bond A (10 years): PV of coupons = 80 × (1 − 1/1.10¹⁰)/0.10 = 80 × 6.144567 = 491.57 PV of par = 1,000 / 1.10¹⁰ = 1,000/2.593742 = 385.54 ─────── Price 877.11 Bond B (2 years): PV of coupons = 80 × (1 − 1/1.10²)/0.10 = 80 × 1.735537 = 138.84 PV of par = 1,000 / 1.10² = 1,000/1.21 = 826.45 ─────── Price 965.29(b) The percentage changes. Both started at 1,000 (par, since coupon = YTM = 8%):
Bond A: (877.11 − 1,000) / 1,000 = −12.29% Bond B: (965.29 − 1,000) / 1,000 = −3.47%Answer (b): Bond A falls 12.29%; Bond B falls 3.47%.
(c) What the comparison demonstrates. The first half of slide 16: for the same coupon and the same change in rates, the longer the time to maturity, the greater the interest rate risk. The two bonds experienced an identical 200-basis-point shock, and one lost three and a half times as much in percentage terms as the other.
The mechanism is the one from Q5. Bond A has ten cash flows spread across a decade, with a large share of its value in the distant par repayment; Bond B has almost all of its value in two nearby coupons and a repayment two years out. When the discount rate rises, the far-dated cash flows lose proportionally far more of their present value.
(d) The coupon question. They have the same coupon — 8% each. The question is a deliberate distraction, and the correct answer is to say so rather than to invent a difference.
But it is worth answering the hypothetical, because it is the second half of slide 16. If Bond A were repriced with a lower coupon — say 4% instead of 8% — its sensitivity would increase further. The reason is the same mechanism running again: a lower coupon shrinks the near-term payments and hands a larger share of the bond's total value to the final par repayment, which is the most rate-sensitive cash flow of all. Carry that logic to its limit — a coupon of exactly zero — and you have the zero-coupon bond, the maximum-sensitivity case from slide 13.
So the two rules combine into one. "Longer maturity" and "lower coupon" are not two independent facts to memorise; both push the bond's value toward distant cash flows, and that is the single thing that makes a bond price-sensitive. Ross Ch. 8 names the combined measure duration — the value-weighted average time to each cash flow — and the approximation %Δprice ≈ −duration × Δy. Duration is not on this course's slides, but it is what slide 16's two conditions are jointly describing.
(a) The spread before the crisis. Corporate yield minus government yield, both of the same maturity — slide 26's definition.
Credit spread = 4.35% − 1.68% = 2.67 pp = 267 BPAnswer (a): 267 BP.
(b) The spread during the crisis.
During: 7.10% − 1.20% = 5.90 pp = 590 BP Change: 590 − 267 = 323 BP of wideningAnswer (b): 590 BP, a widening of 323 BP — the spread more than doubled.
(c) Explaining both movements.
The government yield falling is flight to quality (slide 31). In a crisis, investors sell stocks, real estate, corporate bonds and fund shares and buy treasury bonds and gold. The buying pressure pushes treasury prices up, and since price and yield are inverse, yields fall. The 1.68% → 1.20% move is the risk-free rate compressing as it becomes the only thing anyone wants to hold.
The corporate yield rising is credit risk being repriced (slides 26–27). Lenders have revised upward their estimate of the probability that the issuer does not repay, so they demand more compensation. The 4.35% → 7.10% move is the credit spread widening, exactly as in the shaded windows of slide 27's chart — which is why the slide calls the spread a leading indicator of credit conditions.
And they interact. The two effects are not independent: the corporate yield is built as risk-free rate + credit spread. Here the risk-free rate fell by 48 BP (1.68 → 1.20) while the spread widened by 323 BP, so the net corporate yield still rose by 275 BP. The credit effect dominated the rate effect, which is the normal pattern for a risky issuer in a crisis.
(d) Are the two statements consistent? Yes — they are the same fact seen from two sides.
The confusion is worth naming precisely, because it is where students lose marks. The question says the government bond's price rose and the corporate bond's yield rose, and these sound like opposite statements. They are not, because they are statements about two different bonds using two different measures. The rule is price and yield are inverse for a given bond — it is a statement about one security, not a comparison across securities.
Government bond: yield 1.68% ↓ 1.20% → price ↑ (flight to quality) Corporate bond: yield 4.35% ↑ 7.10% → price ↓ (credit repricing)Both lines obey the inverse rule. What makes them look contradictory is only that the two bonds moved for different reasons: the treasury's yield fell because demand for it rose, while the corporate's yield rose because its perceived default risk rose. A bond can lose value simply because the market has re-rated its issuer, even if the risk-free rate is falling at the same time — and for a high-yield issuer, the re-rating usually wins.
The lesson for slide 32. This is why the bond market refuses the over/underpriced question. An investor holding that corporate bond did nothing wrong — they bought a 267 BP spread and are now looking at 590 BP. The price is not a verdict on the bond; it is a continuously updated forecast of interest rates and credit risk, and in a crisis both forecasts move against a corporate bondholder at once.
Risk and Return
Concept Map
how the lecture buildsLectures 3 to 6 discounted cash flows at a rate handed over without explanation. The syllabus's own gloss for this week is that it "provides the conceptual basis for the discount rate". Here is the chain that turns that into a number, and it is worth seeing in one line before the detail:
risk-free rate + risk premium = the required return on an asset. The risk-free rate is observable — it is the treasury curve from Lecture 6, slide 23. The risk premium is what this lecture measures and what Week 8's CAPM will turn into a formula. Put them together and you have the denominator in every NPV the course has computed.
Slide 9 also names a connection the lecture itself does not labour: the risk premium is "recall credit spread". That is exactly right, and it is the same idea at two levels. A credit spread is the extra yield a bond demands over a treasury for bearing default risk. A risk premium is the extra return an asset demands over the risk-free rate for bearing risk at all. Lecture 6 measured one; this lecture measures the general case.
The arithmetic-versus-geometric distinction on slides 4–5 looks like a technicality about averages. It is actually Lecture 2's compounding theme arriving in a new costume. A geometric average is the compound growth rate: if a fund returns 10%, −5%, 20% and 15% over four years, the geometric average of 9.58% is the constant annual rate that would have produced the same 44.21% four-year outcome. That is the same move as converting a nominal rate into an effective one — collapsing a sequence into a single equivalent rate.
And the gap between the two averages is not a curiosity. The arithmetic average will always be the larger of the two, and the gap widens with volatility. That is the whole reason a fund can advertise a "10% average annual return" while its investors compound at 9.58%. The next section shows both numbers for the same four years.
Measuring Return
slides 2–5Slide 2 defines the return on a stock in the only way that matters for the rest of the course: as cash received plus change in value, expressed as a fraction of what you put in.
The split on the right is worth holding on to, because the two components behave completely differently. A dividend yield is cash the firm chose to hand over — relatively stable, and the reason bond-like stocks attract income investors. A capital gains yield is the market re-rating the firm — volatile, and in a growth company it is the whole return. The same decomposition appears in Lecture 5's dividend model, where r = D₁/P₀ + g is precisely this identity written with the two pieces named.
Holding period return slide 2
Slide 2 then generalises from one period to many. Over T years, the returns chain:
The formula is the one part of this lecture that requires no new intuition, because it is the same compounding from Lecture 2. What it does require is care with the signs: a −5% year multiplies the running total by 0.95, not by 1.05, and the difference compounds.
- The holding period return. Multiply the four growth factors, then subtract one:
HPR = (1.10)(0.95)(1.20)(1.15) − 1
= 1.44210 − 1
= 0.4421 = 44.21%
So $100 becomes $144.21 over the four years. Note that the −5% year contributed the factor 0.95 — a smaller factor multiplies the total down, which is why one bad year is not cancelled by one equally good year.
- The arithmetic average. Add and divide: (10% − 5% + 20% + 15%) / 4 = 40% / 4 = 10.00%
- The geometric average. The fourth root of the growth factor: (1.44210)1/4 − 1 = 1.09586 − 1 = 9.58%
The slide states the conclusion in one line: the fund manager made an average of 9.58% per year, realising a holding period return of 44.21%. Two numbers, both true, and the gap between them is the subject of the next subsection.
Both, in their own place, and the lecture is careful about which is which. Slide 4 gives the definitions: the arithmetic average is "the return earned in an average period over multiple periods", and the geometric average is the "average compound return per period".
The distinction is not hair-splitting, it is a statement about which question is being asked. If you are asking "what did a typical year look like?", the arithmetic average is the answer, and it is the right forecast for a single future year drawn at random — which is why portfolio theory uses it. If you are asking "what did my money actually compound at?", the geometric average is the answer, and it is the right number for a long-horizon outcome.
This is why fund marketing is a minefield. A "10% average annual return" headline is arithmetically true and describes no achievable outcome: nobody can invest for one year per year. The investor's experience is the geometric figure, and it is always the lower of the two whenever the returns vary at all.
The slide states the rule — "the geometric average will be less than the arithmetic average unless all the returns are equal" — and the reason is worth having, because it is the mechanism behind a great deal of investment folklore.
It comes from asymmetry of multiplication. A −50% year halves your money; it takes a +100% year to get back to where you started. Equal-sized up and down moves are not equal in their effect, and the bigger the moves, the worse the asymmetry. The arithmetic average treats +50% and −50% as cancelling; the compounded reality is 1.5 × 0.5 = 0.75, a 25% loss.
So the gap between the two averages is a measure of volatility itself. Hold the arithmetic average fixed and increase the volatility and the geometric average falls — the "volatility drag" that makes a consistently modest return beat a wildly swinging one with the same average. For an investor who is compounding rather than withdrawing, reducing volatility is worth as much as raising the mean, and that is one practical reason the second half of this lecture exists.
Two further notes the slides do not carry. First, when someone quotes a historical "average return on stocks of about 10% a year", ask which average — the arithmetic figure is the larger one, and the geometric figure is what a buy-and-hold investor actually earned. Second, the difference is not an artefact of small samples: it persists over decades, because it is a property of the arithmetic of compounding rather than of the particular data.
Risk, Risk Premium, and the Sharpe Ratio
slides 6–12Slide 6 opens with a disclaimer that is unusually honest for a textbook: there is no universally agreed-upon definition of risk. The lecture then adopts one — and adopting one is what makes the rest of the material possible.
Two things about the definition are worth noticing before the arithmetic. Divisor: the slide divides by T, not T − 1 — this is the variance of the observed population of returns, and the workbench follows the same convention everywhere so that the numbers match the lecture. (Statistical software will often divide by T − 1 instead, giving a slightly larger answer; see the trap note on Homework 2.) Units: variance is in squared percent, which is why the standard deviation — its square root — is the number people actually quote. "$3.36\%$" is interpretable; "$0.0011$" is not.
Why a spread-out distribution is the right definition slide 7
Slide 7 justifies the choice by assuming the expected stock return follows a normal distribution, and the justification is the familiar bell-curve arithmetic from any statistics course:
| Range around the mean | Probability the return lands there | What it means |
|---|---|---|
| ± 1 SD | 68.26% | Roughly two thirds. The slide's own gloss: a yearly return falls within one standard deviation of the mean about 2/3 of the time. |
| ± 2 SD | 95.44% | Nineteen years in twenty. |
| ± 3 SD | 99.74% | The far tails — under three days in a thousand, if returns were truly normal. |
So a larger standard deviation does not make a loss more likely in any direct sense — it makes any deviation from the mean more likely, up or down. That is the honest way to read the definition, and it is why the next ingredient is needed: a spread on its own says nothing about whether you are being paid for it.
The deviations sum to zero, which is always true and is a useful check that the mean was computed correctly. The interpretation: these returns varied by about 3.4 percentage points from their own average, in a typical year.
The slide's own line reads Variance = .0045 / 4 = .0011 Standard Deviation = .0336. The variance is right — .0045/4 = .001125, which rounds to the .0011 printed. The standard deviation is not: √.001125 = .0335410, which is 3.354%, and that rounds to 3.35%, not 3.36%.
This is a rounding slip in the last printed digit, not a structural error, and it is worth a note only because it is the kind of drift that propagates: a reader who copies 3.36% forward into a Sharpe ratio carries a small error into an answer that will then be marked against the slide's own figure. The whole calculation is self-consistent everywhere else — the mean of .105, the deviations, the squared deviations and the variance all check out against the four returns given.
The correct figure is 3.35% (0.0335). Note also that had the slide divided by T − 1 = 3 rather than by T = 4 — the sample convention — the answer would be √(.0045/3) = .03873 = 3.87%, which is a different number again. This workbench uses the lecture's ÷T convention throughout.
Risk premium slide 9
A standard deviation is a description; a return is a reward. Slide 9 connects them by naming the benchmark that separates the two:
The risk-free rate is defined by instrument rather than by number — the interest rate on T-bills, money market funds and bank deposits, the assets where the return is contractual and the probability of default is taken to be zero. This is Lecture 6's treasury curve one more time, and the connection is deliberate: a treasury's YTM is the risk-free rate, which is why Lecture 6 could call it that.
The slide's gloss is the sentence to keep: the risk premium is the added return, over and above the risk-free rate, resulting from bearing risk. Everything else in the lecture is about measuring this quantity and deciding whether it is adequate. Averaged over common stocks it gets its own name — the equity risk premium — and it is the single most consequential estimate in corporate finance, because it is the number that will be multiplied by beta to produce the cost of equity in Week 10.
The tradeoff, and a warning worth reading twice slide 10
Slide 10 plots historical average annual return against annual standard deviation for four asset classes — T-bills, T-bonds, large-company stocks and small-company stocks — and the picture is an upward-sloping scatter. The slide's caption is the lesson in four words: higher risk, higher return.
It also carries a quotation that is not from the textbook at all, and which is more useful than the chart for talking to anyone outside this course. The slide reproduces 郭树清, then chairman of the 中国银行保险监督管理委员会, speaking at the 第十届陆家嘴论坛 on 14 June 2018:
The quotation is worth putting in full, because it is this lecture's chart translated into a sentence a non-specialist can act on:
"高收益意味着高风险，收益率超过6%的就要打问号，超过8%的就很危险，10%以上就要准备损失全部本金。"
Read it alongside the scatter plot and the two say the same thing. The chart says the assets with the highest average return also had the widest spread. The quotation says that a promised return far above the risk-free rate is not a gift — it is a statement about how much risk someone has taken, or about a fraud. The 6% / 8% / 10% thresholds are calibrated to the Chinese market of 2018, where the risk-free rate was around 3–4%, so they should be read as multiples of the risk-free rate rather than as fixed numbers.
There is a second, subtler point in the quotation that connects directly to slide 11. It names the quantity this lecture is built on: excess return. "Exceeding 6%" means nothing in isolation — 6% when the risk-free rate is 5% is a 1-point premium, and 6% when it is 1% is a 5-point premium. The premium, not the headline return, is what is being paid for the risk, and computing it is the first thing the Sharpe ratio does.
Sharpe ratio slide 11
Slide 11 turns the premium into a score by dividing it by the risk taken to earn it:
Three of the slide's statements about it deserve to be separated, because each answers a different question:
The return generated when one undertakes one unit of risk. It is the tradeoff between risk and return compressed into a single number — which is exactly why it, and not the raw return, is the fair way to compare two assets.
Evaluating the performance of financial assets — stocks and bonds alike — and comparing fund managers. Slide 11's wording is deliberate: the same formula that scores an asset scores the person who chose it.
A higher Sharpe ratio indicates superior performance. Higher is better, and unlike a raw return the comparison is meaningful across assets with completely different volatilities.
It scores realised and expected performance on the same axis, so a fund with one spectacular year can look better than a consistent one. And it assumes volatility is the risk that matters — the assumption the whole lecture is built on.
Two securities, and a question with no numerical answer slide 12
Slide 12 sets up the decision that the Sharpe ratio exists to resolve, and it does so with two cases that look similar and are not:
| Case | Stock A | Stock B | Which do you buy? |
|---|---|---|---|
| Case 1 | E(R) = 10%, σ = 5% | E(R) = 15%, σ = 10% | No determinate answer. B pays more and is riskier, so the choice depends on how much risk you are willing to bear. This is the tradeoff, and it has no single solution. |
| Case 2 | E(R) = 10%, σ = 5% | E(R) = 15%, σ = 3% | Stock B, unambiguously. Higher return and lower risk — B dominates A. No preference is needed to decide. |
The distinction between the two cases is not new to this lecture — it is the same structure as Lecture 3's treatment of mutually exclusive projects. There, NPV could rank two projects unambiguously only when one dominated; when the projects differed in scale and timing, the criteria disagreed and a rule was needed to resolve it.
Here it is sharper, because a stock has exactly two attributes that matter. Case 2 is dominance: B is better on both axes, so every investor agrees regardless of taste — that is what an economist means when they say a choice is "unambiguous". Case 1 is a tradeoff: A is better on one axis and worse on the other, so the ranking depends on the investor, and no amount of analysis will settle it.
Everything after this slide is an attempt to get more of the world into Case 2. If you can combine assets so that some portfolio has a higher return and a lower standard deviation than either asset alone, you have manufactured dominance out of a tradeoff — and that is precisely what the diversification result does. Slide 25 demonstrates it with the stock-and-bond portfolio: a 50/50 mix of two assets returns more than the bond and is less risky than either.
Portfolios and Diversification
slides 13–31The second half of the lecture is the part that changes how you think about risk, and it begins by asking how two assets move together — a question that has no analogue in anything so far, because every previous lecture valued one thing at a time.
Covariance and correlation slides 13–14
The covariance is defined exactly as you would guess from the variance: it is the same "squared deviation" recipe, but with the two securities' deviations multiplied together instead of squared. The consequence is the sign convention that makes it useful. If both assets tend to be above their means at the same time, the products are positive and the covariance is positive; if one is up when the other is down, the products are negative and so is the covariance.
Its units, though, are the square of a return, which makes it hard to interpret — a covariance of −0.0117 tells you less than a correlation of −0.998. So the slide normalises it: dividing by the two standard deviations produces the correlation coefficient, a pure number bounded between −1 and +1. That bound is the whole point of the transformation, and it is what makes the four categories on slide 14 possible:
Slide 14 attaches two examples from the Chinese market, and they are chosen to show the two signs rather than to be exotic:
The two-asset example the rest of the lecture runs on slides 15–19
Slide 15 sets up a two-asset world with three equally likely states of the economy and only two securities: a stock fund and a bond fund. Everything from here to the efficient frontier is built on these six numbers.
Expected returns — the probability-weighted average, which with equal probabilities is just the arithmetic mean of the three:
E(rS) = ⅓(−7%) + ⅓(12%) + ⅓(28%) = 11.00% E(rB) = ⅓(17%) + ⅓( 7%) + ⅓(−3%) = 7.00%Variances and standard deviations — the squared deviations from each fund's own mean:
Stock fund Bond fund (−7 − 11)² = 0.0324 (17 − 7)² = 0.0100 (12 − 11)² = 0.0001 ( 7 − 7)² = 0.0000 (28 − 11)² = 0.0289 (−3 − 7)² = 0.0100 ──────── ──────── sum 0.0614 sum 0.0200 ÷3 0.0205 ← var ÷3 0.0067 ← var √ 14.3% ← SD √ 8.2% ← SDCovariance — multiply the two funds' deviations, then weight by probability (slide 19's table):
Scenario Stock dev Bond dev Product × ⅓ weighted Recession −18% +10% −0.0180 −0.0060 Normal +1% 0% 0.0000 0.0000 Boom +17% −10% −0.0170 −0.0057 ─────── Sum / covariance −0.0117Correlation — normalise the covariance by the two standard deviations:
ρ = Cov / (σBσS) = −0.0117 / (0.143 × 0.082) = −0.998A correlation of −0.998 is very close to perfect negative correlation: in this world, whenever the stock fund does badly the bond fund does well. That is a deliberately extreme setup, chosen so the diversification effect in the next subsection is dramatic and easy to see. Real stock and bond funds are negatively correlated but nowhere near this strongly.
Portfolio expected return — the easy half slides 20, 23
The portfolio's expected return is a weighted average of the components — there is no cross-term, no surprise, and nothing to be careful about. If you put half your money in an asset returning 11% and half in one returning 7%, you expect 9%. The slide makes this explicit with the calculator line 9% = 50%×(11%) + 50%×(7%).
Portfolio variance — the hard half, and the whole point slides 20, 24
Read the three terms in order, because the third one does all the work. The first two are the weighted variances — the "average of the parts" you would naively expect. The third is the cross term, and it is the only place the correlation appears. It is positive when the two assets move together, and it is negative when they move oppositely — which means that with a negative correlation, combining two risky assets can produce something riskier than the average of the parts would suggest, or safer.
Formally, the slide states the result in one clause: portfolio SD is less than the weighted average of the SDs of the individual securities as long as ρBS < 1. The inequality is strict and the exemption is narrow — only perfect positive correlation removes the benefit entirely. Even ρ = 0.9 leaves some.
- The portfolio return in each scenario. A weighted average, scenario by scenario: Recession 50%(17%) + 50%(−7%) = 5.0% Normal 50%( 7%) + 50%(12%) = 9.5% Boom 50%(−3%) + 50%(28%) = 12.5%
- The expected return. E(rP) = 50%(11%) + 50%(7%) = 9.0%
- The variance. Squared deviations of the three portfolio returns from 9%: (5.0 − 9.0)² = 0.0016 (9.5 − 9.0)² = 0.000025 (12.5 − 9.0)² = 0.001225 ──────── ÷ 3 0.00095 ≈ 0.0010 ← variance √ 0.0308 = 3.08% ← SD
- Or through the formula, which must give the same answer: σP² = (0.5)²(0.0816)² + (0.5)²(0.143)² + 2(0.5)(0.5)(−0.998)(0.0816)(0.143) = 0.001665 + 0.005112 − 0.005794 = 0.000983 ≈ 0.0010 ✓
The diversification effect slide 25
Now compare the portfolio with its components, because the comparison is the lecture's central result:
| Holding | Expected return | Standard deviation | Comment |
|---|---|---|---|
| 100% bonds | 7.00% | 8.16% | The safer asset on its own. |
| 50 / 50 portfolio | 9.00% | 3.08% | Higher return than bonds, and less than a third of the stock fund's risk. |
| 100% stocks | 11.00% | 14.31% | The higher return, at a much higher risk. |
The slide's own conclusion is worth quoting exactly: an equally weighted portfolio has less risk than either stocks or bonds held in isolation. This is not a small effect. The portfolio's standard deviation of 3.08% is lower than the bond fund's 8.16% — the safer of the two assets — while its expected return is higher than the bond fund's. That combination is the dominance case from slide 12, manufactured where before there was only a tradeoff.
The answer is yes, provided ρ is anything less than exactly +1 — and the reason is that the condition on the formula is an inequality, not an equality. Look again at the cross term 2wBwSρσBσS. What diversification requires is that this term be smaller than the benchmark of perfect co-movement, and every value of ρ below +1 achieves that.
Slide 31 states the boundary precisely: when ρ is exactly 1, no diversification can happen. If not, diversification exists even when ρ > 0. Which means the useful question is never "are these assets negatively correlated?" — that is too strict a test, and almost nothing in a real portfolio passes it. The useful question is "how much less than 1 is the correlation?", because the gap between ρ and +1 is the entire amount of risk reduction available.
This is why the answer to the 美的/格力 question on slide 14 is not the depressing one. Two appliance makers with a strongly positive correlation still diversify each other: a bad quarter for the industry hits both, but a factory fire or an accounting scandal at one is uncorrelated with anything at the other. The residual co-movement is what cannot be removed — and naming that residual is the subject of the last section.
The opportunity set and the efficient frontier slides 26–31
Slide 26 generalises. The 50/50 mix was one choice among many; vary the weights and you get a continuum of achievable risk-return pairs. Slide 26 tabulates it in 5% steps, and the shape of the table is the thing to notice:
| % in stocks | Risk | Return | % in stocks | Risk | Return |
|---|---|---|---|---|---|
| 0% | 8.2% | 7.0% | 55% | 4.2% | 9.2% |
| 10% | 5.9% | 7.4% | 60% | 5.3% | 9.4% |
| 20% | 3.7% | 7.8% | 70% | 7.6% | 9.8% |
| 30% | 1.4% | 8.2% | 80% | 9.8% | 10.2% |
| 35% | 0.4% | 8.4% | 90% | 12.1% | 10.6% |
| 40% | 0.9% | 8.6% | 100% | 14.3% | 11.0% |
| 50% | 3.1% | 9.0% |
Read the risk column from top to bottom and it falls before it rises: 8.2% at all-bonds, down to 0.4% at 35% stocks, then back up to 14.3% at all-stocks. That minimum is the minimum-variance portfolio, and its existence is the whole content of this section. With a strongly negative correlation you can construct a portfolio — 65% bonds, 35% stocks — whose standard deviation is barely a twentieth of either asset's, while its expected return (8.4%) still sits comfortably above the bond fund's 7%.
