Corporate Finance

The big picture

Corporate finance is the study of how a company gets money and puts it to work well. Every firm, from a food truck to a global airline, faces the same three financial questions, and the person who answers them is the financial manager. This first lesson sets the goal that guides every decision in the rest of the course: not to chase profit, but to build value.

Why does this matter? Because the tools you will learn later - discounting, valuation, the accept-or-reject rules for projects - are only useful if you know what they are aiming at. The aim is value, measured carefully, and that is where we begin.

The three big questions

A financial manager spends the whole job answering three questions, and every topic in this course is a tool for one of them.

1. Capital budgeting is the question of what long-term investments the firm should make. Should it build a new plant, launch a product, or buy a rival? Capital budgeting is deciding which long-term projects are worth their cost. Example: a coffee chain deciding whether to spend 400,000 dollars opening a new store is making a capital-budgeting decision. Module 6 is devoted to it.
2. Capital structure is the question of how to pay for those investments. What mix of debt (borrowed money the firm must repay with interest) and equity (owners' money, with no promise to repay) should it use? A firm that funds a 400,000 dollar store with 250,000 of a bank loan and 150,000 of owners' cash has chosen a capital structure.
3. Working capital management is the question of how to handle day-to-day cash, inventory, and short-term bills so the firm never runs dry. Working capital is the short-term money a firm needs to keep operating. Even a profitable firm can fail if it cannot pay this week's payroll.

Key idea: Every finance decision is really one of three questions - which projects to fund, how to fund them, and how to manage cash in the meantime.

The goal: maximize value, not profit

You might guess the goal of a business is to maximize profit, but finance is more precise. The accepted goal of financial management is to maximize the current value of the owners' stake. For a public company, that means maximizing the current share price. Shareholder value is the market value today of the owners' claim on the firm.

Why not just profit? Because profit is slippery, and it ignores three things that value captures.

• Timing. A profit ten years from now is worth less than the same profit this year, because money you have sooner can be reinvested. A dollar today beats a dollar later.
• Risk. A risky profit is worth less than a safe one. Two projects that both "earn 100,000 dollars" are not equal if one is a sure thing and the other a gamble.
• Cash. Accounting profit can differ from the cash actually in the bank. A firm can report a profit while running out of money to pay its bills.

Here is a concrete example of profit and value pulling apart. Suppose a manager cuts the research budget by 2,000,000 dollars this year. Reported profit jumps by 2,000,000 right away, which looks great. But if that research was funding the products of three years from now, the firm's future cash flows shrink, and a well-informed market marks the share price down. Profit rose; value fell. Maximizing value forces the manager to weigh timing, risk, and cash together, which raw profit never does.

Key idea: Value beats profit as a goal because value accounts for when the money arrives, how risky it is, and whether it is real cash.

Whose money is it? The agency problem

In a large corporation, the owners (the shareholders) are usually not the people running the company (the managers). Managers are hired agents of the owners, and their interests can diverge. A manager might prefer a bigger empire, a plush office, or a quiet life over the hard choices that lift the share price. The agency problem is the conflict of interest that arises when the people running a firm are not the people who own it. The costs this conflict creates - wasteful spending, missed opportunities, the money spent monitoring managers - are called agency costs.

Example: a CEO who buys a corporate jet that adds convenience for executives but little for shareholders is imposing an agency cost. Firms fight the agency problem with tools such as tying pay to the stock price (so managers gain when owners gain), oversight by a board of directors, and the discipline of takeover threats (a badly run firm can be bought and its managers replaced).

Key idea: Because managers are agents of the owners, firms must design incentives and oversight so managers actually pursue shareholder value.

Why finance is its own subject

Accounting, which you may have studied, looks backward and records what has already happened. Finance looks forward and asks what a stream of future cash flows is worth today, and whether a decision will add to that value. To do that, we need one master tool that lets us compare dollars arriving at different times. That tool is the time value of money, the idea that a dollar today is worth more than a dollar tomorrow, and it is the heart of this course. Everything from Module 2 onward is built on it.

Key idea: Finance is forward-looking valuation, and its master tool is the time value of money.

Before we start: what this course is and is not

This course teaches the analytical methods of corporate finance. It is education, not investment advice. No lesson recommends buying or selling any security, and no example should be read as a forecast. Every company, price, and rate used here is either hypothetical or a published historical figure used to demonstrate a method. Decisions about your own money belong with a qualified, licensed professional who knows your circumstances.

One more standing rule, because it is where most student errors come from: state your assumptions. Annual or monthly compounding? Cash at the end or the beginning of the period? Nominal or real? Before tax or after? Two people can compute the same problem correctly and get different answers if they assume differently, so the assumptions are part of the answer, not decoration around it.

Key idea: Finance is a set of methods; the answer is only defined once compounding, timing, and tax assumptions are stated.

Putting numbers on profit versus value

The research-cut example above deserves the arithmetic that makes it decisive. Suppose the $2,000,000 of research would have produced $700,000 a year of extra after-tax cash flow, starting in year 4 and running for 12 years, and the firm's investors require a 10% return.

Value the lost cash flows in two steps. First, the value at the end of year 3 of a 12-year stream of $700,000 discounted at 10%: the annuity factor is [1 - 1.10^-12] / 0.10. Since 1.10^12 = 3.13843, its reciprocal is 0.31863, so the factor is (1 - 0.31863) / 0.10 = 6.81369 and the value is $700,000 x 6.81369 = $4,769,584.

Second, bring that back three years to today: $4,769,584 / 1.10^3 = $4,769,584 / 1.331 = $3,583,459.

Now compare. Cutting the research saves $2,000,000 of cash today and destroys $3,583,459 of value, so the net effect on the firm is $2,000,000 - $3,583,459 = -$1,583,459. Reported profit rises by two million dollars and the firm is worth one and a half million dollars less. Both statements are true, and only one of them is the goal.

That single calculation is the whole logic of the course in miniature: identify the cash flows, place them in time, discount at a rate that reflects their risk, and add them up. Everything from Module 2 onward is machinery for doing that carefully.

Key idea: Profit and value diverge whenever a decision trades near cash for distant cash, and only discounting settles which way the trade goes.

What an agency cost actually costs

The corporate jet is a familiar illustration and rarely a quantified one. Suppose the jet costs $4,200,000 to buy and $1,800,000 a year to operate over a ten-year life, while the commercial travel it replaces would cost $250,000 a year. The incremental annual cost is $1,800,000 - $250,000 = $1,550,000.

At a 10% required return, the ten-year annuity factor is [1 - 1.10^-10] / 0.10. With 1.10^10 = 2.59374 and its reciprocal 0.38554, the factor is (1 - 0.38554) / 0.10 = 6.14457. The present value of the operating premium is $1,550,000 x 6.14457 = $9,524,079, and adding the purchase price gives a total of $13,724,079 of shareholder value consumed, before any resale value at the end.

Spread across 40,000,000 shares, that is $13,724,079 / 40,000,000 = $0.34 a share. It sounds small next to a share price, and it is exactly the point: agency costs are usually individually modest, collectively large, and almost never itemized anywhere a shareholder can see them. That is why the response is structural - incentives, boards, disclosure - rather than a line item in a budget.

Key idea: Agency costs are quantifiable as the present value of the value transferred, and they are managed structurally because they never appear as a single visible expense.

The governance machinery

Three mechanisms carry most of the load in aligning managers with owners.

• The board of directors. Elected by shareholders, the board hires and fires the chief executive, sets pay, and approves major transactions. Independence matters here: a board composed largely of the chief executive's colleagues monitors nothing.
• Compensation design. Linking pay to the share price aligns interests, and it introduces new ones. Options reward upside without punishing downside symmetrically, which can encourage excessive risk-taking; short vesting periods can reward moves that lift the price this year and hurt the firm in three. Good design uses long vesting, clawbacks, and multiple performance measures.
• Disclosure and market discipline. Public companies in the United States must file audited annual and quarterly reports and disclose material events, which lets outsiders form their own view. Legislation following the accounting scandals of the early 2000s tightened requirements on internal controls and on executive certification of financial statements. Poorly run firms also face the threat of takeover, which is the bluntest discipline of all.

Key idea: Boards, pay design, and mandatory disclosure exist to close the gap between managers' incentives and owners' interests, and each creates side effects of its own.

The same three questions, one balance sheet

The three questions are not abstract categories; they map directly onto a balance sheet. The left-hand side - the assets - is the capital-budgeting question: what does the firm own, and was it worth buying? The right-hand side - debt and equity - is the capital-structure question: who financed it, and on what terms? The short-term items on both sides, cash and receivables and inventory against payables and short-term debt, are working capital management.

A firm with $8,000,000 of long-term assets, $2,400,000 of current assets, $1,700,000 of current liabilities, $4,000,000 of long-term debt, and $4,700,000 of equity has already answered all three questions, whether or not anyone framed them that way. Net working capital is $2,400,000 - $1,700,000 = $700,000, and the debt share of long-term financing is $4,000,000 / ($4,000,000 + $4,700,000) = 46%. Those two numbers describe the firm's short-term cushion and its financing mix, and the rest of this course is about whether they are the right numbers.

Key idea: Assets are capital budgeting, long-term financing is capital structure, and the short-term items on both sides are working capital.

Common wrong turns

• "The goal is to maximize profit." No. The goal is to maximize value, which accounts for timing, risk, and cash. High reported profit can hide falling value.
• "Debt is bad and equity is safe." Neither is inherently good or bad. They are two ways to fund investment, each with trade-offs studied under capital structure.
• "If a company is profitable, it cannot go broke." False. A profitable firm can still run out of cash and fail, which is why working capital management matters.
• "Managers automatically act in owners' interests." Not automatically. The agency problem means their goals can differ, so incentives and oversight are needed.
• "The research cut is obviously wrong." It is wrong at these numbers. Change the required return to 20% or push the payoff out five more years and it can flip. Compute, do not assert.
• "Agency costs are too small to matter." The jet alone was $13.7 million of present value, or $0.34 a share, and it is one item among many.
• "Tying pay to the share price solves the agency problem." It helps and creates new problems, including incentives for short-horizon risk-taking.
• "An answer is an answer." Not until compounding, timing, and tax assumptions are stated. Two correct methods on different assumptions give different numbers.

Try it

A firm can cancel a $1,500,000 marketing investment this year. Doing so would forfeit $420,000 a year of after-tax cash flow for 8 years beginning in year 3. The required return is 12%. Separately, management proposes a corporate box at a stadium costing $600,000 up front and $95,000 a year for 6 years, against $15,000 a year of client-entertainment costs it replaces. (a) Value the forfeited cash flows as of the end of year 2. (b) Bring that to today and state whether cancelling adds or destroys value. (c) Compute the present value of the corporate box's incremental cost. (d) With 5,000,000 shares outstanding, what is that per share?

Answer: (a) 1.12^8 = 2.47596, so the 8-year annuity factor is (1 - 1/2.47596) / 0.12 = (1 - 0.40388) / 0.12 = 4.96764, and the value at end of year 2 is $420,000 x 4.96764 = $2,086,409. (b) 1.12^2 = 1.2544, so today's value is $2,086,409 / 1.2544 = $1,663,272. Cancelling saves $1,500,000 and costs $1,663,272, a net -$163,272, so it destroys value. (c) Incremental annual cost = $95,000 - $15,000 = $80,000. 1.12^6 = 1.97382, so the factor is (1 - 0.50663) / 0.12 = 4.11141, giving $80,000 x 4.11141 = $328,913, plus $600,000 = $928,913. (d) $928,913 / 5,000,000 = $0.19 a share.

Recap

• Corporate finance answers three questions: capital budgeting (which projects), capital structure (how to fund them), and working capital management (how to manage day-to-day cash).
• The goal of financial management is to maximize the current value of the owners' stake, usually the share price, not short-term profit.
• Value beats profit because it weighs timing, risk, and cash; a research cut can raise profit while lowering value.
• The agency problem is the conflict between managers and shareholders; agency costs are what it costs the firm.
• Finance is forward-looking and rests on the time value of money.
• The research cut raised profit by $2,000,000 and destroyed $3,583,459 of value, a net loss of $1,583,459.
• Boards, incentive pay, and mandatory disclosure are the structural answers to agency conflict.
• This course is education, not investment advice, and every answer depends on stated assumptions.

Sources

1. Dahlquist, J., & Knight, R. (2022). What is finance? In Principles of Finance. OpenStax, Rice University. openstax.org
2. Dahlquist, J., & Knight, R. (2022). The role of finance in an organization. In Principles of Finance. OpenStax, Rice University. openstax.org
3. Dahlquist, J., & Knight, R. (2022). Relationship between shareholders and company management. In Principles of Finance. OpenStax, Rice University. openstax.org
4. Dahlquist, J., & Knight, R. (2022). Agency issues: Shareholders and corporate boards. In Principles of Finance. OpenStax, Rice University. openstax.org
5. Dahlquist, J., & Knight, R. (2022). Role of the board of directors. In Principles of Finance. OpenStax, Rice University. openstax.org
6. U.S. Securities and Exchange Commission. (n.d.). The role of the SEC. Investor.gov ↗ Introduction to Investing. SEC Office of Investor Education and Advocacy. investor.gov
7. Berk, J., & DeMarzo, P. (2020). Corporate finance (5th ed.). Pearson. find source ↗

The big picture

Finance is forward-looking, but the raw material for its forecasts comes from a firm's financial statements. A financial manager must read them fluently, the way a doctor reads a chart. This lesson covers the three core statements, the crucial difference between accounting profit and cash, and the ratios that turn raw numbers into judgments.

Why it matters: every valuation later in the course starts from numbers on these statements. If you cannot tell profit from cash, or a healthy balance sheet from a fragile one, the fanciest discounting formula will lead you astray.

The three financial statements

Three statements matter most, and each answers a different question.

• The balance sheet is a snapshot on one date of what the firm owns and owes. The balance sheet shows assets, liabilities, and equity at a single moment. It obeys the identity Assets = Liabilities + Equity. Assets are listed roughly in order of liquidity (cash first); the right side shows how those assets were financed, by creditors (liabilities) or by owners (equity).
• The income statement measures performance over a period of time: Revenues minus Expenses equals Net Income. Net income, the "bottom line," is the accountant's measure of profit for the period.
• The statement of cash flows tracks the actual cash that moved in and out during the period, split into operating, investing, and financing activities.

Here is a tiny balance sheet to make the identity concrete. If a firm has 300,000 dollars of assets and owes 120,000 dollars, the owners' equity must be the difference: 300,000 minus 120,000 equals 180,000 dollars. The two sides always balance, which is why it is called a balance sheet.

