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.

18 weeks · Tuesday 18:30–20:20 · 仙I 317 Homework 10% · Project 20% · Final 70% Textbook: Ross, Westerfield, Jaffe & Jordan, 11th ed.

How this workbench is organised

the same five blocks in every lecture

1 · Concept Map

How the lecture's ideas build on one another, drawn as a chain rather than a list. Cross-lecture links are marked on the map.

2 · Core Content

Explanatory prose with the slide's own worked examples taken apart step by step, plus the background the slides skip.

3 · Formula Sheet

Every formula from the lecture in one place, with the meaning of each symbol. Printed on its own for revision.

4 · Key Concepts

The vocabulary, defined in English with a Chinese anchor so it maps onto the lecture you actually sat through.

5 · Self-Check

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 colours

🔗 Link

Connects 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.

📖 Beyond the slides

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.

⚠️ Trap

Places where a method is easy to apply wrongly, or where something on a slide is slightly off. Three such errors were found while rebuilding these lectures — a wrong listing date, a rounding slip in an annuity-due calculation, and an intermediate-rounding artefact in a profitability-index table — and each is flagged where it occurs rather than quietly fixed.

💬 Think

The instructor's own discussion questions and quick questions, kept verbatim, with a worked answer below each.

Lectures

weeks 1–4 delivered so far

Progress dashboard

stored locally in this browser

What the final exam asks for

per the syllabus

Closed book, calculator required

Concept 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.

Group project · 20% · from 1 December

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.

Lecture 1 · Week 1

Overview & Financial Statements

26 Aug 2025 Ross Ch. 1–3 Source: Lecture 1 Overview.pdf (27 slides)

This lecture is the setup for everything that follows. It answers three questions: who actually controls a corporation and whose interests they serve; how to read the three statements that describe a firm; and what financial management, as opposed to accounting, is for. The two opening cases — a CEO paid ¥1 a year and a vaccine stock trading at half price in Hong Kong — are not decoration. They are the two problems the rest of this course exists to solve.

Concept Map

how the lecture builds

Separation of ownership and control Shareholders own the firm; managers run it. This gap is the founding problem of corporate finance.

creates two frictions, each with its own remedy

Agency problem Managers may pursue their own objectives. Remedy: corporate governance — boards, independent directors, pay packages.

Information asymmetry Outsiders cannot see inside the firm. Remedy: disclosure and financial analysts who intermediate.

both remedies rest on the same raw material

The financial statements Balance sheet · income statement · cash flow statement — the firm's public description of itself.

which can be compressed into

Financial ratios Liquidity, leverage, turnover, profitability, market value — five lenses on the same three statements.

and the whole apparatus serves one purpose

Financial management Valuation · investment decision · financing decision — pursued to maximise firm value, not profit.

which splits into three recurring questions

Capital budgeting What long-term investments? → Lectures 3–4

Capital structure How to raise the money? → Week 11

Working capital management How to run short-term assets? → Week 18

🔗 Link · why this lecture is not just revision

Nearly 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 7–10

Case 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.

📖 Beyond the slides · the agency problem

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.

⚠️ Trap · the slide's date is wrong

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.

🔗 Link · why this case is here

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 11–18

A 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.

⚠️ Trap · accrual accounting (权责发生制)

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:

Assets = Liabilities + Equity

Net working capital = Current assets − Current liabilities

Capital spending = Ending fixed assets − Beginning fixed assets + Depreciation

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.

Worked Net working capital and capital spending from the table above

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

Net working capital rose by $23m; capital spending was $173m.

🔗 Link · keep both numbers

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

📖 Beyond the slides · why "EBIT" appears twice

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:

Revenue − Expense = Income

EBIT = Sales − Costs − SG&A − Depreciation

Net income = EBIT − Interest − Tax = Retained earnings + Dividends

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

CFO + CFI + CFF = Change in cash balance

CFI = Purchase of fixed assets + Sale of fixed assets

CFF = Net borrowing + Net stock 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.

Balance sheet ↔ Income statement Net income = Retained earnings + Dividends. Of the $86m earned, $43m is paid out and $43m is accumulated — which is why retained earnings on the balance sheet rise from 347 to 390, exactly 43.

and

Balance sheet ↔ Cash flow statement Change in cash balance = CFO + CFI + CFF. Cash on the balance sheet moves from 107 to 140, a change of +33, and the cash flow statement must explain precisely that 33.

and

Income statement ↔ Cash flow statement CFO = Net income + adjustments for non-cash items + adjustments for changes in current assets and liabilities. The bridge from accrual profit to actual cash.

📖 Beyond the slides · the accrual-to-cash bridge, spelled out

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

📖 Beyond the slides · the check that makes this trustworthy

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.

🔗 Link · the two seats of depreciation

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 19

Ratios 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	—

⚠️ Trap · a ratio is a question, not an answer

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?

ROE = Net incomeSales × SalesAssets × AssetsEquity = margin × turnover × leverage

Worked Decomposing the example firm's 10.68% ROE

Net income 86 · Sales 2,262 · Total assets 1,879 · Total equity 805

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

ROE = 10.68% — which recovers the direct calculation (86 / 805 = 10.68%) exactly. The identity holds because Sales and Assets cancel.

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.

📖 Beyond the slides · the cash conversion cycle

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:

CCC = DIO + DSO − DPO

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 20–27

With 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:

Valuation

What is an asset, a project, or a company worth? Everything downstream is an application of this.

Investment decision

How to spend money. Which long-term investments should the firm choose? → Lectures 3–4

Financing decision

How to raise money. How should the firm fund the investments it selected? → Week 11

Impact of policy

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.

📖 Beyond the slides · why not maximise profit?

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.

💬 Think · distinguishing investment from financing

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.

Capital budgeting What long-term investments should the firm choose? Acts on the fixed assets — tangible and intangible. → Lectures 3 and 4

financed by

Capital structure How should the firm raise the money? Chooses the mix of shareholders' equity, long-term debt and current liabilities. → Week 11

while day to day

Short-term asset management How should short-term assets be managed and financed? Acts on net working capital — the gap between current assets and current liabilities. → Week 18

🔗 Link · one number, three lectures

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-friendly

Balance sheet identity

Assets = Liabilities + Equity

The accounting identity. It holds by construction, at every instant.

Net working capital 净营运资本

NWC = Current assets − Current liabilities

2020: 761 − 486 = 275. Becomes ΔWC in Lecture 4, with the sign flipped.

Capital spending 资本性支出

Capex = Ending FA − Beginning FA + Depreciation

2020: 1,118 − 1,035 + 90 = 173. Depreciation is added back because it was already subtracted from the closing balance.

Income statement identity

Revenue − Expense = Income

Accrual basis: revenue is recognised when earned, not when collected.

Operating income 息税前利润

EBIT = Sales − Costs − SG&A − Depreciation

2,262 − 1,655 − 327 − 90 = 190. This is the EBIT that Lecture 4's free cash flow starts from — operating income, not the figure including other income.

Net income 净利润

NI = EBIT − Interest − Tax = Retained earnings + Dividends

170 − 84 = 86 = 43 + 43. Net income is not cash.

Cash flow identity

CFO + CFI + CFF = Δ Cash balance

199 − 173 + 7 = 33, which is exactly 140 − 107.

Investing and financing cash flow

CFI = Purchase of fixed assets + Sale of fixed assets

CFF = Net borrowing + Net stock issuance + Dividends

Dividends enter CFF with a negative sign — they are a use of cash.

Accrual-to-cash bridge

CFO = NI + non-cash items + adjustments to current assets and liabilities

Add back depreciation and deferred tax; subtract increases in current assets; add increases in current liabilities.

DuPont identity 杜邦分解

ROE = NISales × SalesAssets × AssetsEquity

Margin × turnover × leverage. 3.80% × 1.2038 × 2.3342 = 10.68% = 86 / 805.

Liquidity ratios

Current = CACL · Quick = CA − InventoryCL · Cash = CashCL

1.57 · 1.01 · 0.29 for the example firm. The quick ratio strips out the least liquid current asset.

Leverage ratios

Total debt ratio = Assets − EquityAssets · Equity multiplier = AssetsEquity

0.57 and 2.33. Note Equity multiplier = 1 / (1 − total debt ratio).

Coverage ratios

Times interest earned = EBITInterest · Cash coverage = EBIT + DepreciationInterest

3.88 and 5.71. Cash coverage is higher because depreciation is a non-cash charge already deducted from EBIT.

Turnover and period ratios

Inventory turnover = COGSInventory · Days = 365Turnover

Inventory 6.15 → 59.3 days; receivables 7.69 → 47.4 days.

Profitability ratios

Profit margin = NISales · ROA = NIAssets · ROE = NIEquity

3.80% · 4.58% · 10.68%. ROE exceeds ROA only because of leverage.

Cash conversion cycle 现金转换周期

CCC = DIO + DSO − DPO

59.3 + 47.4 − 107.2 = −0.4 days. Suppliers finance the whole operating cycle.

Key Concepts

the vocabulary

Agency problem 代理问题

The conflict of interest between the principals who own the firm (shareholders) and the agents who run it (managers). Contained by incentive contracts, the board, and the market for corporate control. The 刘强东 ¥1-salary case is a textbook incentive contract.

Corporate governance 公司治理

The set of mechanisms — board of directors, independent directors, audit, disclosure rules, executive pay — by which suppliers of capital assure themselves of a return on their investment.

Independent director 独立董事

A board member with no operating role in the firm and no material relationship with it, whose function is to check management rather than to represent it.

Accrual accounting 权责发生制

Income is reported when it is earned or accrued, even though no cash flow has necessarily occurred. The reason net income and cash flow diverge.

Balance sheet 资产负债表

The financial position of a company at one point in time. A photograph. Assets listed in order of how long a going concern would need to convert them to cash.

Income statement 损益表

Measures revenue and expense over a specific period. A film, not a photograph.