The slide's defining sentence reads: "It is part of opportunity set, which offers the highest return given the same risk or offers the lowest return given the same return." The second clause is not self-consistent — "the lowest return given the same return" cannot be a criterion for anything, since the quantity being minimised is the quantity being held fixed.
The intended sentence is "the lowest risk given the same return", and the slide's own two clauses are stated correctly elsewhere in the deck — slide 27 says the upper line is better "since they have higher returns for the same level of risk", and slide 30's opening defines the frontier as the points that "dominate the others (high return but low risk)". The two-sided definition of an efficient frontier is standard: no other portfolio offers more return for the same risk, or less risk for the same return.
This matters for the exam, because the concept-explanation section asks for definitions and a student reproducing the slide verbatim would write a sentence that does not parse. Write the corrected version.
How the correlation shapes the frontier slide 28
Slide 28 is the payoff of the whole section. The formula is unchanged; only ρ moves, and the frontier's shape responds:
| Correlation | Shape of the opportunity set | Diversification |
|---|---|---|
| ρ = +1 | A straight line between the two assets | None possible. Every portfolio is on the line joining the endpoints, and the risk is exactly the weighted average. The frontier is the whole line. |
| ρ = 0.2 | A curve bowing to the left of the line | Most of the benefit already present — the bow is substantial even at mildly positive correlation. |
| ρ = −1 | A curve touching the vertical axis | Complete risk reduction possible. Some mix has exactly zero standard deviation — a genuinely risk-free portfolio built from two risky assets. |
The direction of the bow is the thing to remember: lower correlation pushes the curve further left, which means less risk for the same return. Nothing else about the two assets changed — same means, same standard deviations, same everything. The only moving part is how they move together, and it is worth more than the assets themselves.
Many assets slides 29–31
Slides 29 and 30 lift the restriction to two assets and ask what survives. The answer is: all of it, with one addition. With many risky assets there is still an opportunity set, still a minimum-variance portfolio, and still an efficient frontier above it — the diagram is simply filled with a scatter of individual assets inside the curve rather than with two endpoints on it.
The addition is the floor. Slide 31 names it: there is a minimum level of risk that cannot be diversified away, and that is the systematic portion. Adding more assets closes the gap between the portfolio's risk and that floor, but never breaks through it — which is why the next section splits risk in two.
Slide 31's claim that a portfolio "can substantially reduce the variability of returns without an equivalent reduction in expected returns" is the most important sentence in the lecture, and the reason it holds is worth making concrete.
Much of the risk reduction is visible in the formula itself. For an equally weighted portfolio the variance has n own-variance terms, each contributing σ²/n², and about n²/2 distinct covariance terms, each contributing 2·Cov/n². As n grows, the variance terms add up to σ²/n and vanish, while the covariance terms add up to (n−1)/n · Cov and do not. In the limit, the portfolio's variance is the average covariance and nothing else.
Which gives the floor a precise meaning: the risk that survives diversification is the risk every asset shares — the co-movement itself. The risk that dies is the part unique to each asset, because those parts average out. Own one restaurant and a health-inspection failure is a disaster; own a hundred and it is a line item.
Two practical consequences follow. First, the number of stocks needed for most of the benefit is surprisingly small — the curve flattens well before a hundred, and a few dozen diversified holdings remove the great majority of diversifiable risk. Second, the floor is a floor for a given asset class: a portfolio of Chinese stocks cannot diversify away Chinese market risk, and no amount of adding more Chinese stocks will help. Moving the floor requires adding a different kind of asset — which is exactly what the stock and bond fund in this lecture's example are doing with their −0.998 correlation.
Systematic and Unsystematic Risk
slides 32–34Slide 32 draws the line that the whole of modern portfolio theory is organised around. It is the same line as slide 31's floor, given two names and a test for telling them apart.
Any risk that affects a large number of assets, each to a greater or lesser degree. The slide's examples are the economy's own weather: uncertainty about GDP, interest rates or inflation; policy uncertainty around a presidential election; Covid-19; war.
The reason it cannot be diversified away is stated with unusual precision: the risk arises from outside the "system" or the capital market — meaning from the economy as a whole, not from any security in it. Owning every asset in the market does not help, because every asset is exposed to the same thing.
A risk that specifically affects a single asset or a small group of assets. The slide's example is any announcement specific to one company — an earnings miss, a recall, a fraud, a CEO's resignation.
Also called idiosyncratic risk, or firm-specific risk. This is the half that can be diversified away, and slide 31 gave the mechanism: "worse than expected returns from one asset are offset by better than expected returns from another."
Slide 32's categories are not quite as clean as the table above makes them look, and it is worth seeing where the seam is. A risk is systematic if it affects many assets — but "many" is not a number, and the same event can sit on either side depending on the portfolio.
An oil price spike is systematic for an airline-heavy portfolio and close to idiosyncratic for one holding only software firms. A change in national property policy is systematic for 美的 and 格力 and nearly irrelevant to a drug developer. The classification is relative to the portfolio you actually hold, which is exactly why it is defined by a diversification test rather than by a list of events.
So the operational version of slide 32 is a test, not a taxonomy. Ask: if I held a fully diversified portfolio, would this risk still be there? If yes, it is systematic, and no rearrangement of your holdings will remove it — insurance, hedging or accepting it are the only options. If no, it is unsystematic, and the cheapest remedy is simply to hold more things.
Slide 33 illustrates systematic risk with a quotation and a date: "25 Oct 2020, in the Bund Summit, Jack Ma said…", followed by the line about Chinese finance having no systemic risk because it has no system.
The 第二届外滩金融峰会 ran from 23 to 25 October 2020, and 马云's keynote was delivered on the morning of 24 October 2020 — the middle day, not the last. The date is a day late, in the same way that Lecture 1's listing date was a day early; the substance of the quotation is accurate and the wording matches the reported speech.
The difference between the quotation's 系统性风险 and the slide's framing is worth noting too, since it is a nice illustration of this section's own distinction. 马云's claim is about the financial system's exposure — a macroeconomic, systematic proposition, whether or not one agrees with it. It is not a statement about any individual firm. Which makes it a good example of systematic risk precisely because it cannot be diversified: an investor holding a portfolio of all Chinese financial assets has the same exposure to that claim as one holding a single bank.
Total risk, and what gets priced slide 34
Slide 34 closes the lecture with the accounting identity and the consequence that sets up next week. The standard deviation of returns is a measure of total risk — it is the sum of both components, which is exactly why it was the right measure for a single asset in isolation and the wrong one for a diversified portfolio.
The slide's final sentence is the bridge: for well-diversified portfolios, unsystematic risk is very small. Consequently, the total risk for a diversified portfolio is essentially equivalent to the systematic risk. Slide 34 also says plainly that the measures of the two components are next class's material.
The lecture ends mid-argument, and deliberately. It has established that a diversified investor bears only systematic risk — so only systematic risk should be priced, because an investor who can eliminate a risk for free will not pay to be compensated for bearing it. Portfolios that differ in unsystematic risk but have the same systematic exposure must therefore offer the same expected return.
Which leaves exactly one question unanswered, and it is the one the next lecture answers: how much systematic risk does a given asset carry? The answer will be beta, a measure of how strongly the asset moves with the market, and it replaces the standard deviation in the pricing equation. The security market line then states the whole course's discount rate in one formula: E(R) = rf + β[E(Rm) − rf].
Read that against this lecture's Sharpe ratio and the family resemblance is clear. Both say "risk-free rate plus a premium per unit of risk"; the difference is what counts as a unit. The Sharpe ratio divides by total risk — fine for comparing whole portfolios, wrong for pricing individual assets, because it charges an asset for risk its holder can diversify away. Beta charges only for the part that cannot be escaped.
Formula Sheet
print-friendlyKey Concepts
the vocabularySelf-Check
16 questions · gradedThe examinable core is four things: the two averages and when each applies, the Sharpe ratio, the portfolio variance formula and what the correlation term does to it, and the systematic versus unsystematic split. Two traps account for most lost marks: comparing two assets on return alone without normalising by risk, and treating portfolio risk as a weighted average of the assets' risks — which is precisely the mistake the formula exists to prevent.
A · Concept check
The formula. The variance of a two-asset portfolio is
σP² = wB²σB² + wS²σS² + 2wBwSρBSσBσS └────── weighted variances ──────┘ └──── the cross term ────┘Where the difference comes from. The first two terms are what a "weighted average of risks" would produce. The third term is the only place the correlation appears, and it is what makes the answer differ from the naive one. When ρ is positive the cross term is positive, which raises the variance above the weighted-variance part; when ρ is negative it is negative and lowers it.
Why it is less than the weighted average of the standard deviations. Write the weighted average of the standard deviations and square it:
(wBσB + wSσS)² = wB²σB² + wS²σS² + 2wBwSσBσSThe first two terms are identical to the portfolio's. The difference is entirely in the cross term: the portfolio has 2wBwSρσBσS where the squared weighted average has 2wBwSσBσS. Since ρ ≤ 1, the portfolio's cross term is smaller, so its variance is smaller, so its standard deviation is smaller. That is the whole proof, and it needs only one inequality.
The condition for the effect to disappear. Only at ρ = +1, where the two cross terms are equal, the variance becomes the perfect square (wBσB + wSσS)², and the standard deviation is exactly the weighted average — no benefit at all. For any ρ below +1 the cross term is smaller and some diversification survives, including when ρ is positive. Slide 28 draws the three cases: a straight line at ρ = +1, a curve bowing left at ρ = 0.2, and a curve reaching the vertical axis at ρ = −1, where complete risk reduction is possible.
The economic mechanism. The arithmetic has an intuition behind it. Asset-specific shocks are independent, so a bad surprise in one asset is offset by a good surprise in another — the errors partly cancel. What does not cancel is the co-movement: the part of each asset's return driven by the economy as a whole moves in the same direction in every asset, so no amount of combining removes it. That residual is the systematic risk of the last section, and it is the floor in slide 31's diagram.
The worked numbers. The lecture's own example — stock fund 11% / 14.3%, bond fund 7% / 8.2%, ρ = −0.998 — at equal weights:
Weighted average of the SDs = 50%(14.3%) + 50%(8.2%) = 11.25% Portfolio SD = 3.08%And the portfolio's expected return of 9.0% is higher than the bond fund's 7.0% — so it beats the safer of its own two components on return and on risk.
That last line is the point of the whole exercise. The diversification effect does not trade return for risk; it improves both at once, which is why the lecture calls it a free lunch in all but name.
B · Multiple choice
C · Applied calculation
(a) The holding period return. Chain the five growth factors:
HPR = (1.08)(0.88)(1.22)(1.05)(1.18) − 1 = 1.43661 − 1 = 0.43661 = 43.66%Answer (a): 43.66% over the five years.
(b) The two averages.
Arithmetic = (8 − 12 + 22 + 5 + 18) / 5 = 41 / 5 = 8.20% Geometric = (1.43661)1/5 − 1 = 1.075146 − 1 = 7.51%Answer (b): arithmetic 8.20%, geometric 7.51%.
(c) Which to quote. It depends on the question, and stating that is the answer. For a long-horizon description of what an investor's money did — the usual meaning of "the fund's performance" — quote the geometric average, 7.51%, because it is the compound rate that reproduces the 43.66% five-year outcome. For a forecast of a single future year, the arithmetic average 8.20% is the correct input, and it is why portfolio theory and the CAPM use it.
What you should not do is quote the arithmetic figure to an investor and call it "the average annual return". It is arithmetically true and describes no achievable experience, since nobody invests for one year per year.
(d) What the gap measures. The 0.69-percentage-point gap is a measure of volatility. The geometric average is always at or below the arithmetic, with equality only when every return is identical; with a −12% year in the series the two cannot coincide. Holding the arithmetic average fixed, a wider dispersion of annual returns pushes the geometric average further down — the "volatility drag" that makes a steady return preferable to a swinging one with the same mean. For an investor who is compounding rather than withdrawing, reducing volatility is worth as much as raising the mean.
Check: 1.0751465 = 1.43661 → 43.66% ✓(a) Expected return. A weighted average, nothing more:
E(rP) = 0.70(12%) + 0.30(18%) = 8.4% + 5.4% = 13.80%Answer (a): 13.80%.
(b) Standard deviation. Slide 24's formula, with σ in percentage points:
σP² = (.70)²(18)² + (.30)²(25)² + 2(.70)(.30)(.40)(18)(25) = 158.76 + 56.25 + 75.60 = 290.61 σP = √290.61 = 17.05%Answer (b): 17.05%.
(c) Against the weighted average.
Weighted average of the SDs = .70(18) + .30(25) = 12.6 + 7.5 = 20.10% Portfolio SD = 17.05% Reduction = 20.10 − 17.05 = 3.05 ppThe portfolio is 3.05 percentage points less risky than the weighted average of its components, for the same expected return. This is the diversification effect — and it is the modification of exactly the case-1 tradeoff from slide 12: a mix that beats its own components on risk without giving up return.
(d) The correlation's effect. Only the cross term moves. Holding the weights and the standard deviations fixed:
ρ = +0.40: σP² = 158.76 + 56.25 + 75.60 = 290.61 σP = 17.05% ρ = 0: σP² = 158.76 + 56.25 + 0.00 = 215.01 σP = 14.66% ρ = −0.40: σP² = 158.76 + 56.25 − 75.60 = 139.41 σP = 11.81%The pattern: the expected return is 13.80% in all three cases — the correlation does not touch it. What changes is the risk, and it changes a great deal: from 17.05% down to 11.81% as ρ goes from +0.40 to −0.40, a reduction of more than five percentage points from the correlation alone.
Why. The cross term is 2wBwSρσBσS, which is linear in ρ: everything else is fixed, so moving ρ from +0.4 to −0.4 subtracts twice the ρ = +0.4 cross term from the variance. A negative correlation means the two assets tend to move oppositely, so their deviations partly cancel when combined, and the portfolio inherits less of each one's risk.
The practical reading. Diversification is a property of the combination, not of the assets. The same two assets, in the same proportions, give a portfolio ranging from 11.81% to 17.05% standard deviation depending only on how they co-move. This is why slide 28 plots the opportunity set three times, once for each correlation, and why an investment process that ignores correlation is leaving the cheapest risk reduction on the table.
Note the floor, too. Even at ρ = −0.40 the portfolio's risk does not fall to zero, because a −0.4 correlation is far from −1. Only perfect negative correlation allows complete elimination — and in a world where every asset shares some exposure to the economy, that never quite happens. That residual is the systematic risk of the next section.
| Scenario | Probability | Stock fund | Bond fund |
|---|---|---|---|
| Recession | 1/3 | 0% | 14% |
| Normal | 1/3 | 12% | 8% |
| Boom | 1/3 | 24% | 2% |
(a) Expected returns, variances and standard deviations.
Stock fund Bond fund (0 − 12)² = 144 (14 − 8)² = 36 (12 − 12)² = 0 ( 8 − 8)² = 0 (24 − 12)² = 144 ( 2 − 8)² = 36 ───── ──── sum 288 sum 72 ÷3 96 ← variance ÷3 24 ← variance √ 9.80% ← SD √ 4.90% ← SD E(rS) = (0 + 12 + 24)/3 = 36/3 = 12.00% E(rB) = (14 + 8 + 2)/3 = 24/3 = 8.00%Answer (a): stock fund 12.00% with SD 9.80%; bond fund 8.00% with SD 4.90%.
(b) The covariance. Multiply the two funds' deviations state by state, then weight by probability:
Scenario Stock dev Bond dev Product × ⅓ weighted Recession −12% +6% −72.00 −24.00 Normal 0% 0% 0.00 0.00 Boom +12% −6% −72.00 −24.00 ─────── Covariance −48.00Answer (b): covariance = −48 (in units of percent squared, i.e. −0.0048 in decimal).
(c) The correlation coefficient.
ρ = Cov / (σSσB) = −48.00 / (9.798 × 4.899) = −48.00 / 48.00 = −1.000Answer (c): ρ = −1.00 — perfect negative correlation. The interpretation: the two funds move in exactly opposite directions, in a fixed proportion. Every good state for stocks is a bad state for bonds to precisely the degree that the deviations are exact mirror images: the stock deviations are +12, 0, −12 and the bond deviations are −6, 0, +6, which are the same numbers scaled by −0.5.
(d) A 50/50 portfolio, and the risk-free possibility.
Portfolio return in each state: Recession: .5(0) + .5(14) = 7.0% Normal: .5(12) + .5( 8) = 10.0% Boom: .5(24) + .5( 2) = 13.0% σP² = (.5)²(96) + (.5)²(24) + 2(.5)(.5)(−1)(9.798)(4.899) = 24 + 6 − 24 = 6 σP = √6 = 2.45%The portfolio's standard deviation is 2.45% — dramatically below the 9.80% and 4.90% of the two funds, and below their weighted average of 7.35%. And because ρ = −1, a risk-free combination exists: setting the variance to zero and solving for the weight gives the mix that eliminates risk entirely.
Set σP² = 0: w²(96) + (1−w)²(24) + 2w(1−w)(−1)(9.798)(4.899) = 0 w²(96) + (1−w)²(24) − 48w(1−w) = 0 (9.798w − 4.899(1−w))² = 0 9.798w = 4.899(1 − w) → 14.697w = 4.899 → w = 1/3 Check: ⅓ in stocks, ⅔ in bonds → σP = 0Answer (d): the 50/50 portfolio has SD 2.45%; and with ρ = −1 a mix of one third stocks, two thirds bonds has zero standard deviation — a genuinely risk-free portfolio assembled from two risky assets.
The two things this question is testing. First, that the covariance is computed from paired deviations, and that a negative covariance is what produces a negative correlation — recognising the −48 and dividing by the product of the standard deviations is the whole calculation. Second, what the correlation buys. At ρ = −1 the opportunity set is a curve that touches the vertical axis, which is slide 28's third case, and complete risk elimination is the payoff.
And a warning about the setup. This is a deliberately extreme example — like the lecture's own ρ = −0.998 — constructed so that the arithmetic comes out cleanly at exactly −1.00. Real stock and bond funds are negatively correlated, but an order of magnitude less strongly; a correlation of −1 implies a fixed, exact linear relationship between the two returns that no real pair of securities exhibits. The mechanism being demonstrated is real; the magnitude is a teaching device.
(a) Expected return and standard deviation.
E(rP) = .60(16) + .40(8) = 9.6 + 3.2 = 12.80% σP² = (.60)²(28)² + (.40)²(12)² + 2(.60)(.40)(.20)(28)(12) = 282.24 + 23.04 + 32.256 = 337.536 σP = √337.536 = 18.37%Answer (a): E(R) = 12.80%, SD = 18.37%.
(b) The Sharpe ratio.
Sharpe = (12.80 − 3) / 18.37 = 9.80 / 18.37 = 0.533Answer (b): 0.53.
(c) Against the growth fund alone.
| Holding | Expected return | Standard deviation | Sharpe |
|---|---|---|---|
| 60 / 40 portfolio | 12.80% | 18.37% | 0.53 |
| Growth fund alone | 16.00% | 28.00% | 0.46 |
Answer (c): the portfolio gives up 3.2 percentage points of expected return (16.00 → 12.80) and in exchange removes 9.63 points of risk (28.00 → 18.37). On the risk-adjusted measure the portfolio wins, 0.53 against 0.46 — a portfolio that mixes in the lower-return asset and ends up the better investment.
(d) The floor. Whatever the weights, the portfolio's standard deviation cannot be driven below a certain level, because the two assets are positively correlated at +0.20. The formula's cross term is then positive, so the two variances are only partly offset and some risk always survives the combination. The floor is not zero here — and it is not zero for any real portfolio, because every asset in an economy shares exposure to the economy itself.
That irreducible part is systematic risk — slide 32's market-wide risk, arising "from outside the system" — and the minimum is reached at the minimum-variance portfolio, the lowest point of the opportunity set from which the efficient frontier runs upward. Slide 31 puts it exactly: "there is a minimum level of risk that cannot be diversified away, and that is the systematic portion."
The two lessons this question is testing. First, that higher return is not the same as better performance, and that the Sharpe ratio is the tool that tells them apart — the growth fund wins on return and loses on performance. Second, that diversification has a floor, and what the floor is made of. The unsystematic risk that disappears when you combine assets is a risk nobody needs to bear and so nobody is paid to bear; the systematic risk that remains is the only kind that should command a premium. Questioning which risks deserve compensation is where the next lecture begins.
Cost of Capital
Concept Map
how the lecture buildsLecture 7 ended on the split between systematic and unsystematic risk and said explicitly that measuring the surviving half was next week's job. Slide 8 does exactly that: beta is the measure of systematic risk, and the CAPM converts it into a required return. The security market line drawn here is the same upward-sloping line Lecture 7 described in words — higher beta, higher expected return — now with an equation attached.
It also closes a loop opened in Lecture 3. The NPV rule says accept a project when its return exceeds the return available on a financial asset of comparable risk. Until now "comparable risk" had no operational meaning. After this lecture it does: find the beta, plug it into the SML, and the resulting number is the hurdle rate. The discount rate in every earlier NPV was an assumption; from here on it is a calculation.
Slide 27 says the cost of debt is the bond's yield to maturity, and warns to use the current yield rather than the historical one. That is Lecture 6's central calculation used in reverse: there, the YTM was the discount rate that made the bond's cash flows equal its price; here, it is the return the firm must offer to borrow today. A firm that issued a 3% coupon bond five years ago does not face a 3% cost of new debt if comparable bonds now yield 6% — the old bond's market price has fallen, and its YTM has risen to 6%. The marginal cost is what belongs in the WACC.
The tax adjustment is the other half. Interest is deductible, so a 6% borrowing rate at a 25% tax rate costs the firm 4.5% after tax: the government absorbs a quarter of the interest bill through the tax shield. Nothing analogous happens to equity, because dividends are paid out of after-tax earnings. That asymmetry is the first hint of the capital-structure argument in Week 11.
What the Cost of Capital Is
slides 1–7The lecture opens with two real financings and asks the same question of each: what is the cost to the firm? Slide 2 shows 江苏省广电有线信息网络股份有限公司 issuing a three-year AAA bond — 24苏广K1, ¥500 million, coupon 2.77%. Slide 3 shows 烽火通信 (600498) announcing a private placement of up to ¥1.5 billion of new shares in October 2024, with the proceeds going to working capital.
The bond's cost looks obvious: the firm pays 2.77% a year. The equity's cost looks like nothing at all, because no coupon is written on a share. That asymmetry is the whole reason this lecture exists. The correct answer is the same in both cases, and it is not the coupon:
Slide 4 defines the two return terms carefully. The required return is the minimum return an investment must offer to be worth undertaking; the expected return is what the investor believes the investment will actually generate. They are the same number when the asset is priced correctly — and that equality is worth stating as an identity, because it is the bridge between the firm's decision and the investor's decision:
| Side of the contract | Gives up | Receives | The same number seen twice |
|---|---|---|---|
| Firm (发行人) | Cash today | A claim — stock or bond — and the obligation to pay a return | The cost of capital: what the firm must earn to satisfy the claim. |
| Investor (投资者) | Cash today | The financial asset and its future payoffs | The expected return: what the investor requires for giving up the cash. |
Slide 5 then splits the number along the balance-sheet identity Asset = Liability + Equity. The cost of capital is a weighted blend of the cost of equity and the cost of debt, and the lecture will attack each side separately: equity through the CAPM or the DDM, debt through the bond's YTM.
Why issuing equity is costly — and why firms do it anyway slide 6
Slide 6 makes the dilution argument. New shares give new shareholders a claim on the same firm, so the old shareholders' fractional ownership falls. The cost to the old shareholders is exactly the return the new shareholders require on the money they put in, because that is the slice of future value they now own. The lecture lists the two components of the new shareholders' payoff: dividends and capital gains.
Because the cost is a hurdle, not a fee. Issuing equity is expensive in the same way borrowing at 6% is expensive: it obliges the firm to earn at least the investors' required return. If the firm invests the proceeds in a project whose return exceeds that hurdle, the project's NPV is positive and the old shareholders are better off despite owning a smaller slice of a more valuable firm. Dilution in percentage terms is not the same thing as dilution in value.
There are three further reasons a firm may choose equity even when debt looks cheaper. First, debt capacity is finite: at high leverage the cost of debt rises and financial distress becomes a real cost, so the marginal cost of debt is not the coupon on the last bond. Second, equity shares risk with new investors rather than committing the firm to fixed payments in bad states — valuable for a firm with volatile cash flows. Third, a firm with good investment opportunities but little cash may prefer not to pledge assets to a lender if that would leave future projects unfunded.
The caveat is the one that makes the question interesting. If the new shares are underpriced, the old shareholders transfer value to the new ones: they give away a claim worth more than the cash received. That is why seasoned equity offerings are often read as bad news by the market, and why firms with the strongest prospects are the least likely to need one. The lecture's own framing — the cost is the return to new shareholders — is correct, but the price at which the shares are sold decides who bears it.
The opportunity-cost foundation slide 7
Slide 7 supplies the economic argument that makes the whole lecture more than accounting. A firm with excess cash has two choices: pay it out as a dividend, or invest it in a project. If it pays the dividend, the shareholder can put the cash into a financial asset of her choosing. So the project must offer at least the return she could get on a financial asset of comparable risk; otherwise she would rather have the dividend.