Assets		Financed by
Total assets	300,000	Liabilities (owed to creditors)	120,000
		Equity (owners' stake)	180,000
Total	300,000	Total	300,000

Key idea: The balance sheet is a dated snapshot obeying Assets = Liabilities + Equity, while the income statement and cash-flow statement cover a span of time.

Profit is an opinion; cash is a fact

A finance manager cares about cash flow, not just net income, and the two are not the same. Cash flow is the actual money moving into or out of the firm, which can differ from accounting profit. Net income includes non-cash charges such as depreciation - spreading a long-lived asset's cost over the years it is used - and it records sales made on credit as revenue before the cash arrives.

A rough but useful measure, operating cash flow, adds the non-cash depreciation back to profit:

Operating cash flow (rough) = Net income + Depreciation

Worked example. Suppose a firm reports net income of 120,000 dollars after subtracting 30,000 dollars of depreciation. No cash actually left the building for that depreciation, so cash generated by operations is closer to 120,000 plus 30,000, which equals 150,000 dollars. Verify the logic: depreciation reduced reported profit but not the bank account, so adding it back recovers the cash. A company can look profitable on paper yet run short of cash, which is exactly why finance never loses sight of the cash-flow statement.

Key idea: Net income and cash differ because of non-cash charges like depreciation and credit sales; adding depreciation back to net income gives a rough operating cash flow.

Ratios: turning statements into judgments

Absolute dollars are hard to judge, so we use ratios that divide one figure by another. A financial ratio compares two numbers from the statements to reveal liquidity, leverage, or profitability. Four ratios recur throughout this course.

Ratio	Formula	Measures
Current ratio	Current assets / Current liabilities	Liquidity: can it pay bills soon?
Debt-to-equity	Total debt / Total equity	Leverage: how much borrowing?
Return on equity (ROE)	Net income / Total equity	Profitability per owner dollar
Profit margin	Net income / Sales	Profit kept per sales dollar

Worked example. Take a firm with current assets of 200,000 dollars and current liabilities of 100,000 dollars. Its current ratio is 200,000 divided by 100,000, which equals 2.0, meaning 2 dollars of short-term assets back every 1 dollar of short-term debt. A current ratio above 1 is generally reassuring. Now suppose the same firm earned net income of 90,000 dollars on equity of 600,000 dollars.

Its return on equity (ROE) is 90,000 divided by 600,000, which equals 0.15, or 15 percent, so the owners earned 15 cents of profit per dollar invested. If sales were 1,200,000 dollars, the profit margin is 90,000 divided by 1,200,000, which equals 0.075, or 7.5 percent.

A ratio is only meaningful in context - compared with the firm's own past, a competitor, or an industry norm - but ratios are the vocabulary finance uses to describe a company's health.

Key idea: Ratios standardize the statements: current ratio gauges short-term safety, debt-to-equity gauges leverage, and ROE and profit margin gauge profitability.

One company, one set of statements

Ratios computed from scattered examples never add up to a picture. From here on this lesson uses a single hypothetical firm, Northline Instruments, and every ratio comes from these two statements. Assumptions: the tax rate is 25%, there are 300,000 shares outstanding, the share price is $30.00, and $180,000 of dividends were paid during the year.

Income statement (year ended December 31)	Amount
Sales	$4,800,000
Cost of goods sold	2,880,000
Gross profit	1,920,000
Selling and administrative expenses	960,000
Depreciation	240,000
Operating income (EBIT)	720,000
Interest expense	120,000
Earnings before taxes	600,000
Taxes at 25%	150,000
Net income	$450,000

Balance sheet (December 31)	Amount
Cash	$180,000
Accounts receivable	600,000
Inventory	720,000
Total current assets	1,500,000
Net fixed assets	2,100,000
Total assets	$3,600,000
Accounts payable	420,000
Short-term notes payable	180,000
Total current liabilities	600,000
Long-term debt	1,200,000
Common equity	1,800,000
Total liabilities and equity	$3,600,000

Check the identity first, always: $1,500,000 + $2,100,000 = $3,600,000 of assets, and $600,000 + $1,200,000 + $1,800,000 = $3,600,000 of claims. If those two do not match, stop; nothing computed afterward will mean anything.

Key idea: Compute every ratio from one internally consistent set of statements, and verify the balance sheet balances before starting.

Liquidity and efficiency

• Current ratio = $1,500,000 / $600,000 = 2.50.
• Quick ratio = ($1,500,000 - $720,000) / $600,000 = $780,000 / $600,000 = 1.30. Stripping out inventory matters because inventory is the current asset least likely to convert to cash on demand.
• Cash ratio = $180,000 / $600,000 = 0.30.
• Inventory turnover = COGS / inventory = $2,880,000 / $720,000 = 4.00 times, so days inventory = 365 / 4.00 = 91.3 days.
• Receivables turnover = sales / receivables = $4,800,000 / $600,000 = 8.00 times, so days sales outstanding = 365 / 8.00 = 45.6 days.
• Payables turnover = COGS / payables = $2,880,000 / $420,000 = 6.86 times, so days payables = 365 / 6.86 = 53.2 days.
• Total asset turnover = $4,800,000 / $3,600,000 = 1.33 times.

Combine three of those into the cash conversion cycle: 91.3 + 45.6 - 53.2 = 83.7 days. Northline pays for inventory 83.7 days before it collects the cash from selling it, and it must finance that gap out of working capital every single cycle. Note the sign of the payables term: stretching suppliers shortens the cycle, which is why it is a lever firms reach for under cash pressure and why suppliers resent it.

Key idea: The cash conversion cycle - days inventory plus days receivable minus days payable - measures how long the firm's own money is tied up.

Leverage and coverage

• Debt ratio = total liabilities / total assets = $1,800,000 / $3,600,000 = 0.50.
• Debt-to-equity = $1,800,000 / $1,800,000 = 1.00. Using only interest-bearing debt - the $180,000 of notes plus $1,200,000 of long-term debt - it is $1,380,000 / $1,800,000 = 0.77. Both are used, they are not the same number, and an analyst who does not say which one they mean has said nothing.
• Equity multiplier = total assets / equity = $3,600,000 / $1,800,000 = 2.00.
• Times interest earned = EBIT / interest = $720,000 / $120,000 = 6.00. Operating income could fall by five-sixths before interest went unpaid.

Key idea: Leverage ratios differ depending on whether "debt" means all liabilities or only interest-bearing debt, so define the term before quoting the number.

Profitability, and where ROE comes from

• Gross margin = $1,920,000 / $4,800,000 = 40.0%.
• Operating margin = $720,000 / $4,800,000 = 15.0%.
• Net profit margin = $450,000 / $4,800,000 = 9.38%.
• Return on assets = $450,000 / $3,600,000 = 12.5%.
• Return on equity = $450,000 / $1,800,000 = 25.0%.

The gap between 12.5% and 25.0% is entirely leverage, and the DuPont decomposition makes that explicit by splitting ROE into three drivers:

ROE = Net profit margin x Asset turnover x Equity multiplier

For Northline: 0.0938 x 1.3333 x 2.00 = 0.250, or 25.0%. The first two terms multiply to net income / assets = 12.5%, which is ROA; the third term doubles it because half the assets are financed by someone other than the owners. Two firms can post the same 25% ROE for entirely different reasons - one on fat margins, one on fast turnover, one on heavy borrowing - and DuPont is what tells them apart.

The five-factor version separates the tax and interest effects: ROE = (net income / EBT) x (EBT / EBIT) x (EBIT / sales) x (sales / assets) x (assets / equity). Substituting: 0.75 x 0.8333 x 0.15 x 1.3333 x 2.00 = 0.250 again. Read left to right, those terms are the tax burden (75% of pre-tax profit survives), the interest burden (83.3% of operating profit survives interest), the operating margin, the asset turnover, and the leverage. Now a decline in ROE can be traced to whichever term moved.

Key idea: DuPont decomposes ROE into margin, turnover, and leverage - and the five-factor version adds the tax and interest burdens - so you can see which driver changed.

Market ratios and the two cash flow measures

With 300,000 shares at $30.00: EPS = $450,000 / 300,000 = $1.50; P/E = $30.00 / $1.50 = 20.0; book value per share = $1,800,000 / 300,000 = $6.00; market-to-book = $30.00 / $6.00 = 5.00. Dividends per share are $180,000 / 300,000 = $0.60, so the payout ratio is $180,000 / $450,000 = 40%, the retention ratio is 60%, and the dividend yield is $0.60 / $30.00 = 2.0%.

Finally, two cash measures that get confused. The rough version from earlier gives net income plus depreciation = $450,000 + $240,000 = $690,000. The cleaner operating measure excludes financing, computing EBIT + depreciation - taxes = $720,000 + $240,000 - $150,000 = $810,000. The $120,000 difference is exactly the interest, which the second version deliberately leaves in because interest is a financing decision, not an operating one.

Take it one step further to free cash flow to the firm: EBIT x (1 - tax rate) + depreciation - capital expenditure - increase in net working capital. If Northline spent $320,000 on capital expenditure and its net working capital rose $90,000, then FCFF = $720,000 x 0.75 + $240,000 - $320,000 - $90,000 = $540,000 + $240,000 - $320,000 - $90,000 = $370,000. That $370,000, not the $450,000 of net income, is what is genuinely available to all providers of capital - and it is the number Module 4 will discount to value the firm.

Key idea: Free cash flow to the firm is after-tax operating profit plus depreciation minus capital spending and working-capital growth, and it is usually well below net income.

Common wrong turns

• "Net income is the cash the firm made." No. Net income includes non-cash charges and credit sales; cash flow can be very different.
• "A higher current ratio is always better." Not always. A very high current ratio can mean idle cash or bloated inventory that could be put to better use.
• "Depreciation is a cash outflow each year." No. Depreciation is a non-cash accounting entry; the cash left when the asset was originally bought.
• "A single ratio tells you if a firm is healthy." False. Ratios are only meaningful compared with peers, history, or industry norms.
• "Debt-to-equity is one number." Northline's is 1.00 on all liabilities and 0.77 on interest-bearing debt. Always say which.
• "ROE of 25% means a great business." Half of Northline's ROE came from an equity multiplier of 2.00. Its ROA is 12.5%.
• "Inventory turnover uses sales." It uses cost of goods sold, because inventory is carried at cost. Using sales overstates turnover by the gross margin.
• "Free cash flow is net income plus depreciation." That ignores capital spending and working-capital growth. Northline's FCFF was $370,000 against $690,000 by that shortcut.

Try it

Harbor Tooling reports sales of $6,000,000, COGS of $3,900,000, operating expenses of $900,000, depreciation of $300,000, interest of $150,000, and a 25% tax rate. Its balance sheet shows cash $240,000, receivables $750,000, inventory $900,000, net fixed assets $2,610,000, accounts payable $600,000, notes payable $300,000, long-term debt $1,500,000, and equity $2,100,000. (a) Build the income statement down to net income and confirm the balance sheet balances. (b) Compute current, quick, and times-interest-earned ratios. (c) Compute net margin, asset turnover, equity multiplier, and verify ROE by DuPont. (d) With capital expenditure of $450,000 and net working capital up $120,000, compute FCFF.

Answer: (a) Gross profit = $2,100,000; EBIT = $2,100,000 - $900,000 - $300,000 = $900,000; EBT = $750,000; tax = $187,500; net income = $562,500. Assets = $240,000 + $750,000 + $900,000 + $2,610,000 = $4,500,000; claims = $600,000 + $300,000 + $1,500,000 + $2,100,000 = $4,500,000. Balanced. (b) Current = $1,890,000 / $900,000 = 2.10; quick = $990,000 / $900,000 = 1.10; times interest earned = $900,000 / $150,000 = 6.00. (c) Net margin = $562,500 / $6,000,000 = 9.375%; asset turnover = $6,000,000 / $4,500,000 = 1.3333; equity multiplier = $4,500,000 / $2,100,000 = 2.1429. DuPont: 0.09375 x 1.3333 x 2.1429 = 0.2679, and directly $562,500 / $2,100,000 = 26.79%. They agree. (d) FCFF = $900,000 x 0.75 + $300,000 - $450,000 - $120,000 = $675,000 + $300,000 - $570,000 = $405,000.

Recap

• The balance sheet (a snapshot, Assets = Liabilities + Equity), the income statement (Revenues minus Expenses = Net Income over a period), and the statement of cash flows are the three core statements.
• Cash flow differs from net income because of non-cash charges and credit sales; rough operating cash flow = net income + depreciation.
• The current ratio measures liquidity, debt-to-equity measures leverage, and ROE and profit margin measure profitability.
• Ratios are only meaningful in context: versus the firm's past, a competitor, or an industry norm.
• Northline's cash conversion cycle was 91.3 + 45.6 - 53.2 = 83.7 days.
• DuPont: ROE = margin x turnover x leverage, which for Northline is 0.0938 x 1.3333 x 2.00 = 25.0%.
• The five-factor version adds tax burden and interest burden: 0.75 x 0.8333 x 0.15 x 1.3333 x 2.00 = 25.0%.
• FCFF = EBIT(1 - t) + depreciation - capital expenditure - change in net working capital, which was $370,000 here.

Sources

1. Dahlquist, J., & Knight, R. (2022). The income statement. In Principles of Finance. OpenStax, Rice University. openstax.org
2. Dahlquist, J., & Knight, R. (2022). The balance sheet. In Principles of Finance. OpenStax, Rice University. openstax.org
3. Dahlquist, J., & Knight, R. (2022). The statement of cash flows. In Principles of Finance. OpenStax, Rice University. openstax.org
4. Dahlquist, J., & Knight, R. (2022). Operating cash flow and free cash flow to the firm (FCFF). In Principles of Finance. OpenStax, Rice University. openstax.org
5. Dahlquist, J., & Knight, R. (2022). Ratios: Condensing information into smaller pieces. In Principles of Finance. OpenStax, Rice University. openstax.org
6. Dahlquist, J., & Knight, R. (2022). Profitability ratios and the DuPont method. In Principles of Finance. OpenStax, Rice University. openstax.org
7. Damodaran, A. (n.d.). Operating and net margins by industry. Data: Current. NYU Stern School of Business. pages.stern.nyu.edu

The big picture

Here is the idea the whole course turns on: a dollar today is worth more than a dollar received in the future. A dollar in hand can be invested to earn interest, so it grows; a dollar merely promised next year cannot. This lesson shows how to push a sum forward in time to find its future value, and why compounding makes that growth accelerate.