Cash flow statement 现金流量表

Cash generated and used, split into operating, investing and financing activities. Harder to manipulate than the other two statements, which is why analysts reach for it first.

Non-cash item 非现金项目

An expense charged against income that involves no cash outflow. Depreciation is the clearest example: no firm ever writes a cheque for it. Deferred tax is another.

Net working capital 净营运资本

Current assets minus current liabilities — the short-term capital the firm has committed to operations. Growing it consumes cash; shrinking it releases cash.

Liquidity 流动性

The speed and ease with which an asset converts to cash without significant loss of value, and the ability of the firm to meet short-term obligations.

Book value vs market value 账面价值 / 市场价值

Book value is historical cost net of depreciation — what accounting records. Market value is what a buyer will pay today, and reflects future prospects. Finance values the latter.

Capital budgeting 资本预算

The investment decision: which long-term real assets should the firm buy? Appraised by whether NPV is positive. Lectures 3 and 4.

Capital structure 资本结构

The financing decision: the mix of debt and equity used to fund the firm. Week 11.

ESG 环境、社会与治理

Environmental, social and governance criteria applied alongside value maximisation — the modern qualification the lecture places on the goal of the firm.

Self-Check

15 questions · graded

0 / 0 correct

Group 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

Definitions and the reasoning behind them — the 2×10 section of the paper.

Q1Which of the following best states the goal of financial management as presented in this lecture?

Correct. Value, not profit. Profit is an accounting construct that ignores timing, risk, and the capital employed — all three of which are what owners actually care about.

Not quite. The lecture is explicit that the goal is not to maximise profits, and certainly not revenue. A firm can raise profit by issuing shares and buying bonds, leaving shareholders no better off. The goal is firm value, hence shareholder wealth.

Q2Why is net income not the same as cash flow?

Correct. Two distinct reasons. Accrual accounting books revenue when earned rather than when collected, and depreciation is deducted from income with no cash leaving the firm. Both push reported profit away from cash.

Not quite. The two sources of divergence are the accrual convention and non-cash charges. Depreciation is the clearest: it reduces taxable income without any cash outflow. No firm ever writes a cheque for "depreciation."

Q3A firm reports net income of $86m, of which $43m is paid out as dividends. Its accumulated retained earnings on the balance sheet were $347m at the start of the year. What are they at the end?

Correct. 347 + (86 − 43) = 390. The income statement feeds the balance sheet through retained earnings — this is the first of the three links between the statements.

Not quite. Only the retained portion of net income stays in the firm: 86 − 43 = 43. So 347 + 43 = 390. The other 43 left as dividends and never reaches the balance sheet.

Q4Two firms in different industries report identical ROE of 20%. Firm X has a profit margin of 2% and asset turnover of 5.0. Firm Y has a margin of 20% and turnover of 0.5. Which statement is correct?

Correct. 2% × 5.0 = 10%, and 20% × 0.5 = 10% — the same product from opposite business models. If leverage is equal, both ROEs match. This is why DuPont is more informative than the ROE number alone.

Not quite. Check the DuPont product. X: 2% × 5.0 = 10% of asset return. Y: 20% × 0.5 = 10% too. Same return on assets, reached by completely different routes — a discounter against a luxury brand. A higher margin is not "better" in isolation.

Q5Concept explanation. Explain, in three or four sentences, why the cash flow statement is said to be harder to manipulate than the income statement.

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.

Q6Concept explanation. The lecture states that the goal of financial management is to maximise firm value rather than profit. Give three distinct reasons why profit maximisation is an inadequate objective.

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

Computational and definitional, in the style of the 2×35 single-choice section. All figures refer to the example firm in section 3.

Q7What is the firm's net working capital at the end of 2020?

Correct. Current assets 761 − current liabilities 486 = 275. 761 − 486 = 275

Not quite. Net working capital is current assets minus current liabilities. 761 − 486 = 275 $252m is the 2019 figure (707 − 455); don't confuse the two years.

Q8Using the balance sheet in section 3, what was the firm's capital spending in 2020?

Correct. Ending fixed assets less beginning fixed assets, plus depreciation: 1,118 − 1,035 + 90 = 173. Depreciation is added back because it had already been subtracted from the closing balance.

Not quite. 1,118 − 1,035 + 90 = 173 $83m is the raw change in fixed assets, which ignores depreciation. $90m is depreciation itself. Capital spending is the raw change plus depreciation.

Q9On which statement would a purchase of new factory equipment appear as a cash outflow?

Correct. CFI records purchases and sales of fixed assets. In our reconstruction, capital spending of 173 was the entire investing section.

Not quite. Buying a real asset is an investing activity. Financing (CFF) covers how the money was raised — borrowing, issuing shares, paying dividends; it never covers what the money was spent on.

Q10The firm's EBIT is 190 and interest expense is 49. What is the cash coverage ratio?

Correct. Cash coverage adds back the non-cash charge: (190 + 90) / 49 = 5.71. It is the more forgiving of the two coverage ratios, because depreciation is not a cash cost.

Not quite. (190 + 90) / 49 = 5.71 3.88 is the times-interest-earned ratio, EBIT/interest, which excludes depreciation. Cash coverage adds depreciation back before dividing, giving a higher and more realistic number.

Q11Which of the following is not a financing decision?

Correct. Acquiring a factory buys a real asset — that is an investment decision (capital budgeting), appraised by its NPV. The other three all raise or return capital, which is financing.

Not quite. The test is which side of the balance sheet the decision touches. Bonds, buybacks and IPOs all act on the right side (how the firm is funded). Buying a factory acts on the left side — a real asset — and is therefore an investment decision.

Q12Both the A-share and H-share of CanSino traded on the same day, on the same underlying business, at prices differing by roughly a factor of two. What does this illustrate?

Correct. The shares are not freely convertible between the two markets, so the price gap cannot be arbitraged away. Identical claims, different prices, no contradiction.

Not quite. Arbitrage requires the ability to buy in the cheap market and sell in the expensive one. Convertibility restrictions block exactly that, so the gap persists. Both prices are genuine market outcomes.

Q13A firm's inventory turnover is 3.0 and its receivables turnover is 6.0. Its payables turnover is 4.0. What is its cash conversion cycle?

Correct. Convert each turnover into days, then add the two operating legs and subtract the payables leg: CCC = DIO + DSO − DPO. DIO = 365 / 3.0 = 121.7 days DSO = 365 / 6.0 = 60.8 days DPO = 365 / 4.0 = 91.3 days CCC = 121.7 + 60.8 − 91.3 = 91.2 days

Not quite. DIO = 365 / 3.0 = 121.7 days DSO = 365 / 6.0 = 60.8 days DPO = 365 / 4.0 = 91.3 days CCC = 121.7 + 60.8 − 91.3 = 91.2 days The two commonest slips are forgetting to convert turnover into days before combining, and adding DPO instead of subtracting it. Suppliers' credit shortens the cycle, because it is the one leg where someone else is financing you. About 91 days is the answer.

C · Applied calculation

Work these on paper before revealing the answer — this is the 5×2 subjective section.

Q14Using only the 2020 and 2019 balance sheets and the income statement from section 3, reconstruct the firm's cash flow from operations. State the figure and show each adjustment.

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 199

Why 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.

Q15A firm has a profit margin of 4%, a total asset turnover of 1.5, and an equity multiplier of 2.0. Compute its ROE, then explain what would happen to that ROE if the firm doubled its leverage — and why the resulting figure should be treated with caution.

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.

Lecture 2 · Week 2

Time Value of Money

2 Sep 2025 Ross Ch. 4 Source: Lecture 2 Time Value of Money.pdf (35 slides)

One idea, applied relentlessly: a dollar today is worth more than a dollar tomorrow. Everything else in this lecture — compounding, discounting, perpetuities, annuities, mortgage schedules — is that sentence worked out. The lecture opens with a friend borrowing ¥1,000 and closes with the same question answered, which is a fair signal of what the hour is for: not a formula sheet, but a way of pricing time.

Concept Map

how the lecture builds

¥1 today ≠ ¥1 tomorrow Money has a price, and that price is the interest rate. This is the whole lecture.

compounding forward

Future value FV = PV(1 + r)^T — what today's money becomes

Present value PV = FV / (1 + r)^T — what tomorrow's money is worth today

once you can move a single cash flow in time, you can move any pattern of them

Perpetuity 永续年金 PV = C / r — forever

Annuity 年金 PV = C[1 − 1/(1+r)^T] / r — fixed T

Growing stream Same logic, cash flows grow at g

Delayed & due Shift the whole pattern in time by a factor of (1+r)

all of which is what a loan actually is

Loan amortisation Equal principal vs equal payment — the two schedules behind every Chinese mortgage

and when compounding is not annual

Effective annual rate The rate that makes different compounding conventions comparable — EAR ≥ quoted r, always

🔗 Link · this is the engine room of the course

Nothing 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 2–11

💬 Think · the question the lecture opens and closes with

If 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:

10,000 × (1 + 0.05) = $10,500

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:

10,000 ÷ (1 + 0.05) = $9,523.81

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.

Simple interest 单利

Interest is not reinvested. Invest $1 and you have 1 + 2r after two years — the same $r each year on the original $1.

Compound interest 复利

Interest is reinvested. Invest $1, have 1 + r after one year, reinvest all of it, and end with (1 + r)² after two.

The lecture's generalisation, and it is the formula the entire course runs on:

FV = PV × (1 + r)^T or PV = FV(1 + r)^T

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.

Worked How much must be set aside today to have $20,000 in five years at 15%?

Future value $20,000 · rate 15% · T = 5 years · find PV at t = 0

$20,000

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

Set aside $9,943.53 today.

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.

Worked An investment paying $200, then $400, $600, $800 — at 12%

Cash flows of $200 / $400 / $600 / $800 at the end of years 1–4 · discount rate 12% · find the present value of the whole stream

200

400

600

800

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

Present value of the stream = $1,432.93.

🔗 Link · this is net present value

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.