This is the cleanest statement in the course of why the discount rate is an opportunity cost. It also explains why the rate must match the project, not the firm: the comparison is with an asset of comparable risk, and a safe project should not be held to the standard of a risky one. That distinction returns on slides 24–26, where a conglomerate's single WACC proves to be the wrong hurdle for its divisions.
- What shareholders can earn elsewhere. The CAPM on slide 13 gives the return on a financial asset of Stansfield's risk: RS = RF + β(RM − RF) = 3% + 1.5 × 7% = 3% + 10.5% = 13.5%
- What a project must therefore earn. A project inside Stansfield with the same risk must clear 13.5%. A project expected to return 12% looks profitable against a 10% bank loan and is still a loser: discounting at 13.5% gives a negative NPV. At 12% the firm would be destroying value while reporting an accounting profit.
The two slides are the same idea from opposite ends. Slide 7 states the rule in economic language — match the financial asset's risk — and slide 13 supplies the machine that produces the number. The rest of the lecture is mostly about making the machine's three inputs trustworthy.
The 2.77% coupon on slide 2 is the cash the firm pays, not the return investors require. Three things separate the two. The bond's price may differ from par, so the YTM differs from the coupon. The coupon is pre-tax while the cost of capital is an after-tax hurdle, so the relevant figure is 2.77% × (1 − TC). And the coupon was fixed when the bond was issued; if market rates have moved, the cost of new debt is the current YTM, not the historical coupon.
The same confusion on the equity side — "we paid a ¥0.30 dividend, so equity costs 1.5%" — ignores both the growth in the dividend and the capital gain that shareholders also require. Slide 22's formula, RS = D1/P0 + g, is the correction: the dividend yield is only the first half of the return.
CAPM and Beta
slides 8–13Slide 8 introduces the Capital Asset Pricing Model in the form the exam will use:
Read it as three pieces. RF is what you get for waiting — the risk-free rate. βi is how much market risk the security carries. (RM − RF) is the price the market pays per unit of that risk — the market risk premium. The product is the compensation for risk; add the compensation to the risk-free rate and you have the required return.
| Input | What it measures | Practical proxy | The trap in using it |
|---|---|---|---|
| RF | The return on an asset with no risk | Treasury bond yield; T-bill for short horizons | The proxy must be default-free and should match the horizon of the cash flows. A corporate bond yield includes a credit spread and is not risk-free. |
| RM | The expected return on the market portfolio | S&P 500, 沪深300, 恒生指数; 上证指数 for all SSE stocks | The index must be broad and must match the investor's opportunity set. A narrow sector index is not the market. |
| βi | The security's sensitivity to market movements | Regression of stock returns on index returns | Measured from past data but used to predict a future return; it changes with leverage and with the business. |
Beta as the unit of systematic risk slides 9 and 12
Slide 12 defines beta in covariance form, which is the version that explains what it is:
The numerator is how much the stock moves with the market; the denominator rescales it by the market's own variance, so that the market itself has a beta of exactly one. A stock with β = 2 tends to move twice as far as the market in the same direction; a stock with β = 0.5 moves half as far. The definition also makes clear what beta leaves out: any movement in the stock that is uncorrelated with the market — the firm-specific risk of Lecture 7 — contributes nothing to the covariance and is therefore not priced.
Slide 9's graph is the security market line. It plots expected return against beta and it is a straight line with intercept RF and slope (RM − RF). Three points on it are worth memorising:
| Beta | Expected return | What it describes |
|---|---|---|
| β = 0 | RF | No systematic risk at all — the risk-free asset's position on the line. |
| β = 1 | RM | The market portfolio, by construction. An index fund sits here and earns the market return. |
| β < 0 | Below RF | Possible for a stock that tends to rise when the market falls. It pays a return below the risk-free rate because it is valuable as a hedge — its risk has a negative price. |
Slide 9 adds a warning that cuts against the way beta is usually treated: the model predicts a future return, so all three inputs should be predictions. A beta estimated from the last five years is evidence about the future, not the future itself. Slide 12 makes the same point in the phrase "we need to know three things in the future".
- Apply the CAPM. RS = 3% + 1.5 × 7% = 3% + 10.5% = 13.5%
- Interpret the answer. The 13.5% is the discount rate for an expansion of this firm. It is also the return Stansfield's shareholders could earn on a diversified portfolio of comparable risk, so the expansion must clear it to be worth doing. Because the firm has no debt, the cost of equity is also its WACC — the first and simplest case of slide 29's formula.
Note what the 13.5% is not: it is not the firm's historical return, and it is not a target. It is the return the market requires today for bearing β = 1.5, and it moves when the risk-free rate or the market risk premium moves.
The market portfolio, in theory and in practice slides 10–11
Ideally the market portfolio holds every outstanding asset in the economy, in proportion to its market value. That portfolio is unobservable and prohibitively expensive to hold, so practice substitutes a broad stock index. Slide 10 names the 上证指数 — all stocks traded on the Shanghai exchange — and slide 11 the S&P 500 — the 500 largest listed companies in the United States. The lecture also mentions 沪深300 and 恒生指数 as commonly used proxies for Chinese and Hong Kong exposure.
The choice matters, because beta is measured relative to a benchmark: the same stock has different betas against the 上证指数 and against the S&P 500. The benchmark should be the one that best represents the investor's alternative — the portfolio a marginal investor would otherwise hold. For an A-share investor that is typically a broad A-share index; for a global investor, a global index. There is no universal answer, which is why published betas differ across data providers even for the same company.
Lecture 7 drew the efficient frontier in mean–standard-deviation space. The capital market line — the line from the risk-free rate tangent to the frontier — is also upward-sloping, and it is easy to confuse with the SML. They are different objects. The capital market line prices total risk (σ) and describes efficient portfolios only. The security market line prices systematic risk (β) and applies to every asset, efficient or not.
The reason is Lecture 7's diversification result. In a well-diversified portfolio the unsystematic risk has cancelled, so total risk and systematic risk coincide along the efficient frontier and the two lines look alike. For an individual stock, the unsystematic risk is still there but is not compensated — the SML uses beta precisely because beta ignores it. A stock can have a large standard deviation and a low beta, and it will sit on the SML at a modest expected return.
This distinction answers a question students often ask: why isn't a volatile stock automatically expected to earn more? Because volatility that can be diversified away is not scarce, and the market does not pay for risks investors can eliminate for free.
First, the market risk premium is RM − RF, not RM. Adding the risk-free rate to βRM double-counts it. If the market return is 10% and the risk-free rate is 3%, the premium is 7%, and a β = 1 stock is expected to return 10%, not 13%.
Second, the market's beta is 1 by construction, not by coincidence. If a problem gives you the market return and asks for the beta of the market, the answer is 1; if it gives you a stock's beta and the market premium, do not also add a separate "market risk" term.
Third, expected and realised are different. The CAPM gives the expected return, which is a centre of a distribution; any one year can land far from it. A stock with β = 1.5 and a 13.5% required return can lose 20% in a year without contradicting the model. Reading a single year's outcome as evidence about beta is the same error as judging a coin by one flip.
Estimating the Three Inputs
slides 14–18Slide 14 turns from the model to the data. The CAPM is only as good as its three inputs, and each has a measurement problem. This section takes them in the order the slides do: the risk-free rate, the market return, and beta.
The risk-free rate slide 14
The risk-free rate is the expected return on an asset with no risk. Slide 14 ranks the candidates by riskiness — Treasury bond < corporate bond < stock — and concludes that Treasury securities are the closest proxy. A corporate bond yield is not risk-free: it contains a credit spread for default, the very quantity Lecture 6 measured. A stock is not a proxy at all.
Two practical refinements the slide does not spell out. Match the horizon: a project with ten years of cash flows should be discounted with a rate built on a ten-year government yield, not a three-month bill, or the term structure will leak into the answer. Match the currency and the default-free issuer: for a renminbi project, the Chinese government curve; for a dollar project, the US Treasury curve. Mixing a renminbi-denominated cash flow with a US risk-free rate imports an implicit currency forecast.
The market return — two methods slide 15
Slide 15 gives two ways to estimate the expected market return, and they face in opposite directions in time.
| Method | Inputs | Direction | Strength and weakness |
|---|---|---|---|
| 1 · Historical index returns | Arithmetic or geometric mean of past index returns | Backward looking | Strength: objective and easy to compute. Weakness: the past need not repeat; the choice of window, index and average changes the answer. |
| 2 · Dividend discount model | Aggregate dividend yield plus expected growth: RM = D1/P + g | Forward looking | Strength: uses today's prices and forecasts rather than history. Weakness: requires a stable growth estimate for the whole market, and g is hard to pin down. |
Two details matter for the arithmetic mean. First, the historical risk premium is the average of RM,t − RF,t — the period-by-period excess return — not the difference between two long-run averages. The two are close but not identical, and the former is the correct definition. Second, the arithmetic mean is the right input for a one-period expected return; the geometric mean describes long-run compounding. Lecture 7 drew exactly this distinction for individual stocks, and it carries over unchanged to the market index.
Method 2 is Lecture 5's constant-growth model applied to the whole market. Slide 15 carries the formula Rs = D1/P + g and stresses that every variable is measured at the aggregate market level. The dividend yield of the index plus the expected growth of aggregate dividends is an estimate of the expected market return; subtract the risk-free rate and you have the market risk premium. Analysts' forecasts of aggregate earnings growth are the usual stand-in for g.
Estimating beta, and why the estimate is fragile slides 16–18
Slide 16 says beta is assumed to be stable for the same stock, which is what licenses the use of historical data. The estimate is the sensitivity of the stock's return to the market's: in practice, a regression of monthly or weekly stock returns on index returns, with the slope equal to Cov(Ri, RM)/Var(RM).
Slide 17 then lists three problems with that procedure and three responses. The pairing is worth learning as a set, because exam questions tend to ask for one without naming the other:
| Problem | Why it bites | The slide's solution |
|---|---|---|
| Beta may vary over time | The stability assumption is an assumption. A firm that enters a new line of business, adopts a new technology, is deregulated, or changes its leverage has a different beta. | Use more sophisticated statistical techniques; prefer estimates that allow for time variation. |
| The sample may be inadequate | Too few observations give a noisy slope; too long a window mixes genuinely different regimes. | More sophisticated estimation; choose the window deliberately rather than by habit. |
| Leverage and business risk change | Beta is an equity beta, so it embeds the current capital structure. A refinancing changes it even if the business does not. | Adjust for the change; look at the average beta of comparable firms in the industry. |
Slide 18 adds the important qualification: most analysts believe betas are generally stable for firms that stay in the same industry, but stability is not invariance. Beta changes when the product line changes, when technology changes, when regulation changes, and when financial leverage changes. The last of these gets its own section on slides 19–20, because it is the one a student can compute directly.
- Apply the definition. βi = Cov(Ri, RM) / Var(RM) = 0.0288 / 0.0160 = 1.80
- Interpret. A beta of 1.8 says the stock has tended to move 1.8% for each 1% move in the market. If the risk-free rate is 3% and the market risk premium is 6%, the required return is 3% + 1.8 × 6% = 13.8%. Note that the units cancel: covariance and variance are both in squared percent, so beta is a pure number.
- The common error. Inverting the ratio gives 0.0160/0.0288 = 0.56, which is not a beta. The denominator is always the market's variance, because beta is measured relative to the market.
Because a raw regression beta is noisy, many practitioners shrink it toward 1. The best-known version is the Blume adjustment: βadjusted = 0.67 × βraw + 0.33 × 1. The logic is that betas in the population average to 1, so a noisy estimate should be pulled toward that average; the further a raw beta is from 1, the more of its distance is treated as estimation error.
The effect is easy to see. A raw beta of 1.5 becomes 0.67 × 1.5 + 0.33 = 1.335; a raw beta of 0.6 becomes 0.67 × 0.6 + 0.33 = 0.732. Both are pulled toward 1, and the adjusted numbers are what Bloomberg and many data vendors report as "adjusted beta".
Is the adjustment right? It is a Bayesian shrinkage, and it will be wrong for a firm whose true beta is genuinely far from 1 — a utility at 0.5 or a leveraged tech company at 2.0. The lecture's own advice on slide 17 is the more robust response: use the industry average beta of comparable firms, which pools many noisy estimates and is less sensitive to any single firm's history. The adjustment and the peer-group approach are two answers to the same sampling problem.
Suppose a firm spent the last five years with no debt and has just borrowed heavily to fund a buyback. Its regression beta is a clean estimate of the old capital structure. Plugging it into the CAPM now understates the cost of equity, because leverage has raised the risk of the residual claim — the subject of the next section. The fix is not to re-run the regression on data that do not exist yet, but to unlever the old beta to an asset beta and relever it at the new debt ratio.
The same warning applies to a firm that has changed industry. A retailer that becomes an online platform has a different business risk; its historical beta describes a company that no longer exists.
Financial Leverage and Beta
slides 19–20Slide 19 introduces financial leverage as the sensitivity of the firm's equity to its fixed financing costs, measured by the debt-to-equity or debt-to-asset ratio. The key claim is in the second bullet: leverage always increases the equity beta relative to the asset beta. The reason is priority. Debt holders are paid first, so in bad states the equity holders absorb the shortfall; the equity claim is a levered position on the firm's assets, and a levered position is riskier than the underlying asset.
This is a weighted average, and the weights are the market values of the two claims. The asset beta is the beta of the operating business viewed from the asset side; it has nothing to do with how the business is financed, and slide 19 says it is usually fixed across an industry. The equity beta is what shareholders actually bear, and it is the beta that enters the CAPM for the stock.
Slide 19 makes two simplifying assumptions that are standard in this course. Debt beta is zero, on the grounds that the return on debt does not vary with the stock market. That is a good approximation for investment-grade debt and a poor one for distressed debt, where bondholders bear business risk too. Asset beta is fixed across an industry, which is what allows a firm to estimate its own asset beta from comparable companies.
With βDebt = 0, the formula inverts into the relevering equation used throughout practice:
The derivation is one line. Set βDebt = 0 in the weighted average: βA = E/(D+E) × βE, so βE = βA × (D+E)/E = βA(1 + D/E). Leverage enters as a multiplier: at D/E = 0 it disappears, at D/E = 1 it doubles the asset beta, and it rises without bound as the firm approaches 100% debt.
- The asset beta does not change. The business is the same, so βAsset stays 0.90. This is the step students skip: leveraging changes who bears the risk, not how much risk the assets contain.
- Relever. At 50/50, D/E = 1, so βAsset = 0.90 = [1/(1+1)] × βEquity βEquity = 2 × 0.90 = 1.80 The slide writes the equity weight as 1/(1+1), which is E/(D+E) at D/E = 1. The algebra is the same as the inverted formula above.
- What it means. The equity beta doubles, so the cost of equity rises: if RF = 3% and the market risk premium is 7%, the cost of equity goes from 3% + 0.9 × 7% = 9.3% to 3% + 1.8 × 7% = 15.6%. The firm's asset cost of capital is unchanged; the capital structure has merely shifted risk from debt holders (who bear none, under the zero-beta assumption) to equity holders.
This is the first half of the Modigliani–Miller argument that Week 11 will complete. Without taxes, changing the mix of debt and equity does not change the cost of capital — it only changes who bears each slice of it. With taxes, the interest tax shield changes the total, which is where the WACC formula's (1 − TC) comes from.
The lecture's relevering formula deliberately ignores corporate taxes. The version that includes them is the Hamada equation:
The difference is the tax shield. Debt interest is deductible, so part of the debt's cost is borne by the government, and the equity claim is less risky than it would be in a no-tax world. At D/E = 1, βA = 0.90 and TC = 25%, Hamada gives 0.90 × (1 + 0.75 × 1) = 1.575, below the lecture's 1.80.
Which formula should you use? Follow the question. If it supplies a tax rate and asks for a relevered beta, use Hamada; if it follows the lecture's own convention and ignores taxes on the beta calculation, use βA(1 + D/E). The two are not contradictory — they answer different models — but mixing them (using the no-tax formula and then claiming the result is after tax) is wrong.
Debt beta is not always zero. For a highly levered or distressed firm, bondholders bear business risk: when the market falls, default risk rises and the bonds fall too. In that case βDebt > 0, and the equity beta is lower than βA(1 + D/E) predicts. The zero-beta short-cut is safest for investment-grade issuers and least safe for the firms whose leverage most needs analyzing.
The D/E ratio must be a market-value ratio. Book equity can be tiny or negative while market equity is large; book debt misses the interest-rate sensitivity of long bonds. Using book values in the relevering formula can produce a beta that has nothing to do with the risk shareholders actually bear. The same market-value rule governs the WACC weights on slide 29.
DDM, and Project versus Firm Risk
slides 21–26The dividend discount route to the cost of equity slides 21–22
Slide 21 recalls Lecture 5. A constant-growth stock is a growing perpetuity, so its price is the next dividend divided by the excess of the required return over growth:
Slide 22 says the model can recover the expected return on a single stock from its dividend and price, and that the same formula applies to the whole market using aggregate dividends and prices. The symbols are worth pinning down: D1 is the dividend next period — not the one just paid — P0 is the current price, g is the constant growth rate, and RS is the expected return. Slide 21 adds the reminder in bold: Not D0. A common error is to divide the most recent dividend by the price and add g, which mixes a past cash flow with a forward-looking formula; the correct numerator is D0(1 + g).
The formula is a decomposition as well as a calculation. The first term, D1/P0, is the dividend yield; the second, g, is the capital gains yield. Together they are the shareholder's total expected return — exactly the return decomposition from Lecture 7, now solved for R instead of used to compute a price.
| Comparison | CAPM | DDM | Which to use |
|---|---|---|---|
| Inputs | RF, market risk premium, beta | D1, P0, g | CAPM needs market data; DDM needs a dividend history and a growth forecast. |
| Risk adjustment | Explicit, through beta | Implicit, through the price and the growth rate | The CAPM's transparency is why academics prefer it. |
| Non-dividend payers | Works — beta exists regardless of payout | Fails — no D1 to put in the numerator | A large share of growth companies pay nothing, so the DDM's usable range is narrower. |
| Sensitivity | Moderate; beta is an estimate | High; a 1-point error in g moves RS one-for-one | The DDM is only as good as the growth forecast, which is the hardest input to defend. |
Slide 23's verdict: the two are internally consistent and give similar answers for a stable dividend payer, but academics generally favour the CAPM and companies use it more consistently. The CAPM's decisive advantage is that it adjusts explicitly for risk and works for firms that pay no dividend. The DDM remains useful as a cross-check — if the two disagree sharply, at least one input is wrong, and the disagreement is informative.
Project risk is not firm risk slides 24–26
Slide 24 draws the security market line with the firm's own beta marked on the horizontal axis and its WACC marked on the vertical axis. That point is the firm's hurdle rate. The slide's warning is that a single rate applied to every project inside a multi-business firm systematically misallocates capital:
| Project position | Project beta | IRR relative to the firm's hurdle | What a single-WACC rule does |
|---|---|---|---|
| Below the SML, left of the firm | Lower than βFIRM | Above its own required return but below the firm's WACC | Rejects a positive-NPV project. The safe division is held to a rate that reflects someone else's risk. |
| Above the SML, right of the firm | Higher than βFIRM | Below its own required return but above the firm's WACC | Accepts a negative-NPV project. The risky division gets a subsidy from the safe one. |
The cumulative effect is the sentence at the bottom of slide 24: a firm that uses one discount rate for all projects may over time increase the risk of the firm while decreasing its value. It keeps the projects that should have been rejected and loses the ones that should have been kept — a selection effect in exactly the wrong direction.
- The firm's WACC. R = 2% + 1.3 × 7% = 11.1%
- Check the weighted-average beta. The divisions are equal-sized, so (2.0 + 1.3 + 0.6)/3 = 1.3 — the firm beta is the average of its parts, which is the asset-side logic of slide 19 applied across divisions.
- The right hurdle for electrical generation. Use the utility's own beta, not the firm's: Rutility = 2% + 0.6 × 7% = 6.2% A generation project with an IRR of, say, 8% is clearly worth doing at a 6.2% hurdle and would be wrongly rejected by the 11.1% firm WACC.
- The other two divisions. Rhard drive = 2% + 1.3 × 7% = 11.1% Rauto retail = 2% + 2.0 × 7% = 16.0% A hard-drive project at 12% clears its own hurdle; an auto-retailing project at 12% does not, even though both clear nothing more than the firm average in the first case and fail it in the second. The 12% number is judged against 11.1% for one division and 16.0% for the other — the same cash flow, two different verdicts.
Slide 26 draws the three points on the SML: 6.2% at β = 0.6, 11.1% at β = 1.3, and 16.0% at β = 2.0. The visual is the argument — the divisions lie on one line, and the firm's WACC is only one point on it.
How does a firm with no traded comparables estimate a division's asset beta? The pure-play method: find listed firms whose only business is the division's business, unlever each one's equity beta to an asset beta, average them, and relever to the firm's own capital structure.
The steps, in order: (1) collect the comparables' equity betas; (2) unlever each using βA = βE / (1 + D/E) with each comparable's own market D/E; (3) average the asset betas — this is the industry asset beta; (4) relever at the division's or the firm's target D/E. Step 2 is where comparables with different leverage are made comparable; skipping it and averaging raw equity betas builds the comparables' financing into the estimate.
A worked shape: comparables with βE = 1.4 at D/E = 0.5 and βE = 1.1 at D/E = 0.2 unlever to 1.4/1.5 = 0.933 and 1.1/1.2 = 0.917. Their average asset beta is about 0.925. At the division's target D/E = 0.8, the relevered equity beta is 0.925 × 1.8 = 1.665. The numbers are illustrative, but the sequence — unlever, average, relever — is the examinable part.
Slide 31 opens with: "Good Food is seeking to acquire the Happy Meals. The discount rate and cash flows should be based on which company?" The intuitive answer — "Good Food, because it is doing the buying" — is half right at best. The cash flows must be Happy Meals': they are what the acquisition delivers. The discount rate must reflect the risk of those cash flows. When acquirer and target are in the same business, the acquirer's WACC is a reasonable proxy for that risk, which is why the slide proceeds with Good Food's 6%. When they are in different businesses, using the acquirer's WACC is exactly slide 24's error at the level of a whole company.
The same logic governs the cash-flow side: a target's cash flows should be forecast on a standalone basis, without the acquirer's synergies unless the synergies are being valued explicitly. Mixing target cash flows with acquirer discount rates, or the reverse, is the most common valuation error in an M&A problem.
Cost of Debt, WACC, and Firm Valuation
slides 27–33The cost of debt, after tax slide 27
When a firm issues debt, the cost is the coupon or the required interest rate; for a corporate bond the yield to maturity is the best proxy. Slide 27 gives two practical warnings. Use the current yield, not the historical one — the cost of new money is what matters. And obtain it from the dealer or the OTC bond market, because the last trade may be stale.
The step the slide stresses is the tax adjustment. Interest is deductible from taxable income, so the government bears TC of every dollar of interest:
At a 7% YTM and a 25% tax rate, the after-tax cost is 5.25%. That is the number that enters the WACC, because the WACC is built from after-tax cash flows — the same convention as Lecture 4's capital-budgeting problems, where operating cash flow was EBIT(1 − T) plus depreciation. Mixing a pre-tax debt cost with after-tax cash flows overstates the WACC and rejects good projects.
The cost of preferred stock slide 28
Preferred stock pays a constant dividend in perpetuity and does not share in the firm's capital gains, so it behaves more like a bond than like common stock. Its price is the dividend divided by the required return, and solving for the return gives:
There is no tax adjustment for preferred dividends: they are paid out of after-tax earnings, so the firm gets no deduction. This is a frequent exam trap — the debt term in WACC is multiplied by (1 − TC) and the preferred term is not. A preferred stock with a ¥5 dividend and a ¥50 price has a cost of 10%, used as is.
Slide 28 also says preferred stock does not dilute the ownership of old shareholders. Read that as a statement about voting control, not about economic priority: preferred shares usually carry no votes, so they do not dilute control, but they are senior to common stock in dividends and liquidation and therefore do dilute the common shareholders' residual claim. The precise statement is that preferred is a hybrid — debt-like in its fixed claim, equity-like in its tax treatment and its lack of a maturity date.
Weighted average cost of capital slide 29
With the components in hand, the firm's overall cost of capital is their weighted average:
Three properties are worth stating explicitly, because each is a question waiting to be asked. The weights are market values (S is the market value of equity, B the market value of debt), because the cost of capital measures what investors require on the capital at its current value, not its historical book value. The debt term is after tax; the equity term is not, because dividends are not deductible. The result is a blended rate, between the after-tax cost of debt and the cost of equity, weighted toward whichever source is larger.
| Component | Market value | Weight | Cost | Weighted |
|---|---|---|---|---|
| Debt | $4 billion | 4/6 = 2/3 | 5% × (1 − .2) = 4% | 2/3 × 4% = 2.667% |
| Equity | $2 billion | 2/6 = 1/3 | 10% | 1/3 × 10% = 3.333% |
| Total | $6 billion | 1 | — | WACC = 6.000% |
The debt weight is 2/3, not 1/3, because the firm is financed mostly with debt: $4 billion of $6 billion. The after-tax cost of debt is 4%, and the weighted sum is exactly 6%. That 6% is the discount rate used for the rest of the example.