Why it matters: future value is one half of the master tool of finance. Once you can grow a sum forward, the next lesson reverses the operation to value future cash today, and from there you can value bonds, stocks, and entire projects.

The time value of money

The time value of money is the principle that money available now is worth more than the same amount later, because it can earn a return. The rate at which money grows over time is the interest rate, often written r and also called the discount rate or the required return. Give a sum time and a positive interest rate, and it grows.

Key idea: Money has a time value: with a positive interest rate, a sum today is always worth more than the same sum in the future.

Compounding: earning interest on interest

Put 1,000 dollars in an account paying 10 percent per year. After one year you have 1,000 times 1.10, which equals 1,100 dollars. In year two you earn 10 percent not just on the original 1,000 but on the whole 1,100, giving 1,100 times 1.10, which equals 1,210 dollars. That extra 10 dollars - interest earned on last year's interest - is the magic of compounding. Compounding means earning interest on both the original principal and the interest already accumulated.

The general formula for the future value of a single sum is:

FV = PV times (1 + r)^n

where PV is the present amount, r is the interest rate per period, and n is the number of periods. Future value is what a present sum grows to after earning interest for n periods.

Key idea: Compounding earns interest on interest, and future value is found by multiplying the present sum by (1 + r) raised to the number of periods.

Worked example: growing a single sum

What will 1,000 dollars grow to in 3 years at 8 percent per year? State the formula, plug in, and verify.

FV = 1,000 times (1.08)^3 = 1,000 times 1.259712 = 1,259.71

The 1,000 dollars becomes 1,259.71 dollars. Verify by stepping year by year: 1,000 times 1.08 = 1,080.00; then 1,080 times 1.08 = 1,166.40; then 1,166.40 times 1.08 = 1,259.71. It checks. Of the 259.71 dollars of growth, 240 dollars is simple interest (3 years times 80 dollars) and the extra 19.71 dollars is interest earned on interest.

Simple versus compound interest

Simple interest pays only on the original principal, never on accumulated interest. So 1,000 dollars at 8 percent simple for 3 years earns a flat 1,000 times 0.08 times 3, which equals 240 dollars, reaching 1,240 dollars. Compound interest reaches 1,259.71 dollars. The 19.71-dollar gap is small over 3 years but explodes over long horizons: at 8 percent for 40 years, simple interest would triple your money while compounding multiplies it more than twentyfold.

Year	Simple interest balance	Compound balance
1	1,080.00	1,080.00
2	1,160.00	1,166.40
3	1,240.00	1,259.71

Key idea: Simple interest grows in a straight line; compound interest curves upward because it earns interest on interest, and the gap widens with time.

Compounding more often than once a year

Interest is often added more frequently than yearly. If a stated annual rate of 8 percent is compounded semiannually, each half-year earns 4 percent, over 6 periods in 3 years: FV = 1,000 times (1.04)^6 = 1,265.32 dollars, a little more than the 1,259.71 from annual compounding. The more often interest compounds, the faster money grows. The effective annual rate (EAR) captures this: at 8 percent compounded semiannually, EAR = (1.04)^2 minus 1 = 0.0816, or 8.16 percent. The effective annual rate is the true yearly rate once compounding within the year is counted.

Key idea: More frequent compounding grows money faster, and the effective annual rate states the true yearly return after that compounding.

The Rule of 72

A handy shortcut: money roughly doubles in 72 divided by r years, where r is the percentage rate. The Rule of 72 estimates the doubling time of an investment as 72 divided by its percentage interest rate. At 8 percent, that is about 72 divided by 8, which equals 9 years. Check it: 1,000 times (1.08)^9 is about 1,999 dollars, almost exactly double. The Rule of 72 lets you gauge growth in your head.

Key idea: The Rule of 72 gives a quick mental estimate of how long money takes to double at a given rate.

Every compounding frequency, side by side

Compounding frequency is the single most common source of wrong answers in time-value problems, so work it exhaustively once. Take the same $1,000 at a stated annual rate of 8% for 3 years, and change only how often interest is added. The general formula is FV = PV x (1 + r/m)^(m x n), where m is the number of compounding periods per year.

Compounding	Periods per year (m)	Rate per period	Number of periods	Future value	Effective annual rate
Annual	1	8.0000%	3	$1,259.71	8.0000%
Semiannual	2	4.0000%	6	$1,265.32	8.1600%
Quarterly	4	2.0000%	12	$1,268.24	8.2432%
Monthly	12	0.6667%	36	$1,270.24	8.3000%
Daily	365	0.021918%	1,095	$1,271.22	8.3278%
Continuous	infinite	-	-	$1,271.25	8.3287%

Two things are worth noticing. First, the whole range from annual to continuous is only $11.54 on $1,000 over three years - about 1.2%. Compounding frequency matters, and it matters less than students often fear on short horizons. Second, the gains shrink fast: going from annual to semiannual adds $5.61, from semiannual to quarterly adds $2.92, and from daily to continuous adds three cents. There is a ceiling, and continuous compounding is it.

That ceiling has a clean formula. Continuous compounding gives FV = PV x e^(r x n), so here FV = $1,000 x e^(0.08 x 3) = $1,000 x e^0.24 = $1,000 x 1.271249 = $1,271.25, matching the table. The corresponding effective annual rate is e^0.08 - 1 = 8.3287%.

Key idea: FV = PV x (1 + r/m)^(mn) covers every frequency, and continuous compounding, FV = PV x e^(rn), is the upper limit as m grows.

APR is not a rate you can compound with

The annual percentage rate (APR) quoted on a loan or account is a stated rate: it is the periodic rate multiplied by the number of periods per year, with no compounding inside it. The effective annual rate (EAR) is what you actually earn or pay once the within-year compounding is counted:

EAR = (1 + APR / m)^m - 1

A credit card quoting 21.99% APR compounded monthly has a monthly rate of 0.2199 / 12 = 0.018325, so its EAR = (1.018325)^12 - 1 = 1.24350 - 1 = 24.35%. The gap of 2.36 percentage points is real money, and it is why two products with the same APR can cost different amounts.

The comparison runs both ways. Which is better: 7.90% compounded monthly or 8.00% compounded annually? The first has EAR = (1 + 0.0790/12)^12 - 1 = (1.0065833)^12 - 1 = 8.19%, so the lower stated rate is the better deal. Never compare stated rates with different compounding frequencies; convert both to EAR first. This is the practical form of the "state your assumptions" rule from Lesson 1.

Key idea: Compare rates only after converting them all to effective annual rates, because a stated APR hides its compounding.

Solving for the rate and for the time

The future-value formula has four variables, and any one can be the unknown. Rearranging is simple algebra and it appears constantly in practice.

Solving for r. You have $1,000 and need it to become $2,500 in 7 years. What annual return is required? From FV = PV x (1 + r)^n, r = (FV / PV)^(1/n) - 1 = (2.5)^(1/7) - 1. Since ln 2.5 = 0.916291 and 0.916291 / 7 = 0.130899, we have e^0.130899 = 1.139855, so r = 13.99%. Verify: $1,000 x 1.139855^7 = $1,000 x 2.5000 = $2,500. It checks.

Solving for n. How long does $1,000 take to reach $1,800 at 6% annually? From the same formula, n = ln(FV / PV) / ln(1 + r) = ln(1.8) / ln(1.06) = 0.587787 / 0.058269 = 10.09 years. Verify: $1,000 x 1.06^10.09 = $1,800. The fractional year is not a rounding artifact; it is the honest answer, and rounding it down to 10 leaves you $8 short.

Key idea: r = (FV/PV)^(1/n) - 1 and n = ln(FV/PV) / ln(1 + r), and both should be verified by substituting back.

How good is the Rule of 72?

The exact doubling time is ln(2) / ln(1 + r) = 0.693147 / ln(1 + r). Compare that with the shortcut across a range of rates:

Rate	Rule of 72 estimate	Exact doubling time	Error
2%	36.0 years	35.00 years	+1.00 year
6%	12.0 years	11.90 years	+0.10 year
8%	9.0 years	9.01 years	-0.01 year
12%	6.0 years	6.12 years	-0.12 year
20%	3.6 years	3.80 years	-0.20 year

The rule is remarkably good between roughly 4% and 12%, drifts high at low rates, and drifts low at high rates. Use it for mental arithmetic and a sanity check; never use it in a report.

Key idea: The Rule of 72 is accurate to within about 1% for rates between 4% and 12% and degrades outside that band.

Growing in nominal dollars is not growing in purchasing power

Every figure so far is nominal - measured in future dollars, which buy less than today's. The exact relationship between nominal rates, real rates, and inflation is (1 + nominal) = (1 + real) x (1 + inflation), so real = (1 + nominal) / (1 + inflation) - 1.

At a nominal 8% with 3% inflation, the real rate is 1.08 / 1.03 - 1 = 4.854%. The familiar approximation, 8% - 3% = 5%, overstates it by 0.15 percentage points - trivial over one year and not trivial over twenty.

See it: $1,000 invested at a nominal 8% for 20 years grows to $1,000 x 1.08^20 = $4,660.96. Deflate that by 20 years of 3% inflation, dividing by 1.03^20 = 1.80611: $4,660.96 / 1.80611 = $2,580.66 in today's purchasing power. Compounding the real rate directly gives the same answer, $1,000 x 1.04854^20 = $2,580.7 to rounding. The money grew 4.66 times in dollars and 2.58 times in what those dollars buy, and only the second number tells you whether you are better off.

Key idea: Real = (1 + nominal) / (1 + inflation) - 1; the subtract-the-inflation-rate shortcut is close for one year and misleading over decades.

Common wrong turns

• "Doubling the years doubles the money." No. Growth is exponential, not linear; because of compounding, the second stretch of years adds more than the first.
• "Simple and compound interest are about the same." Only over very short horizons. Over decades the compound total dwarfs the simple total.
• "A higher stated rate always means faster growth." Not necessarily. Compounding frequency matters too; 7.90% compounded monthly beats 8.00% compounded annually.
• "The Rule of 72 is exact." It is only an approximation, best for rates roughly between 4 and 12 percent.
• "Use the annual rate with monthly periods." The rate per period is r/m and the number of periods is m x n. Mixing them is the most common arithmetic error in this course.
• "APR and EAR are the same thing." A 21.99% APR compounded monthly is a 24.34% EAR.
• "Real return equals nominal minus inflation." That is an approximation. The exact figure at 8% and 3% is 4.854%, not 5%.
• "Round the answer to whole years." $1,000 reaches $1,800 at 6% in 10.09 years, and 10 years leaves you short.

Try it

(a) What does $2,500 grow to in 5 years at 9% compounded quarterly, and what is the EAR? (b) A savings account advertises 5.40% APR compounded monthly; a bond offers 5.55% compounded annually. Which pays more? (c) What annual return turns $4,000 into $10,000 in 9 years? (d) At a nominal 7% with 2.5% inflation, what is the exact real rate, and what is $5,000 worth in today's dollars after 15 years?

Answer: (a) Rate per quarter = 0.09 / 4 = 2.25%; periods = 20; FV = $2,500 x 1.0225^20 = $2,500 x 1.560509 = $3,901.27. EAR = 1.0225^4 - 1 = 9.308%. (b) Savings EAR = (1 + 0.0540/12)^12 - 1 = (1.0045)^12 - 1 = 5.536%, below the bond's 5.55%, so the bond pays slightly more. (c) r = (10,000 / 4,000)^(1/9) - 1 = 2.5^(1/9) - 1; ln 2.5 / 9 = 0.101810, e^0.101810 = 1.107173, so r = 10.72%. (d) Real rate = 1.07 / 1.025 - 1 = 4.390%. Nominal FV = $5,000 x 1.07^15 = $5,000 x 2.759032 = $13,795.16; deflated by 1.025^15 = 1.448298 gives $13,795.16 / 1.448298 = $9,525.08 in today's dollars.

Recap

• The time value of money says a sum today beats the same sum later because it can earn a return.
• Future value of a single sum is FV = PV times (1 + r)^n; 1,000 at 8 percent for 3 years grows to 1,259.71.
• Compounding earns interest on interest, so it beats simple interest, and the gap grows with time.
• More frequent compounding raises the effective annual rate; 8 percent compounded semiannually is an EAR of 8.16 percent.
• The Rule of 72 estimates doubling time as 72 divided by the percentage rate.
• The general formula is FV = PV x (1 + r/m)^(mn), with continuous compounding FV = PV x e^(rn) as the limit.
• EAR = (1 + APR/m)^m - 1, and rates are only comparable once converted to EAR.
• Real = (1 + nominal) / (1 + inflation) - 1, so 8% nominal with 3% inflation is a 4.854% real return.

Sources

1. Dahlquist, J., & Knight, R. (2022). Now versus later concepts. In Principles of Finance. OpenStax, Rice University. openstax.org
2. Dahlquist, J., & Knight, R. (2022). Time value of money (TVM) basics. In Principles of Finance. OpenStax, Rice University. openstax.org
3. Dahlquist, J., & Knight, R. (2022). Methods for solving time value of money problems. In Principles of Finance. OpenStax, Rice University. openstax.org
4. Dahlquist, J., & Knight, R. (2022). Stated versus effective rates. In Principles of Finance. OpenStax, Rice University. openstax.org
5. U.S. Securities and Exchange Commission. (n.d.). Compound interest calculator. Financial Tools and Calculators. Investor.gov ↗. investor.gov
6. Dahlquist, J., & Knight, R. (2022). Historical picture of inflation. In Principles of Finance. OpenStax, Rice University. openstax.org
7. Board of Governors of the Federal Reserve System. (n.d.). H.15 selected interest rates. Statistical Releases. Federal Reserve. federalreserve.gov

The big picture

Future value pushes money forward in time. Present value does the reverse: it pulls a future cash flow back to today, telling you what it is worth right now. This process is called discounting, and it is the workhorse of finance, because valuing anything - a project, a bond, a company - means discounting the cash it will produce.

Why it matters: nearly every remaining topic in this course is an application of present value. Master it here and bond pricing, stock valuation, and net present value become the same move applied to different cash flows.

The present value formula

Present value is what a future dollar is worth today. For example, 100 dollars a year from now, discounted at 5 percent, is worth about 95.24 dollars now, because 95.24 times 1.05 equals 100. To find present value in general, just rearrange the future-value formula to solve for PV:

PV = FV / (1 + r)^n

The term 1 / (1 + r)^n is the discount factor, always less than 1, which shrinks a future amount down to its worth today. The discount factor is the multiplier that converts one future dollar into its present value. The rate r used here is the discount rate: the discount rate is the required return used to bring future cash flows back to the present.