⚠️ Trap · spreadsheet conventions

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, never 5.
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 13–18

Everything 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.

FV = C₀ × (1 + rm)^mT m = compounding periods per year

r — always the annual (quoted) rate · m — times per year compounding occurs · T — years

⚠️ Trap · r is an annual rate, always

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.

Worked $50 invested for 3 years at 12%, compounded semi-annually

Principal $50 · quoted annual rate 12% · T = 3 years · compounding m = 2

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 investment grows to $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.

Worked Finding the EAR of the semi-annual investment above

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

EAR = 12.36%. Compounding annually at 12.36% is identical to compounding semi-annually at 12%.

The closed form is much shorter, and it is the version to memorise:

EAR = (1 + rm)^m − 1

⚠️ Trap · the key takeaway, and it has a direction

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.

💬 Think · which is better?

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

Worked ¥10,000 at a 7-day annualised rate of 2%

Principal ¥10,000 · quoted 7-day annualised rate 2% · daily compounding (m = 365)

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

One year: ¥10,200. Seven days: ¥10,003.84, an interest gain of ¥3.84.

📖 Beyond the slides · the 7-day annualised rate is not an EAR

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 20–28

Discounting 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

PV = C1 + r + C(1 + r)² + C(1 + r)³ + ⋯

converges, because the terms shrink geometrically. The result is startlingly simple:

PV = Cr perpetuity

Worked A bond paying $15 every year, forever, at 10%

Constant cash flow $15 per year · discount rate 10% · no maturity

£15

⋯

Apply PV = C / r PV = 15 / 0.10
Evaluate 15 / 0.10 = 150

Present value = $150.

⚠️ Trap · the perpetuity formula is impatient

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.

PV = C [ 1 − 1(1 + r)^T r ] annuity

The bracketed expression is the annuity factor — the present value of $1 per period for T periods. Everything about annuities follows from it.

Worked How much car can you afford with a $400 monthly payment?

Monthly payment $400 · 36 months · annual rate 7%, monthly compounding · first payment one month from now

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

You can afford a car costing $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:

PV_{annuity due} = PV_ordinary × (1 + r)

Worked The same car loan, but paid at the start of each month

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

Annuity due = $13,030.15 — you can afford $75.57 more car.

⚠️ Trap · a small arithmetic slip in the slide

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.

💬 Think · interest-free instalments, or pay in full?

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):

PV = 333 × 1 − 1(1 + 0.00115)²⁴ 0.00115 = 7,878.25

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.

Worked A four-year $100 annuity whose first payment is two years from today, at 9%

Payments $100 at t = 2, 3, 4, 5 · discount rate 9% · find PV at t = 0

$100

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

Present value = $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.

📖 Beyond the slides · two patterns the exam may still expect

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 28–34

This 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:

Pure discount loan 一次性还本付息

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.

Interest-only loan 月付利息到期还本

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.

Amortised loan 等额本金 / 等额本息

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%:

C = PV × r 1 − 1/(1 + r)^T = 5,000 × 0.09 1 − 1/(1.09)⁵ = 1,285.46

Worked Building the first row, then the rest follows

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

Note how the split moves: interest falls from $450.00 to $106.14 while principal rises from $835.46 to $1,179.32, even though the payment never changes.

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

⚠️ Trap · the incentive the loan officer has, and it is not yours

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.

🔗 Link · the amortisation table is a bond, seen from inside

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-friendly

Future value

FV = PV × (1 + r)^T

Compound interest assumed throughout. 10,000 at 5% for 1 year → 10,500.

Present value

PV = FV(1 + r)^T

The same relation rearranged. 10,000 due in 1 year at 5% → 9,523.81.

Discount factor 现值系数

1(1 + r)^T

Always strictly less than 1. Five years at 15% → 0.497177.

Compounding m times a year

FV = C₀ × (1 + rm)^mT

r is always the annual quoted rate. $50 at 12% semi-annual for 3 years → $70.93.

Effective annual rate 有效年利率

EAR = (1 + rm)^m − 1

If m > 1 then EAR > r. 12% semi-annual → 12.36%; 12% monthly → 12.68%.

Perpetuity 永续年金

PV = Cr

$15 forever at 10% → $150. Values the stream one period before the first payment.

Growing perpetuity

PV = Cr − g

Requires r > g. This is the Gordon growth model of Week 5.

Annuity 年金

PV = C [ 1 − 1(1 + r)^T r ]

The bracket is the annuity factor. $400 × 36 months at 7% → $12,954.59.

Growing annuity

PV = C × 1 − ((1+g)/(1+r))^T r − g

Setting g = 0 recovers the ordinary annuity.

Annuity due 即付年金

PV_due = PV_ordinary × (1 + r)

Every payment has moved one period earlier, so each PV is scaled by (1 + r). → $13,030.15.

Delayed annuity

Step 1: value as an ordinary annuity at t_first − 1
Step 2: discount that single sum to t = 0

$100 × 4 years starting at t = 2, at 9% → 323.97 → /1.09 = $297.22.

Loan payment 等额本息月供

C = PV × r 1 − 1/(1 + r)^T

The annuity formula solved for C. $5,000 over 5 years at 9% → $1,285.46.

Equal principal interest 等额本金利息

Interest_t = (PV − PVT × (t − 1)) × r

Charged on the declining balance: 450, 360, 270, 180, 90 → total $1,350.

Key Concepts

the vocabulary

Time value of money 货币的时间价值

One dollar today is worth more than one dollar tomorrow, because today's dollar can be invested and earn a return. The interest rate is the price of time.

Future value 终值

The total amount due at the end of an investment — today's money compounded forward.

Present value 现值

The amount that must be set aside today to meet a promised future payment — future money discounted back.

Discount rate 折现率

The rate used to convert future cash flows into present value. Called the required return when it reflects what an investor demands, and the cost of capital when it reflects what the firm pays — the same number seen from two sides (Week 10).

Discount factor 现值系数

1/(1+r)^T. Always less than 1, so discounting always shrinks a number. Its value is what a present-value table lists.

Simple interest 单利

Interest is not reinvested. $1 grows to 1 + 2r after two years.

Compound interest 复利

Interest is reinvested and itself earns interest. $1 grows to (1+r)² after two years. Assumed throughout this course.

Quoted (annual) rate 名义利率

The headline annual rate, before any adjustment for how often compounding actually occurs. Called the APR in lending. It is the r that goes on top of the r/m fraction.

Effective annual rate 有效年利率

The rate of annually-compounded interest that would produce the same end-of-year wealth as the actual compounding convention. The only correct basis for comparing two quoted rates.

Perpetuity 永续年金

A constant stream of cash flows lasting forever. PV = C / r. Values the stream one period before the first payment.

Annuity 年金

A constant stream of cash flows with a fixed maturity. An annuity is a perpetuity with the tail removed.

Annuity due 即付年金

An annuity whose payments fall at the beginning of each period. Worth exactly (1+r) times an ordinary annuity.

Delayed annuity 递延年金

An annuity whose first payment is deferred. Solve in two steps: value it as an ordinary annuity, then discount that single sum to today.

Amortised loan 摊销贷款

A loan requiring repayment of principal over time in addition to interest. Contrasted with pure discount and interest-only loans, neither of which reduces the principal before maturity.

Equal principal 等额本金

Fixed principal repayment, declining interest. Payment falls over time; total interest lower.

Equal payment 等额本息

Fixed total payment, shifting split between interest and principal. Total interest higher, because principal is repaid more slowly.

Self-Check

20 questions · graded

0 / 0 correct

This 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

The 2×10 concept-explanation section.

Q1A friend offers you a choice: ¥1,000 today, or ¥1,050 in one year. The risk-free interest rate is 5%. Which should you take, and why?

Correct. PV of ¥1,050 one year out, at 5%, is 1,050 / 1.05 = ¥1,000. The two offers are identical in value — this is the definition of a fair interest rate.

Not quite. PV = 1,050 / 1.05 = 1,000 Discount the ¥1,050 at the 5% risk-free rate and you get exactly ¥1,000. Neither offer is better; the 5% rate is precisely the rate at which you are indifferent.

Q2At an interest rate of 12% compounded monthly, what is the effective annual rate?

Correct. More frequent compounding always pushes the EAR above the quoted rate. EAR = (1 + 0.12/12)¹² − 1 = (1.01)¹² − 1 = 0.126825

Not quite. EAR = (1 + 0.12/12)¹² − 1 = (1.01)¹² − 1 = 0.126825 Monthly compounding at a 12% quoted rate produces an EAR of 12.68% — higher than 12%, never lower. Option D, 11.39%, is what you get if you divide instead of compounding.

Q3What distinguishes a perpetuity from an annuity?

Correct. Both pay a constant C. The only difference is whether the stream ever stops — and that difference is what turns C/r into the annuity factor.

Not quite. Both pay a constant cash flow, and both are discounted at the same rate. The single distinguishing feature is maturity: a perpetuity never ends, an annuity stops after T periods.

Q4A perpetuity pays $80 per year forever. The discount rate is 6%. What is its present value?

Correct. PV = 80 / 0.06 = 1,333.33 Despite being an infinite sum, the series converges because the discount factors shrink geometrically.

Not quite. PV = C / r = 80 / 0.06 = 1,333.33 The sum of infinitely many terms is finite here, because each term is smaller than the last by a constant factor.

Q5Concept explanation. Explain why the effective annual rate is always greater than or equal to the quoted annual rate, and why the distinction matters to a borrower.

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.

Q6Concept explanation. Why does the equal-payment method (等额本息) produce a higher total interest than the equal-principal method (等额本金) on the same loan?

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

Computational, in the style of the 2×35 single-choice section.

Q7You deposit $1,000 at 5% for two years. What is the future value?

Correct. The extra $2.50 against simple interest is interest earned on the first year's interest. 1,000 × (1.05)² = 1,000 × 1.1025 = 1,102.50

Not quite. 1,000 × (1.05)² = 1,102.50 $1,100 would be simple interest (1,000 + 50 + 50). Compounding adds $2.50 in the second year — interest on the first year's interest.