Valuing the firm slides 30 and 32–33
Slide 30 states the valuation rule. The value of the whole firm is the present value of its expected future free cash flow, discounted at the WACC. Free cash flow is the cash available to all capital providers — debt and equity together — which is why the WACC, a blend of both costs, is the matching discount rate. To find the equity value, subtract the value of the debt from the firm value. Slide 30 adds a practical shortcut: the market value of debt is much more stable than the market value of equity, so its book value is often an acceptable approximation.
Slide 32 applies this to Happy Meals. EBIT is $150 million in year 1 and grows 10% a year for five years. Capital spending and the increase in net working capital are each 24% of EBIT; depreciation is 8% of EBIT; the tax rate is 20%. The cash-flow build is the Lecture 4 format, line by line:
| Year | 1 | 2 | 3 | 4 | 5 | How it is built |
|---|---|---|---|---|---|---|
| EBIT | 150.0 | 165.0 | 181.5 | 199.7 | 219.6 | 150 growing at 10% |
| − Taxes (20%) | 30.0 | 33.0 | 36.3 | 39.9 | 43.9 | 20% of EBIT |
| = Earnings after taxes | 120.0 | 132.0 | 145.2 | 159.8 | 175.7 | EBIT × (1 − .20) |
| + Depreciation | 12.0 | 13.2 | 14.5 | 16.0 | 17.6 | 8% of EBIT |
| − Capital spending | 36.0 | 39.6 | 43.6 | 47.9 | 52.7 | 24% of EBIT |
| − Increase in NWC | 36.0 | 39.6 | 43.6 | 47.9 | 52.7 | 24% of EBIT |
| = Net cash flow | 60.0 | 66.0 | 72.6 | 79.9 | 87.8 | The FCF available to all providers |
Slide 33 then values the stream in two parts. The five explicit years are discounted individually, and everything after year 5 is collapsed into a terminal value using the growing-perpetuity formula with a 2% perpetual growth rate:
- Terminal value at the end of year 5. TV5 = FCF5(1 + g) / (RWACC − g) = 87.8 × 1.02 / (0.06 − 0.02) = 89.556 / 0.04 = 2,238.9
- Discount every cash flow to today. PV0 = 60.0 / 1.06 + 66.0 / 1.06² + 72.6 / 1.06³ + 79.9 / 1.06⁴ + 87.8 / 1.06⁵ + 2,238.9 / 1.06⁵ = 56.60 + 58.74 + 60.96 + 63.29 + 65.61 + 1,673.04 = 1,978.2
- Read the answer correctly. $1,978.2 million is the value of the whole firm — the enterprise value, not the equity value. If Happy Meals had debt outstanding, the equity value would be $1,978.2 million minus that debt. The slide does not supply a debt figure for the target, so the example stops at enterprise value.
The terminal value is 84.6% of the total (1,673.0 of 1,978.2), which is typical of a five-year explicit forecast with perpetual growth: most of a going concern's value lives beyond year five. That is worth remembering when a valuation is presented — the terminal value is not a rounding detail, it is the main event, and it rests on a growth rate that is very hard to defend.
The lecture's WACC has two terms because it treats only common equity and debt. With preferred stock the formula gains a third term, with no tax adjustment:
A numerical shape: S = $500m, P = $100m, B = $400m, RS = 12%, RP = 8%, RB = 6%, TC = 25%. Then WACC = 0.50 × 12% + 0.10 × 8% + 0.40 × 4.5% = 6.0% + 0.8% + 1.8% = 8.6%.
There are two equivalent valuation routes, and consistency is what matters. The entity route discounts free cash flow to the firm (FCFF) at the WACC and subtracts debt to get equity. The equity route discounts free cash flow to equity (FCFE) — after interest and net borrowing — at the cost of equity, and the present value is the equity value. Both should give the same answer if the assumptions are consistent. The error to avoid is crossing them: discounting FCFE at the WACC, or FCFF at the cost of equity, mixes a cash flow defined before financing with a rate that reflects financing.
Book weights instead of market weights. A firm with old, low-coupon debt may have a book debt ratio far above its market debt ratio, because the market value of its equity has grown. Using the book ratio overweights debt and understates the WACC.
A pre-tax cost of debt. Entering 5% rather than 4% at a 20% tax rate overstates the WACC by roughly the debt weight times 1 percentage point. The debt term is the one place the tax rate belongs.
The coupon instead of the YTM. The coupon is the historical contract rate; the YTM is today's market rate for that credit risk. Only the latter is the marginal cost of borrowing.
One WACC for a multi-division firm. This is slide 24's error in a different costume. A conglomerate's WACC is a weighted average of its divisions' costs; applying it to a division whose beta differs from the average misprices every project in that division.
The table on slide 32 rounds to one decimal place. Recomputing the whole chain without rounding gives FCF5 = $87.846 million rather than $87.8 million, and a terminal value of 87.846 × 1.02 / 0.04 = 2,240.1 rather than the slide's 2,238.9. The difference — 0.05% — is a rounding artefact, not an error in the method; the slide's 1,978.2 is exactly what its printed cash flows produce. This workbench reproduces the slide's printed figures and flags the rounding rather than silently substituting the unrounded chain.
Formula Sheet
print-friendlyKey Concepts
the vocabularySelf-Check
16 questions · gradedThe examinable core is four things: the CAPM and its three inputs, relevered beta, the WACC with its after-tax cost of debt, and the distinction between project and firm risk. Two traps account for most lost marks: using the coupon as the cost of debt instead of the current YTM, and applying the firm's WACC to a project whose risk differs from the firm's. The valuation section adds a third: using book weights, or forgetting the terminal value's growing-perpetuity form.
A · Concept check
The investor's side. An investor who buys a security gives up cash today and receives a claim on future payoffs. The return she requires is the minimum compensation for that delay and risk. In a competitive capital market, the price of the security adjusts until the expected return it offers equals that required return. At that price the security's NPV to the investor is exactly zero: she pays fair value and expects to earn her required return, no more.
The firm's side. The firm is the issuer of the same claim. The return it must deliver to the investor is its cost of raising the money. If a project earns more than that return, the firm keeps the surplus and the project has positive NPV for the existing shareholders. If it earns less, the firm is transferring value to the new investors — the project has negative NPV even if its cash flows are positive.
Why the two numbers coincide. They are two views of one contract. The investor's asset is the firm's liability; the return the investor expects is the return the firm must pay. The balance-sheet identity in slide 5 — Asset = Liability + Equity — is the accounting version of the same statement. When the security is fairly priced, both sides use the same rate, and the NPV is zero on both sides; any positive NPV is a deviation from fair pricing that accrues to the firm's existing shareholders.
Why it is an opportunity cost specifically. The number is not the firm's actual historical return and not a target chosen by management. It is what investors could earn on an alternative investment of the same risk. That is why slide 7's dividend-versus-project comparison is the right frame: the firm is competing with the capital market for the shareholder's money. A project that cannot beat the market's offer at the same risk should not be funded, however profitable it looks in isolation.
The practical consequence. The cost of capital must match the risk of the cash flow being discounted. A safe project should be compared with a safe financial asset, not with the firm's average opportunity set. That is exactly the argument slides 24–26 make against a single firm-wide WACC, and it is why the hurdle rate is a project-level, not a firm-level, concept.
B · Multiple choice
C · Applied calculation
Cost of equity — CAPM.
R_S = R_F + β(R_M − R_F) = 3% + 1.4 × 6% = 3% + 8.4% = 11.4%After-tax cost of debt.
R_B(1 − T_C) = 5% × (1 − 0.25) = 5% × 0.75 = 3.75%Weights and WACC. Total capital is $1,000 million, so equity is 70% and debt 30%.
WACC = 0.70 × 11.4% + 0.30 × 3.75% = 7.98% + 1.125% = 9.105% ≈ 9.11%Check the answer sits in the right range. The WACC must lie between the 3.75% after-tax cost of debt and the 11.4% cost of equity; 9.11% does. A result outside that range means the weights or the tax adjustment was applied to the wrong term.
What not to do. Applying (1 − TC) to the equity term as well gives 0.70 × 8.55% + 0.30 × 3.75% = 7.11%, a materially wrong answer. Dividends are not tax-deductible, so the equity cost is used as given.
Relever the beta.
β_E = β_A(1 + D/E) = 0.80 × (1 + 0.6) = 0.80 × 1.6 = 1.28New cost of equity.
R_S = 3% + 1.28 × 6% = 3% + 7.68% = 10.68%Weights. With D/E = 0.6, total capital is 1.6 units per 1 unit of equity: E/V = 1/1.6 = 0.625 and D/V = 0.6/1.6 = 0.375.
After-tax cost of debt and WACC.
R_B(1 − T_C) = 5% × 0.80 = 4.0% WACC = 0.625 × 10.68% + 0.375 × 4.0% = 6.675% + 1.500% = 8.175% ≈ 8.18%The checks. The new cost of equity, 10.68%, is above the all-equity cost of 3% + 0.80 × 6% = 7.8%, which is what leverage should do. The WACC of 8.18% lies between the 4.0% after-tax cost of debt and the 10.68% cost of equity, so the weighting is sensible. The question follows the lecture's convention for beta — βD = 0 and no tax term in the relevering formula — while applying the tax shield to the debt cost, exactly as slides 19 and 27 do separately. If a question instead says to use Hamada's equation, relever with βE = βA[1 + (1 − T)D/E] first; do not mix the two beta formulas.
Firm value — a growing perpetuity.
EV = FCF₁ / (WACC − g) = 50 / (0.08 − 0.03) = 50 / 0.05 = $1,000 millionEquity value.
Equity = EV − Debt = 1,000 − 200 = $800 millionValue per share.
Price = 800 / 100 = $8.00What the calculation assumes. The $50 million is FCF to the firm — before interest — which is why it is discounted at the WACC. Subtracting debt once gives the equity value; the debt is not subtracted from each year's cash flow, because FCF is already defined before financing. The alternative route discounts free cash flow to equity — after interest and net borrowing — at the cost of equity, and its present value is the equity value directly. Both routes agree when the assumptions are consistent.
Sanity check. The equity is 80% of firm value, matching the $800/$1,000 split; the implied share price is positive and the EV is finite because WACC (8%) exceeds g (3%). If g were above the WACC the growing-perpetuity formula would break down, which is why terminal growth is normally set at or below the long-run growth rate of the economy.
(a) Good Food's WACC. Total capital is $6 billion, so debt is 4/6 = 2/3 and equity is 2/6 = 1/3. The after-tax cost of debt is 5% × (1 − 0.20) = 4%.
WACC = (2/3) × 4% + (1/3) × 10% = 2.667% + 3.333% = 6.00%(b) Terminal value.
TV₅ = 87.8 × 1.02 / (0.06 − 0.02) = 89.556 / 0.04 = 2,238.9 ($ million)Value of Happy Meals.
PV₀ = 60.0/1.06 + 66.0/1.06² + 72.6/1.06³ + 79.9/1.06⁴ + 87.8/1.06⁵ + 2,238.9/1.06⁵ = 56.60 + 58.74 + 60.96 + 63.29 + 65.61 + 1,673.04 = 1,978.2 ($ million)Why Good Food's WACC is the right rate here. The cash flows are Happy Meals', so the discount rate must reflect their risk. Acquirer and target are in the same business, so Good Food's WACC is a reasonable proxy for that risk — which is why the slide proceeds with 6%. If Happy Meals operated in a different industry, the correct rate would be a divisional or pure-play rate for its business, not Good Food's corporate average. This is slide 24's lesson applied to an acquisition.
Why the answer stops at firm value. $1,978.2 million is the enterprise value of Happy Meals. To get its equity value you would subtract Happy Meals' own debt; the slide gives no target debt figure, so the example legitimately stops here. Subtracting Good Food's $4 billion would be an error: that debt belongs to the acquirer, not to the acquired cash flows.
Efficient Capital Markets
Concept Map
how the lecture buildsLecture 8 estimated beta by regressing past returns on an index, took the bond's yield from its market price, and used a historical market risk premium. All three assume that prices aggregate information reasonably well. The EMH is the justification — and the lecture is careful about what it does and does not claim. It does not say the market's forecast is right; the 2008 crisis and the 2015 A-share crash were not forecast by prices. It says the forecast is hard to beat with information everyone already has. A market can be efficient and still spectacularly wrong ex post, because efficiency is about available information, not omniscience.
The link to Lecture 7 is the same one: diversification is free, so the market does not pay for diversifiable risk. The EMH is the information-side version of that argument. If a fact is public and cheap to act on, its value is competed away, just as a free diversification benefit cannot earn a premium.
Lecture 3 said a project has positive NPV only if it earns more than a financial asset of comparable risk. In an efficient market, buying or issuing a security is a zero-NPV transaction: you pay fair value. So a firm cannot create value simply by choosing how to raise money — only by investing in real assets with positive NPV. That is the logical foundation of Lecture 10's Modigliani–Miller propositions, which say capital structure is irrelevant once taxes and frictions are removed.
Prices and Information
slides 2–5The lecture opens with a fact and a question. Slide 2 shows Tesla rising about 14% after a US presidential election result; slide 3 asks what that has to do with financing decisions. The answer is that every financing decision is a decision about the price at which the firm sells a claim, and every price is a summary of information. Before the EMH can be stated, the lecture needs to show why information moves prices at all.
Three ways a firm might lower its cost of capital slide 3
Slide 3 lists the channels through which a financing decision could add value:
| Channel | The claim | What has to be true for it to work |
|---|---|---|
| Fool investors | Raise capital at prices above fair value. | Some investors must persistently misprice securities. The lecture's own question — if it happens, why are investors so stupid persistently? — is the EMH in one line. |
| Reduce financing costs | Use a form of financing with a tax advantage or subsidy. | A real friction. This is the interest tax shield that returns in Lecture 10, and the one channel the EMH does not rule out. |
| Create a new security | Design a claim a particular clientele wants and will pay a premium for. | Demand must be unmet and not easily replicated. Financial innovation is the real-world version — but competition erodes the premium. |
Only the first depends on inefficiency. The second and third are about real costs and real clienteles, and they survive even if markets are efficient. That distinction matters for the exam: an efficient market does not make all financing irrelevant — it makes fooling people irrelevant.
When should the firm issue? slide 4
Slide 4 shows an index running from about 20,000 to 30,000 and marks a peak A and a trough B. If the firm knew it was at A, it would issue there: selling a claim for more than it is worth transfers value from new investors to old ones. At B it would rather buy back shares or wait.
In a truly efficient market, the question is a trap: the answer is that it does not matter. At A the price is high because the information says the future is bright; at B it is low for the same reason in reverse. Selling at A gets more cash per share but gives up a correspondingly larger claim on that brighter future. The NPV of the issue is zero at both points.
The question becomes real only if you believe the price is wrong. Then issuing at A is a positive-NPV financing decision — the firm sells overpriced paper. The empirical pattern that firms issue equity after price run-ups is called market timing, and it is one of the facts that keep the efficiency debate alive. Notice what it implies: managers either have private information the market lacks (a violation of the strong form) or they are reading public information the market has mispriced (a violation of the semi-strong form). The slide's chart is therefore an argument by contradiction — in an efficient market it should be impossible to say which point is better.
Why expectations make prices move slide 5
Slide 5 writes out the discounting identity that every earlier lecture used:
The cash flows are expected cash flows — investors' personal predictions. Two consequences follow directly. Different investors predict differently, so the price is an aggregation of heterogeneous beliefs, not a single analyst's forecast. And predictions change when new information arrives, so the price changes even when nothing about the firm's operations has yet changed. That is the entire mechanism behind the Tesla example: an election result changes expected future cash flows, or the rate at which they are discounted, or both.
Slide 11 gives the rule for deciding whether a piece of news is good or bad: check whether it raises or lowers expected cash flows, or lowers or raises the discount rate. The lecture sorts news into three levels and deliberately leaves some signs ambiguous:
| Level | Examples from slide 11 | Why the sign is what it is |
|---|---|---|
| Macro | War breaks out (−); an election result (?); the central bank cuts the benchmark rate (+) | A rate cut lowers the discount rate, so it is good news for almost every stock. An election result is ambiguous because its effect on cash flows and on rates can point in opposite directions. |
| Industry | Subsidies for electric vehicles are cut (−); the government announces support for semiconductors (+) | Industry-wide news moves all firms in the sector together — the common component of returns, which is exactly the part the CAPM prices. |
| Firm | Earnings beat forecasts (+); a firm announces it is seeking to buy another (?) | For the acquirer the sign is ambiguous: synergies raise cash flows, but the premium paid may transfer value to the target's shareholders. The market's verdict is in the announcement return — often negative for the buyer. |
Only surprise news moves the price slide 12
A common error is to read a price move as a verdict on the level of a number. The slide's example makes the point sharply: analysts forecast earnings of ¥10,000; the firm reports ¥10,000 and the price falls. Nothing went wrong. The price had already incorporated the forecast, so only the difference between the outcome and the expectation — the surprise — carries new information. A firm can grow earnings 30% year on year and still fall if the market expected 50%.
This is why the EMH is stated in terms of available information rather than "the news". By the time an announcement is published, the information it contains is already public and already in the price if the market is efficient. What moves the price is the part nobody had priced.
A large improvement is not automatically good news. The benchmark is the expectation, not last year's number. A firm whose earnings rise 20% when the consensus expected 35% has delivered a negative surprise. Conversely, a firm whose earnings fall 10% when the consensus expected −25% has delivered a positive surprise and can rise on "bad" numbers.
Ambiguous announcements are not a failure of the model. Slide 11 marks the election and the acquisition with a question mark because the direction depends on which channel dominates. The correct exam answer is to name both channels — cash flow and discount rate — rather than to guess a sign.
Price moves are not value moves. A change in expectations changes the price; it does not by itself change the firm's assets or its future cash flows. Confusing the two is the error that makes bubbles feel justified.
The Efficient Market Hypothesis
slides 6–7Slide 6 gives the definition the exam will ask for:
Every word matters. Stock prices, not returns, not trading volume. Fully reflect, which means the adjustment is fast and complete enough that no rule based on that information earns an abnormal profit. And available information, which is why the hypothesis comes in three versions: "available" can mean past prices, all public information, or everything anyone knows.
The lecture draws two implications for investors and two for firms:
| Who | Implication | The reasoning |
|---|---|---|
| Investors | Knowing information when it is released does an investor little good. | By the time you read the announcement, the price has already moved. The information is public and therefore already priced. |
| Firms | They should expect to receive fair value for the securities they sell. | No persistent overpricing to exploit. Issuing equity is zero-NPV in the sense that the cash received equals the value of the claim given up. |
| Firms | They cannot profit from fooling investors. | Investors learn across repeated issues; an overpriced security would be arbitraged. This is the efficient-market answer to slide 3's first channel. |
| Everyone | The stock market is a barometer of the economy (股市是经济的晴雨表). | Prices aggregate dispersed information about future cash flows. The barometer can be wrong about the level and still be the best available forecast. |
Slide 7 credits the idea to Eugene Fama of Chicago Booth, who shared the 2013 Nobel Prize in economic sciences for it. The attribution is worth knowing for the concept question, and so is the caveat that comes with it: Fama's own later work emphasises that efficiency is unavoidably joint with an asset-pricing model. To test whether prices reflect information you need a model of what the price should be; if the test fails, you cannot tell whether the market is inefficient or your model is wrong. That is the "joint hypothesis problem", and it is why the debate has lasted fifty years.
What efficiency does not claim
Three misunderstandings cause most of the confusion in this topic:
An efficient price can be wrong. In 2020 the market priced many travel stocks as if the pandemic would last a few months; it lasted years. The prices were wrong ex post, but at the time the information available arguably justified them. Efficiency says the error was not predictable and exploitable, not that the forecast was accurate.
A price can be volatile and still be efficient. Prices move because new information arrives, and new information is by definition unpredictable. Volatility is what an efficient market looks like from the inside, not evidence against it.
Efficiency is a statement about abnormal returns, not about zero returns. Investors still earn the required return; they just cannot earn more than the required return using available information. A fund that beats the index by 10% is not automatically evidence against efficiency — it could be luck, and with thousands of funds some will be lucky. The test is whether the outperformance persists and survives costs.
In the exam, define efficiency with the slide's words — prices fully reflect available information — and then add the operational meaning: no trading rule using that information earns an abnormal profit after costs. Do not write "prices are always fair" or "investors are rational". The first is too strong (prices can be wrong), and the second is a behavioural assumption that the EMH does not need. A market can be efficient with irrational investors if their errors cancel, or if arbitrageurs correct them.
The Three Forms of Efficiency
slides 13–18Slide 13 splits the hypothesis by information set. The three forms are not rival theories to choose between: they are progressively stronger statements, and the stronger one contains the weaker ones.
| Form | Information reflected | What becomes useless | The named activity |
|---|---|---|---|
| Weak 弱式 | All historical price and volume information | Technical analysis | Not fundamental analysis. Public accounting data can still be used if the weak form is all that holds. |
| Semi-strong 半强式 | All publicly available information: past prices, published statements, annual reports, news | Fundamental analysis on public data; forecasting from public information | Does not forbid insider profits. Trading on material non-public information is illegal but, if it works, it is evidence against the strong form. |
| Strong 强式 | All information, public and private | Even insider trading | The strongest claim, and the one the evidence rejects. Slide 26: insider trading is abnormally profitable. |
Weak form and the random walk slides 14–15
The weak form says prices already reflect everything contained in past prices and volume. If that is true, studying the past cannot predict the future: whatever pattern you think you see has already been traded on by people who saw it first. Technical analysis (技术分析) — predicting price from past price and volume — is therefore of no value.
Slide 15 makes the argument self-defeating in the nicest way. If a reliable pattern existed, everyone would buy at the signal and sell at the top. The buying would push the price up immediately, destroying the profit the pattern promised. Competition is the mechanism that removes the pattern, and it is the same mechanism that removes free diversification benefits in Lecture 7.
The random walk is not a claim that prices are meaningless. It is a claim that changes are driven by new information, and new information is by definition unpredictable. Slide 20 draws the practical conclusion: financial managers cannot "time" stock and bond sales, and people who claim to see patterns are probably seeing optical illusions. Slide 21 illustrates the point with a plot of randomly selected numbers — it looks enough like a price chart to make the lesson stick.
A random walk in prices implies returns are unpredictable. But if prices had a random walk in levels including drift, the expected return would be positive and predictable — which is exactly what the CAPM says. The "randomness" is in the deviations from expected return, not in the expected return itself. A stock trading at ¥100 is not equally likely to be ¥50 or ¥150 tomorrow; the surprise component is what has mean zero.
This is also why "the market is random, so analysis is pointless" is too strong. Fundamental analysis can still be useful under the weak form; it is the semi-strong form that rules it out, and even then only to the extent it uses public data.
Semi-strong form slide 16
The semi-strong form extends the information set to everything publicly available: historical prices, published accounting statements, annual reports, news releases. If it holds, then no strategy based on analysing public information earns an abnormal return — fundamental analysis is as useless as chart-reading.
The slide is careful on one point that students routinely get wrong: insider trading can still make profits. Private information is not in the public information set, so the semi-strong form says nothing about it. Trading on material non-public information is illegal in most jurisdictions, but illegality and profitability are different questions — and the profitability of insider trading is precisely the evidence that rejects the strong form.
Strong form and the information-set nesting slides 17–18
The strong form says prices reflect all information, public and private. Even an insider cannot profit: as soon as she tries to trade, the market would detect the order flow and the price would move before her trade completed. Slide 18's concentric circles make the nesting explicit — past prices are a subset of public information, which is a subset of all information:
The nesting gives the logical relations that exam questions love. Strong ⇒ semi-strong ⇒ weak. If all information is in the price, then public information certainly is; if public information is in the price, past prices certainly are. The converses do not hold. Rejecting the strong form does not reject the semi-strong form, and rejecting the semi-strong form does not reject the weak form. This is why the evidence section treats each form separately rather than declaring efficiency "true" or "false".
The defensible answer is: at most the weak form, and even that is contested. The strong form is clearly violated — insider trading and information leakage are documented, and the pre-announcement price moves in slide 10's chart are the general pattern. The semi-strong form is doubtful on several grounds: price limits and the T+1 settlement rule create predictable microstructure patterns; retail investors dominate turnover; and policy announcements move prices in ways that look like delayed rather than instantaneous adjustment. The weak form is the least controversial, but even there the evidence is mixed — short-term reversal and momentum effects have both been documented in A-shares, which is awkward for a pure random walk.