Key idea: Present value reverses compounding by dividing a future sum by (1 + r)^n, so a future dollar is always worth less than a dollar today.

Worked example: discounting a single sum

How much must you invest today at 6 percent to have 5,000 dollars in 4 years? State the formula, plug in, and verify.

PV = 5,000 / (1.06)^4 = 5,000 / 1.262477 = 3,960.47

You need 3,960.47 dollars today. Verify by compounding it forward: 3,960.47 times (1.06)^4 = 3,960.47 times 1.262477, which equals 5,000.00. It checks. Put another way, a promise of 5,000 dollars in 4 years is worth only 3,960.47 dollars to you now, if 6 percent is the return you could otherwise earn - your opportunity cost of capital.

What moves present value

Two forces push present value down.

• A higher discount rate lowers PV. At 10 percent instead of 6 percent, that same 5,000 dollars in 4 years is worth only 5,000 / (1.10)^4 = 3,415.07 dollars. The more you could earn elsewhere, the less a fixed future sum is worth today.
• A longer wait lowers PV. The farther off the money, the more the discount factor shrinks it. 5,000 dollars in 10 years at 6 percent is worth just 5,000 / (1.06)^10 = 2,791.97 dollars.

5,000 dollars received in...	At 6 percent	At 10 percent
4 years	3,960.47	3,415.07
10 years	2,791.97	1,927.72

Key idea: Present value falls when the discount rate rises or the horizon lengthens; both shrink the discount factor.

Solving for the rate or the time

The same equation solves for a missing rate or horizon. If 3,960.47 dollars grows to 5,000 dollars in 4 years, the implied rate satisfies (1 + r)^4 = 5,000 / 3,960.47 = 1.2625, so r = 1.2625^(1/4) minus 1 = 0.06, confirming 6 percent. Likewise, given a rate and the two amounts, you could solve for n. This one relationship - four quantities (PV, FV, r, n) with any three giving the fourth - underlies almost every calculation in the rest of the course.

You know	You want	Use
PV, r, n	FV	FV = PV times (1 + r)^n
FV, r, n	PV	PV = FV / (1 + r)^n
PV, FV, n	r	r = (FV / PV)^(1/n) minus 1

Key idea: The present-value relationship links four quantities, and knowing any three lets you solve for the fourth.

Discounting a stream of unequal cash flows

Real investments rarely pay one lump sum. The method extends with no new ideas: discount each cash flow separately and add. Assumption stated up front - each amount arrives at the end of its year, and the discount rate is 9% annually.

Year	Cash flow	Discount factor 1/(1.09)^n	Present value
1	$12,000	0.917431	$11,009.17
2	$18,000	0.841680	$15,150.24
3	$25,000	0.772183	$19,304.59
4	$9,000	0.708425	$6,375.83
5	$30,000	0.649931	$19,497.94
Total	$94,000	-	$71,337.77

The stream totals $94,000 in raw dollars and $71,337.77 in today's money - a difference of $22,662.23, which is the price of waiting. Notice the year-5 payment of $30,000 contributes only $19,497.94, less than the year-3 payment of $25,000 in raw terms would suggest, because it sits two years deeper in the discount.

Change nothing but the rate and the answer moves substantially: at 6% the same stream is worth $77,877.76, and at 12% it is worth $65,600.75. A six-point range in the discount rate moved the value by $12,277, or 17%. This is why arguing about the discount rate is not pedantry - it is often the largest single assumption in a valuation.

Key idea: Discount each cash flow at its own horizon and sum; the total is highly sensitive to the rate you choose.

How fast distant money stops mattering

Present value decays exponentially, and the decay is faster than intuition suggests. Here is $30,000 seen from four horizons at three rates.

Received in	At 4%	At 8%	At 12%
5 years	$24,657.81	$20,417.50	$17,022.81
10 years	$20,266.93	$13,895.80	$9,659.20
20 years	$13,691.61	$6,436.45	$3,110.00
30 years	$9,249.56	$2,981.32	$1,001.34

At 12%, $30,000 arriving in 30 years is worth $1,001.34 today - about 3.3 cents on the dollar. A useful mental model is the half-life of value: present value halves every ln(2) / ln(1 + r) years, which is the same expression as doubling time. At 9% that is 8.04 years, so a cash flow 24 years out (three half-lives) is worth roughly one eighth of its face amount.

Two practical consequences follow. First, forecasting years 15 through 30 in fine detail is usually wasted effort at ordinary discount rates. Second, when a valuation does depend heavily on distant cash flows - as growth-stock valuations often do - it is depending on the part of the forecast that is least knowable, and it should be treated with corresponding suspicion.

Key idea: Present value halves roughly every ln(2)/ln(1+r) years, so distant forecasts carry little weight unless the discount rate is very low.

Match real cash flows to real rates

The most common serious error in discounting is mixing conventions: forecasting cash flows in today's purchasing power and then discounting them at a nominal rate that already contains inflation. That double-counts inflation and understates value.

There are two internally consistent routes, and they give identical answers. Suppose a project produces $50,000 a year in today's dollars for 5 years, inflation runs at 3%, and the nominal required return is 8%.

• Nominal route: inflate the cash flows, so year t receives $50,000 x 1.03^t, then discount at 8%. Summing the five terms gives $217,348.07.
• Real route: leave the cash flows at $50,000 and discount at the real rate, 1.08 / 1.03 - 1 = 4.854%. Summing gives $217,348.07.

Identical to the cent, because the inflation factor in the numerator cancels the inflation embedded in the denominator. The wrong route - $50,000 flat cash flows discounted at the nominal 8% - gives $199,635, understating value by $17,713, or 8%. Pick a convention, state it, and use it in both the numerator and the denominator.

Key idea: Discount nominal cash flows at nominal rates and real cash flows at real rates; mixing them double-counts inflation.

When cash does not arrive on December 31

The end-of-period convention used throughout is a simplification. Two adjustments come up often.

Mid-year convention. If cash flows in evenly through the year, discounting at t - 0.5 is more accurate than at t. The whole stream is then multiplied by (1 + r)^0.5, which at 9% is 1.044031 - a 4.40% uplift. The stream above rises from $71,337.77 to $74,478.82. Neither treatment is more "correct" in the abstract; what matters is knowing which you used and applying it consistently across the projects you compare.

Non-annual periods. The rule from the previous lesson holds in reverse: use the rate per period and the number of periods. What is $20,000 received in 6 years worth today at 10% compounded quarterly? Rate per quarter = 0.10 / 4 = 2.5%, periods = 24, so PV = $20,000 / 1.025^24 = $11,057.51. Discounting annually at 10% would give $20,000 / 1.10^6 = $11,289.48, overstating the value by $231.97 because it ignores the extra compounding.

Key idea: Mid-year discounting raises value by (1 + r)^0.5, and non-annual compounding requires the periodic rate with the matching period count.

Where the discount rate comes from

Nothing so far explains how to choose r, and the answer shapes everything. The discount rate is the opportunity cost of capital: the return available on an alternative investment of similar risk. Those last three words carry the weight. Discounting a risky project's cash flows at the government bond rate treats a gamble as though it were certain and will approve projects that should be rejected.

A rough hierarchy: government bond yields for near-certain cash flows, corporate borrowing rates for contractual payments a firm expects to make, and higher rates still for equity-like cash flows whose amount is genuinely uncertain. Module 6 makes this precise by building a weighted average cost of capital from the returns debt and equity holders require. For now, the discipline is simply to notice that r is a risk judgment wearing the costume of a number, and to test how much the answer changes when it moves - as the 6% to 12% comparison above did.

Key idea: The discount rate is the return available elsewhere at similar risk, so riskier cash flows must be discounted harder.

Common wrong turns

• "Present value just subtracts the interest." No. You divide by (1 + r)^n, which is not the same as subtracting a flat interest amount.
• "The discount rate is a fee." No. It is the return you could earn elsewhere on money of similar risk, the opportunity cost of capital.
• "A higher discount rate raises present value." Backward. A higher rate lowers present value, because future dollars are discounted harder.
• "Doubling the years halves the present value." Not exactly. The relationship is exponential, not linear, so the effect compounds.
• "Add the cash flows, then discount the total." Each flow must be discounted at its own horizon. Discounting $94,000 as a lump sum at year 5 gives $61,094, not $71,338.
• "Use today's dollars and the market interest rate." That mixes real cash flows with a nominal rate and understated this project's value by 8%.
• "Use the same rate for every project." The rate must match the risk. A safe cash flow and a speculative one do not share a discount rate.
• "The discount rate is a detail." Moving from 6% to 12% changed this stream's value by $12,277, or 17%.

Try it

A project pays $8,000, $14,000, $11,000, and $22,000 at the ends of years 1 to 4. (a) Compute the present value at 7%, showing each discount factor. (b) Recompute at 13% and state the percentage change. (c) Using the mid-year convention at 7%, what is the value? (d) The same $22,000 in year 4 arrives instead in year 12. What is it worth at 7%, and roughly how many value half-lives is that?

Answer: (a) Factors are 1/1.07 = 0.934579, 1/1.07^2 = 0.873439, 1/1.07^3 = 0.816298, 1/1.07^4 = 0.762895. Present values: $7,476.64 + $12,228.14 + $8,979.28 + $16,783.69 = $45,467.75. (b) At 13%: $7,079.65 + $10,964.05 + $7,623.55 + $13,493.01 = $39,160.26, a fall of 13.9%. (c) Multiply by 1.07^0.5 = 1.034408: $45,467.75 x 1.034408 = $47,032.21. (d) $22,000 / 1.07^12 = $22,000 / 2.252192 = $9,768.26. The half-life is ln(2)/ln(1.07) = 10.24 years, so 12 years is about 1.17 half-lives, consistent with the value falling slightly below half.

Recap

• Discounting pulls a future cash flow back to today; PV = FV / (1 + r)^n.
• The discount factor 1 / (1 + r)^n is always below 1, so a future dollar is worth less than a dollar today.
• 5,000 dollars in 4 years at 6 percent is worth 3,960.47 dollars today; verify by compounding it back up to 5,000.
• Present value falls as the discount rate rises or the horizon lengthens.
• PV, FV, r, and n are linked, so any three determine the fourth.
• An uneven stream is valued flow by flow: $94,000 of cash was worth $71,337.77 at 9%.
• Value halves every ln(2)/ln(1+r) years, so distant cash flows carry little weight.
• Real cash flows need real rates and nominal cash flows need nominal rates; mixing them double-counts inflation.

Sources

1. Dahlquist, J., & Knight, R. (2022). Time value of money (TVM) basics. In Principles of Finance. OpenStax, Rice University. openstax.org
2. Dahlquist, J., & Knight, R. (2022). Applications of TVM in finance. In Principles of Finance. OpenStax, Rice University. openstax.org
3. Dahlquist, J., & Knight, R. (2022). Timing of cash flows. In Principles of Finance. OpenStax, Rice University. openstax.org
4. Dahlquist, J., & Knight, R. (2022). Unequal payments using a financial calculator or Microsoft Excel. In Principles of Finance. OpenStax, Rice University. openstax.org
5. Dahlquist, J., & Knight, R. (2022). Interest rates. In Principles of Finance. OpenStax, Rice University. openstax.org
6. Board of Governors of the Federal Reserve System. (n.d.). Data. Federal Reserve Board. Federal Reserve. federalreserve.gov
7. Brealey, R. A., Myers, S. C., & Allen, F. (2020). Principles of corporate finance (13th ed.). McGraw-Hill Education. find source ↗

The big picture

Many real cash flows are not one lump sum but a series of equal payments: a car loan, a mortgage, a pension, rent. This lesson gives compact formulas that value such streams in one step, instead of discounting each payment separately, and extends them to payments that last forever.

Why it matters: loans, retirement savings, and even the stock-valuation model in Module 4 are all annuities or perpetuities in disguise. These formulas turn a page of arithmetic into a single calculation.

What is an annuity?

An annuity is a stream of equal cash payments made at equal time intervals. Rent of 1,000 dollars a month for a year, or a loan payment of 300 dollars a month for five years, are annuities. In an ordinary annuity, payments arrive at the end of each period; this is the standard case for most loans.

Key idea: An annuity is a set of equal payments at equal intervals, and by default the payments fall at the end of each period.

Present value of an ordinary annuity

Rather than discount each payment one by one, use the closed-form formula:

PV = PMT times [ 1 minus (1 + r)^(-n) ] / r

where PMT is the payment per period. PMT is the equal cash amount paid or received each period.

Worked example. You will receive 2,000 dollars at the end of each year for 5 years, and the discount rate is 7 percent. State, plug in, verify.

PV = 2,000 times [1 minus (1.07)^(-5)] / 0.07 = 2,000 times 4.100197 = 8,200.39

The five payments are worth 8,200.39 dollars today - less than the 10,000 dollars of raw payments, because most of the money arrives years from now. Verify by discounting each payment: 1,869.16 + 1,746.88 + 1,632.60 + 1,525.79 + 1,425.97 = 8,200.39 (the individual figures are rounded), which matches the formula.

Key idea: The present value of an ordinary annuity is PMT times the annuity factor, and it is smaller than the sum of the raw payments because of discounting.

Future value of an ordinary annuity

If instead you save 2,000 dollars at the end of each year for 5 years at 7 percent, the accumulated future value is:

FV = PMT times [ (1 + r)^n minus 1 ] / r

FV = 2,000 times [(1.07)^5 minus 1] / 0.07 = 2,000 times 5.750739 = 11,501.48

Saving 10,000 dollars in deposits grows to 11,501.48 dollars; the extra 1,501.48 dollars is compound interest earned along the way.

Key idea: The future value of an annuity accumulates the equal deposits plus interest, so it exceeds the total dollars deposited.

Annuity due: payments at the start

An annuity due pays at the beginning of each period, so every payment sits one period longer and earns one more period of interest. Just multiply the ordinary-annuity answer by (1 + r). The 2,000-dollar, 5-year, 7 percent stream as an annuity due is worth 8,200.39 times 1.07 = 8,774.42 dollars today - more than the ordinary annuity, because the money arrives sooner. Leases and rent are often annuities due.

Key idea: Because its payments come one period earlier, an annuity due is worth (1 + r) times the matching ordinary annuity.

Perpetuities: payments forever

A perpetuity pays a fixed amount every period forever. Remarkably, an infinite stream has a finite value, because distant payments discount to almost nothing:

PV = PMT / r

A perpetuity paying 100 dollars a year when the discount rate is 5 percent is worth 100 / 0.05 = 2,000 dollars. If the payment grows at a constant rate g each year, it is a growing perpetuity worth PMT / (r minus g). A growing perpetuity is a forever stream whose payment rises at a constant rate g, valued as PMT divided by (r minus g). A 100-dollar payment growing 3 percent a year at a 8 percent discount rate is worth 100 / (0.08 minus 0.03) = 2,000 dollars. This growing-perpetuity formula returns in Module 4 to value a share of stock.