Q8What is the present value of $10,000 to be received in three years, at a discount rate of 8%?

Correct. 10,000 / (1.08)³ = 10,000 / 1.259712 = 7,938.32

Not quite. 10,000 / (1.08)³ = 10,000 / 1.259712 = 7,938.32 $7,600 discounts by simple interest over three years. $9,259.26 discounts for a single year, and $12,597.12 compounds forward instead of discounting back.

Q9An annuity pays $500 at the end of each year for 10 years. The discount rate is 6%. What is its present value?

Correct. Annuity factor = [1 − 1/(1.06)¹⁰] / 0.06 = 7.360087 PV = 500 × 7.360087 = 3,680.04

Not quite. Annuity factor = [1 − 1/(1.06)¹⁰] / 0.06 = 7.360087 PV = 500 × 7.360087 = 3,680.04 $3,900.85 is the annuity-due figure (3,680.04 × 1.06). $8,333.33 is the perpetuity value at 6% (500/0.06), which ignores the ten-year cutoff. $6,590.40 compounds forward instead of discounting.

Q10Take the annuity in Q9. What is its present value if the payments fall at the beginning of each period instead?

Correct. Every payment has moved one period earlier, so multiply the ordinary-annuity PV by (1 + r) once — not by (1 + r) ten times. 3,680.04 × 1.06 = 3,900.85

Not quite. 3,680.04 × 1.06 = 3,900.85 An annuity due is worth more, not less — you receive the money sooner. And the factor is applied once, because every payment shifts by exactly one period; $3,472.68 would be discounting further.

Q11What is the present value of $200 per year for three years, where the first payment arrives three years from today, at a discount rate of 10%?

Correct. Two steps: value the annuity, then discount the lump sum. It is valued at t = 2, one period before the first payment at t = 3, and then discounted two periods to today. Step 1: 200 × [1 − 1/(1.1)³]/0.1 = 200 × 2.486852 = 497.37 (at t = 2) Step 2: 497.37 / (1.1)² = 497.37 / 1.21 = 411.05 (at t = 0)

Not quite. Step 1: 200 × [1 − 1/(1.1)³]/0.1 = 497.37 (at t = 2) Step 2: 497.37 / (1.1)² = 411.05 (at t = 0) $497.37 is the value at t = 2 — stopping there is the commonest error. $373.73 discounts for three periods instead of two; the annuity formula already places the value one period before the first payment.

Q12A firm borrows $10,000 for four years at 8%. It makes four equal annual payments, the first one year from now. What is the annual payment?

Correct. C = PV × r / [1 − 1/(1+r)^T] = 10,000 × 0.08 / [1 − 1/(1.08)⁴] = 800 / 0.264970 = 3,019.21

Not quite. C = 10,000 × 0.08 / [1 − 1/(1.08)⁴] = 800 / 0.264970 = 3,019.21 $2,500 is just principal divided by four, ignoring interest entirely. $3,300 adds one year's interest to it. The correct payment is higher than $2,500 because it must also cover interest on the declining balance: 10,000 × 4 = ... 3,019.21 × 4 = 12,076.84 total paid, of which $2,076.84 is interest.

Q13You invest $10,000 for five years at a quoted rate of 6%. Which compounding convention gives you the most money at the end?

Correct. More frequent compounding means interest starts earning interest sooner, so the effective annual rate is higher. Annual : 10,000 × (1.06)⁵ = 13,382.26 (EAR 6.000%) Semi-annual: 10,000 × (1.03)¹⁰ = 13,439.16 (EAR 6.090%) Monthly : 10,000 × (1.005)⁶⁰ = 13,488.50 (EAR 6.168%)

Not quite. Semi-annual: 10,000 × (1.03)¹⁰ = 13,439.16 Monthly : 10,000 × (1.005)⁶⁰ = 13,488.50 The quoted rate is the same but the effective rate is not. Monthly compounding gives $106.24 more than annual — because interest is reinvested twelve times a year rather than once.

Q14Two loans each quote 9% annual interest. Loan X compounds semi-annually; Loan Y compounds annually. Which statement is correct?

Correct. Loan X: EAR = (1 + 0.09/2)² − 1 = (1.045)² − 1 = 0.092025 Loan Y: EAR = 0.09 The more frequent compounding makes X the more expensive loan, despite the identical quoted rate.

Not quite. Loan X: EAR = (1 + 0.09/2)² − 1 = 9.20% Loan Y: EAR = 9.00% The loan with more frequent compounding is the more expensive one. Compare only on an EAR basis — the quoted 9% is identical and therefore tells you nothing.

Q15A five-year loan of $5,000 at 9% can be repaid on either an equal-principal or an equal-payment basis. Which is true?

Correct. Equal payment: $1,427.31 of interest against $1,350.00 — higher, because principal is repaid more slowly. But its first payment is $1,285.46 against $1,450.00.

Not quite. Equal payment costs more interest ($1,427.31 vs $1,350.00) precisely because it defers principal repayment, leaving a larger balance on which interest accrues. Larger early payments reduce total interest rather than increasing it.

C · Applied calculation

The 5×2 subjective section. Work each on paper before revealing.

Q16You can afford a $350 monthly car payment. Interest rates are 6% on 48-month loans. How much car can you afford? Then state how the answer changes if the payments fall at the beginning of each month instead.

Step 1 — monthly basis. r = 0.06/12 = 0.005; T = 48.

Step 2 — annuity factor.

[1 − 1/(1.005)⁴⁸] / 0.005 = 42.580318

Step 3 — present value.

350 × 42.580318 = 14,903.11

Step 4 — annuity due. Every payment moves one month earlier, so multiply by (1 + r) once:

14,903.11 × 1.005 = 14,977.63

Answer. $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.

Q17A phone is offered at ¥7,999 in full, or as 24 monthly instalments of ¥333 with no stated interest. Your money currently earns 1.38% annualised. Which should you choose, and what would change your mind?

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 month

Step 2 — present value of the 24 instalments.

PV = 333 × [1 − 1/(1.00115)²⁴] / 0.00115 = 333 × 23.658412 = 7,878.25

Step 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.

Q18An investment costs $5,000 today and returns $1,500, $2,000 and $2,500 at the end of years 1, 2 and 3. The discount rate is 8%. Compute the present value of each cash flow, the total present value, and the net present value. Would you accept the investment?

PV of 1,500 at t=1: 1,500 / (1.08)¹ = 1,388.89 PV of 2,000 at t=2: 2,000 / (1.08)² = 1,714.68 PV of 2,500 at t=3: 2,500 / (1.08)³ = 1,984.58 ───────────────────────── Total present value of inflows = 5,088.15 Less initial cost = (5,000.00) ───────────────────────── Net present value = 88.15

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.

Q19Construct the complete amortisation schedule for a $5,000 loan over five years at 9% on the equal-principal basis. State the total interest paid, and explain in one sentence why it differs from the equal-payment figure of $1,427.31.

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 6,350.00

Total 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.

Q20Your bank offers 1.9% compounded semi-annually. A money-market fund quotes a 7-day annualised rate of 2%. Using an effective annual rate basis, which is the better home for your money?

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.

Lecture 3 · Week 3

Net Present Value

9 Sep 2025 Ross Ch. 5 Source: Lecture 3 Net Present Value.pdf (32 slides)

Lecture 2 built the machinery. This lecture points it at a decision: should the firm buy this asset? It introduces four answers to that question — net present value, internal rate of return, payback period and the profitability index — and then spends most of its time showing why three of them are inferior to the first. The lecture's own conclusion, stated on its summary slide, is blunt: use NPV, and when the others disagree with it, use NPV anyway.

Concept Map

how the lecture builds

Should I approve this project? Cash flows arrive at different dates, so they cannot be compared directly. That is the whole problem.

discount everything to today and subtract the cost

Net present value NPV = PV of future cash flows − initial cost. Accept if positive. The only rule with no serious flaws.

the other three criteria are attempts to answer the same question more cheaply

Internal rate of return The discount rate that makes NPV = 0. Intuitive, and unreliable.

Payback period How long until the cost is recovered. Ignores the time value of money entirely.

Profitability index PV per dollar invested. Useful only under capital rationing.

IRR fails in four distinct ways

Multiple IRRs Non-conventional cash flows can produce several roots

Borrowing vs lending The rule reverses direction for financing cash flows

The scale problem A percentage ignores how much capital is at stake

The timing problem IRR implicitly assumes reinvestment at the IRR itself

all four appear only when projects are mutually exclusive

When IRR and NPV disagree, NPV wins Because NPV is measured in the thing shareholders actually receive: value.

🔗 Link · the bridge between the two halves of the course

This 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 2–10

💬 Think · should I approve the project?

You 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 = −76.19

Reject 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

NPV = −C₀ + C₁(1 + r) + C₂(1 + r)² + ⋯ + C_T(1 + r)^T

C_t — the cash flow at time t (negative at t = 0 for the investment) · r — the discount rate

NPV rule

Accept the project when NPV > 0.
Reject the project when NPV < 0.

Ranking rule

Among mutually exclusive projects, choose the one with the highest NPV.

Worked example: Finance.com

Worked A high-speed computer costing $50,000

Finance.com can invest $50,000 in a computer generating cost savings of $25,000, $20,000 and $15,000 at the end of years 1, 2 and 3. The machine is worthless afterwards. The appropriate discount rate is 7%. Should Finance.com invest?

$50,000

25,000

20,000

15,000

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

NPV = $3,077.73 > 0, so Finance.com should invest.

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.

⚠️ Trap · Excel's NPV function does not do what its name says

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, …) + CF0

with 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:

1 · Estimate future cash flows

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.

2 · Estimate the discount rate

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.

3 · Estimate the initial cost

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.

📖 Beyond the slides · the discount rate is where the risk lives

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 11–21

The 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.

Minimum acceptance criteria

Accept if the IRR exceeds the required return.