The honest exam answer names the evidence rather than picking a label. "China's market is weak-form efficient at best; strong-form is rejected by insider trading, and semi-strong efficiency is undermined by momentum and policy-driven drift" is a better answer than "it is semi-strong efficient".
What the Evidence Shows
slides 8–12, 19–26The lecture tests each form with a different instrument. The weak form is tested by asking whether price changes are random; the semi-strong form by event studies and the performance of active funds; the strong form by the profitability of insider trading. This section takes them in that order, and adds the slide 8–10 charts that frame the whole question: what an efficient reaction looks like compared with the alternatives.
Three shapes of a price reaction slides 8–9
Slides 8 and 9 plot the stock price around good news and bad news. There are three candidate paths, and only one of them is consistent with semi-strong efficiency:
| Path | What it looks like | What it implies |
|---|---|---|
| Efficient response | A jump at the announcement, then a flat line | The information is incorporated immediately and completely. This is the benchmark. |
| Overreaction with reversion | Overshoot at the announcement, then drift back | The market initially overestimates the news. A contrarian strategy would profit — an anomaly. |
| Delayed response | Slow drift in the direction of the news for weeks | Underreaction. Momentum would profit — the anomaly that behavioural finance explains with conservatism. |
The lesson of the two charts is methodological: you cannot tell efficiency from a price level. You need the path relative to the announcement, and you need a benchmark for what the price would have done without the news. That benchmark is what an event study supplies.
Event studies slides 22–24
An event study (事件研究法) measures the price reaction to a specific event — an earnings announcement, a merger, a new issue, a dividend change. The core object is the abnormal return: the part of the return not explained by the market.
The expected return can be estimated with the market model, E(Rt) = α + βRM,t, fitted over a window before the event. The cumulative abnormal return (CAR) sums the daily abnormal returns over the event window, so the vertical axis of slide 23 is a CAR in percent. The test asks whether the path shows under-reaction, overreaction, early reaction, or delayed reaction — the four alternatives to the efficient benchmark.
Reading slide 23: dividend omissions slide 23
Slide 23 is the textbook's classic event study: cumulative abnormal returns for companies announcing a dividend omission. The numbers on the chart are the CAR in percent at each event day:
| Event day | CAR (%) | What happened |
|---|---|---|
| −2 | −0.483 | Still essentially flat relative to the market. |
| −1 | −3.619 | The big move. Most of the bad news is already in the price before the public announcement. |
| 0 | −5.015 | Announcement day. A further 1.4 percentage points. |
| +1 | −5.411 | The bottom — 5.4% below the pre-event level. |
| +8 | −4.490 | A partial rebound of about 0.9 points. The price does not stay at the bottom. |
Two facts stand out. First, the reaction is fast. About 92% of the eventual fall from the pre-event level to the bottom had happened by the end of the announcement day (−4.532 of −4.928 percentage points). That is the semi-strong benchmark: public information is incorporated almost immediately. Second, the price fell before the announcement. The drop from −0.483 on day −2 to −3.619 on day −1 is larger than the announcement-day move itself. That is leakage — some traders knew, or inferred, before the rest of the market did. Leakage is evidence against the strong form, not against the semi-strong form, because private information is exactly what the strong form says cannot be traded on.
Slide 24 generalises. Event studies have been run on dividend changes, earnings announcements, mergers, capital spending, and new issues. The general finding is that the US market is semi-strong form efficient, with the qualification that news tends to leak in advance. The leakage is visible in the pre-event drift, which is why the strongest statement the evidence supports is "fast adjustment plus some anticipation", not "prices jump only at the announcement".
- The announcement-day abnormal return. Subtract the previous CAR: AR(0) = CAR(0) − CAR(−1) = −5.015 − (−3.619) = −1.396%
- The pre-announcement abnormal return. The same calculation one day earlier: AR(−1) = CAR(−1) − CAR(−2) = −3.619 − (−0.483) = −3.136% The day before the announcement is more than twice as large as the announcement day itself.
- How much had happened by day 0. Measure from the pre-event level, which is the right base for a decline: total fall = −5.411 − (−0.483) = −4.928 points fall by day 0 = −5.015 − (−0.483) = −4.532 points share = −4.532 / −4.928 = 0.920 → 92.0% Dividing the day-0 CAR by the bottom instead gives 5.015 / 5.411 = 92.7%, but that ratio is measured from zero, not from the pre-event level; the two numbers answer slightly different questions.
- The post-announcement path. From the bottom at +1 to +8 the CAR rises by −4.490 − (−5.411) = +0.921 points That is a 17.0% retracement of the 5.411-point decline — a partial reversal, not a full one.
The interpretation matters more than the arithmetic. Fast adjustment supports the semi-strong form; the pre-announcement drop is leakage; the partial rebound is mild overreaction. A single chart can therefore support one form and undercut another. If you are asked whether this evidence is consistent with efficiency, the correct answer is yes for semi-strong, no for strong, with the caveat that the leakage and rebound show the market is not perfectly efficient.
Insider trading and the strong form slides 10 and 26
Slide 10 plots the same kind of event window for a takeover announcement and shows the price rising before the news is released. The annotation is blunt: information is leaked before the public announcement. Slide 26 then reports the systematic finding: studies of insider trading show it is abnormally profitable. Since those profits come from private information, the strong form — which says even insiders cannot profit — is not substantiated.
This is the cleanest asymmetry in the lecture. The same evidence that rejects the strong form is consistent with the semi-strong form, because the semi-strong form never claimed private information was priced. The exam question "does insider trading prove markets are inefficient?" has a precise answer: it proves the strong form false, not the semi-strong form.
Do active funds beat the market? slide 25
Slide 25 distinguishes passive funds (被动基金), which track an index such as 沪深300, from active funds (主动基金), which research and select stocks to outperform. If the semi-strong form holds, then no matter what public information active managers use, their average return should match the average investor's — before costs. After costs it must be worse, because the costs are real and the gross returns are the same on average.
The empirical record is broadly consistent: few active funds beat their benchmark persistently, and the survivors are hard to distinguish from luck. But the comparison is a test of the semi-strong form only if you accept that fund managers' information set is public. Managers do private research — site visits, expert networks, order flow — so their failure to beat the index is also consistent with the strong form being false and the semi-strong form being roughly true.
The slide gives 沪深300 and 上证1000 as examples of indices that passive funds track. There is no such index. The Shanghai Stock Exchange's main published indices are the 上证综指 (SSE Composite), the 上证50, the 上证180 and the 上证380; the widely used small-cap index is the 中证1000 (CSI 1000), which is published by 中证指数公司, not the SSE. The intended example was almost certainly 中证1000 — or possibly the 上证50, if the point was large-cap exposure. This does not affect the argument, but do not reproduce "上证1000" in an exam answer.
Slide 21 plots a sequence of random numbers and notes that it looks like a price chart. The point is a good one but the chart alone proves nothing: human beings are pattern-seeking machines and will see trends in any sufficiently noisy series. The formal test is statistical — autocorrelation of returns, runs tests, variance ratios — and the results are mixed. Daily returns are close to uncorrelated, which supports the weak form; weekly and monthly returns show some momentum and reversal, which does not. The honest summary is that the weak form is a good approximation, not an exact law.
Behavioural Finance and the Challenges
slides 27–29Slide 27 turns from prices to the people who set them. Traditional finance assumes investors are rational; behavioural finance (行为金融学) starts from the observation that they are not. It is an interdisciplinary field — psychology applied to finance — and its claim is not that markets are always irrational, but that the EMH's rationality assumptions are too clean to describe actual behaviour.
Six named biases slide 28
| Bias | Definition | What it does to prices |
|---|---|---|
| Herding 羊群效应 | Buy when others buy, sell when others sell — 追涨杀跌. | Amplifies moves and creates bubbles and crashes. The 2015 A-share crash is the lecture's example. |
| Overconfidence | Excessive confidence in the companies one buys. | Over-trading, under-diversification, and underestimating downside risk. |
| Disposition effect 处置效应 | Hold on to losers but sell winners. | The opposite of tax-loss selling; it slows the incorporation of bad news and can produce momentum. |
| Familiarity | Buy stocks you are familiar with — for example, your own employer's. | Under-diversification. Employees holding employer stock bear both labour income risk and equity risk in the same firm. |
| Representativeness 代表性 | Drawing conclusions from too little data — overreaction. | A few good quarters are extrapolated into a growth story. Reversal follows. |
| Conservatism 保守主义 | Adjusting beliefs to new information too slowly — underreaction. | Post-earnings-announcement drift: prices keep moving in the direction of the surprise for weeks. |
Notice that the last two point in opposite directions: representativeness produces overreaction and conservatism produces underreaction. That is not a contradiction — different investors, different situations, different horizons. It is why behavioural finance is a collection of models rather than a single alternative to the EMH.
Four challenges to the EMH slide 29
| Challenge | The evidence | Why it is not the end of efficiency |
|---|---|---|
| Limits to arbitrage | Short-selling restrictions compromise market efficiency. | This cuts both ways. It explains why mispricing can persist — but the EMH only needs the mispricing to be hard to exploit, not absent. "Limits to arbitrage" is an argument that markets can be inefficient and still efficient enough to defeat you. |
| Earnings surprises | Prices adjust slowly to earnings announcements; behaviouralists call it conservatism. | The strongest direct evidence against semi-strong efficiency: a public announcement followed by predictable drift. |
| Value vs growth | Low-P/E stocks outperform high-P/E stocks. | Could be a risk premium for distress rather than a free lunch. The Fama–French factors are the standard rationalisation. |
| Crashes and bubbles | The dot-com bubble and the 2015 Chinese crash. | Hard to reconcile with prices equal to fundamentals. Herding is the behavioural explanation; the rational version relies on heterogeneous beliefs and short-sale constraints. |
Because exploiting them is risky and costly. A mispriced stock can get more mispriced before it converges, and a fund that is right in the long run can be closed by investors in the short run — Keynes's point that markets can stay irrational longer than you can stay solvent. Short-selling is restricted or expensive, especially in A-shares. And the anomaly may be compensation for a risk you cannot diversify: value stocks are often distressed, and the value premium may be the return for bearing that distress risk.
This is why the EMH and behavioural finance are not simply enemies. The behaviouralists supply the biases; the efficient-market camp insists that a bias which cannot be profitably exploited is not an inefficiency in the economic sense. The exam can ask you to state both sides, and the best answers do.
Implications for Corporate Finance
slide 30Slide 30 closes the loop opened on slide 3. If information is reflected in prices quickly, three things follow for the firm:
| Implication | Why | What it rules out |
|---|---|---|
| Investors earn only a normal return | Abnormal returns require information that is not yet in the price; in an efficient market there is none. | A strategy that reliably beats the market using public information. This is the investor-side statement of the EMH. |
| Firms sell securities at fair value | The cash received equals the value of the claim given up, so the issue is zero-NPV. | Financing as a source of value. The firm cannot create wealth by issuing overpriced shares if investors price correctly. |
| Discount rate = expected return = required return | If the price is fair, the return the investor expects equals the return the investor requires. | The three terms Lecture 8 used interchangeably are the same number here — and this is the assumption that makes WACC meaningful. |
The slide then turns the logic around: managers can learn from the market. A price move is a disagreement with management's own forecast, and the direction of the move tells you what the market believes about the decision. Three examples from the slide, each with a reading:
| Event | Price reaction | The information in the reaction |
|---|---|---|
| The CEO dies suddenly | The stock rises | The market expected the CEO's departure to create value — perhaps the successor is better, or the CEO was overpaid and entrenched. A fall would have said the opposite. |
| Company A announces it will acquire B | A's stock falls | The market thinks A is overpaying: the premium to B's shareholders exceeds the synergies. This is the classic acquirer reaction and a warning about hubris. |
| A candidate wins an election | The market rises | The market expects the winner's policies to raise expected cash flows or lower the discount rate. The size of the move measures the surprise, not the candidate's quality. |
There is a subtlety in the last row that the slide leaves implicit. The election was partly anticipated, so the move reflects the difference between the outcome and the expected outcome — not the full effect of the winner's programme. If the market had fully priced a win, a win would produce no move at all. This is the same surprise logic as slide 12, applied to politics.
This is the hinge between the two lectures. If a firm cannot fool investors, then the choice between debt and equity does not change the total value of the firm by itself — issuing either claim is zero-NPV. Lecture 10 will state this as the Modigliani–Miller irrelevance proposition and then relax it: taxes make debt valuable through the interest tax shield, and distress costs make too much debt costly. Both are real frictions, not attempts to fool anyone. The EMH does not say capital structure never matters; it says the only reasons it matters are taxes, costs, and information — the frictions, not the financing label.
Reading the market's reaction to a decision is not the same as predicting the market's next move. The first is using the price as a summary of dispersed information; the second is claiming to know more than the price already contains, which is exactly what efficiency denies. A manager who issues equity because the price "looks high" is making a bet against the EMH. If the market is efficient, the price looks high for a reason, and the issue is fairly priced.
Formula Sheet
print-friendlyKey Concepts
the vocabularySelf-Check
14 questions · gradedThe examinable core is four things: the definition of an efficient market, the three forms and what each one rules out, the event-study logic, and the implications for firms. Three traps account for most lost marks: saying efficiency means prices are correct, mixing up which form insider trading violates, and treating a large earnings increase as unambiguously good news. A fourth is the nested-set logic: strong implies semi-strong implies weak, never the reverse.
A · Concept check
What efficiency claims. Prices fully reflect the information that is available at the time. The operational version is that no trading rule using that information earns an abnormal return after costs. The claim is about the relationship between price and information, not about the relationship between price and some unknowable true value.
Why the price can be wrong. The information available may itself be incomplete or misleading. Investors form expectations from the data they have; if the data understate a future shock — a pandemic, a war, a technological breakthrough — the price will be wrong ex post. Efficiency does not require omniscience. It requires only that the error not be systematically predictable from available information.
The economic meaning. A mispricing that no one can identify in advance, or that no one can profitably exploit because of risk and transaction costs, is not an inefficiency in the sense that matters for capital allocation. The price can be wrong and still be the best estimate available — like a weather forecast that is wrong but was the best forecast given the data.
The link to the evidence. This is why event studies matter. A jump at the announcement followed by a flat line is evidence of efficient incorporation, even if the new level later turns out to be wrong. The test is the speed and completeness of the adjustment, not the accuracy of the forecast.
What would actually contradict efficiency. A trading rule that uses available information and earns a persistent abnormal return after costs — post-earnings-announcement drift, a reliable technical pattern, or a fund manager who beats the market for decades. Those are the observations that would count as evidence; a wrong price by itself does not.
B · Multiple choice
C · Short answer
Weak form. Information set: all historical prices and trading volume. It rules out technical analysis. Evidence: daily returns are close to uncorrelated and most trading rules do not survive costs, which supports the form — though momentum effects complicate the picture.
Semi-strong form. Information set: all publicly available information — past prices, accounting statements, annual reports, news. It rules out fundamental analysis on public data. Evidence: event studies show prices adjust within a day or two of announcements, and active funds rarely beat passive benchmarks after costs. Against it: post-earnings-announcement drift and the value premium.
Strong form. Information set: all information, public and private. It rules out insider trading. Evidence: insider trades earn abnormal profits and prices move before some announcements — both reject the form.
The nesting. The sets are nested, so the forms are too: strong ⇒ semi-strong ⇒ weak. Rejecting the strong form says nothing about the semi-strong form, which is why the evidence can support one and reject another at the same time.
Announcement-day abnormal return. AR(0) = CAR(0) − CAR(−1) = −5.0 − (−3.6) = −1.4%.
Pre-announcement abnormal return. AR(−1) = CAR(−1) − CAR(−2) = −3.6 − (−0.5) = −3.1%. The day before the announcement is more than twice as large as the announcement day itself.
Post-announcement path. From the bottom at +1 to +8 the CAR rises by −4.5 − (−5.4) = +0.9 percentage points, a partial reversal of about 17% of the 5.4-point decline.
Interpretation. The reaction is largely complete by day 0 — about 93% of the eventual decline — which is consistent with semi-strong efficiency. The large pre-announcement drop is evidence of leakage, so the strong form fails. The partial post-announcement rebound suggests mild overreaction or noise, not a systematic exploitable drift. The honest conclusion is that the market is broadly efficient at the semi-strong level but not perfectly so.
What would change the conclusion. A continued drift in the same direction after day 0 — underreaction — would challenge semi-strong efficiency directly. A full reversal back to the pre-event level would indicate overreaction. A flat line after day 0, with no pre-announcement move, is the cleanest efficient benchmark.
Capital Structure
Concept Map
how the lecture buildsLecture 1 drew the split between investment decisions (which assets to buy) and financing decisions (how to pay for them). Lectures 2–9 were all investment-side: NPV, capital budgeting, the discount rate. This is the first lecture on the other side of the balance sheet. Its answer, in one sentence: financing cannot change the size of the operating pie, but it can change how much of the pie goes to the government and how much is lost in distress — and those are real effects.
The link to Lecture 9 is direct. In an efficient market, issuing a security is a zero-NPV transaction, so the only way the debt-equity mix can matter is through a friction: the tax code, bankruptcy costs, or asymmetric information. Every theory in the second half of the lecture is a story about a specific friction.
Lecture 4 computed the depreciation tax shield as TC × depreciation and discounted the stream. The interest tax shield is the same arithmetic: TC × interest each year, and for perpetual debt the present value collapses to TC × B because the stream is a level perpetuity at the same rate as the discount rate. The difference is that depreciation is attached to an asset, while the interest shield is attached to a financing choice — which is why this lecture is about capital structure and Lecture 4 was about project cash flows.
Capital Structure and the Leverage Effect
slides 2–7Slide 2 defines the object. Capital structure is the amount of debt and equity a firm uses, usually expressed as a debt-to-equity ratio. The value of the firm is the sum of the market values of its debt and equity:
The question is which ratio maximises shareholder wealth. The tempting answer is "all debt", because debt is cheaper than equity: the interest rate on a bond is usually far below the return shareholders require. The lecture spends the next 29 slides explaining why that temptation is wrong in the world without taxes and only half-right in the world with them.
A firm contemplating leverage slides 3–5
Slides 3–5 set up a running example. An all-equity firm has $20,000 of assets, 400 shares outstanding, and a share price of $50. It proposes to borrow $8,000 at 8% and use the proceeds to repurchase shares — perhaps because some original shareholders want to cash out.
| Item | Current (all equity) | Proposed (with debt) | How the proposed column is built |
|---|---|---|---|
| Assets | $20,000 | $20,000 | The asset base does not change; the financing does. |
| Debt | $0 | $8,000 | The new borrowing. |
| Equity | $20,000 | $12,000 | Assets − debt = 20,000 − 8,000. |
| Debt / equity | 0.00 | 2/3 | 8,000 / 12,000 = 0.667 = 2/3. |
| Interest rate | n/a | 8% | The coupon on the new debt. |
| Shares outstanding | 400 | 240 | 8,000 / $50 = 160 shares repurchased; 400 − 160 = 240. |
| Share price | $50 | $50 | The repurchase is at the market price, so the price does not jump on the announcement in this simplified example. |
Slides 4 and 5 compare the two capital structures across three states of the economy. The operating earnings are the same in both; only the split between interest, taxes and shareholders changes.
| State | EBIT | All-equity EPS | With-debt EPS | Who wins |
|---|---|---|---|---|
| Recession | $1,000 | $2.50 | $1.50 | All-equity wins. Interest of $640 has to be paid before shareholders see anything. |
| Expected | $2,000 | $5.00 | $5.67 | Leverage wins slightly: the firm earns 10% on assets and pays 8% on debt, so the difference accrues to shareholders. |
| Expansion | $3,000 | $7.50 | $9.83 | Leverage wins decisively: the same 10% spread is earned on a smaller equity base. |
The full slide tables also report ROA and ROE. Without debt, ROA and ROE are identical — there is no leverage to separate them. With debt, ROA falls in every state because the numerator is net income after interest, while the asset base is unchanged; ROE rises in the good states and falls in the recession state. The gap between ROA and ROE is the leverage effect in accounting form.
| State | EBIT | Interest | Net income | ROA | ROE |
|---|---|---|---|---|---|
| Recession | $1,000 | $640 | $360 | 1.8% | 3.0% |
| Expected | $2,000 | $640 | $1,360 | 6.8% | 11.3% |
| Expansion | $3,000 | $640 | $2,360 | 11.8% | 19.7% |
The arithmetic is worth checking once, because every later claim depends on it. Interest is 8% of $8,000 = $640. In the expected state, net income is 2,000 − 640 = $1,360. EPS is 1,360 / 240 = $5.667. ROE is 1,360 / 12,000 = 11.33%. The all-equity comparison is 2,000 / 400 = $5.00 and 2,000 / 20,000 = 10%. Leverage adds 0.667 per share in the expected state and 2.33 in the expansion state, but subtracts 1.00 in the recession.
The break-even point slide 6
Slide 6 plots EPS against EBIT for the two structures. The lines cross at the break-even EBIT, the level at which the two capital structures produce the same EPS. Set the two EPS expressions equal:
Cross-multiplying gives 240 × EBIT = 400 × (EBIT − 640), so 240 EBIT = 400 EBIT − 256,000, and 160 EBIT = 256,000, which gives EBIT = $1,600. At that level both structures produce EPS = 1,600 / 400 = $4.00. Below $1,600 the all-equity structure wins; above it, the levered structure wins. The chart labels the region above the crossing point "advantage to debt" and below it "disadvantage to debt".
Slide 7 draws the lesson: debt is a leverage (财务杠杆). It raises EPS in good states and lowers it in bad states, and more debt means more leverage and higher firm risk. Because leverage increases the probability and cost of bankruptcy, the optimal structure is not 100% debt. Managers should choose the structure that maximises firm value, because that is what benefits shareholders in the end — not the structure that maximises EPS in the expected state.
Maximising EPS is not the same as maximising shareholder wealth. Leverage can raise EPS in the expected state while raising the risk of the shares, which raises the return shareholders require. If the discount rate rises by enough, the share price can fall even as EPS rises. The lecture's example holds the share price at $50 in both structures to isolate the arithmetic; in reality the announcement of leverage would move the price, and the direction is the empirical question in slide 30.
The ROA in slide 5 is net income / total assets, so it falls when interest rises. That is a definitional choice, not evidence that leverage makes the assets less productive. Operating return on assets — EBIT / assets — is 5%, 10% and 15% in every state and in both structures, because the assets are identical. The decline in the slide's ROA is purely the interest expense, and it is why ROE and ROA diverge.
The share repurchase does not change firm value in the example. The firm swaps $8,000 of equity for $8,000 of debt, so V stays at $20,000. That is MM Proposition I in miniature, before the formal statement on slide 8.
MM Proposition I Without Taxes
slides 8–10Slide 8 states the first Modigliani–Miller proposition in its cleanest form:
where VL is the value of the levered firm and VU is the value of the unlevered firm. In a world without taxes, the choice of capital structure does not change firm value. The pie is the same size however it is sliced.
The assumptions slide 9
| Assumption | What it rules out | Which later theory relaxes it |
|---|---|---|
| Homogeneous expectations | Investors disagree about future cash flows. | The pecking order and signalling theories introduce asymmetric information. |
| Homogeneous business-risk classes | Firms differ in operating risk within a class. | Needed to compare VL and VU for the same assets; not a major source of real-world failure. |
| Perpetual cash flows | Finite project lives and changing debt levels. | Keeps the tax shield a level perpetuity, which is why TCB is so clean. |
| Perfect capital markets | Taxes, transaction costs, borrowing-rate differences, information advantages. | The big one. Relaxing taxes gives MM with taxes; relaxing the rest gives distress costs and the other theories. |
The perfect-market assumptions deserve a second look because they are where the real world enters. Firms and investors can borrow and lend at the same rate; everyone has equal access to information; there are no transaction costs; and there are no taxes. Each assumption is a friction the lecture will later restore, one at a time. MM is not a description of reality — it is a benchmark that tells you which friction is doing the work when the mix does matter.
Why Proposition I holds slide 10
The proof on slide 10 runs through free cash flow. Recall that firm value is the present value of the free cash flows generated by the operating assets. The only difference between the levered and unlevered firm is the interest expense, and interest is a financing cash flow: it splits the operating cash flow between debt holders and shareholders. It does not change the total operating cash flow, and without taxes it does not change the total cash flow available to all capital providers.
The second half of the argument is about the WACC. One might think debt is cheaper, so more debt lowers the WACC and raises value. The slide blocks that reasoning: as the firm issues more debt, the equity becomes riskier, so the cost of equity rises. In the no-tax world the rise is exactly enough to offset the shift toward cheaper debt, and the WACC stays at R0, the cost of capital of the unlevered firm. Business risk and financial risk are different things: leverage does not change the risk of the operating assets, it concentrates that risk on a smaller equity claim.
Because the WACC is the cost of the operating assets, and the operating assets have not changed. The WACC is a weighted average of the costs of the claims against those assets. Replacing equity with debt changes the weights and the cost of each claim: debt is cheaper, but the remaining equity is riskier. In the no-tax world the two effects cancel exactly. The formal version is Proposition II, on the next slide.