Key idea: A level perpetuity is worth PMT / r, and a growing perpetuity is worth PMT / (r minus g), provided g is below r.

A quick application: a loan payment

The annuity formula also finds a loan payment. Borrow 200,000 dollars for a 30-year mortgage at 6 percent annual, compounded monthly (0.5 percent per month for 360 months). Solve the PV formula for PMT: PMT = 200,000 / [ (1 minus (1.005)^(-360)) / 0.005 ] = 1,199.10 dollars per month. The same relationship values a stream or, run backward, finds the payment that produces a given value.

Key idea: Loan payments are annuities, so the same present-value formula, solved for PMT, gives the monthly payment.

Inside a loan payment: the amortization schedule

The $1,199.10 mortgage payment above is constant, and what it buys changes every month. An amortization schedule splits each payment into interest and principal. Assumptions: $200,000 borrowed, 6% nominal annual rate compounded monthly, so 0.5% a month for 360 months, with payments at the end of each month.

The rule for each row is: interest = beginning balance x 0.005; principal = payment - interest; ending balance = beginning balance - principal.

Month	Beginning balance	Payment	Interest	Principal	Ending balance
1	$200,000.00	$1,199.10	$1,000.00	$199.10	$199,800.90
2	$199,800.90	$1,199.10	$999.00	$200.10	$199,600.80
3	$199,600.80	$1,199.10	$998.00	$201.10	$199,399.71
4	$199,399.71	$1,199.10	$997.00	$202.10	$199,197.61
5	$199,197.61	$1,199.10	$995.99	$203.11	$198,994.50

Read the shape of it. In month 1, $1,000.00 of the $1,199.10 payment is interest and only $199.10 reduces the debt - 16.6% of the payment. Five months in, the borrower has paid $5,995.50 and cut the balance by $1,005.50. The principal portion grows every month, slowly at first and then quickly, because each dollar of principal repaid permanently removes 0.5% a month of future interest. The schedule is front-loaded with interest not by design or unfairness but by arithmetic: interest is charged on what is still owed, and early on that is nearly everything.

Key idea: Each payment is interest on the outstanding balance plus whatever is left over, so early payments are mostly interest and late payments mostly principal.

What the term costs

Over 360 months the borrower pays 360 x $1,199.10 = $431,676.00 for a $200,000 loan, so total interest is $231,676.00 - more than the amount borrowed.

Shorten the term to 15 years and rerun the formula: PMT = $200,000 x 0.005 / [1 - 1.005^-180] = $200,000 / 118.5035 = $1,687.71 a month. Total paid is 180 x $1,687.71 = $303,788.46, so total interest is $103,788.46.

Paying $1,687.71 - $1,199.10 = $488.61 more each month saves $231,676.00 - $103,788.46 = $127,887.54 of interest. That is not a free lunch - the extra $488.61 a month could have been invested elsewhere, and comparing the two properly means discounting both streams - but it does show how strongly interest cost depends on how long the balance stays outstanding.

Key idea: Total interest depends on how long principal is outstanding, so shortening the term cuts interest far more than proportionally.

Growing annuities and deferred annuities

Two variants cover most remaining cases.

A growing annuity pays an amount that rises at rate g for n periods:

PV = PMT / (r - g) x [ 1 - ((1 + g) / (1 + r))^n ]

Value a career: $60,000 in the first year, rising 3% annually for 25 years, discounted at 7%. PV = $60,000 / (0.07 - 0.03) x [1 - (1.03 / 1.07)^25] = $1,500,000 x [1 - 0.38578] = $1,500,000 x 0.61422 = $921,335. Note the structure: as n grows very large the bracketed term approaches 1 and the expression collapses to $60,000 / 0.04 = $1,500,000, the growing perpetuity. The finite version is always the perpetuity minus a tail.

A deferred annuity starts later than next period. Suppose $5,000 a year arrives at the ends of years 6 through 15, with r = 8%. Value it in two steps. First, the ten-year annuity factor is [1 - 1.08^-10] / 0.08 = 6.710081, so the stream is worth $5,000 x 6.710081 = $33,550.41 as of the end of year 5 - one period before the first payment, which is where the annuity formula always places its answer. Second, discount that back five years: $33,550.41 / 1.08^5 = $33,550.41 / 1.469328 = $22,833.84.

The step that trips students is the timing of the intermediate answer. The annuity formula returns a value one period before the first payment, not at the first payment. Getting that wrong here would multiply the answer by 1.08 and overstate it by $1,827.

Key idea: The annuity formula values a stream as of one period before its first payment, which is what makes deferred annuities a two-step problem.

Solving for the number of payments

Sometimes the payment is fixed by what you can afford and the question is how long it takes. Rearranging the present-value formula gives:

n = -ln(1 - PV x r / PMT) / ln(1 + r)

Borrow $20,000 at 9% APR compounded monthly (0.75% a month) and pay $450 a month. Then PV x r / PMT = $20,000 x 0.0075 / $450 = $150 / $450 = 0.33333, so n = -ln(0.66667) / ln(1.0075) = 0.405465 / 0.0074720 = 54.26 months, or about four and a half years.

The formula also shows when a loan never gets repaid. If the monthly payment is less than or equal to PV x r - here $150 - the term inside the logarithm is zero or negative and no finite n exists: the payment does not even cover the interest and the balance grows. That is the arithmetic behind negative amortization, and it is worth being able to spot in one line.

Key idea: A loan is only repayable if the payment exceeds the interest charged on the opening balance; otherwise the balance grows without limit.

Why a perpetuity is not worth infinity

The claim that an endless stream has a finite value deserves a demonstration rather than an assertion. At a 5% discount rate, the perpetuity factor is 1 / 0.05 = 20.0. The 30-year annuity factor is [1 - 1.05^-30] / 0.05 = 15.37. So the first 30 years of an infinite stream account for 15.37 / 20.0 = 76.9% of its entire value, and everything from year 31 to eternity accounts for the remaining 23.1%.

This is the same exponential decay from the previous lesson, and it explains why perpetuity formulas are practical rather than fanciful. It also carries a warning that matters in Module 4: the perpetuity value is PMT / r, and when r and g are close, the denominator (r - g) becomes small and the value explodes. At r = 8% and g = 3%, a $100 payment is worth $2,000; at g = 7% it is worth $10,000; at g = 7.9% it is worth $100,000. Small changes in an unknowable growth rate produce enormous changes in value, which is a fact about the formula and a caution about how much weight it can bear.

Key idea: Perpetuities are finite because value decays exponentially, but the growing version is dangerously sensitive when g approaches r.

Common wrong turns

• "An annuity's present value equals the sum of the payments." No. Because later payments are discounted, PV is less than the raw total.
• "An infinite stream must be worth infinity." No. Distant payments discount toward zero, so a perpetuity has a finite value, PMT / r.
• "Ordinary annuity and annuity due give the same answer." They differ by a factor of (1 + r); the due version is larger because payments come earlier.
• "The growing-perpetuity formula works for any g." Only when g is less than r. If g is at or above r, the formula breaks down.
• "The annuity formula gives the value at the first payment." It gives the value one period before the first payment, which is the whole difficulty of deferred annuities.
• "Half the payment is interest and half is principal." In month 1 of the mortgage, 83.4% was interest. The split changes every month.
• "Use the annual rate for a monthly loan." Use 0.5% for 360 periods, not 6% for 30. The second gives a payment of $14,529 a year, a different question entirely.
• "A small change in g barely matters." Moving g from 3% to 7.9% with r = 8% raised a perpetuity's value fiftyfold.

Try it

A car loan of $35,000 runs 5 years at 6.6% APR compounded monthly. (a) Compute the monthly payment. (b) Build the first three rows of the amortization schedule. (c) Compute total interest over the loan. (d) Separately, value a 30-year income stream starting at $42,000 and growing 2.5% a year, discounted at 8%.

Answer: (a) Monthly rate = 0.066 / 12 = 0.55%; n = 60. PMT = $35,000 x 0.0055 / [1 - 1.0055^-60] = $686.46. (b) Month 1: interest = $35,000 x 0.0055 = $192.50, principal = $493.96, balance = $34,506.04. Month 2: interest = $34,506.04 x 0.0055 = $189.78, principal = $496.68, balance = $34,009.37. Month 3: interest = $187.05, principal = $499.41, balance = $33,509.97. (c) Total paid = 60 x $686.46 = $41,187.35, so interest = $6,187.35. (d) PV = $42,000 / (0.08 - 0.025) x [1 - (1.025 / 1.08)^30] = $763,636 x [1 - 0.20835] = $604,456.

Recap

• An annuity is a stream of equal payments; ordinary annuities pay at period end.
• PV of an ordinary annuity = PMT times [1 minus (1 + r)^(-n)] / r; 2,000 for 5 years at 7 percent is 8,200.39.
• FV of an ordinary annuity = PMT times [(1 + r)^n minus 1] / r; the same stream saved is 11,501.48.
• An annuity due is worth (1 + r) times the ordinary annuity, because payments come earlier.
• A perpetuity is worth PMT / r, and a growing perpetuity PMT / (r minus g).
• An amortization schedule splits each payment into interest on the balance and principal; month 1 of the mortgage was 83.4% interest.
• The 30-year mortgage cost $231,676 of interest against $103,788 for the 15-year version.
• A growing annuity is PMT/(r-g) x [1 - ((1+g)/(1+r))^n], and a deferred annuity is a two-step discount.

Sources

1. Dahlquist, J., & Knight, R. (2022). Perpetuities. In Principles of Finance. OpenStax, Rice University. openstax.org
2. Dahlquist, J., & Knight, R. (2022). Annuities. In Principles of Finance. OpenStax, Rice University. openstax.org
3. Dahlquist, J., & Knight, R. (2022). Loan amortization. In Principles of Finance. OpenStax, Rice University. openstax.org
4. Dahlquist, J., & Knight, R. (2022). Equal payments with a financial calculator and Excel. In Principles of Finance. OpenStax, Rice University. openstax.org
5. Consumer Financial Protection Bureau. (n.d.). What is negative amortization? Ask CFPB. CFPB. consumerfinance.gov
6. Consumer Financial Protection Bureau. (n.d.). Understand the different kinds of loans available. Owning a Home. CFPB. consumerfinance.gov
7. Dahlquist, J., & Knight, R. (2022). Applications of TVM in finance. In Principles of Finance. OpenStax, Rice University. openstax.org

The big picture

We can now value a whole investment. The method is discounted cash flow: forecast every cash flow a project will produce, discount each back to today, and add them up. Compared against what the project costs, this gives its net present value, the single most important tool in capital budgeting.

Why it matters: NPV converts a messy stream of future cash into one number in today's dollars that answers the manager's core question - does this project add value? Every accept-or-reject decision in the course rests on it.

Discounted cash flow

Discounted cash flow (DCF) values an asset by forecasting its future cash flows and discounting them to the present. It is simply present value applied to a whole series of cash flows, which we already know how to handle from the annuity lesson. The only new step is subtracting the cost.

Key idea: DCF is present value applied to every cash flow a project produces.

The NPV formula

Net present value is the present value of a project's cash inflows minus its cost; it is the value the project creates. If a project costs CF0 today (a cash outflow, so a negative number) and returns cash flows CF1, CF2, and so on through CFn over its life, then at discount rate r:

NPV = CF0 + CF1/(1+r) + CF2/(1+r)^2 + ... + CFn/(1+r)^n

Key idea: NPV sums the discounted inflows and subtracts the upfront cost, all in today's dollars.

Worked example

A project costs 10,000 dollars today and is expected to generate 4,000, 5,000, and 6,000 dollars at the end of years 1, 2, and 3. The firm's discount rate is 10 percent. Discount each inflow, then subtract the cost.

Year	Cash flow	Discount factor at 10 percent	Present value
1	4,000	1 / 1.10 = 0.9091	3,636.36
2	5,000	1 / 1.10^2 = 0.8264	4,132.23
3	6,000	1 / 1.10^3 = 0.7513	4,507.89
Total present value of inflows			12,276.48

Now subtract the cost:

NPV = minus 10,000 + 12,276.48 = 2,276.48

The NPV is positive 2,276.48 dollars. Verify the three present values add up: 3,636.36 + 4,132.23 + 4,507.89 = 12,276.48, and 12,276.48 minus 10,000 = 2,276.48. It checks.

The NPV rule

The decision rule is beautifully simple. The NPV rule says accept a project if its NPV is positive and reject it if its NPV is negative.

• If NPV is greater than 0, accept the project - it is worth more than it costs and adds value.
• If NPV is less than 0, reject it - it destroys value.
• If NPV equals 0, the project exactly earns its required return; you are indifferent.

Because our project has NPV of positive 2,276.48 dollars, the firm should accept it. That figure is not abstract: it is the dollar amount by which the project is expected to increase the value of the firm today, in current dollars, above and beyond the 10 percent return the firm demanded. NPV directly measures value created, which is exactly the manager's goal from Module 1.

Key idea: Accept positive-NPV projects and reject negative-NPV ones, because NPV is the value added in today's dollars.

The profitability index

A closely related measure, the profitability index (PI), divides the present value of inflows by the initial cost. The profitability index is the present value of inflows divided by the upfront cost; above 1 means a positive NPV. For our project, PI = 12,276.48 / 10,000 = 1.23, so each dollar invested returns about 1.23 dollars of present value. A PI above 1 always corresponds to a positive NPV, so the two rules agree for a single project. PI is handy when comparing projects of different sizes under a fixed budget.

Key idea: The profitability index scales NPV per dollar invested; PI above 1 is the same signal as NPV above 0.

Why the discount rate matters

NPV depends on the rate you discount at. Raise the rate and future inflows are worth less, so NPV falls; lower it and NPV rises. Discount the same project at 9 percent instead of 10 percent and the inflows are worth more, lifting NPV to about 2,511 dollars. This is why choosing the right discount rate - the firm's cost of capital, which Module 6 develops - is as important as forecasting the cash flows.

Key idea: A higher discount rate lowers NPV and a lower one raises it, so the choice of rate is central.