Ranking criteria

Select the alternative with the highest IRR.

⚠️ Trap · the reinvestment assumption, stated honestly

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

Worked Finding the IRR of −$200, $50, $100, $150

Cash flows of −$200 at t = 0, then $50, $100, $150 at t = 1, 2, 3. Find the rate at which NPV = 0.

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.

IRR = 19.44%.

💬 Think · the three borrowing plans (the slide leaves this open)

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.

Worked A project with two internal rates of return

Cash flows of −$200, +$200, +$800, −$800 at t = 0, 1, 2, 3. Note the signs: −, +, +, −. That is two sign changes.

$200

$800

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 2^t −200 + 200/2 + 800/4 − 800/8 = −200 + 100 + 200 − 100 = 0 → IRR = 100% is also a solution

Two IRRs: 0% and 100%. Which one should we use? IRR gives no answer to that question.

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.

Worked The timing problem

Two mutually exclusive projects, each costing $10,000. Required return 10%.

10,000

1,000

Project A — money back early

10,000

1,000

12,000

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 rate at which the two NPVs are equal — the crossover rate — is 10.55%. Below it, B wins; above it, A wins.

📖 Beyond the slides · the crossover rate and the NPV profile

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

Advantages of IRR

Easy to understand and to communicate. A percentage is intuitive in a way that a dollar NPV is not, which is why IRR survives in practice despite its flaws.

Disadvantages of IRR

Does not distinguish between investing and borrowing. The IRR may not exist, or there may be several. It is unreliable for mutually exclusive investments.

📖 Beyond the slides · NPV and IRR usually agree

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 22–32

Payback 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.

Worked A $50,000 project returning $30,000 / $20,000 / $10,000

Time	Cash flow	Cumulative
0	−50,000	−50,000
1	+30,000	−20,000
2	+20,000	0
3	+10,000	+10,000

Payback period = 2 years — the cumulative flow reaches zero at t = 2.

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.

Advantages

Easy to understand. Biased toward liquidity — it favours projects that return cash quickly, which is genuinely valuable for a firm with tight finances.

Disadvantages

Ignores the time value of money. Ignores all cash flows after the payback date. Biased against long-term projects. Requires an arbitrary acceptance cutoff. And a project accepted on payback grounds may well have a negative NPV.

⚠️ Trap · payback ignores everything interesting

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:

PI = Total PV of future cash flows Initial investment

Acceptance and ranking

Accept if PI > 1. Among alternatives, choose the highest PI.

Where it earns its place

Capital rationing — when the firm cannot fund every positive-NPV project, PI measures value created per dollar of scarce capital.

Worked Capital rationing — you have only $20 to invest

Three independent projects, discount rate 12%, total budget $20 million. Figures in $m.

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 PI criterion picks a combination worth $68.65m against $50.47m from the highest-NPV project — a difference of $18.18m.

⚠️ Trap · a small rounding difference from the slide

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.

Advantages of PI

Useful when investment funds are limited. Easy to understand and communicate. Gives the correct decision when evaluating independent projects.

Disadvantages of PI

Because it is a ratio, it suffers the scale problem for mutually exclusive investments — it will happily prefer a tiny project with a high ratio over a large one that creates far more value.

Everything at once: the lecture's capstone example

Worked Two projects compared on all four criteria, required return 10%

Two mutually exclusive projects. Compute IRR, NPV, PI and payback for each.

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 CF_1–3	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.

Choose Project B — and note that this is the only project of the two for which all four criteria even produce a coherent answer.

📖 Beyond the slides · which method do firms actually use?

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-friendly

Net present value 净现值

NPV = −C₀ + C₁(1+r) + C₂(1+r)² + ⋯ + C_T(1+r)^T

Accept if NPV > 0. Among mutually exclusive projects, take the highest NPV.

Internal rate of return 内部收益率

Find r such that NPV = 0

Accept if IRR > required return — for conventional (investing) cash flows only. The rate that makes the project break even in present-value terms.

Profitability index 盈利指数

PI = Total PV of future cash flowsInitial investment

Accept if PI > 1. Use it to rank under capital rationing, where the plain NPV rule fails.

Payback period 回收期

Years until cumulative cash flow ≥ 0

Cutoff set by management. Ignores the time value of money and everything after the payback date.

Discounted payback 折现回收期

Years until cumulative discounted cash flow ≥ 0

Ross's fix for payback's most obvious defect. Still ignores cash flows beyond the cutoff.

Crossover rate

Find r such that NPV_A = NPV_B

The discount rate below which one mutually exclusive project wins and above which the other does. In the lecture's timing example, 10.55%.

Equivalent annual annuity 等额年金法

EAA = NPVAnnuity factor(r, T)

Converts a project's NPV into an equivalent annual cash flow, making projects of unequal lives comparable.

Excel construction

=NPV(rate, CF1, CF2, …) + CF0

The function computes a present value, not a net present value. CF₀ goes outside, as a negative number.

Key Concepts

the vocabulary

Net present value 净现值

The difference between an investment's market value and its cost — equivalently, the present value of all future cash flows net of the initial outlay. The addition to firm value from accepting the project.

Internal rate of return 内部收益率

The discount rate that makes NPV equal to zero. The compound return the project earns on the capital invested in it, assuming interim cash flows are reinvested at the IRR itself.

Reinvestment assumption

IRR's silent assumption that every interim cash flow can be redeployed at the IRR. NPV makes no such assumption — it discounts at the market rate. This is the root of the timing problem.

Mutually exclusive projects 互斥项目

Projects where only one of several can be chosen. IRR is unreliable here. Contrast with independent projects, where accepting one does not affect the decision on the others — and where IRR is perfectly sound.

Conventional cash flows 常规现金流

Cash flows that change sign exactly once — typically an outflow followed by inflows. A non-conventional pattern, with two or more sign changes, is what produces multiple IRRs.

Multiple IRRs

When cash flow signs change more than once, the NPV equation can have several roots and IRR cannot say which to use. Worked example: (−200, +200, +800, −800) has IRRs at both 0% and 100%.

The scale problem

IRR and PI are percentages or ratios, so they ignore how much capital is at stake. A 40% return on $100 loses to a 30% return on $1,000 in value terms, but IRR ranks it first.

The timing problem

When two projects of similar scale return cash at different dates, IRR and NPV can rank them oppositely. Arises from the reinvestment assumption.

Crossover rate

The discount rate at which two projects have equal NPV. Below it one wins, above it the other. A narrow gap between the crossover and the required return means a fragile decision.

Payback period 回收期

The length of time until the initial investment is recovered. A liquidity screen rather than a valuation criterion — it cannot see value created after the cutoff.

Profitability index 盈利指数

Present value of future cash flows per dollar of initial investment. The correct ranking tool when capital is rationed, and a scale-blind one otherwise.

Capital rationing 资本限额

A situation where the firm cannot fund every positive-NPV project available to it. The ordinary NPV rule then fails, because it assumes unlimited capital.

Equivalent annual annuity 等额年金法

The constant annual cash flow with the same NPV as the project over its life. Used to compare projects with different lifespans, where raw NPV unfairly favours the longer one.

Hurdle rate 最低要求收益率

The minimum return a project must earn to be accepted. In this lecture it is given; in Week 10 it is derived as the cost of capital.

Self-Check

18 questions · graded

0 / 0 correct

The 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

The 2×10 concept-explanation section.

Q1An investment has a net present value of exactly zero at the required rate of return. What does this mean?

Correct. A zero NPV means the project returns precisely what the capital could earn elsewhere at the same risk. Nothing is lost and nothing is gained — hence the indifference.

Not quite. Zero NPV is not zero cash. It means the project's return exactly equals the required return for its risk, so the firm is neither better nor worse off. The project should not be rejected on value grounds — it simply does not add anything.

Q2Which statement about the internal rate of return is correct?

Correct. That is the definition. Its consequences — the reinvestment assumption, the multiple-root problem — follow from it rather than qualifying it.

Not quite. IRR is by definition the rate that sets NPV to zero. It is a property of the project, not of the firm's assets (that is ROA); it assumes reinvestment at the IRR itself, not at the cost of capital; and it does not always agree with NPV.

Q3A project with a 15% IRR has a payback period of 2 years. Which of the following is the most serious weakness of relying on the payback figure?

Correct. Two distinct defects, and both matter: everything after the cutoff is invisible, and the flows before it are not discounted. Option D has the bias backwards — payback favours short-term projects.

Not quite. Payback is if anything easier to calculate than IRR, and it handles uneven cash flows fine (you just accumulate them). Its real defects are that it ignores the time value of money and ignores everything that happens after the payback date.

Q4A firm has a fixed capital budget and must choose among independent positive-NPV projects. Project X has the highest NPV. Why might the firm still not choose it?

Correct. This is the capital rationing problem exactly. The NPV rule assumes you can fund every positive-NPV project; when you cannot, you are choosing a portfolio under a constraint, and the profitability index is the right ranking tool.

Not quite. It is not that NPV is unreliable — the NPVs are all correct. The issue is that the rule "take every positive-NPV project" is infeasible, so the decision becomes a constrained portfolio choice. The profitability index measures value created per dollar of scarce capital.

Q5Concept explanation. Explain why NPV is regarded as the superior investment criterion, and why firms nevertheless keep using IRR.

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.

Q6Concept explanation. Explain how a project can have more than one internal rate of return, and what a manager should do when this happens.

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

Computational, in the style of the 2×35 single-choice section.

Q7A computer costs $50,000 and generates cost savings of $25,000, $20,000 and $15,000 at the end of years 1, 2 and 3. The discount rate is 7%. What is the NPV?

Correct. 25,000/1.07 = 23,364.49 20,000/(1.07)² = 17,468.77 15,000/(1.07)³ = 12,244.47 ────────── PV of inflows = 53,077.73 − initial cost = (50,000.00) ────────── NPV = 3,077.73

Not quite. PV of inflows = 23,364.49 + 17,468.77 + 12,244.47 = 53,077.73 NPV = 53,077.73 − 50,000 = 3,077.73 $53,077.73 is the present value of the inflows alone — the cost has not been subtracted. $10,000 is the undiscounted sum of the savings (60,000) less the cost, which ignores the time value of money. The NPV is positive, so accept.