The intuition is the same as the lecture's pie metaphor. The size of the pie is determined by the operating cash flows and their risk. Cutting it into more debt slices and fewer equity slices does not bake a bigger pie. The only way to make the pie bigger is to reduce what someone outside the firm takes — the government, through taxes — or to reduce the costs of slicing it, which is what distress costs and issuance costs are.
There is a second, more intuitive proof of Proposition I that the slides skip. An investor who wants the payoff of a levered firm can create it herself: buy the unlevered firm's shares on margin, borrowing at the same rate the firm would pay. Her personal balance sheet then replicates the levered firm's payoffs. If levered firms traded at a premium, she would simply replicate the leverage at home and avoid the premium. Competition would drive the premium to zero.
This is homemade leverage, and it is why the assumption that firms and investors borrow at the same rate matters. If investors can borrow more cheaply than the firm, corporate leverage has value; if they borrow more expensively, it destroys value. In practice margin rates are usually higher than corporate bond yields for large firms, so this argument cuts in favour of some corporate debt — but that is a financing-cost argument, not a tax argument.
MM Proposition II Without Taxes
slides 11–13Proposition II explains what happens to the cost of equity as leverage rises. Slide 11 states it:
where RS is the cost of levered equity, R0 is the cost of capital of the unlevered firm, RB is the cost of debt, B is the value of debt, and SL is the value of levered equity. Read it as a base plus a premium: the equity holder earns R0 for bearing business risk, plus a premium for bearing financial risk. The premium is the debt-to-equity ratio times the spread between the asset return and the debt rate, because leverage lets equity earn that spread on borrowed money.
The derivation slide 12
Slide 12 derives Proposition II from the definition of the WACC. Start with the weighted average:
Set RWACC = R0, because Proposition I says the firm's overall cost of capital is unaffected by leverage. Multiply both sides by (B + S)/S. The algebra (shown in full on the slide) reduces to
The key step is the assumption RWACC = R0: the WACC is pinned to the asset return by Proposition I, and the equation then forces RS to rise with B/S. If R0 > RB — the usual case, because equity is riskier than debt — the slope is positive. Debt is cheaper than equity, but using more of it makes the remaining equity more expensive, and the two movements offset.
The graph and the beta version slide 13
Slide 13 plots the three costs against the debt-to-equity ratio. RB is flat, RS rises linearly, and RWACC is flat at R0. The flat WACC is the visual statement of Proposition I: leverage moves money between the two lines without moving the total.
The slide's quick question — how does the stock beta change after issuing debt? — has the same answer in beta language. Start from the asset beta identity used in Lecture 8:
With riskless debt, βB = 0, so βS = β0(1 + B/S). A firm with an asset beta of 1.0 that moves to B/S = 2/3 has an equity beta of 1.667. The equity beta rises for exactly the same reason the cost of equity rises: the residual claim is levered.
- The cost of equity. Apply Proposition II: RS = R0 + (B/S)(R0 − RB) = 10% + (2/3)(10% − 8%) = 10% + 1.333% = 11.333%
- The WACC. With B = 2, S = 3, the weights are B/(B+S) = 40% and S/(B+S) = 60%: RWACC = 0.40 × 8% + 0.60 × 11.333% = 3.2% + 6.8% = 10.0% The WACC is unchanged at R0 = 10%, as Proposition I requires.
- The equity beta. With riskless debt: βS = 1.0 × (1 + 2/3) = 1.667 The CAPM would then price the equity at 10% + 1.667 × market risk premium — higher than the asset's required return, because the equity is levered.
This example is constructed, not from the slides, and the numbers are chosen to divide cleanly. The point is the cancellation: the equity cost rises from 10% to 11.33%, but the equity weight falls from 100% to 60%, and the product stays at 10%. The next section adds taxes and the cancellation stops being exact.
RS in the formula is the return shareholders require for bearing the levered risk, not the return they will actually earn. In a good state the realised return can be far above 11.33%; in a bad state far below. The formula prices the risk; it does not predict the outcome.
And B/S must be market values, not book values. The weights in the WACC and in Proposition II are the market values of debt and equity. Using the balance-sheet debt and the book equity gives the wrong cost of equity, often badly wrong for a firm with a large market-to-book ratio. Lecture 8 made the same point about the WACC's weights; here it reappears inside the leverage formula.
MM With Taxes
slides 14–22Slide 14 adds the one friction the tax code makes unavoidable. Interest is deductible from corporate income; dividends are not. So debt produces a tax shield, and the value of the firm rises with leverage:
where TC is the corporate tax rate and TCB is the present value of the tax shield. The formula assumes the debt is perpetual and the tax shield is as risky as the debt, so it is discounted at RB. Each year the firm saves TC × RBB in tax; the present value of that level perpetuity is (TCRBB)/RB = TCB.
Why Proposition I holds with taxes slide 16
Slide 16 proves the result by decomposing the total cash flow to all stakeholders. For a levered firm:
Expand the first term and collect:
The first term is the after-tax operating cash flow of an unlevered firm, whose present value is VU. The second term is the annual tax saving, whose present value is TCB. So VL = VU + TCB. The government's claim is smaller when the firm has debt, and the difference accrues to the firm's investors.
The numbers on slide 17 slide 17
Slide 17 makes the decomposition concrete with the same $8,000 of debt at 8%, a 35% tax rate, and three EBIT states. The levered firm pays $640 of interest, and the government takes 35% of what remains.
| State | EBIT | Interest | EBT | Tax (35%) | Cash to S/H + B/H |
|---|---|---|---|---|---|
| Recession | $1,000 | $640 | $360 | $126 | $874 |
| Expected | $2,000 | $640 | $1,360 | $476 | $1,524 |
| Expansion | $3,000 | $640 | $2,360 | $826 | $2,174 |
The identity from slide 16 reproduces the same total. In the recession state, EBIT(1 − TC) = 1,000 × 0.65 = $650, and the tax shield is 0.35 × 640 = $224; their sum is $874, exactly the cash that reaches shareholders and bondholders. In the expected state, 1,300 + 224 = $1,524; in the expansion state, 1,950 + 224 = $2,174. The all-equity comparison pays $650, $1,300 and $1,950 to shareholders — the levered firm pays $224 more in every state, which is the annual tax saving.
Proposition II with taxes slides 15 and 18
With taxes, the cost of equity rises more slowly with leverage, because part of the extra risk is absorbed by the tax shield:
Slide 18 derives it from VL = VU + TCB and the balance-sheet identity VL = S + B. The cash flows on the two sides must match: SRS + BRB = VUR0 + TCBRB. Substituting VU = S + B(1 − TC) and dividing by S gives the formula above. The factor (1 − TC) is what slows the rise: only the after-tax cost of debt is offset by the equity premium.
WACC falls with leverage slide 19
Slide 19 plots the with-tax case. RS rises more slowly than in the no-tax case, while the WACC now declines with leverage:
The declining WACC is the formal version of "debt is cheap". It is also the source of the 100% debt problem: if the WACC falls with every unit of debt, value rises with every unit of debt, and the optimal structure is all debt. That conclusion is absurd in the real world, which tells you a cost is missing.
No. In the no-tax world it is the same — that is Proposition I. In the with-tax world the levered firm's WACC is lower than the unlevered firm's, because the government subsidises part of the interest bill. The all-equity firm's WACC is R0; the levered firm's is below R0. Slide 20's pie diagram is the picture: the levered firm's pie (S + B) is larger than the unlevered firm's pie (S) because the government's slice (G) is smaller. The pie is not bigger because the firm is more productive; it is bigger because the tax collector takes less.
- The cost of equity. Apply Proposition II with taxes: RS = 10% + (2/3)(1 − 0.35)(10% − 8%) = 10% + 0.8667% = 10.867% Compare the no-tax value of 11.333%: the tax shield absorbs 0.467 percentage points of the equity premium.
- The after-tax cost of debt. RB(1 − TC) = 8% × 0.65 = 5.2%
- The WACC. With weights 40% debt and 60% equity: RWACC = 0.40 × 5.2% + 0.60 × 10.867% = 2.08% + 6.52% = 8.6% The WACC has fallen from 10% to 8.6%, and the 1.4-point fall is the after-tax value of the tax shield in the discount rate.
- The equity beta. With the tax-adjusted relevering formula from Lecture 8: βS = 1.0 × [1 + (1 − 0.35)(2/3)] = 1.0 × 1.4333 = 1.433 Lower than the no-tax 1.667, for the same reason the cost of equity is lower.
The pattern to remember: with taxes, leverage lowers the WACC but raises the cost of equity by less than in the no-tax world. The difference is the tax shield. If the debt is perpetual, the total value added is TCB = 0.35 × B; with B/S = 2/3 and S = 12,000 in the earlier example, B = 8,000, so the shield is worth $2,800.
The line reads Interest ($800 @ 8%) and reports $640 of interest. But 8% of $800 is $64, not $640. The debt is $8,000, the same figure as slide 3's proposed borrowing: 8% × 8,000 = 640. The "$800" is a dropped digit, and the rest of the table — $360 of EBT, $126 of tax, $874 of total cash flow — is built on the $640 figure and is correct. Use $8,000 in an exam answer; do not reproduce "$800 @ 8%" with $640 of interest.
The slide's line is: "Without taxes, it affects a firm's operating cash flow, but does not affect free cash flows." Read strictly, the first clause is wrong. Interest is a financing cash flow: it is paid after EBIT and does not enter operating cash flow, which is built from EBIT, depreciation and working-capital changes. What interest does is split the operating cash flow between debt holders and shareholders. The total cash flow to all capital providers — the free cash flow the firm's value depends on — is unchanged. That is the correct intuition for Proposition I, and it is the opposite of what the sentence literally says.
Summary: the two worlds side by side slides 21–22
| No taxes | With taxes | The difference | |
|---|---|---|---|
| Proposition I | VL = VU | VL = VU + TCB | The tax shield is the only new term. |
| Proposition II | RS = R0 + (B/S)(R0 − RB) | RS = R0 + (B/S)(1 − TC)(R0 − RB) | The (1 − TC) factor slows the rise in the cost of equity. |
| WACC | Flat at R0 | Falls with leverage | The declining WACC is why debt creates value with taxes. |
| Optimal debt | Indeterminate — the mix does not matter | 100%, in the model | The absurd implication that motivates the trade-off theory. |
Beyond MM: Four Theories
slides 23–29Slide 23 is the pivot. The MM model builds a foundation, but it rests on assumptions that are false. Academic research relaxes them one at a time and produces competing theories of capital structure: trade-off (bankruptcy costs), signalling (debt as a signal of profitability), the free cash flow hypothesis (agency conflicts), and the pecking order (information asymmetry). No single theory dominates, and the evidence on slide 31 refuses to settle the contest.
Trade-off theory slides 24–25
The MM model with taxes implies that firms should borrow as much as possible. In reality, more debt means larger fixed obligations, a higher probability of missing them, and therefore higher financial distress costs. The ultimate distress is bankruptcy.
Slide 24 makes a subtle but important point about what bankruptcy costs are: bankruptcy itself does not lower the firm's cash flows; the costs associated with bankruptcy do. A reorganization can consume legal fees, force asset sales at fire-sale prices, and drive away customers and employees before the process is complete. Those are real deadweight losses, and they reduce the size of the pie.
Slide 25 plots the two effects. The blue line is the MM-with-taxes value, rising linearly with debt. The red curve subtracts the present value of distress costs, which grows slowly at first and then steeply as leverage rises. The difference reaches a maximum at B*, the optimal amount of debt. The firm trades off the tax subsidy against the distress cost, and the optimum is interior — not zero, not 100%.
Signalling theory slide 26
Signalling starts from a different asymmetry: managers know more about the firm's prospects than investors do. A firm that expects strong future profits can afford the fixed interest payments of debt; a firm that expects weak profits cannot. So debt is a credible signal of profitability. Firms with low anticipated profits take on little debt; firms with high anticipated profits take on more.
The slide adds the discipline that makes the signal credible. A manager who knows how investors interpret debt has an incentive to take on more-than-optimal debt to imitate a strong firm. But the imitation is costly: the firm must actually service the debt, and investors eventually discover the truth. The long-run cost is what deters the imitation and keeps the signal informative.
Free cash flow hypothesis slide 27
The free cash flow hypothesis is an agency story. Managers may want perquisites — lavish offices, private jets, empire-building acquisitions — but they need the cash to pay for them. Free cash flow supplies the opportunity. Debt reduces free cash flow by committing cash to interest and principal, which leaves less for wasteful spending.
The slide compares debt with dividends. An increase in dividends also reduces the free cash available to managers, so it should benefit shareholders. But debt is the stronger discipline: dividends are discretionary and can be cut, while interest must be paid or the firm defaults. A manager who wants to keep spending cannot simply skip a coupon. That is why the free cash flow hypothesis predicts a positive value for debt beyond its tax shield — it constrains the manager, not just the tax bill.
Pecking order theory slides 28–29
The pecking order starts from information asymmetry: managers know the true value of the firm and investors do not. When equity is overpriced, managers want to issue equity; when debt is overpriced, they want to issue debt. Because debt is less risky, its mispricing is relatively small — the value of debt is bounded by the contract, while equity's value depends on the entire future of the firm.
Investors know this strategy, so they treat an equity issue as a signal that the shares are overpriced. They discount the issue price, which makes equity expensive for the firm. The result is a hierarchy:
Slide 29 lists the predictions that distinguish the pecking order from the trade-off theory:
| Prediction | Pecking order | Trade-off theory | Why they differ |
|---|---|---|---|
| Target D/E ratio | None | A target B* exists | The pecking order is a financing sequence, not an optimisation toward a ratio. |
| Profitable firms | Use less debt | Use more debt | Profitable firms have more internal funds, so they need less external financing. Trade-off says they have more taxable income to shield, so they should borrow more. |
| Financial slack | Firms value it | Less central | Cash and unused debt capacity let a firm fund projects without issuing underpriced equity. |
Slide 31 attributes the profitability evidence to Fama and French (2000). The paper that established the leverage–profitability result is "Testing Trade-Off and Pecking Order Predictions About Dividends and Debt", published in the Review of Financial Studies in 2002. The 2000 Fama–French paper is "Forecasting Profitability and Earnings", which is about the time-series behaviour of profitability, not about capital structure. Cite 2002 if you use the result in an exam answer.
Capital Structure in the Real World
slides 30–31Slide 30 lists what the data show about actual capital structures:
| Fact | What it means | Which theory it supports |
|---|---|---|
| Most corporations have low debt-to-asset ratios | Most firms are not close to 100% debt, and many carry little debt at all. | Consistent with the trade-off theory's interior optimum, and with the pecking order's preference for internal funds. |
| Leverage changes affect firm value | Stock prices tend to rise when firms lever up and fall when they delever. | Consistent with MM with taxes — but also with signalling: a lever-up can signal confidence. |
| Differences across industries and over time | Capital structures are not random; they cluster by industry and drift with conditions. | Distress costs differ by industry: a firm with tangible assets can carry more debt than one whose value is in intangible growth options. |
| Firms behave as if they had a target D/E ratio | They appear to move toward a preferred ratio when they deviate. | The strongest evidence for the trade-off theory, because the pecking order predicts no target. |
The profitability puzzle slide 31
Slide 31 closes with the test that separates the theories. Should profitable firms use more or less debt?
| Theory | Prediction | The logic |
|---|---|---|
| Trade-off | More debt | Profitable firms have more taxable income, so the interest tax shield is worth more. |
| Pecking order | Less debt | Profitable firms generate internal funds and do not need external financing. |
| Signalling | More debt | Profitable firms can afford the fixed payments, so debt signals strength. |
The empirical answer is that profitable firms are less levered. Fama and French documented the pattern, and it favours the pecking order over the trade-off and signalling theories — at least on this dimension. The result is not decisive: profitability is correlated with intangible assets and with stable cash flows, both of which affect the trade-off in ways the simple test does not control for. Slide 31's conclusion is the honest one: many theories exist, none dominates, and the empirical evidence is inconclusive.
Use the theories as a checklist rather than a formula. Taxes say debt is valuable up to the point where distress costs bite. Distress says borrow less if cash flows are volatile or assets are intangible. Signalling says the market reads a lever-up as confidence, so do not lever up to fund a weak project. Agency says debt can discipline a management team with too much cash. Pecking order says keep financial slack and issue equity only when you have to.
The practical synthesis is the trade-off framework with a pecking-order overlay: aim for a target range, not a point; fund with internal cash first; use debt when the tax shield is worth more than the distress risk; issue equity when leverage is already at the top of the range. That is not a theorem, but it is what the evidence supports.
The no-tax proposition says the mix does not matter given its assumptions. The assumptions are false in exactly the places the rest of the lecture explores. The correct exam sentence is: "In a world without taxes, distress costs or information asymmetry, capital structure is irrelevant (MM I); with corporate taxes, debt adds TCB of value; with distress costs, the optimum is interior; with information asymmetry, the pecking order describes financing choices." Each clause relaxes one assumption, and the sequence is the lecture.
Formula Sheet
print-friendlyKey Concepts
the vocabularySelf-Check
16 questions · gradedThe examinable core is four things: MM I and II in both tax worlds, the tax shield and its present value, the trade-off optimum, and the four theories' predictions — especially the profitability puzzle. Three traps account for most lost marks: applying the no-tax proposition in a with-tax problem, forgetting that the WACC's weights are market values, and treating the pecking order's "less debt" prediction as if it were the trade-off's "more debt".
A · Concept check
The arithmetic. The WACC is a weighted average of the cost of debt and the cost of equity. Adding debt lowers the weight on the expensive claim, which by itself would lower the average. But it also makes the remaining equity riskier, which raises RS. In the no-tax world the two effects cancel exactly, and the WACC stays at R0.
The formal version. Set RWACC = R0 and solve for RS. The result is MM Proposition II: RS = R0 + (B/S)(R0 − RB). The equity cost rises linearly with leverage, and substituting back into the WACC returns R0.
The economic version. The WACC is the return required on the operating assets. Leverage does not change the assets or their risk; it changes how the claims on those assets are divided. The total risk is the same, so the total required return is the same. Debt holders take the safer slice and equity holders take the riskier residual, but the average does not move.
The pie metaphor. Slicing a pie differently does not bake a bigger pie. The only ways to make the pie bigger are to take less from someone outside the firm — the government, through taxes — or to reduce the deadweight costs of slicing, which is what distress costs are. The with-tax case is exactly the first of these; the trade-off theory is the second.
Why the intuition fails in practice. Debt is cheaper per unit, but adding it makes each remaining unit of equity more expensive. The error is to look at the price of debt and ignore the price of the equity that remains. That is why the no-tax WACC is flat and the with-tax WACC declines only because the government absorbs part of the interest bill.
B · Multiple choice
C · Short answer
No taxes. Proposition I: VL = VU. Proposition II: RS = R0 + (B/S)(R0 − RB). The WACC is flat at R0.
With taxes. Proposition I: VL = VU + TCB. Proposition II: RS = R0 + (B/S)(1 − TC)(R0 − RB). The WACC falls with leverage.
What changes. The tax deductibility of interest creates a shield worth TCB in present value. It raises the value of the levered firm, slows the rise in the cost of equity, and makes the WACC decline with leverage — but it also implies 100% debt, which is why the trade-off theory adds distress costs.
The two forces. Debt creates a tax shield worth TCB, which raises firm value. Debt also raises the probability and cost of financial distress: legal fees, fire sales, lost customers and employees. The present value of those costs rises with leverage, slowly at first and then steeply.
The optimum. Value is V = VU + TCB − PV(distress). The tax benefit is linear in B; the distress cost is convex. At low leverage the tax benefit dominates, so value rises with debt. At high leverage the distress cost dominates, so value falls. The maximum is at B*, where the marginal tax saving equals the marginal distress cost.
Why the optimum is interior. It is neither zero nor 100% debt. Zero debt would leave the tax shield unused; 100% debt would make distress certain and its cost enormous. The interior optimum is the firm trading off the two, which is why the theory is called the trade-off theory.
The empirical support. Most firms have low debt ratios and appear to move toward a target — both consistent with an interior optimum. The theory struggles with the profitability result: profitable firms should have more taxable income to shield, yet they borrow less.
Target ratio. The trade-off theory predicts a target B* and movement toward it. The pecking order predicts no target: financing is a sequence — internal funds, then debt, then equity — driven by the cost of information asymmetry, not by an optimal ratio.
Profitability. Trade-off predicts that profitable firms use more debt, because they have more taxable income to shield. The pecking order predicts they use less, because internal funds reduce the need for external financing. The evidence favours the pecking order: profitable firms are less levered.
Financial slack. The pecking order values slack — cash and unused debt capacity — because it lets the firm fund projects without issuing underpriced equity. The trade-off theory treats slack as less central; what matters is the distance from the target ratio.
The synthesis. Most empirical work finds elements of both: firms do appear to have target ranges, but they also follow a financing hierarchy and value slack. Neither theory dominates, which is slide 31's conclusion.
Without taxes. R_S = 12% + 1 × (12% − 6%) = 18.0% Weights: B/(B+S) = 50%, S/(B+S) = 50% WACC = 0.50 × 6% + 0.50 × 18% = 12.0% β_S = 1.2 × (1 + 1) = 2.4
With taxes. R_S = 12% + 1 × (1 − 0.25) × 6% = 16.5% After-tax debt = 6% × 0.75 = 4.5% WACC = 0.50 × 4.5% + 0.50 × 16.5% = 10.5% β_S = 1.2 × [1 + 0.75 × 1] = 2.1
The comparison. Without taxes the WACC stays at R0 = 12%, exactly as Proposition I requires. With taxes it falls by 1.5 percentage points to 10.5%, and the cost of equity rises by less (16.5% versus 18%) because the tax shield absorbs part of the financial risk. The equity beta is lower with taxes for the same reason: 2.1 instead of 2.4.
What the numbers mean. The 1.5-point fall is the tax advantage of debt at this leverage. It is the discount-rate version of the TCB value increase. The equity beta of 2.1 is what you would plug into the CAPM to get the 16.5% cost of equity — consistent with Lecture 8's relevering formula.
Homework 1 — Worked in Full
What Each Question Is Really Testing
and where it comes fromBefore working any of them, it is worth seeing the shape of the set. Every question maps onto one lecture, and every one of them is a question type the final exam uses.
| Q | Topic | From | The actual test |
|---|---|---|---|
| 1 | Debt-equity ratio, equity multiplier, ROE, net income | L1 | Recognising that ROE = ROA × equity multiplier — the DuPont identity — so that net income never has to be computed from a projected income statement. The ratio section of Lecture 1. |
| 2 | Present value of a delayed annuity | L2 | The payments do not start at t = 1, so the annuity factor lands you at t = 5, not at t = 0. One further discounting step. Also: counting the payments correctly — 6 to 25 inclusive is 20, not 19 or 25. |
| 3 | Income statement → net income → retained earnings | L1 | Interest is deducted before tax, and it is not part of EBIT. Then the dividend is subtracted from net income, not from anything else. |
| 4 | Solving for the rate: APR and EAR | L2 | Slightly harder than Lecture 2's examples, because r is the unknown. Requires iteration or a financial calculator — and then the APR→EAR conversion, which is the part the lecture actually emphasised. |
| 5 | Payback · NPV · IRR on mutually exclusive projects | L3 | The three criteria disagree. The question is not the arithmetic; it is knowing that NPV decides, and being able to say why. |
| 6 | Full capital budgeting appraisal | L4 | One paragraph of facts, converted into a cash flow series, discounted. The Lecture 4 workflow in miniature — and note what is absent: there is no working capital, no salvage, no sunk cost. Not every project has them. |
Q1 and Q3 are Lecture 1. Q2 and Q4 are Lecture 2. Q5 is Lecture 3. Q6 is Lecture 4. If one of them is giving trouble, it is worth going back to the corresponding lecture page rather than to another problem set — the technique is the same, and the lecture page carries the derivation the homework assumes.
One thing the set does not contain, and the final exam may: free cash flow. Q6 computes an operating cash flow using the bottom-up form (net income + depreciation), which is a special case that works only because the project has no working-capital or capex movements after t = 0. If the exam question adds a working-capital commitment or an asset sale, Q6's method alone will not be enough — Lecture 4's full formula will be needed.
The Six Questions, Worked in Full
work each on paper firstEach answer below shows the full working, the intermediate figures, and — where it matters — the route a typical wrong answer takes. The problems are stated as they appear on the sheet.
A · Statement arithmetic
Step 1 — the equity multiplier, straight from the debt-equity ratio. The two ratios are the same fact written two ways: D/E measures debt against equity, the equity multiplier measures assets against equity, and assets are the two added together.
EM = 1 + D/E = 1 + 0.70 = 1.70Answer: equity multiplier = 1.70.
Step 2 — total assets. The multiplier is assets per dollar of equity:
TA = EM × Equity = 1.70 × 840,000 = 1,428,000Which implies debt of 1,428,000 − 840,000 = 588,000. Check: 588,000 / 840,000 = 0.70 ✓
Step 3 — return on equity by the DuPont route. This is the step the question is really testing. You are given ROA, not ROE, and you are given nothing at all about sales or margins.