The NPV profile

Rather than argue about a single discount rate, plot NPV against a range of them. The NPV profile of the example project looks like this:

Discount rate	NPV
0%	$5,000.00
5%	$3,527.70
10%	$2,276.48
15%	$1,204.08
20%	$277.78
25%	-$528.00
30%	-$1,233.50

Three readings. At 0% the NPV is simply the sum of the raw cash flows, -$10,000 + $15,000 = $5,000, which is a useful check on your spreadsheet. The curve slopes down and it is convex, not straight - the drop from 0% to 5% is $1,472 while the drop from 25% to 30% is $706. And it crosses zero somewhere between 20% and 25%; the exact crossing is 21.65%, which is the internal rate of return that the next lesson takes up.

The profile also tells you how much comfort you have. This project stays positive as long as the true cost of capital is below 21.65%, which is a wide margin against a 10% estimate. A project whose NPV turned negative at 11% would be a very different proposition even with the same NPV at 10%.

Key idea: The NPV profile shows how the verdict depends on the rate, and the rate at which it crosses zero measures your margin for error.

Which cash flows count

Getting the discounting right is the easy half. The hard half is deciding what belongs in the cash flows at all. The governing principle is incremental cash flow: include every cash flow that changes because the project goes ahead, and nothing else.

Consider a firm evaluating a new product line costing $500,000 up front and generating $180,000 a year of operating cash flow for 6 years at an 11% cost of capital. Four items are in dispute:

Item	Include?	Why
$80,000 market study, already paid	No	Sunk. The money is gone whatever is decided, so it cannot change the decision.
$30,000 a year of warehouse rent forgone	Yes	Opportunity cost. The space could earn $30,000 elsewhere, so using it costs that much.
$45,000 a year of lost contribution on an existing line	Yes	Erosion. The firm's total cash flow rises by less than the new line's own cash flow.
$60,000 a year of interest on the loan	No	Financing cost. It is already captured in the discount rate; subtracting it too would double-count.

The correct incremental cash flow is $180,000 - $30,000 - $45,000 = $105,000 a year. The 6-year annuity factor at 11% is [1 - 1.11^-6] / 0.11 = 4.230538, so NPV = -$500,000 + $105,000 x 4.230538 = -$500,000 + $444,206 = -$55,794. Reject.

Now see what carelessness costs. Using the raw $180,000 instead gives NPV = -$500,000 + $180,000 x 4.230538 = +$261,497. Accept. The two errors - forgetting the opportunity cost and forgetting the erosion - swing the answer by $317,290 and reverse the decision. Discounting was performed correctly in both cases.

Key idea: Include opportunity costs and erosion, exclude sunk costs and financing costs, and note that only the second pair feels counterintuitive.

NPVs add up

A property that gets used constantly without being named: value additivity. The NPV of two projects taken together equals the sum of their separate NPVs, because present value is a linear operation.

Split the example project into two pieces. Project A costs $6,000 and returns $3,000 a year for 3 years; project B costs $4,000 and returns $1,000, $2,000, and $3,000. At 10%, NPV(A) = $1,460.56 and NPV(B) = $815.93, summing to $2,276.48. Combining the cash flows first, -$10,000 with inflows of $4,000, $5,000, and $6,000, gives $2,276.48. Identical.

This is why a firm can evaluate projects one at a time rather than having to value every possible combination, and it is why "this project only makes sense as part of the bigger plan" deserves scrutiny. If the pieces genuinely interact - shared equipment, cannibalized sales - then the interaction is itself an incremental cash flow and belongs in the analysis explicitly, not smuggled in as a reason not to compute.

Key idea: NPVs are additive, so projects can be evaluated separately unless a genuine interaction exists, which must then be modelled as an incremental cash flow.

Payback, and why NPV replaced it

Payback period is the time until cumulative cash flows recover the initial cost. For the example project, $4,000 and $5,000 in years 1 and 2 leave $1,000 outstanding against a year-3 flow of $6,000, so payback is 2 + $1,000 / $6,000 = 2.17 years.

The measure has three defects, in ascending order of seriousness. It ignores the time value of money, treating a dollar in year 3 as equal to a dollar today. It ignores everything after the cutoff, so a project returning $1,000,000 in year 4 scores identically to one returning nothing. And its cutoff is arbitrary: nothing in finance says three years is the right threshold, so the rule can be tuned to approve whatever was already wanted.

Discounted payback fixes only the first. Cumulative present values here are $3,636.36 after year 1 and $7,768.60 after year 2, leaving $2,231.40 to recover from a year-3 present value of $4,507.89, so discounted payback is 2 + $2,231.40 / $4,507.89 = 2.50 years. Better, and still blind to everything after the cutoff.

Payback survives in practice because it is a rough liquidity measure - how long is our money exposed? - and surveys of chief financial officers find it still widely used alongside NPV and IRR. Treat it as a supplementary risk indicator, never as the decision rule.

Key idea: Payback ignores discounting, ignores cash after the cutoff, and uses an arbitrary threshold; discounted payback repairs only the first defect.

Testing the forecast

NPV is only as good as the cash flows fed into it, so the professional habit is to shake each input and see what moves. Suppose the year-3 inflow of $6,000 turns out 20% lower, at $4,800. The loss in present value is $1,200 / 1.10^3 = $901.58, so NPV falls from $2,276.48 to $1,374.90 - still comfortably positive.

Compare that with the discount rate. Moving from 10% to 20% cut NPV from $2,276.48 to $277.78, a fall of 88%. In this project the discount rate is a far more dangerous input than any single cash-flow estimate, which is a common pattern and a good reason to spend as much effort on the cost of capital as on the revenue forecast. Sensitivity analysis, scenario analysis, and break-even analysis are all versions of this one habit: find out which assumption the answer actually rests on.

Key idea: Vary each input to find which one the decision hinges on; it is often the discount rate rather than the cash flows.

Common wrong turns

• "A big NPV means a big percentage return." Not necessarily. NPV is a dollar amount; a huge project can have a large NPV yet a modest percentage return.
• "Add up the raw cash flows and subtract the cost." No. You must discount each inflow first; undiscounted sums ignore the time value of money.
• "NPV of zero means the project is bad." No. NPV of zero means the project earns exactly the required return, so it neither adds nor destroys value.
• "The discount rate barely affects the answer." It can flip the decision; small rate changes move NPV noticeably.
• "We already spent $80,000 on the study, so we should proceed." Sunk costs are irrelevant. The $80,000 is gone whichever way the decision goes.
• "Subtract the loan interest from the project's cash flows." That double-counts financing, which is already in the discount rate.
• "The new line will make $180,000, so count $180,000." Count what the firm gains, which here was $105,000 after opportunity cost and erosion.
• "Payback under three years means accept." Payback ignores discounting and everything after the cutoff, and the cutoff itself is arbitrary.

Try it

A machine costs $24,000 and produces $9,000, $9,500, $8,000, and $7,000 at the ends of years 1 to 4. (a) Compute NPV at 8%. (b) Compute NPV at 14%. (c) Compute the profitability index at 8%. (d) Compute plain payback, and state the rate at which NPV becomes zero.

Answer: (a) Present values are $8,333.33 + $8,144.00 + $6,350.66 + $5,145.93 = $27,973.92, so NPV = $27,973.92 - $24,000 = $3,973.92. (b) At 14% the present values total $24,749.01, so NPV = $749.01. (c) PI = $27,973.92 / $24,000 = 1.166, above 1, consistent with the positive NPV. (d) After 2 years cumulative cash is $18,500, leaving $5,500 against a year-3 flow of $8,000, so payback = 2 + $5,500 / $8,000 = 2.69 years. NPV reaches zero at 15.58%, which is the project's internal rate of return.

Recap

• DCF forecasts and discounts every project cash flow; NPV then subtracts the cost.
• NPV = CF0 + CF1/(1+r) + ... + CFn/(1+r)^n, all in today's dollars.
• The example project (cost 10,000; inflows 4,000, 5,000, 6,000 at 10 percent) has NPV of positive 2,276.48.
• Accept if NPV is positive, reject if negative; a profitability index above 1 gives the same verdict.
• NPV falls as the discount rate rises, so the rate matters as much as the cash flows.
• The NPV profile crosses zero at 21.65% for this project, which measures the margin for error in the discount rate.
• Include opportunity costs and erosion, exclude sunk and financing costs; here that swung NPV by $317,290.
• Payback of 2.17 years and discounted payback of 2.50 years are liquidity indicators, not decision rules.

Sources

1. Dahlquist, J., & Knight, R. (2022). Net present value (NPV) method. In Principles of Finance. OpenStax, Rice University. openstax.org
2. Dahlquist, J., & Knight, R. (2022). Payback period method. In Principles of Finance. OpenStax, Rice University. openstax.org
3. Dahlquist, J., & Knight, R. (2022). Choosing between projects. In Principles of Finance. OpenStax, Rice University. openstax.org
4. Dahlquist, J., & Knight, R. (2022). Timing of cash flows. In Principles of Finance. OpenStax, Rice University. openstax.org
5. Graham, J. R., & Harvey, C. R. (2001). The theory and practice of corporate finance: Evidence from the field. Journal of Financial Economics, 60(2-3), 187-243. faculty.fuqua.duke.edu
6. Damodaran, A. (n.d.). Data: Current year. Useful Data Sets. NYU Stern School of Business. pages.stern.nyu.edu
7. Lo, A. W. (2008). Finance theory I (MIT 15.401). MIT OpenCourseWare. ocw.mit.edu

The big picture

NPV answers "how much value?" in dollars. Managers often also want a percentage return, and that is the internal rate of return. This lesson defines the IRR, shows how to apply it against a hurdle rate, and explains the traps that make finance treat NPV as the gold standard.

Why it matters: the IRR is the single most quoted number in investing - venture funds, private equity, and corporate project reviews all speak in IRR. Knowing what it does and does not tell you keeps you from being fooled by a big percentage.

What IRR means

The internal rate of return is the discount rate that makes a project's NPV exactly zero. It is the break-even rate at which the project's inflows just cover its cost. Formally, the IRR is the rate r-star that solves:

0 = CF0 + CF1/(1+r-star) + ... + CFn/(1+r-star)^n

Take the same project from the last lesson: cost 10,000 dollars, inflows of 4,000, 5,000, and 6,000. We found its NPV is positive at 10 percent. As we raise the discount rate, NPV falls; the rate at which it hits zero is the IRR. Solving (by trial or a financial calculator) gives an IRR of about 21.65 percent. We can check it near the answer: at 21 percent the NPV is a small positive 107.70, and at 22 percent it is a small negative 57.76, so the crossing point sits between - about 21.65 percent.

Key idea: IRR is the discount rate that zeroes out NPV, found where the NPV curve crosses zero.

A cleaner example

IRR is easiest to see with one future cash flow. If you invest 1,000 dollars today and receive 1,120 dollars in one year, the IRR solves 1,000 = 1,120 / (1 + r-star), giving r-star = 1,120 / 1,000 minus 1 = 0.12, or exactly 12 percent. Verify: at 12 percent, the present value of 1,120 is 1,120 / 1.12 = 1,000, so NPV is zero. The project earns a 12 percent return.

Key idea: With a single future cash flow, the IRR is just the growth rate that links the outlay to the payoff.

The IRR rule

Compare the IRR to the firm's required return, also called the hurdle rate or cost of capital. The hurdle rate is the minimum acceptable return a project's IRR must beat.

• If IRR is greater than the required return, accept - the project earns more than the firm demands.
• If IRR is less than the required return, reject.

Our project's IRR of 21.65 percent comfortably beats the 10 percent hurdle, so we accept - the same verdict NPV gave. The IRR rule is to accept a project when its IRR exceeds the required return. For a single, standard project, NPV and IRR always agree, because "NPV positive at 10 percent" and "IRR above 10 percent" are two ways of saying the same thing.

Key idea: Accept when IRR beats the hurdle rate; for one ordinary project this matches the NPV verdict exactly.

Where IRR can mislead

IRR is intuitive but has traps that NPV avoids.

• Scale. A tiny project can have a huge IRR yet add little dollar value. A 50 percent return on 100 dollars adds less than a 15 percent return on 1,000,000 dollars.
• Ranking mutually exclusive projects. When you must choose one of several projects, IRR can rank them differently from NPV; when they conflict, trust NPV, because it measures value directly. Mutually exclusive projects are ones where choosing one rules out the others.
• Unusual cash flows. A project whose cash flows switch sign more than once can have multiple IRRs or none at all. Multiple IRRs arise when cash flows change sign more than once, producing more than one break-even rate.

For these reasons finance treats NPV as the gold standard and IRR as a useful companion. When the two disagree, follow NPV.

Key idea: IRR can mislead on scale, on ranking, and on projects with sign changes, so NPV is the tiebreaker.

Estimating an IRR by trial

Without a calculator, you can bracket the IRR. Consider a project costing 8,000 dollars that returns 3,000 dollars a year for 4 years. Try 18 percent: NPV is slightly positive. Try 19 percent: NPV is slightly negative. So the IRR is between, near 18.45 percent. At exactly 18.5 percent the NPV is about negative 7.65 dollars, so the true IRR is just under 18.5 percent. Since 18.45 percent beats a 12 percent hurdle, accept. Trial-and-error bracketing is exactly what a calculator automates.

Key idea: You can find an IRR by testing rates until NPV changes sign, then narrowing the bracket.

Conflict one: scale

The abstract warning about scale becomes vivid with numbers. A firm with a 10% cost of capital must choose one of two mutually exclusive projects, each lasting a single year.

Project	Cost today	Cash in one year	IRR	NPV at 10%
S (small)	$50,000	$70,000	40.00%	$13,636.36
L (large)	$500,000	$650,000	30.00%	$90,909.09

IRR ranks S first at 40% against 30%. NPV ranks L first, by $77,272.73. Only one of these rankings can be followed, and NPV is right, because the firm's owners are made better off by dollars of value, not by percentages. A 40% return on $50,000 is a smaller prize than a 30% return on $500,000.

IRR can be rescued if you use it correctly, through incremental analysis. Ask not "which project has the higher IRR?" but "is the extra investment worth making?" The incremental project L - S costs an extra $450,000 today and returns an extra $580,000 in a year, so its incremental IRR is $580,000 / $450,000 - 1 = 28.89%. That comfortably beats the 10% hurdle, so the extra investment is worthwhile and L wins - the same answer NPV gave. The incremental NPV is $77,272.73, exactly the gap between the two NPVs.

Key idea: With mutually exclusive projects of different size, compare the incremental cash flows, not the raw IRRs.

Conflict two: timing

Scale is not the only source of disagreement. Two projects of identical size can be ranked differently because of when their cash arrives. Both of these cost $100,000, and the cost of capital is 10%.