Q8A project costs $1,000 and returns $600 at the end of each of the next two years. What is its IRR?

Correct. 0 = −1,000 + 600/(1+r) + 600/(1+r)² Solving: r = 0.13066

Not quite. 0 = −1,000 + 600/(1+r) + 600/(1+r)² → r = 13.07% 20% is the naive answer: total inflow of 1,200 against a cost of 1,000 looks like 20%. But the second $600 arrives a year late and must be discounted, so the true return is lower. IRR is not total profit divided by cost.

Q9A project costs $1,000 and returns $400 at the end of each year for three years. What is its payback period?

Correct. Cumulative: −1,000 → −600 → −200 → +200 After 2 years, $200 of the initial cost remains unrecovered. Year 3 brings in $400, so it takes 200/400 = 0.5 of a year. Payback = 2 + 0.5 = 2.50 years

Not quite. Cumulative: −1,000 → −600 → −200 → +200 Payback = 2 + (200/400) = 2.50 years The cumulative flow is −200 after two years, so the project has not broken even yet. It breaks even halfway through year 3. Note that payback here ignores the time value of money entirely.

Q10Project Small costs $100 and returns $140 next year. Project Large costs $1,000 and returns $1,300 next year. They are mutually exclusive and the required return is 10%. Which should be chosen?

Correct. This is the scale problem. IRR (40% vs 30%) and PI (1.27 vs 1.18) both rank Small first because both are ratios that cannot see how much capital is employed. NPV can, and Large creates $154.55 more value.

Not quite. Both B and C rank Small first, and both are wrong. IRR and PI are ratios: they measure return per dollar invested and ignore the number of dollars. In value terms Large is far better — $181.82 against $27.27. For mutually exclusive projects of different scale, use NPV.

Q11A project has cash flows of +$100 at t = 0 and −$130 at t = 1, giving an IRR of 30%. The required return is 12%. Should it be accepted?

Correct. NPV = 100 − 130/1.12 = 100 − 116.07 = −16.07 The NPV is negative, so reject. This cash flow is borrowing: you receive 100 now and repay 130. A 30% rate is the cost of the money, not its return — and 30% is worse than the 12% you could borrow at elsewhere.

Not quite. NPV = +100 − 130/1.12 = −16.07 The naive answer, A, applies the IRR rule in the wrong direction. When the initial cash flow is positive the project is financing, not investing, and a high IRR is bad. The NPV confirms: negative, so reject.

Q12Which of the following cash-flow patterns guarantees that a project will have exactly one IRR?

Correct. One sign change gives one root. Two or more sign changes open the possibility of additional roots — though they do not guarantee them, which is why the multiple-IRR problem is intermittent rather than predictable.

Not quite. The relevant property is the number of sign changes, not the size or evenness of the cash flows. One sign change (− then +) guarantees a single IRR; more than one introduces the possibility of several. A positive NPV or large total inflows say nothing about the number of roots.

Q13An investment costs $1,000 and returns $500, $400 and $300 at the end of years 1, 2 and 3. The discount rate is 8%. What is the NPV?

Correct. 500/1.08 = 462.96 400/(1.08)² = 342.94 300/(1.08)³ = 238.15 ─────── PV inflows = 1,044.05 NPV = 1,044.05 − 1,000 = 44.05

Not quite. PV of inflows = 462.96 + 342.94 + 238.15 = 1,044.05 NPV = 1,044.05 − 1,000 = 44.05 $200 is the undiscounted total (1,200 − 1,000). $1,044.05 is the present value of the inflows before subtracting the cost. The NPV is positive, so accept — but only just.

Q14The lecture states that one criterion "has no serious problems" and is therefore the preferred decision rule. Which is it?

Correct. NPV is the only one of the four that survives every case the lecture tests it against: non-conventional cash flows, mutually exclusive projects, differing scale and timing, and capital rationing.

Not quite. IRR fails on all four counts; payback ignores the time value of money and everything after the cutoff; PI suffers the scale problem with mutually exclusive projects. Only NPV is free of serious defects, which is why the lecture calls it the preferred criterion.

Q15Two mutually exclusive projects each cost $10,000. Project A returns $10,000 in year 1, then $1,000 in years 2 and 3. Project B returns $1,000 in years 1 and 2, then $12,000 in year 3. At a 10% required return, what should be done?

Correct. This is the timing problem. IRR prefers A because its cash returns arrive earlier; NPV prefers B because B creates more value. The two disagree, and NPV wins.

Not quite. A and C both apply the wrong criterion. IRR ranks A higher and payback certainly prefers A — but the two projects are mutually exclusive, so one must be given up. B creates $82.64 more shareholder value, and value is what the decision is for.

C · Applied calculation

The 5×2 subjective section. Work each on paper before revealing.

Q16A project costs $1,200 and returns $500, $500 and $400 at the end of years 1, 2 and 3. The required return is 9%. Compute the NPV, the IRR, the profitability index and the payback period, and state whether the project should be accepted.

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 = −11.57

IRR. 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 = 0.9904

Payback period.

Cumulative: −1,200 → −700 → −200 → +200 Payback = 2 + (200/400) = 2.50 years

Decision: 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.

Q17Two mutually exclusive projects each cost $10,000 and have a required return of 10%. Project A returns $10,000, $1,000, $1,000; Project B returns $1,000, $1,000, $12,000. Compute both IRRs and both NPVs, identify the conflict, and explain which criterion should govern and why.

IRR NPV at 10% Project A 16.04% $668.67 Project B 12.94% $751.31

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.

Q18A firm must choose between two machines, both of which perform the same function. Machine A costs $200 and lasts 2 years, saving $140 a year. Machine B costs $400 and lasts 5 years, saving $130 a year. The discount rate is 10%. Machine A has the lower NPV — so does that settle it? Explain, and compute the equivalent annual annuity for each.

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 = 92.80

Equivalent 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 year

Answer: 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 = 150.43

A 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.

Lecture 4 · Week 4

Capital Budgeting

15 Sep 2025 Ross Ch. 6–7 Source: Lecture 4 Capital Budgeting.pdf (23 slides)

Lecture 3 gave the rule: accept positive-NPV projects. This lecture confronts the hard part — working out what the cash flows actually are. In a real project, the accounting numbers sitting in front of you are the wrong numbers: they include costs that are already spent, omit costs you never pay in cash, and ignore what you gave up. The Baldwin Company example that occupies half the slides is the canonical demonstration, and it is worked through here line by line.

Concept Map

how the lecture builds

Capital budgeting Evaluating major projects — new equipment, a new plant, a new product — and deciding whether to undertake them. The lecture's own definition.

the decision rule comes from Lecture 3; the difficulty is the inputs

Which cash flows count? Incremental ones — the difference between the firm with the project and the firm without it.

which resolves into six specific rules

Sunk costsAlready spent → excluded

Opportunity costsWhat you give up → included

Side effectsErosion and synergy → included

TaxesAfter-tax cash flows → included

InflationNominal with nominal, real with real

Financing costs→ excluded, and in the discount rate instead

which assemble into a single quantity

Free cash flow EBIT − Tax + Depreciation − ΔWC − Capex. The cash left for shareholders and debtholders.

whose two halves are

Part 1 — operations Revenue − costs − taxes. Interest excluded.

Part 2 — supporting outlays Capital spending and the change in working capital

discounted at the right rate, which is where risk re-enters

Risk analysis Sensitivity · scenario · Monte Carlo

Cost of capital Where the discount rate comes from → Week 10

🔗 Link · this lecture closes a loop opened in Lecture 1

Look 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 3–8

The 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.

Incremental cash flows

Not accounting earnings. The difference between the cash flows with the project and the cash flows without it.

Sunk costs

Do not matter. Already incurred, whatever you decide.

Opportunity costs

Do matter. Using an asset forgoes whatever else it could have earned.

Side effects

Do matter — both erosion and synergy.

Taxes

We want incremental after-tax cash flows. Tax is a cash flow, and it is often the largest single line.

Inflation

Does matter. Compare real cash flows at real rates, or nominal at nominal — never mix them.

⚠️ Trap · the fundamental rule, and it is not about accounting

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.

⚠️ Trap · sunk costs are the hardest rule to obey

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.

💬 Think · is forgone investment return an opportunity cost?

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.

Firm's financing cost = required return of investors

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:

Higher financing cost → higher discount rate → lower NPV

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 侵蚀

A new product reduces the sales of existing ones.

The lecture's example: iPhone 17 cannibalising iPhone 16. Only the net increase in sales counts. If the new model sells 1m units but existing models lose 300,000, the project's incremental revenue comes from 700,000.

Synergy 协同

A new product increases the sales of other ones.

The lecture's example: the iPhone driving Apple Music subscriptions. This is value the project creates beyond its own cash flows, and it belongs in the analysis too.

📖 Beyond the slides · why erosion is not always bad

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 9–10

Free 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.

📖 Beyond the slides · the going concern assumption

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.

FCF = EBIT − Tax + Depreciation − ΔWC − Capex

The two parts

Part 1 · cash flow from the project

EBIT − Tax

The operating cash the project generates. Interest expense is not included, since it is a financing cost and is reflected in the discount rate instead — the point made in the previous section.

Part 2 · expenditures required to support it

Capex + ΔWC

Capital spending is investment in fixed assets; the change in net working capital is investment in working capital. Both are subtracted — they are cash the project absorbs.

📖 Beyond the slides · why depreciation is added back, and why the formula still works

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:

Part 1 = (Sales − Costs) × (1 − T) + Depreciation × T

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 6.2

This 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.