ROE = ROA × EM = 0.084 × 1.70 = 0.1428 → 14.28%Answer: ROE = 14.28%.
Step 4 — net income. Either route works, and doing both is the check:
From ROE: NI = 0.1428 × 840,000 = 119,952 From ROA: NI = 0.084 × 1,428,000 = 119,952 ✓Answer: net income = $119,952.
Why this is a DuPont question in disguise. Lecture 1 decomposed ROE into three multiplicative components — profit margin × asset turnover × equity multiplier. ROA is the first two of those multiplied together, so ROE = ROA × EM falls straight out, and the question is solvable without a single figure from the income statement. Recognising an identity is worth more than any amount of arithmetic here: students who try to build an income statement from these three numbers have nothing to build it from.
Note the leverage, too. The firm earns 8.4% on its assets but 14.28% on its equity, and the entire gap is the 0.70 of debt. That is financial leverage working in the firm's favour — and it would work against it just as hard if ROA fell below the interest rate on the debt. This is the mechanism Week 11's capital structure material is built on.
Step 1 — build the income statement top-down. The order matters, and it is the whole content of the question. Depreciation is an operating cost and comes out before EBIT; interest is a financing cost and comes out after.
Sales 435,000 − Costs −216,000 − Depreciation −40,000 ───────── EBIT (operating income) 179,000 − Interest expense −21,000 ───────── EBT (income before taxes) 158,000 − Tax at 35% −55,300 ───────── Net income 102,700Answer: net income = $102,700.
Step 2 — the addition to retained earnings. Net income is allocated between the shareholders and the firm's own reserves. Whatever is not paid out is retained.
Addition to retained earnings = NI − Dividends = 102,700 − 30,000 = 72,700Answer: addition to retained earnings = $72,700.
The two traps. First, taxing EBIT instead of EBT: 179,000 × 0.35 = 62,650 tax, giving net income of 116,350 — wrong, because it ignores the fact that interest is tax-deductible. Interest is paid with pre-tax money, which is why it sits above the tax line. Getting this wrong overstates net income by 21,000 × 0.35 = 7,350.
Second, treating dividends as an expense. Dividends are a distribution of profit, not a cost of earning it, so they are not deducted on the income statement — the $30,000 never appears above. It only reduces what the firm keeps. Net income is 102,700 whether or not the dividend is paid; the retained earnings figure is what changes.
Cross-check against Lecture 1. The retained earnings walk on Lecture 1's balance sheet is the same identity over time: ending retained earnings = beginning + net income − dividends. Here the question gives you one year's increment; the balance sheet version strings them together.
B · Time value of money
Step 1 — count the payments, before doing anything else. This is where most of the lost marks are. Payments fall at t = 6, 7, 8, …, 25.
Number of payments = 25 − 6 + 1 = 20Not 25 (the last payment date) and not 19 (the difference). The "+1" is because both endpoints are included. Twenty payments of $7,500 is $150,000 of nominal cash — worth remembering as a sanity check on the answer at the end.
Step 2 — value the annuity at the point one period before the first payment. The ordinary annuity factor assumes the first cash flow arrives at t = 1 and returns a value at t = 0. Here the first payment is at t = 6, so the factor returns a value at t = 5.
AF(7.1%, 20) = [1 − 1/(1.071)²⁰] / 0.071 = 10.512172 PV at t = 5 = 7,500 × 10.512172 = 78,841.29Step 3 — discount that lump back to today. Five years between t = 5 and t = 0:
PV at t = 0 = 78,841.29 / (1.071)⁵ = 78,841.29 / 1.409396 = 55,950.81Answer: PV = $55,950.81.
Why it must be five years and not six. The most common error in this question is dividing by (1.071)6, on the reasoning that the first payment arrives in six years. That confuses the valuation date of the annuity with the date of a payment. Discounting an annuity factor back six years would value the stream as though its first payment fell at t = 7. The factor's valuation point is always one period before its first cash flow — here 6 − 1 = 5.
The check. An alternative route: the present value of 25 payments less the present value of the five payments you do not receive.
AF(7.1%, 25) = 11.549350 AF(7.1%, 5) = 4.089242 Difference = 7.460108 PV = 7,500 × 7.460108 = 55,950.81 ✓The two routes agree, which is the test that the five-year discounting step was right. Worth seeing because the second route never requires you to think about valuation dates at all — it treats the problem as "all 25 payments, minus the 5 you don't get", which is sometimes the easier way to see it.
Step 1 — the loan amount. Note that the mortgage covers 80% of the purchase price, not the whole of it. The remaining 20% is the down payment and plays no part in the loan arithmetic.
Loan = 0.80 × 5,200,000 = 4,160,000Step 2 — set up the equation. Thirty years of monthly payments is n = 30 × 12 = 360 periods. The unknown is the monthly rate r:
4,160,000 = 27,500 × [1 − 1/(1 + r)³⁶⁰] / rThere is no algebraic solution for r — it has to be found by iteration, or with the I/Y key on a financial calculator. Entering PV = 4,160,000, PMT = −27,500, n = 360, FV = 0 and solving gives:
r = 0.578060% per monthStep 3 — the APR. The APR is the monthly rate multiplied by twelve — by definition, and with no compounding applied:
APR = 0.578060% × 12 = 6.9367% → 6.94%Answer: APR = 6.94%.
Step 4 — the EAR. Now compound the monthly rate twelve times, which is what actually happens over a year:
EAR = (1 + r)¹² − 1 = (1.00578060)¹² − 1 = 1.0716156 − 1 = 0.0716156 → 7.1616%Answer: EAR = 7.16%.
The 0.22-point gap is the whole point of the question. The APR says 6.94%; the rate you actually bear is 7.16%. The difference is compounding, and the APR is legally allowed to ignore it. Lecture 2 made exactly this point with the 12% example — APR 12%, EAR 12.36% monthly, 12.68% daily — and the reason the gap here looks smaller is only that the rate itself is smaller. In China the equivalent disclosure rule requires the APR to be quoted, which is why comparing loan offers means asking for the EAR.
One observation worth carrying away. Over the 30 years you will pay 360 × 27,500 = $9,900,000 to borrow $4,160,000. The interest is $5,740,000 — more than the principal, and more than the warehouse cost. That is not a trap or a scandal; it is what a 30-year amortising loan does, because most of the early payments are interest and the balance stays large for a long time. Lecture 2's amortisation schedules show the mechanism.
C · Investment criteria and capital budgeting
| Year | Dry Prepreg | Solvent Prepreg |
|---|---|---|
| 0 | −1,700,000 | −750,000 |
| 1 | 1,100,000 | 375,000 |
| 2 | 900,000 | 600,000 |
| 3 | 750,000 | 390,000 |
(a) Payback period. Count how long the cumulative cash flow takes to turn positive, then interpolate within the year it does.
Dry Prepreg (outlay 1,700,000) Year 1 cumulative: 1,100,000 still −600,000 Year 2 cumulative: 1,100,000 + 900,000 = 2,000,000 positive Payback = 1 + 600,000 / 900,000 = 1.667 years Solvent Prepreg (outlay 750,000) Year 1 cumulative: 375,000 still −375,000 Year 2 cumulative: 375,000 + 600,000 = 975,000 positive Payback = 1 + 375,000 / 600,000 = 1.625 yearsAnswer (a): Solvent Prepreg — 1.63 years against Dry's 1.67. Both are under three years, so payback cannot reject either; it only ranks them.
(b) Net present value at 8%.
Discount factors at 8%: 1.0800 1.1664 1.2597 Dry Prepreg 1,100,000 / 1.0800 = 1,018,518.52 900,000 / 1.1664 = 771,604.94 750,000 / 1.2597 = 595,374.18 ─────────── PV of inflows 2,385,497.64 − Initial outlay −1,700,000.00 ─────────── NPV 685,497.64 Solvent Prepreg 375,000 / 1.0800 = 347,222.22 600,000 / 1.1664 = 514,403.29 390,000 / 1.2597 = 309,594.57 ─────────── PV of inflows 1,171,220.09 − Initial outlay −750,000.00 ─────────── NPV 421,220.09Answer (b): Dry Prepreg — NPV of $685,498 against Solvent's $421,220. Both are positive, so both are worth doing if you could do both — but the question says they are mutually exclusive, so only one can be chosen, and Dry adds more value.
(c) Internal rate of return. The IRR is the discount rate at which NPV = 0; it must be solved for iteratively. Solving each:
Dry Prepreg: IRR = 30.90% Solvent Prepreg: IRR = 36.51%Answer (c): Solvent Prepreg — 36.51% against Dry's 30.90%. On a rate-of-return basis, Solvent is the better project.
(d) Which should actually be taken? Dry Prepreg.
The three criteria disagree, and the disagreement is resolved by a rule, not by a vote: for mutually exclusive projects, use NPV. NPV answers the question the firm is actually asking — by how much will this project increase the value of the firm? — and it is measured in dollars, which is the unit the shareholders care about.
Why IRR misleads here — three reasons, in order of importance.
1 · IRR is a rate, not an amount. A 36.51% return on a $750,000 outlay is $421,220 of value; a 30.90% return on $1,700,000 is $685,498. The firm does not eat percentages; it eats dollars. IRR systematically rewards projects that are small, and this is the classic scale problem that Lecture 3 identified.
2 · The two projects are of very different size, so the comparison is not like-for-like. If the extra $950,000 that Dry requires could earn more elsewhere than the 25.5% incremental return shown below, Solvent would be the better choice. NPV implicitly assumes the additional capital earns the 8% discount rate — and at 8%, Dry's incremental investment is comfortably worth making.
3 · The crossover rate makes the rule explicit. The difference between the two projects' cash flows is an investment of 950,000 returning 725,000, 300,000 and 360,000. Its IRR — the rate at which the two projects have the same NPV — is:
Incremental cash flows (Dry − Solvent): −950,000 725,000 300,000 360,000 Crossover rate = 25.52% Below 25.52%: Dry has the higher NPV ← the 8% case here Above 25.52%: Solvent has the higher NPVAt the question's 8% discount rate we are well below the crossover, so Dry wins on NPV while Solvent wins on IRR. The conflict is not a paradox — it is a predictable consequence of the crossover rate lying above the cost of capital. Whenever a project's IRR ranking and NPV ranking disagree, this is the structure producing it, and drawing the two NPV profiles against a common rate axis is the way to show it.
Why payback is the weakest of the three. Payback ranks Solvent first for a reason that has nothing to do with value: Solvent recovers its (much smaller) outlay marginally faster. Payback ignores everything after the cutoff date — Dry's year-3 cash flow of $750,000 is invisible to it — and it ignores the time value of money entirely, treating a dollar in year 3 as equal to a dollar today. Lecture 3 lists it as a screening device, not a decision rule, and this question demonstrates why.
Step 1 — annual depreciation. Straight-line to zero over three years. Note the phrase "after which it will be worthless": there is no salvage value and no terminal cash flow, which removes an entire complication from the problem.
Depreciation = 1,650,000 / 3 = 550,000 per yearStep 2 — the annual cash flow. Build it from EBIT down, then convert to cash:
Sales 1,240,000 − Costs −485,000 − Depreciation −550,000 ───────── EBIT 205,000 − Tax at 35% −71,750 ───────── Net income 133,250 + Depreciation (non-cash) +550,000 ───────── Operating cash flow 683,250Cross-check with the tax-shield form: (1,240,000 − 485,000) × 0.65 + 550,000 × 0.35 = 490,750 + 192,500 = 683,250 ✓ — the same figure, which is the standard verification that the tax treatment is right.
Step 3 — assemble the cash flow series. Here is the step worth pausing on. The project has no working capital investment (it is not mentioned, so it is zero) and no salvage value, so cash flows are the operating figure in years 1 to 3 and nothing else.
t = 0: −1,650,000 t = 1: 683,250 t = 2: 683,250 t = 3: 683,250Step 4 — discount at 12%. This is an ordinary annuity — a level stream for three periods — so the annuity factor is the efficient route.
AF(12%, 3) = [1 − 1/(1.12)³] / 0.12 = [1 − 0.711780] / 0.12 = 2.401831 PV of cash flows = 683,250 × 2.401831 = 1,641,051.21 NPV = 1,641,051.21 − 1,650,000 = −8,948.79Answer: NPV = −$8,949. Reject the project.
What makes this question worth doing carefully. The NPV is negative by less than 0.55% of the initial outlay — $8,949 against $1,650,000. That is not a project to reject with confidence; it is a project balanced on a knife edge, and every estimate in the paragraph is a forecast. A 1% error in the sales figure moves the NPV by roughly $19,400 — more than twice the size of the gap. Identifying this as a marginal case is the answer to the question, and Lecture 4's sensitivity analysis is the tool for saying so properly.
The lesson for the exam. A question that says "costs of $485,000" and produces an NPV of −$8,949 is telling you to be careful with the tax line. Note what the answer is not: it is not EBIT(1 − T) − outlay = 205,000 × 0.65 × 2.401831 − 1,650,000 = −1,329,956. Forgetting to add back depreciation, or forgetting that it is a tax shield rather than a cash cost, is the error this problem is constructed to catch. Depreciation never leaves the firm, but the 35% of it that the tax authority would otherwise have taken does stay — and that is $192,500 a year.
Formula Sheet for This Set
only what these six questions needEverything below appears in one of the six questions. The fuller formula sheets live on the lecture pages.
EBT = EBIT − Interest
NI = EBT × (1 − T)
The Traps This Set Contains
one per questionEvery one of the six questions has a specific, predictable error built into it. They are collected here because the exam sets the same traps.
Given only D/E, ROA and equity, there is no path to net income through sales, costs or margins — none of them are given, and the problem is not missing information. The route is the identity ROE = ROA × EM. When a question gives you two ratios where you expected a statement, look for the relationship that connects them rather than for the missing data.
Interest sits above the tax line because it is tax-deductible, so the tax is 35% of 158,000, not of 179,000. Getting this wrong inflates net income by 21,000 × 0.35 = 7,350. The second half of the question has its own trap: dividends reduce retained earnings, not net income — they are never on the income statement.
Two errors, and both are common. Counting the payments as 25 or 19 instead of 20 — the endpoints are both included. Discounting by six years instead of five, because the annuity factor values the stream one period before its first payment, not on the date of that payment. Six years would value it as though the first payment fell at t = 7.
The mortgage covers 80% of $5,200,000, so the loan is $4,160,000 — using the full price will not solve at all. And the two required outputs are different numbers: the APR (6.94%) is the monthly rate times twelve with no compounding; the EAR (7.16%) is what you actually pay. Quoting one when asked for the other costs the mark.
The question is constructed so the three criteria give different answers. Only one of them is a decision rule for mutually exclusive projects, and it is NPV. A common half-answer is to report all three rankings and then choose Solvent because two of the three agree — that is voting, not analysis, and it is wrong. The supporting work is the crossover rate of 25.52%, which shows the conflict is a consequence of the discount rate lying below it.
Depreciation is subtracted to reach EBIT and then added straight back — which looks like it cancels, and does not, because in between it reduced the tax bill. The correct annual cash flow is $683,250; the figure you get by forgetting the add-back is $133,250, and the NPV swings from −$8,949 to −$1,329,956. The tax-shield form — (Sales − Costs)(1 − T) + Dep × T — makes the mechanism visible and is worth using as the cross-check in every such question.
The NPV is −$8,949 against an outlay of $1,650,000 — a margin of 0.54%. In an exam, the mechanically correct answer is "reject, NPV is negative". The better answer says so and notes that the decision turns on estimates: sales of $1.24m, costs of $485,000 and a 12% required return are all forecasts, and a small revision to any of them would flip the sign.
This is the situation Lecture 4's risk analysis exists for. A sensitivity analysis on the sales figure would show that the project breaks even at sales of about $1,245,700 — roughly 0.5% above the estimate. Once you know the margin is that thin, "reject" becomes a statement about the quality of the forecast as much as about the project.
There is a second, subtler point. Rejecting a marginally negative-NPV project also forgoes whatever option value it carries — the ability to expand if demand is stronger than expected, or to abandon if it is weaker. Plain NPV assumes the decision is made once and never revisited. Real options, which Ross Ch. 7 covers, is the correction.
Self-Check
6 questions · new numbers, same skillsThese are not the set questions with the answers filled in. The numbers are different and the setups are changed slightly, so the only way through them is the technique. If you can do these, the homework is done.
A · Drill
Step 1 — depreciation.
Depreciation = 900,000 / 3 = 300,000 per yearStep 2 — route one: bottom-up, from net income.
Sales 760,000 − Cash costs −310,000 − Depreciation −300,000 ───────── EBIT 150,000 − Tax at 30% −45,000 ───────── Net income 105,000 + Depreciation +300,000 ───────── Operating cash flow 405,000Step 3 — route two: the tax-shield form, as the check.
OCF = (Sales − Costs) × (1 − T) + Dep × T = (760,000 − 310,000) × 0.70 + 300,000 × 0.30 = 450,000 × 0.70 + 90,000 = 315,000 + 90,000 = 405,000 ✓The two routes agree. Notice what the second one makes explicit: of the $405,000, $315,000 is the after-tax operating margin and $90,000 is the depreciation tax shield on its own — 22% of the annual cash flow exists only because the tax system permits the write-off.
Step 4 — the NPV. No working capital, no salvage, so the cash flows are simply −900,000 at t = 0 and 405,000 in each of years 1 to 3. That is an ordinary annuity.
AF(10%, 3) = [1 − 1/(1.10)³] / 0.10 = [1 − 0.751315] / 0.10 = 2.486852 PV of cash flows = 405,000 × 2.486852 = 1,007,175.06 NPV = 1,007,175.06 − 900,000 = 107,175.06Answer: OCF = $405,000 per year (by both routes); NPV = +$107,175. Accept the project.
The comparison with Q6 is the point of this question. Q6 had a negative NPV of −$8,949; this one is positive by $107,175. Run the two side by side and the difference in the inputs is small — but look at the margins:
Q6: Sales 1,240,000, costs 485,000 → margin 60.9% Q12: Sales 760,000, costs 310,000 → margin 59.2%Almost identical. The decisive difference is capital intensity: Q6 spends $1,650,000 of assets to generate $755,000 of annual margin; Q12 spends $900,000 to generate $450,000. Q6's assets are 2.19 times its annual margin, Q12's are 2.00 times. Q6's project is bigger, and at a 12% required return the extra capital is not quite earning its keep. This is why NPV is the right criterion and why a project can look excellent on the income statement and still destroy value: the margin is not the decision — the margin relative to the capital it requires is.
Homework 2 — Worked in Full
What Each Question Is Really Testing
and where it comes fromBefore working any of them, it is worth seeing the shape of the set. Every question maps onto one lecture, and every one of them is a question type the final exam uses.
| Q | Topic | From | The actual test |
|---|---|---|---|
| 1 | Pricing a bond with semiannual coupons | L6 | The quoted YTM is a bond-equivalent annual figure, so the period rate is 5.2%/2 = 2.6%. Halve the rate, double the periods, halve the coupon. Coupon 5.5% > YTM 5.2%, so the price must exceed par — a free direction check. |
| 2 | Arithmetic average · standard deviation · Sharpe ratio | L7 | Two measures for the fund and two for the market, then a comparison that disagrees with itself: she wins on return and loses on the Sharpe ratio. Deciding which one answers "did she do better" is the question. |
| 3 | Bond holding-period return when the yield moves | L6 + L7 | The bond is a vehicle for a return question. Note the two halves: if the yield is unchanged the return equals the yield exactly; if it falls, the capital gain is added on top. The same identity as L6's slide 17, with the numbers arranged to prove it. |
| 4 | Free-cash-flow valuation → share price → P/E | L5 | A multi-stage DCF: explicit forecasts, then a perpetuity once growth stops. The terminal value supplies about 77% of the answer, which is the honest thing to notice about terminal-value models. |
| 5 | P/E ratio and NPVGO | L5 | The P/E of a no-growth firm is 1/r — an identity, not a coincidence. A new project adds its NPV per share to the numerator while the denominator is unchanged, so the P/E rises even though nothing about the existing business changed. |
| 6 | Portfolio return, variance, and the correlation | L7 | The portfolio return is a weighted average; the portfolio risk is emphatically not. The question then moves ρ twice to show what the correlation term is doing. |
Four of the six questions are Lecture 7 material or Lecture 7 material applied to a security: Q2 and Q6 are squarely risk and return, and Q3 is a return calculation on a Lecture 6 instrument. That is the natural shape of a problem set issued in week 7 — it consolidates the lecture just given, using the securities from the two before it as the material.
What the set does not contain, and what the exam may: CAPM and beta. Every risk question here is answered with the standard deviation, which is correct for a single asset or a whole portfolio but wrong for pricing an individual security inside a diversified portfolio — because it charges that security for risk its holder can diversify away. If the exam asks for a required return on a stock, the answer will need rf + β[E(Rm) − rf] and a beta, not a σ. Q2's Sharpe ratio is the closest this set comes — and it is a performance measure, not a pricing model.
The Six Questions, Worked in Full
work each on paper firstEach answer below shows the full working, the intermediate figures, and — where it matters — the route a typical wrong answer takes. The problems are stated as they appear on the sheet.
A · Bond valuation and bond returns
Step 1 — the direction check, before any arithmetic. The coupon rate (5.5%) exceeds the yield to maturity (5.2%), so this is a premium bond and the answer must be greater than $1,000. Any result below par is wrong before you compute it, and the check costs nothing.
Step 2 — adjust the three inputs to a semiannual basis. This is the whole content of the question. The quoted YTM is a bond-equivalent annual rate, which is the market convention for a semiannual bond — so the rate per period is half of it.
Coupon payment = 5.5% × 1,000 / 2 = 27.50 per period Periods = 10 years × 2 = 20 Rate per period = 5.2% / 2 = 2.6%Step 3 — discount the par repayment.
1.026²⁰ = 1.670888 PV of par = 1,000 / 1.670888 = 598.48Step 4 — discount the coupon stream, using Lecture 2's annuity factor:
AF(2.6%, 20) = (1 − 1/1.026²⁰) / 0.026 = (1 − 0.598484) / 0.026 = 15.44291 PV of coupons = 27.50 × 15.44291 = 424.68Step 5 — add them.
Bond value = 424.68 + 598.48 = 1,023.16Answer: $1,023.16. And the direction check passes: 1,023.16 > 1,000, as a premium bond requires.
The trap this question is built around. Using the quoted 5.2% as the period rate instead of halving it. That single slip changes the answer from $1,023.16 to about $699.79 — a discount bond far below par, and a result that contradicts the direction check you could have run in step 1. The mismatch between the check and the answer is the signal that a frequency adjustment was missed.
Why 5.2% / 2 and not a compounding adjustment. A subtlety worth stating, because it looks inconsistent with Lecture 6's EAR material. The quoted yield on a semiannual bond is defined as twice the semiannual rate — 2 × 2.6% = 5.2% — which is a nominal convention rather than a compounding one. The effective annual rate the bond actually delivers is 1.026² − 1 = 5.2676%, slightly above the quoted 5.2%. Both numbers are correct; they answer different questions. The market quotes the nominal one so that bonds of different payment frequencies can be compared directly.
Step 0 — price the bond today. Both parts need the purchase price, so compute it once. Coupon 5% × 1,000 = 50 a year, six years, discounted at 3%:
PV of coupons = 50 × (1 − 1/1.03⁶) / 0.03 = 50 × 5.417191 = 270.86 PV of par = 1,000 / 1.03⁶ = 1,000 / 1.194052 = 837.48 ─────── Price today 1,108.34A premium bond (coupon 5% > yield 3%, price > par). Note the price for part (b) is needed too — it is the same bond, so the same 1,108.34 is the denominator in both parts.
Step 1 — the holding-period return formula. Over one year the bondholder receives the coupon and experiences a price change, so the return has two pieces:
Return = (Coupon + Price change) / Purchase price = (50 + P₁ − 1,108.34) / 1,108.34(a) The yield is unchanged at 3%. One year has passed, so five years remain. Reprice at 3%:
P₁ = 50 × AF(3%, 5) + 1,000/1.03⁵ = 50 × 4.579707 + 1,000/1.159274 = 228.99 + 862.61 = 1,091.59 Return = (50 + 1,091.59 − 1,108.34) / 1,108.34 = 33.25 / 1,108.34 = 0.0300 = 3.00%Answer (a): 3.00% — exactly the yield to maturity.
(b) The yield falls to 2%. Same five years remaining, repriced at the new rate:
P₁ = 50 × AF(2%, 5) + 1,000/1.02⁵ = 50 × 4.713460 + 1,000/1.104081 = 235.67 + 905.73 = 1,141.40 Return = (50 + 1,141.40 − 1,108.34) / 1,108.34 = 83.06 / 1,108.34 = 0.07494 = 7.49%Answer (b): 7.49%.