Project	Year 0	Year 1	Year 2	Year 3	Total cash	IRR	NPV at 10%
E (early)	-$100,000	$80,000	$40,000	$10,000	$130,000	20.20%	$13,298.27
T (late)	-$100,000	$5,000	$25,000	$135,000	$165,000	19.85%	$26,634.11

IRR prefers E, by 0.35 percentage points. NPV prefers T, by $13,335.84 - twice the value. This is a genuine, unavoidable conflict, and it happens because the two rules answer different questions.

The reconciliation is again incremental. T - E has cash flows of $0, -$75,000, -$15,000, and +$125,000, and its IRR is 19.49%. That is the crossover rate: below it T is worth more, above it E is worth more. Check the boundary: at 19.49% the two NPVs are $828.07 and $823.70, essentially equal. Since the firm's actual cost of capital is 10%, well below 19.49%, the extra investment in T pays and T should be chosen.

Discount rate	NPV of E	NPV of T	Winner
5%	$21,110.03	$44,055.72	T
10%	$13,298.27	$26,634.11	T
15%	$6,386.13	$12,016.11	T
19.49%	$828.07	$823.70	tie
22%	-$2,044.66	-$4,759.65	E

Key idea: Timing conflicts flip the ranking at a crossover rate, which is the IRR of the incremental cash flows.

Why IRR favors front-loaded projects

The reason for the disagreement is a hidden assumption. Setting NPV to zero at rate r implicitly treats every intermediate cash flow as reinvested at r until the project ends. So project E's 20.20% IRR assumes its $80,000 year-1 receipt earns 20.20% for two more years, which is only true if the firm has another 20.20% opportunity waiting. NPV makes the more modest assumption that intermediate cash is reinvested at the cost of capital, 10%, which is by definition what the firm can actually get.

The modified internal rate of return (MIRR) makes the assumption explicit. Compound every inflow forward to the final year at the reinvestment rate, then find the single rate linking the initial outlay to that terminal value.

• Project E: terminal value = $80,000 x 1.10^2 + $40,000 x 1.10 + $10,000 = $96,800 + $44,000 + $10,000 = $150,800, so MIRR = (150,800 / 100,000)^(1/3) - 1 = 14.67%.
• Project T: terminal value = $5,000 x 1.21 + $25,000 x 1.10 + $135,000 = $6,050 + $27,500 + $135,000 = $168,550, so MIRR = (168,550 / 100,000)^(1/3) - 1 = 19.01%.

MIRR now ranks T above E, agreeing with NPV. Notice also that both MIRRs sit well below the corresponding IRRs, which is the general result whenever the IRR exceeds the cost of capital: the headline IRR flatters the project by assuming reinvestment on terms the firm does not have.

Key idea: IRR implicitly assumes reinvestment at the IRR itself; MIRR replaces that with the cost of capital and generally agrees with NPV.

When there is no single IRR

A project whose cash flows change sign more than once can have several IRRs. Consider a mining project: it costs $16,000 to open, yields $100,000 of ore in year 1, and requires $100,000 of site restoration in year 2. Two sign changes, minus to plus to minus.

Setting NPV to zero and writing x = 1 / (1 + r) gives -16,000 + 100,000x - 100,000x^2 = 0, which simplifies to x^2 - x + 0.16 = 0. The quadratic formula gives x = (1 +/- 0.6) / 2, so x = 0.8 or x = 0.2 - that is, r = 25% or r = 400%. Both are genuine IRRs and neither means anything on its own.

Discount rate	NPV
10%	-$7,735.54
25%	$0.00
100%	$9,000.00
400%	$0.00
500%	-$2,111.11

At the firm's actual 10% cost of capital the NPV is -$7,735.54 and the project should be rejected - even though someone quoting "an IRR of 25%, well above our 10% hurdle" would recommend it. NPV never has this ambiguity, because there is one NPV for one discount rate. A useful rule of thumb: count the sign changes in the cash flows; if there is more than one, do not quote an IRR at all.

Key idea: Each sign change can add an IRR; with more than one sign change the IRR rule is undefined and only NPV is safe.

What practitioners actually do

Despite all of this, IRR remains extremely common in practice. Large surveys of chief financial officers have found NPV and IRR used at broadly similar rates, with payback still widely used alongside them, especially at smaller firms. The reason is communicative rather than analytical: "this project returns 21.65%" travels through an organization more easily than "this project has an NPV of $2,276.48 at a 10% cost of capital," even though the second statement is the one that answers the question.

The workable professional position is to compute NPV to decide, quote IRR to communicate, use incremental IRR whenever choosing between alternatives, and refuse to quote any IRR on a project with multiple sign changes.

Key idea: Decide with NPV, communicate with IRR, compare with incremental IRR, and stay silent about IRR when the cash flows change sign more than once.

Common wrong turns

• "The higher the IRR, the better the project." Not for dollar value. A small project can post a giant IRR yet create little wealth; NPV captures size.
• "IRR and NPV always agree." They agree for one standard project, but can conflict when ranking mutually exclusive projects.
• "Every project has exactly one IRR." Projects with more than one sign change can have multiple IRRs or none.
• "IRR already accounts for the cost of capital." No. IRR is a property of the cash flows; you must compare it to the hurdle rate yourself.
• "Pick the higher IRR when choosing between two projects." That chose the $13,636 project over the $90,909 one on scale, and the $13,298 project over the $26,634 one on timing.
• "IRR is a return you will actually earn." Only if intermediate cash can be reinvested at the IRR. MIRR at the cost of capital gave 14.67% where IRR gave 20.20%.
• "An IRR above the hurdle means accept." The mining project had an IRR of 25% against a 10% hurdle and an NPV of -$7,735.54.
• "Take the difference in IRRs to size the advantage." IRRs do not subtract meaningfully. Take the difference in cash flows and compute the incremental IRR.

Try it

A firm with a 12% cost of capital must choose between one-year projects A (cost $90,000, returns $120,000) and B (cost $300,000, returns $375,000). Separately it is offered project C with cash flows of -$5,000, +$30,000, and -$40,000 in years 0 to 2. (a) Compute each IRR and NPV for A and B. (b) Which does IRR pick, which does NPV pick, and what is the incremental IRR? (c) Compute the MIRR of project B, reinvesting at 12%. (d) Find both IRRs of project C and its NPV at 20%.

Answer: (a) A: IRR = $120,000 / $90,000 - 1 = 33.33%; NPV = -$90,000 + $120,000 / 1.12 = $17,142.86. B: IRR = $375,000 / $300,000 - 1 = 25.00%; NPV = -$300,000 + $375,000 / 1.12 = $34,821.43. (b) IRR picks A, NPV picks B. Incremental B - A is -$210,000 now and +$255,000 in a year, so incremental IRR = $255,000 / $210,000 - 1 = 21.43%, which beats 12%, confirming B. (c) With one inflow at year 1 there is nothing to reinvest, so MIRR equals the IRR, 25.00%. (d) Writing x = 1/(1+r): -5,000 + 30,000x - 40,000x^2 = 0, or 40,000x^2 - 30,000x + 5,000 = 0, so x^2 - 0.75x + 0.125 = 0 and x = 0.5 or 0.25, giving IRRs of 100% and 300%. NPV at 20% = -$5,000 + $25,000 - $27,777.78 = -$7,777.78, so reject.

Recap

• IRR is the discount rate that makes NPV zero; for the example project it is about 21.65 percent.
• With one future cash flow (1,000 in, 1,120 out), the IRR is exactly 12 percent.
• The IRR rule accepts a project when IRR exceeds the required return (hurdle rate).
• IRR can mislead on scale, ranking, and sign changes, so NPV is the gold standard.
• An IRR can be found by trial: bracket the rate where NPV flips sign.
• Scale conflict: IRR chose a $13,636 NPV over a $90,909 one; incremental IRR of 28.89% resolves it.
• Timing conflict: E and T cross over at 19.49%, and below that rate the later-paying project wins.
• Multiple sign changes give multiple IRRs; the mining project had IRRs of 25% and 400% and an NPV of -$7,735.54 at 10%.

Sources

1. Dahlquist, J., & Knight, R. (2022). Internal rate of return (IRR) method. In Principles of Finance. OpenStax, Rice University. openstax.org
2. Dahlquist, J., & Knight, R. (2022). Alternative methods. In Principles of Finance. OpenStax, Rice University. openstax.org
3. Dahlquist, J., & Knight, R. (2022). Choosing between projects. In Principles of Finance. OpenStax, Rice University. openstax.org
4. Dahlquist, J., & Knight, R. (2022). Net present value (NPV) method. In Principles of Finance. OpenStax, Rice University. openstax.org
5. Graham, J. R., & Harvey, C. R. (2001). The theory and practice of corporate finance: Evidence from the field. Journal of Financial Economics, 60(2-3), 187-243. faculty.fuqua.duke.edu
6. Lo, A. W. (2008). Finance theory I (MIT 15.401). MIT OpenCourseWare. ocw.mit.edu
7. Brealey, R. A., Myers, S. C., & Allen, F. (2020). Principles of corporate finance (13th ed.). McGraw-Hill Education. find source ↗

The big picture

A bond is a loan sliced into a tradable security. This lesson identifies a bond's cash flows, prices it by discounting them, and explains the single most important fact about bonds: prices and interest rates move in opposite directions.

Why it matters: bonds are how governments and large firms borrow trillions of dollars, and their pricing is pure discounted cash flow. Once you can price a bond, you understand a huge slice of the financial markets and the cost of debt used later in the course.

What a bond is

When a firm or government issues a bond, it promises the holder two things: a fixed interest payment called the coupon each period, and repayment of the face value (or par value, usually 1,000 dollars) at the maturity date. The coupon is the fixed periodic interest payment; the face value is the amount repaid at maturity. Because a bond is just a set of future cash flows, we value it the way we value everything else - by discounting.

Key idea: A bond pays fixed coupons over its life and returns its face value at maturity, so it is a stream of known cash flows.

The bond pricing formula

A bond's price is the present value of its coupons (an annuity) plus the present value of its face value (a single sum), both discounted at the market's required return, the yield to maturity. The yield to maturity (YTM) is the market's required return on the bond, the discount rate that sets its price.

Price = C times [1 minus (1 + y)^(-n)] / y + Face / (1 + y)^n

where C is the coupon per period, y is the yield per period, and n is the number of periods.

Key idea: A bond's price is the discounted value of its coupon annuity plus the discounted value of its face amount.

Worked example

Value a 3-year bond with a 1,000-dollar face value and a 5 percent annual coupon (so C = 50 dollars) when the market yield is 6 percent. Discount the two pieces, then add.

Component	Calculation	Present value
Coupons (50 for 3 yrs)	50 times [1 minus (1.06)^(-3)] / 0.06	133.65
Face value (1,000 in 3 yrs)	1,000 / (1.06)^3	839.62
Bond price		973.27

The bond is worth 973.27 dollars, below its 1,000-dollar face value. Verify the pieces add: 133.65 + 839.62 = 973.27. It trades at a discount because its 5 percent coupon is stingier than the 6 percent the market now demands, so buyers will only pay less than par to make up the difference. A discount bond is priced below par because its coupon rate is below the market yield.

Rates and prices move in opposite directions

This is the central fact of bond investing: when interest rates rise, bond prices fall, and vice versa. The coupon is fixed, so the only way an old bond can offer a competitive return when rates change is for its price to adjust. The pattern is:

• Coupon rate greater than yield gives a price above par (a premium bond). A premium bond is priced above par because its coupon exceeds the market yield. An 8 percent coupon at a 6 percent yield on our 3-year bond prices at 1,053.46 dollars.
• Coupon rate equal to yield gives a price equal to par. A 6 percent coupon at a 6 percent yield prices at exactly 1,000 dollars.
• Coupon rate less than yield gives a price below par (a discount bond), as in our example.

Coupon vs. yield	3-year bond price	Trades at
8 percent coupon, 6 percent yield	1,053.46	Premium
6 percent coupon, 6 percent yield	1,000.00	Par
5 percent coupon, 6 percent yield	973.27	Discount

Key idea: Bond prices move inversely to yields; a coupon above the yield sells at a premium, below it at a discount, and equal to it at par.

Maturity, current yield, and semiannual coupons

Longer-maturity bonds swing more when rates change, because more of their cash flow is discounted over more periods. A useful quick measure is the current yield, the annual coupon divided by the price: for our discount bond, 50 / 973.27 = 5.14 percent, above the 5 percent coupon rate but below the 6 percent YTM (the YTM also counts the gain toward par at maturity).

Most bonds pay coupons twice a year; you simply halve the coupon and yield and double the number of periods. For example, an 8 percent coupon bond yielding 6 percent over 3 years, paid semiannually, uses C = 40, y = 3 percent, n = 6, and prices at 1,054.17 dollars. The logic never changes: a bond is worth the discounted value of the cash it will pay.

Key idea: Longer maturities are more rate-sensitive, current yield is a rough coupon-to-price gauge, and semiannual bonds just split the period.

The whole price-yield curve, one bond

The inverse relationship deserves to be seen across a range rather than asserted. Here is the same 3-year, $1,000-face, 5%-annual-coupon bond priced at six different market yields.

Market yield	Price	Change from previous	Trades at
3%	$1,056.57	-	Premium
4%	$1,027.75	-$28.82	Premium
5%	$1,000.00	-$27.75	Par
6%	$973.27	-$26.73	Discount
7%	$947.51	-$25.76	Discount
8%	$922.69	-$24.82	Discount

The price falls with every rate rise, as expected, and the size of each fall shrinks: $28.82, then $27.75, down to $24.82. The price-yield relationship is a curve, not a line, and it is convex - bowed toward the origin. Practically, this means a bond loses less when rates rise than it gains when rates fall by the same amount, which is a small structural advantage to owning bonds.

Key idea: The price-yield relationship is inverse and convex, so equal rate rises cause progressively smaller price falls.

Maturity is the lever on rate sensitivity

Take three bonds, all with a 5% coupon, all priced at par when the yield is 5%, and move the yield to 6%.

Maturity	Price at 5%	Price at 6%	Percentage change
3 years	$1,000.00	$973.27	-2.67%
10 years	$1,000.00	$926.40	-7.36%
30 years	$1,000.00	$862.35	-13.76%

The same one-point rate rise costs the 30-year holder five times what it costs the 3-year holder. The reason is straightforward: the long bond's cash flows sit further out, so each is discounted by a factor that changes more. A zero-coupon bond is the extreme case, since it has only one cash flow at the very end - a 10-year zero at a 6% yield is worth $1,000 / 1.06^10 = $558.39 and moves more than any coupon bond of the same maturity.