⚠️ Trap · units

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

Given Everything the analysis starts from

Marketing test, already spent: $250,000 (250K)
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:

Excluded · sunk cost

The $250,000 marketing test. Already spent, regardless of the decision, so it is not incremental. It appears nowhere in the analysis below — and its absence is the single most commonly examined point about this example.

Included · opportunity cost

The factory site. Taking the project forgoes selling it, so $150,000 is a cash outflow at t = 0. The lecture assumes the site's market value is unchanged at the end and that the company sells it then — so $150,000 also comes back at t = 5. Two cash flows, not one.

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

⚠️ Trap · why year 1 shows a blank in rows (6) and (7)

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.

Worked Building row (5), the working capital balance

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 changes sum to zero across the project's life — +10.00 and −21.22 net against +6.32 + 8.65 − 3.75. That is a general property worth remembering: working capital is committed and later released, not consumed. Its entire effect is a matter of timing.

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%:

Revenue_t = Units_t × $20 × (1.02)^t−1

Worked Two rows worked in full, then the series

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

Revenue: 100.00 · 163.20 · 249.70 · 212.24 · 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.

Operating cost_t = Units_t × $10 × (1.10)^t−1

Worked The same two rows, then the series

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

Operating costs: 50.00 · 88.00 · 145.20 · 133.10 · 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

⚠️ Trap · the missing year 6, and why the salvage works out

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

📖 Beyond the slides · what this net income line is doing here

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

🔗 Link · three ways to the same number

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

NPV = −260 + 39.80(1.10) + 54.19(1.10)² + 66.85(1.10)³ + 59.90(1.10)⁴ + 224.65(1.10)⁵

Worked The discounted cash flows

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

NPV = $51,594.80 — accept the project. In the table's own units, 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.

Worked Selling the machine at the end of year 5

Original cost $100,000 · accumulated depreciation after 5 years $94,240 · ending market value $30,000 · tax rate 34%

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

After-tax salvage value = $21,758 — the figure appearing at the top right of the Step 1 table.

So the year-5 Cash Flow Part 2 of 192.98 is:

21.76 (after-tax salvage) + 150.00 (factory site sold) + 21.22 (working capital released) = 192.98

📖 Beyond the slides · why the tax is on the gain, not the proceeds

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.

🔗 Link · the numbers worth taking to the exam

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 20–23

Three 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.

📖 Beyond the slides · why MACRS beats straight-line, in one number

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.

(1 + Nominal rate) = (1 + Real rate) × (1 + Inflation rate) Fisher equation

The lecture also gives the approximation that everyone actually uses in conversation:

Nominal rate ≈ Real rate + Inflation rate

Worked How much does the approximation cost you?

Nominal rate 10% · inflation 5% · find the real rate

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

The approximation overstates the real rate by about 0.24 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.

⚠️ Trap · the consistency rule, with the arithmetic that shows why

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 = 267.30

Route 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 = 267.30

The 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.

Sensitivity analysis 敏感性分析

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.

Scenario analysis 情景分析

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.

Monte Carlo simulation 蒙特卡洛模拟

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.

Method Monte Carlo simulation, in five steps

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

The output is not a single NPV but a distribution — "expected NPV of $51.6k, with a 22% probability of a negative NPV" carries far more information than "$51.6k".

📖 Beyond the slides · the three techniques are not equally useful

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-friendly

Free cash flow 自由现金流

FCF = EBIT − Tax + Depreciation − ΔWC − Capex

The number discounted. Interest never appears — it is in the discount rate.

Part 1 · top-down

Sales − Costs − Taxes

Do not subtract depreciation — it is not a cash flow. Taxes are computed on EBIT.

Part 1 · bottom-up

Net income + Depreciation

Valid only when there is no interest expense, which is the standing assumption.

Part 1 · tax shield

(Sales − Costs)(1 − T) + Depreciation × T

Isolates the value of the depreciation deduction. Baldwin year 1: 50 × 0.66 + 20 × 0.34 = 39.80.

Part 2 · supporting outlays

Capex + ΔWC

Both subtracted from FCF. Working capital changes sum to zero over a project's life — only the timing costs (or creates) value.

Book value

Book value = Cost − Accumulated depreciation

Baldwin machine at sale: 100.00 − 94.24 = 5.76.

After-tax salvage value

ATS = MV − T × (MV − Book value)

Tax falls on the gain, not the proceeds. Baldwin: 30 − 0.34 × 24.24 = 21.758.

MACRS 5-year schedule

20.00% · 32.00% · 19.20% · 11.52% · 11.52% · 5.76%

Sums to 100% over six years. The final year is rarely claimed because the asset is sold first — which is what creates the book value and hence the taxable gain.

Fisher equation 费雪方程

(1 + Nominal) = (1 + Real) × (1 + Inflation)

Exact. The approximation Nominal ≈ Real + Inflation drops the cross-product term and always overstates the real rate: at 10% and 5%, by 0.24 points.

Consistency rule

Real cash flows ÷ real rate · Nominal cash flows ÷ nominal rate

Never mix. Discounting nominal flows at the real rate overstated the worked example by 8.1%.

Break-even (Ross Ch. 7)

Solve for the sales level at which NPV = 0

Distinguish from accounting break-even, where net income is zero — a much less useful number, since a project can break even on paper and still destroy value.

Equivalent annual cost

EAC = PV of costsAnnuity factor

The cost analogue of the EAA in Lecture 3. Used to choose between machines of unequal life when the benefit is the same either way.

Key Concepts

the vocabulary

Capital budgeting 资本预算

Evaluating major projects and investments — new equipment, a new plant, a new product — and deciding whether to undertake them.

Incremental cash flow 增量现金流

The difference between the firm's cash flows with the project and without it. The only cash flows that matter, and the test that disposes of sunk costs, opportunity costs and side effects alike.

Sunk cost 沉没成本

A cost already incurred regardless of the decision. Not incremental, therefore irrelevant. Baldwin's $250,000 marketing test is the canonical example.

Opportunity cost 机会成本

The value of what is given up when an asset is used in the project — a forgone asset sale, for instance. A real cost even though no cash is ever paid out.

Financing cost 融资成本

The cost of the capital used to fund the project, equal to investors' required return. Reflected in the discount rate, never as a cash flow — including it in both places double-counts it.

Erosion / cannibalisation 侵蚀

A new product reducing sales of existing ones. Only the net increase counts — but erosion is only a genuine cost to the extent that the lost sales would not have been lost to a competitor anyway.

Synergy 协同效应

A new product increasing sales of other ones. Value created beyond the project's own cash flows, and equally part of the incremental analysis.

Free cash flow 自由现金流

Cash left to shareholders and debtholders after funding the investments the going concern requires. "Free" means the firm is free to spend it.

Going concern assumption

The assumption that the firm continues operating. It is why Capex and ΔWC must be subtracted: a continuing operation has to replace assets and carry working capital.

Capital spending 资本性支出

Investment in fixed assets. Ending fixed assets − beginning fixed assets + depreciation (Lecture 1).

ΔWC 营运资本变动

The change in net working capital. A use of cash when it grows, a source when it shrinks. Sums to zero over a project's life.

MACRS

Modified Accelerated Cost Recovery System — a tax depreciation schedule that front-loads deductions. Accelerating depreciation does not change the total, but shifts the tax shield earlier, which is worth real money.

Depreciation tax shield

Depreciation × T — the cash saved purely because depreciation is deductible. Often the difference between a positive and a negative NPV for capital-intensive projects.

After-tax salvage value 税后残值

Sale proceeds less tax on the capital gain, where the gain is market value minus book value. Selling below book value generates a tax credit instead.

Nominal vs real

Nominal cash flows are in the actual money of the day; real cash flows are in constant purchasing power. Discount each at its own rate and never mix them.

Sensitivity, scenario, Monte Carlo

Three escalating responses to estimation risk: vary one input; vary several coherently with probabilities; or draw from full distributions thousands of times to obtain a distribution of NPVs.

Self-Check

16 questions · graded

0 / 0 correct

The 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 2×10 concept-explanation section.

Q1A firm has already spent $250,000 on a marketing study for a new product. The study is complete and the firm is now deciding whether to build the factory. How should the $250,000 be treated?

Correct. The $250,000 is spent whether the factory is built or not, so it does not differ between the two scenarios. Non-incremental means irrelevant. This is the single most examined point in Lecture 4.

Not quite. The test for relevance is whether the cash flow differs depending on the decision. This one does not — the money is gone either way. It is a sunk cost and is excluded from the analysis, however painful that feels.

Q2A firm owns a warehouse it could sell today for $150,000. It decides to use the warehouse for a new project instead. How should the $150,000 be treated?

Correct. Using the warehouse forgoes the $150,000 sale, so that forgone cash is a genuine cost of the project. In the Baldwin analysis the same treatment applies to the factory site — and note that the site also returns $150,000 at t = 5 when it is finally sold.

Not quite. Ownership does not make an asset free. The warehouse has an alternative use — selling it — and giving that up costs $150,000. Option D confuses opportunity cost with sunk cost: the purchase price may be sunk, but the current market value is very much a live opportunity.

Q3How should the interest expense on debt raised to finance a project be treated in the capital budgeting analysis?

Correct. Financing costs are reflected in the required return, which is the discount rate. The discount rate is the cost of capital, expressed as a rate. Including interest in the cash flows as well would charge the project twice for the same money.

Not quite. This is the double-counting trap. The discount rate already represents what the capital costs — it is the investors' required return. Subtracting interest from the cash flows and discounting at a rate that includes it charges the project twice, and biases the decision against acceptance.

Q4A firm launches a new phone model. It sells 1 million units, but sales of the previous model fall by 300,000 units. How should the analysis treat this?

Correct. Erosion is a side effect and must be netted off. But note the qualification from the lecture's own discussion: if a competitor would have taken those 300,000 units anyway, then eroding them yourself is not a real cost at all.

Not quite. Incremental thinking requires the lost sales to be netted off — the firm is only 700,000 units better off. But option C overcorrects: launching and cannibalising yourself is better than letting a competitor do it, so erosion alone is rarely a reason to reject.