The decomposition in (b), which is the point of the question. Split the 7.49% into its two sources:
Coupon yield = 50 / 1,108.34 = 4.51% Capital gain = (1,141.40 − 1,108.34) / 1,108.34 = 2.98% ───── Total return 7.49% ✓So the fall in the yield contributed roughly three percentage points of capital gain on top of the coupon income — a 100-basis-point move in rates producing a 298-basis-point price gain on a five-year bond. That ratio (about 3 to 1) is a fair instinct for a five-year bond's sensitivity, and it is the number a bond trader is paid to forecast.
Why (a) comes out at exactly 3%. This is not a rounding coincidence, it is an identity, and it is worth understanding rather than memorising. If the yield is unchanged, the bond is priced on the same yield curve one year later, and the entire return is the coupon plus the pull to par implied by holding a premium bond one year closer to maturity. Those two components together reproduce the yield by construction. Formally: the return equals the yield whenever the yield does not move.
Which makes the two answers a diagnostic. If a holding-period return calculation returns the YTM in the no-change case, the machinery is right; if it does not, something is wrong with the repricing step. Here the (a) figure lands on 3.00% to two decimals, which is the arithmetic's own confirmation that the cash-flow treatment is correct.
The sign of the rate change matters both ways. The question only asks about a fall, but the same machinery gives the answer for a rise — and it is the asymmetry from Lecture 6 that matters. A 100 bp rise to 4% would produce a price fall, and because of convexity the loss would be slightly smaller than the 298 bp gain here. The bondholder is better off with volatility than without it, which is the one piece of good news in interest rate risk.
B · Risk, return, and the portfolio
| Year | 2010 | 2011 | 2012 | 2013 | 2014 |
|---|---|---|---|---|---|
| Ms. Sauros | +24.9 | −0.9 | +18.6 | +42.1 | +15.2 |
| S&P 500 | +17.2 | +1.0 | +16.1 | +33.1 | +12.7 |
Step 1 — Ms. Sauros's arithmetic average.
(24.9 − 0.9 + 18.6 + 42.1 + 15.2) / 5 = 99.9 / 5 = 19.98%Step 2 — her standard deviation. Deviations from 19.98%, then squares:
Year Return Deviation Squared deviation 2010 24.9 +4.92 24.2064 2011 −0.9 −20.88 435.9744 2012 18.6 −1.38 1.9044 2013 42.1 +22.12 489.2944 2014 15.2 −4.78 22.8484 ───────── Sum 974.2280 Variance = 974.2280 / 5 = 194.8456 Standard deviation = √194.8456 = 13.96%Answer (a): arithmetic average 19.98%, standard deviation 13.96%.
Step 3 — her Sharpe ratio.
Sharpe = (19.98 − 3) / 13.96 = 16.98 / 13.96 = 1.216 → 1.22Answer (b): 1.22.
Step 4 — the market's figures, for comparison. Same arithmetic on the S&P 500 column:
Average = (17.2 + 1.0 + 16.1 + 33.1 + 12.7) / 5 = 80.1 / 5 = 16.02% Deviations +1.18 −15.02 +0.08 +17.08 −3.32 Squares 1.3924 225.6004 0.0064 291.7264 11.0224 Sum = 529.7480 Variance = 529.7480 / 5 = 105.9496 Standard deviation = 10.29% Sharpe = (16.02 − 3) / 10.29 = 13.02 / 10.29 = 1.265 → 1.26Step 5 — the comparison.
| Measure | Ms. Sauros | S&P 500 | Winner |
|---|---|---|---|
| Arithmetic average return | 19.98% | 16.02% | Sauros — by 3.96 points |
| Standard deviation | 13.96% | 10.29% | Market — she took 36% more risk |
| Sharpe ratio | 1.22 | 1.26 | The market wins on a risk-adjusted basis |
Answer (c): the honest answer is that she did better on one measure and worse on the other, and the risk-adjusted measure is the one that should decide. She earned a higher average return (19.98% against 16.02%) but accepted substantially more risk (13.96% against 10.29%), and once the return is normalised by the risk taken to earn it, the market's Sharpe ratio is the higher of the two — 1.26 against 1.22.
So: better in raw terms, worse in risk-adjusted terms, and the second is the fairer verdict. An investor who simply wanted her 19.98% average could have obtained something close to it by holding a leveraged position in the index — and the comparison being made here is precisely whether she added value beyond what that leverage would have delivered.
Why the ranking is robust despite the divisor. This question divides by T = 5, following Lecture 7's convention. If you use the sample convention, T − 1 = 4, both standard deviations rise by the same factor of √(5/4) = 1.118 — Sauros to 15.61% and the market to 11.51% — and both Sharpe ratios fall by that same factor, to 1.09 and 1.13. The market still wins. The choice of divisor is worth a sentence in an exam answer, but it never changes this conclusion, because it scales numerator and denominator of every Sharpe ratio identically.
Two further observations worth carrying. First, look at 2011: the fund returned −0.9% while the market returned +1.0%. That single year is almost the whole difference between the two Sharpe ratios, and it is a reminder that this is a five-observation sample — one year's performance is doing a great deal of work. A real assessment would want far more history. Second, note that her return was higher and her worst year was worse, which is what a higher-beta portfolio looks like: it amplifies the market in both directions. Whether that is skill or leverage is the question the Sharpe ratio cannot answer, and which the next week's CAPM is designed to.
| Share A | Share B | |
|---|---|---|
| Expected return (%) | 15 | 20 |
| Standard deviation (%) | 20 | 22 |
| Correlation between returns | 0.5 | |
Step 1 — the portfolio's expected return. A weighted average, and the correlation plays no part in it:
E(rP) = 0.60(15%) + 0.40(20%) = 9.0% + 8.0% = 17.00%Step 2 — the portfolio's variance at ρ = 0.5. Lecture 7's three-term formula, with the correlation in the cross term:
σP² = (.60)²(20)² + (.40)²(22)² + 2(.60)(.40)(0.5)(20)(22) = 144.00 + 77.44 + 105.60 = 327.04 σP = √327.04 = 18.08%Answer (a): E(R) = 17.00%, SD = 18.08%.
Step 3 — the same calculation at the other two correlations. Nothing changes except the cross term, which is linear in ρ:
ρ = +0.5: cross = +2(.60)(.40)(+0.5)(20)(22) = +105.60 σP² = 144.00 + 77.44 + 105.60 = 327.04 σP = 18.08% ρ = 0 : cross = 0.00 σP² = 144.00 + 77.44 + 0.00 = 221.44 σP = 14.88% ρ = −0.5: cross = −105.60 σP² = 144.00 + 77.44 − 105.60 = 115.84 σP = 10.76%Answer (b): the expected return stays at 17.00% in all three cases — the correlation does not touch it. The standard deviation falls from 18.08% to 14.88% to 10.76% as ρ goes from +0.5 to 0 to −0.5.
Step 4 — the comparison in (c).
| Holding | Expected return | Standard deviation |
|---|---|---|
| Share A alone | 15.00% | 20.00% |
| 60 / 40 portfolio (ρ = 0.5) | 17.00% | 18.08% |
Answer (c): better — and unambiguously so. The portfolio has a higher expected return (17% against 15%) and a lower standard deviation (18.08% against 20%) than share A held on its own. That is the dominance case from Lecture 7's slide 12: one option is better on both axes, so no investor needs a preference to decide, and the choice is not a tradeoff at all.
Where the improvement came from — and it is not the diversification. This is the subtlety worth pausing on, because the obvious explanation is the wrong one. Share B has a higher expected return (20% against 15%) and a higher standard deviation than A — but the correlation is only 0.5, so adding B does not only import risk. The 7.35-point weighted average of the two standard deviations, 20.8%, is dragged down to 18.08% by the cross term, and the return rises from 15% to 17% simply because B pays more.
So the portfolio beats A for two independent reasons, and it is worth separating them: (i) B's higher expected return lifts the portfolio's return toward B's; (ii) the imperfect correlation means B's extra risk arrives at a discount — it does not add 22/20 times as much standard deviation as it contributes in return. Free lunch (ii) is the diversification effect. Reason (i) is just a better asset, and it would work even at ρ = 1.
What part (b) adds. The three correlations are the same two assets combined the same way, and they produce standard deviations spanning 10.76% to 18.08% — a range wider than the gap between A and B themselves. The correlation is more important than the individual volatilities. And note the asymptote: even at ρ = −0.5 the portfolio is not risk-free, because complete elimination needs ρ = −1 exactly. There is always a floor, and that floor is the systematic risk Lecture 7 ended on.
C · Equity valuation
| ($ millions) | 2017 | 2018 | 2019 | 2020 | 2021 |
|---|---|---|---|---|---|
| Net income | 1.0 | 2.0 | 3.2 | 3.7 | 4.0 |
| Investment | 1.0 | 1.0 | 1.2 | 1.4 | 1.4 |
| Free cash flow | 0 | 1.0 | 2.0 | 2.3 | 2.6 |
Step 1 — the shape of the cash flows. Five years of explicit forecasts, then a level perpetuity from 2022 onward. The only thing to be careful about is where the perpetuity sits in time.
Step 2 — discount the five explicit years. Note that the first forecast is for 2017, one year from the 2016 valuation date, so no shifting is needed:
Year FCF ÷ factor PV 2017 0.0 1.09¹ 0.0000 2018 1.0 1.09² = 1.1881 0.8417 2019 2.0 1.09³ = 1.295029 1.5444 2020 2.3 1.09⁴ = 1.411582 1.6294 2021 2.6 1.09⁵ = 1.538624 1.6898 ─────── PV of explicit years 5.7053Step 3 — the terminal value. From 2022 the cash flow is a level 2.6 forever, which is a perpetuity — Lecture 2's PV = C/r — valued as of the end of 2021. Then discount it back five years like any other 2021 cash flow:
Value at end of 2021 = 2.6 / 0.09 = 28.8889 Discount factor 1/1.09⁵ = 0.649931 PV of terminal value = 28.8889 × 0.649931 = 18.7758Step 4 — add the two pieces.
PV = 5.7053 + 18.7758 = 24.4810 ≈ 24.48Answer (a): $24.48 million.
Step 5 — the price per share. Divide the total equity value by the share count:
Price per share = 24.4810 / 12 = 2.0401 → $2.04Answer (b): $2.04 per share.
Step 6 — the P/E ratio. Market value divided by net income — and note which net income the question supplies: the 2016 figure of $1 million, which is the year before the forecast begins:
P/E = 24.4810 / 1.0 = 24.48 Alternatively per share: 2.0401 / (1.0/12) = 2.0401 / 0.0833 = 24.48 ✓Answer (c): P/E ≈ 24.5.
The thing worth noticing: the terminal value is 77% of the answer.
Terminal value 18.7758 = 76.7% of the total Explicit years 5.7053 = 23.3%Three quarters of Phoenix's value sits in a forecast that says, in effect, "and then nothing interesting ever happens again". That is not a flaw in the arithmetic — it is how multi-stage DCF models behave, and it is the honest caveat to attach to any of them. The five years the analyst actually thought about contribute a quarter of the answer; the assumption about the infinite future contributes the rest. Change the long-run growth rate from 0% to 1% and the terminal value alone rises by about 12.5%.
Two checks on the terminal value. First, the perpetuity starts in 2022 and is valued at the end of 2021 — the same "one period before the first payment" rule as Homework 1's delayed annuity, so dividing by 1.09 five times, not six, is the correct discounting. Second, the free cash flow in the perpetuity is the level from 2021 (2.6), not a grown figure, because the question says explicitly that there is no further growth. A model that grew the perpetuity at, say, 2% would need C/(r − g) = 2.6/0.07 = 37.14 instead, more than doubling the answer — which is exactly the sensitivity that makes terminal-value assumptions contentious.
Why the P/E looks high. A P/E of 24.5 is roughly two and a half times the 9.09 that a no-growth firm would earn at this discount rate (see Q5 for that calculation). The difference is the market's expectation of growth — the same distinction Q5 makes explicitly with NPVGO.
Step 1 — value a no-growth firm. With no investment, every dollar of earnings is paid out, so earnings are the dividend stream — a perpetuity. At 11%:
Price = 630,000 / 0.11 = 5,727,272.73Step 2 — the P/E ratio.
P/E = 5,727,272.73 / 630,000 = 9.0909 → 9.09Answer (a): 9.09 for both firms.
Both, because both are the same firm financially — same earnings, same payout, same required return — and because of an identity worth isolating:
P/E = (E/r) / E = 1/r 1 / 0.11 = 9.0909A no-growth firm's P/E is simply 1/r. The earnings cancel, so the ratio tells you nothing about the firm's size or its profits and everything about the discount rate the market applies to it. That is the single most useful thing to take from this question, and it is the baseline against which every other P/E should be read.
Step 3 — the new project. Pacific Energy adds a stream of $100,000 a year in perpetuity. Value the project on its own, then add it:
NPV of the new project = 100,000 / 0.11 = 909,090.91 Price (no growth) 5,727,272.73 + NPV of growth opportunities 909,090.91 ──────────── Price (with the project) 6,636,363.64Step 4 — the new P/E ratio. The denominator is unchanged — still the existing earnings of 630,000 — because the question asks for the P/E of the company, and P/E is by convention current price over current earnings:
New P/E = 6,636,363.64 / 630,000 = 10.5339 → 10.53Answer (b): 10.53.
The decomposition, which is the whole point. Write the result in the two pieces it is actually made of:
P/E = 1/r + NPVGO / E = 9.09 + 909,090.91 / 630,000 = 9.09 + 1.4430 = 10.53 ✓This is the standard decomposition: a P/E ratio has a "no-growth" floor of 1/r and a growth premium equal to the NPV of future opportunities per dollar of current earnings. Pacific Energy's extra 1.44 points of P/E are entirely the new project — the existing business has not changed at all. U.S. Bluechips, with no new project, stays at 9.09.
Why the numerator moved and the denominator did not — the trap here. A tempting alternative is to add the new earnings to the denominator: 6,636,363.64 / 730,000 = 9.09. That is wrong, and it is wrong in an instructive way, because it produces exactly the no-growth ratio again and so looks like a satisfying confirmation. What it actually does is assume the market pays nothing for the opportunity to earn the extra 100,000 — it values the firm as though the extra earnings arrived without any value being created. If the project required no investment and its earnings were certain, the market would indeed capitalise it, and the correct numerator is the one that includes its NPV.
Which raises the question the question leaves implicit: why is the new project worth exactly its perpetuity value, with no cost deducted? Because the sheet specifies no investment outlay. In a fuller problem the project would require capital, the NPV would be the perpetuity minus that outlay, and the growth premium would be smaller — possibly negative, if the firm invested in projects earning less than the 11% cost of capital. At that point the firm would be destroying value by growing, which is the situation the phrase "growth is not always good" refers to.
And the link to Q4. Phoenix Corp. had a P/E of 24.5 at a 9% cost of equity, and the equivalent no-growth floor for that firm would be 1/0.09 = 11.11. So 13.4 points of its multiple — more than half — is growth premium. Whether that is justified is the judgement call Lecture 5's comparable-method section is about, and Q5 supplies the frame for making it.
Formula Sheet for This Set
only what these six questions needEverything below appears in one of the six questions. The fuller formula sheets live on the lecture pages.
The Traps This Set Contains
one per questionEvery one of the six questions has a specific, predictable error built into it. They are collected here because the exam sets the same traps.
The 5.2% is an annual bond-equivalent quote; the period rate is 2.6%. Using 5.2% as the period rate gives $699.79 instead of $1,023.16 — and, more usefully, contradicts the direction check: the coupon (5.5%) exceeds the quoted yield (5.2%), so the answer must exceed par. Running that check first turns a silent arithmetic error into an obvious one.
Part (a)'s answer is 3.00%, the yield — which is not a coincidence and not a bug in the question. If the yield does not move, the holding-period return equals the yield, by construction. Part (b) then shows what a yield change adds: a 100 bp fall in rates produces 7.49%, of which 2.98 points is capital gain and 4.51 points is the coupon yield. The trap is forgetting the coupon's contribution when the yield falls, or forgetting that the price change is measured from 1,108.34 and not from par.
Ms. Sauros earned more (19.98% vs 16.02%) and took more risk (13.96% vs 10.29%). The question is therefore not answerable from the return column alone — which is exactly why part (b) asked for a Sharpe ratio first. The market wins on a risk-adjusted basis, 1.26 to 1.22. The half-answer to avoid is "she did better because her return was higher"; a full answer reports both measures and says which decides. A second, smaller trap: dividing the variance by T − 1 rather than T. It changes both Sharpe ratios, but it does not change the ranking — say so rather than agonising over it.
The terminal value is calculated as of the end of 2021 and must then be discounted five years, not six — the same "one period before the first payment" rule as Homework 1's delayed annuity. And note the magnitude: the terminal value supplies 76.7% of the $24.48m. A model whose answer is three-quarters assumption deserves that fact being stated.
The new project raises the price, not the earnings used in the ratio. Adding 100,000 to the denominator gives 6,636,364 / 730,000 = 9.09 — the no-growth ratio again, which reads like a confirmation and is actually a mistake, because it implies the market pays nothing for the opportunity. The correct structure is P/E = 1/r + NPVGO/E = 9.09 + 1.44 = 10.53.
The weighted average of the two standard deviations is 20.8%; the actual portfolio standard deviation is 18.08%. The 2.7-point gap is the entire content of the question. And be careful with part (c): the portfolio beats share A for two separate reasons — B's higher expected return, and the imperfect correlation — only the second of which is diversification. Then notice what part (b) demonstrates: varying ρ alone moves the risk from 18.08% down to 10.76%, more than the difference between the two shares themselves.
The homework sheet reads "Calculate her sharp ratio". The measure is the Sharpe ratio, named for William F. Sharpe — who shared the 1990 Nobel Memorial Prize for precisely this body of work on risk-adjusted performance. The sheet's spelling is a typo, not an alternative name, and an exam answer that reproduces it will read as a transcription error. Worth knowing because the wrong spelling is common enough that it sometimes appears in print.
Three of the six questions end in the same place by different routes. Q2 finds a manager who earned more and performed worse. Q6 finds a portfolio that beats its own best component. Q3 finds a bond whose yield did not change and whose return did not either — but only because nothing moved.
The common thread is that a return number, on its own, is not evidence of anything. It is evidence only once you know what risk produced it and what the alternatives were paying. That is what the Sharpe ratio does for a manager (Q2), what the portfolio formula does for a mix of assets (Q6), and what the risk-free rate does for everything (all six).
Which is also why the set stops where it does. Every risk measurement here is a standard deviation, and a standard deviation charges an asset for all of its variability — including the part a diversified holder never bears. The next lecture replaces it with beta, which charges only for the systematic half. If Q2 had asked whether Ms. Sauros's fund was a good holding rather than whether it was good performance, the Sharpe ratio would not be the right tool at all.
Self-Check
6 questions · new numbers, same skillsThese are not the set questions with the answers filled in. The numbers are different and the setups are changed slightly, so the only way through them is the technique. If you can do these, the homework is done.
A · Drill
(a) The no-growth P/E. Every dollar of earnings is paid out, so earnings are a perpetuity:
Price = 800,000 / 0.10 = 8,000,000 P/E = 8,000,000 / 800,000 = 10.00 Or directly: P/E = 1/r = 1/0.10 = 10.00Answer (a): 10.00 — and note the shortcut, because it is the whole idea. For a no-growth firm the earnings cancel out of the ratio and what is left is 1/r. The P/E says nothing about the firm's size; it reveals the discount rate the market applies to it.
(b) With the new project. Value the project as a level perpetuity and add it to the existing business:
NPV of new project = 150,000 / 0.10 = 1,500,000 Price (no growth) 8,000,000 + NPV of growth opportunities 1,500,000 ─────────── Price (with the project) 9,500,000 New P/E = 9,500,000 / 800,000 = 11.875 → 11.88Answer (b): 11.88. The denominator is still the existing earnings of 800,000 — the question asks for the firm's P/E, and the convention is current price over current earnings.
(c) The decomposition.
P/E = 1/r + NPVGO / E = 10.00 + 1,500,000 / 800,000 = 10.00 + 1.875 = 11.875 ✓Answer (c): a no-growth floor of 10.00 plus a growth premium of 1.875 points. All of the increase is the new project; nothing about the existing business changed. This is exactly the structure of Q5 in the homework set, where Pacific Energy's floor was 9.09 and its premium 1.44.
(d) The project with an investment outlay. Now the project is not free, so its value to shareholders is the perpetuity minus the cash the firm must put in:
PV of the new cash flows 150,000 / 0.10 = 1,500,000 − Upfront investment −2,000,000 ─────────── NPV of the project −500,000 Price = 8,000,000 − 500,000 = 7,500,000 P/E = 7,500,000 / 800,000 = 9.375 → 9.38Answer (d): 9.375 — below the no-growth ratio of 10.00.
Why (d) is the part of the question that matters. The project still generates $150,000 a year, and the firm's reported earnings will rise by that amount. It nevertheless destroys value, and the P/E ratio falls below the no-growth floor — which is the signal that the market has recognised it. The reason is the return on the investment: $150,000 a year forever on $2,000,000 of capital is 150,000 / 2,000,000 = 7.5%, against a 10% cost of capital. The firm is investing at less than its required return, so growth costs shareholders money.
Return on the new capital = 150,000 / 2,000,000 = 7.5% Required return (cost of equity) = 10.0% Shortfall = −2.5% → value destroyedThe general rule, which is the lesson of the whole question. Growth is worth something only if the firm earns more than its cost of capital on the new investment. When it does, NPVGO is positive and the P/E sits above 1/r; when it does not, NPVGO is negative and the P/E sits below it. The ratio is therefore not a measure of how fast a firm is growing — it is a measure of how profitably. A high P/E and a low P/E can each be justified, and the question to ask is never "is that multiple high?" but "what return on capital is it pricing in?"
And the trap. The tempting error in (d) is to notice the higher future earnings and conclude the firm is better off. Reported earnings do rise — by 150,000 a year — but the shareholders paid 2,000,000 for them and would have been better off with the cash. This is why Lecture 5 values free cash flow rather than earnings: FCF nets the investment off, so value-destroying growth shows up as a lower number instead of a higher one. Reading (d) against (c) — 9.38 against 11.88, for the same project cash flows — is the cleanest demonstration in the course that earnings and value are not the same thing.
Roadmap · Weeks 8–16
Where the course goes next
and what each week will connect back toTutorial
- Problem session; no new material
Raising Capital
- Venture capital, IPO, seasoned equity offerings
- Underpricing, the winner's curse, and the long-run underperformance puzzle
- Back to the 康希诺 A/H price gap from Lecture 1's opening case
Payout Policy
- Dividends vs share repurchases; the irrelevance proposition
- Dividend clienteles, signalling, and the tax argument
- Reads directly off the retained-earnings / dividend split in the Lecture 1 income statement
Tutorial
- Problem session; groups prepare presentations
Review & Student Presentations
- Group presentations (20% of the grade)
- Course review ahead of the closed-book final
Delivered
these are written up in fullOverview & Financial Statements
Time Value of Money
Net Present Value
Capital Budgeting
Stock Valuation
Bond Valuation
Risk and Return
Cost of Capital
Efficient Capital Markets
Capital Structure
Weeks 1–4 in one problem set
Weeks 5–7 in one problem set
Group project · 20%
presentations begin 1 DecemberThe syllabus asks each group to pick a course-related topic that was not taught in class, and structure the presentation around three things: what the concept is, how it is used in the real world, and how it connects back to this course. Ten to twenty minutes. The instructor's own suggestions are LPR, fintech, independent directors, ESG and generative AI.
The third requirement — the link back to the course — is the one groups routinely under-prepare, and it is also the one the marking most rewards. Each of these topics lands on a specific piece of machinery from Lectures 1–10:
The loan prime rate is the risk-free benchmark that sits inside every discount rate. A presentation here can trace one number from the PBoC's fixing through to the WACC used to appraise a project — the LPR feeds the cost of debt, which feeds the discount rate, which decides whether the NPV in Lecture 3 is positive.
Where technology lowers the cost of the information that Lecture 1 argued investors are short of. Mobile payment, credit scoring and robo-advice all attack the same friction: the analyst's role as an information intermediary.
Directly the corporate-governance block of Lecture 1. China's 独立董事 regime is a regulatory answer to the agency problem between controlling shareholders and minority shareholders — the 刘强东 case, with the mechanism spelled out.
ESG is the syllabus's own amendment to the goal of the firm — maximise value, but observed through environmental, social and governance screens rather than profit alone. Generative AI, meanwhile, arrives on the analyst's desk: it is a tool for the valuation work of Lectures 2–4, and worth a candid assessment of where it helps and where it feeds you confident nonsense.
Open with a number, not a definition. "LPR" is a definition; "a 35-basis-point cut in the 5-year LPR lowers the monthly payment on a ¥1m 30-year mortgage by ¥210, which is a transfer of roughly ¥75,000 of NPV from banks to households" is an argument — and it uses Lecture 2's annuity formula to make it. Then say which lecture each piece comes from. That is the link requirement, discharged.