Q5Concept explanation. Explain why a project's cash flows must be forecast directly rather than taken from its projected net income, and identify the specific items that create the difference.

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

Computational, in the style of the 2×35 single-choice section.

Q6A project has sales of $500, cash operating costs of $200, depreciation of $50, a tax rate of 30%, an increase in net working capital of $20 and capital spending of $100. What is its free cash flow?

Correct. EBIT = 500 − 200 − 50 = 250 Tax = 250 × 0.30 = 75 FCF = 250 − 75 + 50 − 20 − 100 = 105

Not quite. EBIT = 500 − 200 − 50 = 250 Tax = 250 × 0.30 = 75 FCF = 250 − 75 + 50 − 20 − 100 = 105 $250 is EBIT (no tax, no add-back, no capex). $175 is EBIT − tax, but stops before adding back depreciation and subtracting ΔWC and Capex. $225 adds the depreciation back but still omits ΔWC and Capex. All five terms are needed.

Q7A project generates $80 of depreciation in a year. The tax rate is 25%. What is the cash value of the depreciation tax shield that year?

Correct. Depreciation × T = 80 × 0.25 = 20 Depreciation is not itself a cash flow, but it reduces taxable income by $80, saving $20 of tax. That $20 is a cash flow — the only way depreciation affects cash.

Not quite. Shield = Depreciation × T = 80 × 0.25 = 20 Option A treats the full depreciation as cash, which it is not. Option D is half right — depreciation itself is not a cash flow — but it misses that the tax saving it generates very much is. $60 is the after-tax depreciation (80 × 0.75), which is not the shield.

Q8A machine costing $200,000 is depreciated under the 5-year MACRS schedule. What is its book value at the end of year 3?

Correct. Year 1: 200,000 × 0.2000 = 40,000 Year 2: 200,000 × 0.3200 = 64,000 Year 3: 200,000 × 0.1920 = 38,400 Accumulated depreciation = 142,400 Book value = 200,000 − 142,400 = 57,600

Not quite. Accumulated depreciation after 3 years = 40,000 + 64,000 + 38,400 = 142,400 Book value = 200,000 − 142,400 = 57,600 $142,400 is the accumulated depreciation, not the book value. $80,000 is the straight-line book value after three years of 20% a year — MACRS is accelerated, so the true figure is lower.

Q9Equipment with a book value of $40,000 is sold at the end of a project for $70,000. The tax rate is 30%. What is the after-tax salvage value?

Correct. Gain = 70,000 − 40,000 = 30,000 Tax on gain = 30,000 × 0.30 = 9,000 After-tax = 70,000 − 9,000 = 61,000

Not quite. Taxable gain = 70,000 − 40,000 = 30,000 Tax = 9,000; ATS = 70,000 − 9,000 = 61,000 Option A forgets tax entirely. Option B taxes the full proceeds (70,000 × 0.70 = 49,000) — wrong, because tax falls only on the gain above book value. Option C taxes the whole book value, which has already been depreciated.

Q10The nominal interest rate is 8% and inflation is 3%. What is the real interest rate, to the nearest basis point?

Correct. (1 + real) = 1.08 / 1.03 = 1.048544 real = 4.8544% Note it is below the 5% approximation — the approximation always overstates, because it drops the real × inflation cross term.

Not quite. real = (1.08 / 1.03) − 1 = 4.8544% 5.00% is the approximation (8 − 3), which is close but not exact. 11.24% comes from multiplying the factors (1.08 × 1.03 − 1) instead of dividing. Use the exact Fisher form for arithmetic.

Q11A project's net working capital rises from $50,000 at the start to $65,000 at the end of year 1. What is the effect on that year's free cash flow?

Correct. ΔWC = 65,000 − 50,000 = 15,000 (an increase) FCF effect = −ΔWC = −15,000 Growing working capital means cash has been absorbed — into inventory, or into receivables the firm has not yet collected. It is subtracted in the free cash flow formula.

Not quite. Working capital appears in the free cash flow formula as −ΔWC, and the sign follows the direction of travel. Growing working capital absorbs cash, so it reduces FCF. Only the change matters, not the level — and when the project ends and the working capital is released, the sign reverses.

Q12Which of the following is least likely to be treated as an incremental cash flow in a capital budgeting analysis?

Correct. The consulting fee is sunk — paid last year regardless of the decision. The other three are all incremental: erosion is a side effect, the building is an opportunity cost, and the inventory is a investment in working capital.

Not quite. A, B and D are all incremental. The consulting fee is sunk — it was paid last year whether or not the project proceeds, so it cannot affect the decision. Note the phrase "last year": had the fee been payable only if the project went ahead, it would be incremental and would count.

Q13In the Baldwin Company case, the $250,000 marketing test is excluded and the $150,000 factory site is included. Which pair of concepts explains this treatment?

Correct. The marketing test is sunk — spent regardless. The factory site has an alternative use — selling it — so using it forgoes $150,000. Both are applications of the same incremental principle, in opposite directions.

Not quite. This is the sunk-cost / opportunity-cost pair, and it is worth being able to state both directions fluently. Erosion and synergy concern other products; financing cost concerns the discount rate; working capital and capex are the Part 2 outlays.

C · Applied calculation

The 5×2 subjective section. Work each on paper before revealing.

Q14A project requires $300,000 of equipment with a three-year life, depreciated straight-line to zero, and an immediate investment of $25,000 in net working capital which is recovered at the end of year 3. It generates sales of $400,000 and cash operating costs of $150,000 each year. The tax rate is 30% and the discount rate is 10%. Compute the free cash flow each year and the NPV.

Step 1 — annual depreciation. Straight-line over three years:

300,000 / 3 = 100,000 per year

Step 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 year

Cross-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 = 230,000

Step 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 = 203,587.53

Answer. 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.

Q15A machine costing $500,000 is depreciated on the 5-year MACRS schedule. It is sold at the end of year 5 for $120,000, and the tax rate is 34%. Compute its book value at sale, the tax due, and the after-tax salvage value. Then explain what would change if the machine sold for only $20,000.

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 = 28,800

Step 2 — taxable gain.

Gain = 120,000 − 28,800 = 91,200

Step 3 — tax due and after-tax salvage.

Tax = 91,200 × 0.34 = 31,008 After-tax salvage = 120,000 − 31,008 = 88,992

Answer. 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 = 22,992

The 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.

Q16A project generates a real cash flow of $100 a year for three years. The real discount rate is 6% and inflation is 4%. Compute the present value two ways — real cash flows at the real rate, and nominal cash flows at the nominal rate — and confirm they agree. Then state what goes wrong if the nominal cash flows are discounted at the real rate.

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 = 267.30

Route 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 = 267.30

The 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 = 288.82

That 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.

Syllabus agenda

Roadmap · Weeks 5–16

4 of 16 weeks delivered Chapter numbers: Corporate Finance 11e, 31-chapter edition

Weeks 5–16 are scaffolded but not yet written: the lecture PDFs have not been released. Each placeholder shows the syllabus topic, the matching textbook chapters, and the concepts this workbench will need to pick up from where Lecture 4 left off. When a new deck arrives, the placeholder is replaced and the build is re-run — the rest of the workbench, including the cross-lecture links, updates with it.

Where the course goes next

and what each week will connect back to

Week 5

Stock Valuation

Ross Ch. 9

Dividend discount model; the growing perpetuity first met in Lecture 2
Constant-growth (Gordon) model and where it breaks
NPVGO — growth opportunities as positive-NPV projects, tying back to Lecture 3
Price–earnings and market-to-book, the market ratios already introduced in Lecture 1

PDF pending

Week 6

Bond Valuation

Ross Ch. 8

Coupon bonds as an annuity plus a lump sum — Lecture 2's two building blocks
Yield to maturity as an IRR, with the same reinvestment caveat as Lecture 3
Term structure, spot rates, and why the Fisher equation from Lecture 4 matters here
Interest-rate risk, duration, and the premium/discount price relation

PDF pending

Week 7

Return and Risk

Ross Ch. 10–12

Holding-period returns, arithmetic vs geometric averages
Variance, covariance, diversification — the portfolio frontier
CAPM, beta, and the security market line
This is where the discount rate used throughout Lectures 3–4 finally gets its justification

PDF pending

Week 8

Tutorial

—

Problem session; no new material

PDF pending

Week 10

Cost of Capital & Efficient Capital Markets

Ross Ch. 13–14

Cost of equity via CAPM; cost of debt; WACC as a weighted average
WACC is the number the "discount rate" in Lectures 3–4 has been standing in for
Flotation costs, divisional cost of capital, and when WACC is the wrong rate
Market efficiency: the random walk, event studies, and the behavioural critique

PDF pending

Week 11

Capital Structure

Ross Ch. 16–17

Modigliani–Miller without taxes, then with taxes
Interest tax shield — the mirror image of the depreciation tax shield in Lecture 4
Financial distress costs, agency costs, pecking order, trade-off theory
Answers the "financing decision" left open at the end of Lecture 1

PDF pending

Week 12

Raising Capital

Ross Ch. 20

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

PDF pending

Week 13

Payout Policy

Ross Ch. 19

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

PDF pending

Week 14

Tutorial

—

Problem session; groups prepare presentations

PDF pending

Week 15–16

Review & Student Presentations

—

Group presentations (20% of the grade)
Course review ahead of the closed-book final

PDF pending

Delivered

these are written up in full

Week 1

Overview & Financial Statements

Ross Ch. 1–3 Written up · Lecture 1

Week 2

Time Value of Money

Ross Ch. 4 Written up · Lecture 2

Week 3

Net Present Value

Ross Ch. 5 Written up · Lecture 3

Week 4

Capital Budgeting

Ross Ch. 6–7 Written up · Lecture 4

Group project · 20%

presentations begin 1 December

The 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 1–4:

LPR reform

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.

Fintech

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.

Independent directors

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 & generative AI

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.

💬 A framing that tends to mark well

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.