Key idea: Longer maturity means greater price sensitivity to rates, and a zero-coupon bond is the most sensitive of all at a given maturity.

Duration: measuring the sensitivity

Macaulay duration is the weighted average time until a bond's cash flows arrive, weighting each period by the share of total present value it represents. Compute it for the 3-year, 5% bond at a 6% yield, whose price is $973.27.

Year	Cash flow	Present value at 6%	Weight (PV / price)	Year x weight
1	$50	$47.17	0.048465	0.048465
2	$50	$44.50	0.045722	0.091444
3	$1,050	$881.60	0.905813	2.717438
Total	-	$973.27	1.000000	2.8573

Macaulay duration is 2.857 years - less than the 3-year maturity, because two coupons arrive earlier. Modified duration converts this into a sensitivity: 2.857 / 1.06 = 2.696, meaning a one-percentage-point rise in yield should cut the price by about 2.696%.

Test it. Predicted change = -2.696% x $973.27 = -$26.24. The actual price at 7% is $947.51, an actual change of -$25.76. Duration overstated the loss by $0.48, and it always will for a rate rise, because duration is the straight-line approximation to a curved relationship. That gap is exactly the convexity noted above. For small rate moves duration is an excellent shortcut; for large ones it needs a convexity correction.

Key idea: Modified duration is the percentage price change per one-point yield change, and it slightly overstates losses because the true relationship is convex.

Working backward: finding the yield to maturity

In practice you observe the price and want the yield. Suppose the same 3-year, 5% bond trades at $960. The YTM solves $960 = $50 x [1 - (1+y)^-3] / y + $1,000 / (1+y)^3, which has no closed-form solution.

A quick approximation is available: YTM is roughly [C + (F - P) / n] / [(F + P) / 2] = [$50 + ($1,000 - $960) / 3] / [($1,000 + $960) / 2] = [$50 + $13.33] / $980 = 6.463%. Iterating properly gives 6.511%, so the approximation is 4.8 basis points low - close enough to start the search and not close enough to quote.

Note also the relationship among the three yield measures for this bond: the coupon rate is 5.00%, the current yield is $50 / $960 = 5.21%, and the yield to maturity is 6.51%. They rank in that order for any discount bond, because YTM alone counts the $40 capital gain earned as the price pulls toward par at maturity. For a premium bond the order reverses.

Key idea: YTM must be found iteratively; for a discount bond, coupon rate is below current yield, which is below YTM.

Semiannual coupons, done properly

Most corporate and government bonds pay twice a year, and the convention is worth stating exactly: halve the annual coupon, halve the quoted annual yield, and double the number of years. Price a 10-year, $1,000-face bond with a 6% annual coupon when the quoted yield is 7%.

Set C = $30, y = 3.5%, n = 20. The annuity factor is [1 - 1.035^-20] / 0.035 = 14.212403, so the coupons are worth $30 x 14.212403 = $426.37. The face value is worth $1,000 / 1.035^20 = $502.57. The price is $928.94.

Two cautions. First, the quoted 7% is a bond-equivalent yield - twice the semiannual rate - not an effective annual rate. The effective annual yield is 1.035^2 - 1 = 7.1225%. Second, using the annual figures instead (C = $60, y = 7%, n = 10) gives $929.76, which is $0.82 different here and can be far more on longer or higher-coupon bonds. Say which convention you are using.

Key idea: Semiannual pricing halves the coupon and yield and doubles the periods, and the quoted yield is twice the semiannual rate, not an effective annual rate.

The other risks in a bond

Interest-rate risk is only one of several, and a bondholder faces at least three more.

• Reinvestment risk. The YTM assumes every coupon is reinvested at the YTM until maturity. If rates fall, coupons are reinvested at less and the realized return falls below the promised YTM. Interest-rate risk and reinvestment risk work in opposite directions, which is the idea behind duration matching.
• Default (credit) risk. The issuer may not pay. The market charges for this through a credit spread over comparable government debt: if a 10-year Treasury yields 4.20% and a corporate bond of the same maturity yields 6.10%, the spread is 190 basis points. Spreads widen sharply in recessions, so corporate bond prices fall for two reasons at once.
• Yield-curve and inflation risk. Rates differ by maturity, and the shape of that yield curve changes over time; a bond's price depends on the rate at its own maturity, not on a single market rate. Inflation erodes the real value of fixed payments, which is why a fixed 5% coupon is a very different proposition at 2% inflation than at 6%.

Key idea: A bond's yield compensates for interest-rate, reinvestment, default, and inflation risk together, and the credit spread isolates the default portion.

Common wrong turns

• "A bond always sells for its face value." No. It sells at par only when the coupon rate equals the yield; otherwise it trades at a premium or discount.
• "Rising interest rates are good for existing bondholders." Backward. Rising rates push existing bond prices down, hurting current holders.
• "The coupon rate is the return you earn." Not if you buy above or below par. The yield to maturity, not the coupon rate, is the return from buying at the market price and holding to maturity.
• "Current yield equals yield to maturity." They differ whenever price is not par, because YTM also includes the pull toward face value.
• "All bonds fall by the same amount when rates rise." A one-point rise cost the 3-year bond 2.67% and the 30-year bond 13.76%.
• "Duration is the same as maturity." The 3-year bond had a duration of 2.857 years. Only a zero-coupon bond has duration equal to maturity.
• "A quoted 7% semiannual yield is a 7% annual return." It is twice the semiannual rate; the effective annual yield is 7.1225%.
• "YTM is the return you will get." Only if you hold to maturity, the issuer pays, and every coupon is reinvested at the YTM.

Try it

A 5-year, $1,000-face bond pays a 4.5% annual coupon. (a) Price it at a 6% market yield and at a 3% market yield, and say which is a premium and which a discount. (b) Compute its current yield at the 6% price. (c) Price it at 6% assuming semiannual coupons instead. (d) Its Macaulay duration at the 6% yield is 4.572 years. Compute modified duration and predict the price change if yields rise one point, then compare with the actual price at 7%.

Answer: (a) At 6%: coupons = $45 x [1 - 1.06^-5] / 0.06 = $45 x 4.212364 = $189.56, face = $1,000 / 1.06^5 = $747.26, price = $936.81, a discount. At 3%: $45 x 4.579707 = $206.09 plus $862.61 = $1,068.70, a premium. (b) Current yield = $45 / $936.81 = 4.80%. (c) C = $22.50, y = 3%, n = 10: $22.50 x 8.530203 = $191.93 plus $1,000 / 1.03^10 = $744.09, giving $936.02. (d) Modified duration = 4.572 / 1.06 = 4.313, so the predicted change is -4.313% x $936.81 = -$40.40. The actual price at 7% is $897.50, a change of -$39.31. Duration overstates the loss by $1.09, the convexity effect.

Recap

• A bond pays fixed coupons plus its face value at maturity; its price is the present value of those cash flows at the YTM.
• The 3-year, 5 percent-coupon bond at a 6 percent yield prices at 973.27, a discount.
• Prices move inversely to yields: coupon above yield gives a premium, below gives a discount, equal gives par.
• Longer maturities are more rate-sensitive; current yield is coupon over price.
• Semiannual bonds halve the coupon and yield and double the periods.
• The price-yield curve is convex, so successive one-point rate rises cost progressively less.
• Modified duration of 2.696 predicted a $26.24 fall against an actual $25.76 for a one-point rise.
• For a discount bond, coupon rate is below current yield, which is below yield to maturity.

Sources

1. Dahlquist, J., & Knight, R. (2022). Characteristics of bonds. In Principles of Finance. OpenStax, Rice University. openstax.org
2. Dahlquist, J., & Knight, R. (2022). Bond valuation. In Principles of Finance. OpenStax, Rice University. openstax.org
3. Dahlquist, J., & Knight, R. (2022). Using the yield curve. In Principles of Finance. OpenStax, Rice University. openstax.org
4. Dahlquist, J., & Knight, R. (2022). Risks of interest rates and default. In Principles of Finance. OpenStax, Rice University. openstax.org
5. U.S. Securities and Exchange Commission. (n.d.). Bonds. Investor.gov ↗ Glossary. SEC Office of Investor Education and Advocacy. investor.gov
6. U.S. Department of the Treasury. (n.d.). Daily treasury par yield curve rates. Interest Rate Statistics. U.S. Department of the Treasury. home.treasury.gov
7. TreasuryDirect. (n.d.). Treasury bonds. Marketable Securities. U.S. Department of the Treasury. treasurydirect.gov

The big picture

A share of common stock is part ownership of a company. This lesson shows why a stock's value is the present value of its future dividends, applies the constant-growth model that makes an endless dividend stream tractable, and marks where the model strains.

Why it matters: stock valuation is where the growing-perpetuity formula from the time-value module pays off, and it is the foundation of the cost of equity used in Module 6. The same discounting logic that priced a bond now prices a share.

Why a stock is worth its future dividends

Common stock is a security representing partial ownership of a corporation and a claim on its dividends. Its cash flows to an investor are the dividends the company pays, plus whatever price the share can eventually be sold for. A dividend is a cash distribution a company pays to shareholders out of earnings. But that future selling price itself depends on the dividends the next owner expects. Follow the logic to its end and a share's value today is simply the present value of all future dividends, discounted at the return r that investors require for the stock's risk.

Key idea: Because any future sale price is itself the value of later dividends, a share is worth the present value of all the dividends it will ever pay.

The constant-growth (Gordon) model

Forecasting an endless dividend stream sounds impossible, but one assumption tames it: suppose dividends grow at a constant rate g forever. Then the stream is a growing perpetuity, and its value collapses to the elegant Gordon growth model, also called the dividend discount model:

P0 = D1 / (r minus g)

Here D1 is next year's expected dividend, r is the required return, and g is the constant dividend growth rate (which must be less than r for the formula to work). The Gordon growth model prices a stock as next year's dividend divided by the required return minus the growth rate.

Key idea: If dividends grow at a constant rate g forever, the stock is worth D1 / (r minus g), the growing-perpetuity formula applied to dividends.

Worked example

A company is expected to pay a dividend of 2.00 dollars next year. Investors require a 10 percent return, and the dividend is expected to grow 4 percent per year forever. State, plug in, and interpret.

P0 = 2.00 / (0.10 minus 0.04) = 2.00 / 0.06 = 33.33

The share is worth 33.33 dollars. Notice how sensitive this is to the inputs: because we divide by the small number (r minus g), a growth estimate that is even one point too high sharply inflates the value. If D0 (this year's dividend just paid) were given instead of D1, we would first grow it: D1 = D0 times (1 + g). For instance, a just-paid dividend of 2.00 and 4 percent growth gives D1 = 2.00 times 1.04 = 2.08 dollars.

Growth g	D1 / (r minus g) at r = 10 percent, D1 = 2.00	Price
0 percent	2.00 / 0.10	20.00
4 percent	2.00 / 0.06	33.33
6 percent	2.00 / 0.04	50.00

Key idea: The Gordon price is very sensitive to the gap (r minus g); a higher growth assumption raises the price sharply.

Zero-growth stock

If dividends never grow (g = 0), the growing perpetuity becomes an ordinary perpetuity and the formula simplifies to P0 = D / r. A stock paying a level 3-dollar dividend forever, with a required return of 12 percent, is worth 3 / 0.12 = 25 dollars per share. Preferred stock, which pays a fixed dividend, is often valued exactly this way.

Key idea: With zero growth the price is simply D / r, the same formula used for a level perpetuity and for preferred stock.

The required return implied by price

Rearranging the Gordon model shows where a stock's expected return comes from: r = D1 / P0 + g. The first term, D1 / P0, is the dividend yield; the second, g, is the expected capital-gain rate. The dividend yield is next year's dividend divided by the current price. For our 33.33-dollar stock, the dividend yield is 2.00 / 33.33 = 6 percent, and with 4 percent growth the total required return is 6 + 4 = 10 percent, exactly the r we started with. So return equals income plus growth.

Key idea: The required return splits into a dividend yield plus a growth rate: r = D1 / P0 + g.

Where the model strains

The dividend discount model is powerful but assumes a company pays dividends and grows them steadily - untrue for a young firm that pays nothing yet, or one whose growth is lumpy. In those cases analysts forecast the early years explicitly and apply the constant-growth formula only to the stable years that follow, or turn to other approaches such as discounting free cash flow. The core idea, though, is unshaken: a stock is worth the present value of the cash it will return to its owners.

Key idea: The constant-growth model fails for non-dividend or irregular-growth firms, but the underlying present-value principle still holds.

How dangerous the growth assumption is

The sensitivity mentioned above is worth seeing in full, because it is the reason serious analysts distrust single-stage models. Holding D1 = $2.00 and r = 10%:

Growth g	r - g	Price	Change from previous
3%	0.07	$28.57	-
4%	0.06	$33.33	+$4.76
5%	0.05	$40.00	+$6.67
6%	0.04	$50.00	+$10.00
7%	0.03	$66.67	+$16.67
8%	0.02	$100.00	+$33.33
9%	0.01	$200.00	+$100.00

The first extra point of growth adds $4.76 and the last adds $100.00. As g approaches r the price rises without limit, and at g equal to or above r the formula produces a negative or infinite number - not a very expensive stock, but a broken equation. Any valuation whose answer depends on the difference between two estimated rates, each uncertain by a percentage point, is reporting the analyst's assumptions rather than the company's value.

The economic discipline behind this is simple: no firm can grow faster than the economy forever, because it would eventually become the economy. A long-run g above nominal GDP growth is not a forecast, it is an arithmetic impossibility with a decimal point in it.

Key idea: Gordon prices explode as g approaches r, so a perpetual growth rate above long-run economy-wide growth is never defensible.

Where growth comes from

Rather than guess g, derive it. The sustainable growth rate is what a firm can grow at using only retained earnings, without changing its leverage:

g = ROE x retention ratio

Return to Northline Instruments from Lesson 2: ROE was 25.0% and the payout ratio 40%, so the retention ratio is 60% and g = 0.25 x 0.60 = 15.0%. That is far above any plausible required return, which tells you immediately that the constant-growth model cannot be applied to Northline as it stands. The firm may well grow 15% for a few years; it cannot do so forever.

The formula also connects back to DuPont. Since ROE = margin x turnover x leverage, growth is ultimately funded by profitability, asset efficiency, and borrowing. A firm that wants faster growth without new equity must improve one of those three or pay out less.

Key idea: g = ROE x retention ties the growth assumption to the firm's actual profitability and payout instead of leaving it as a guess.

The two-stage model

The standard fix is to forecast a high-growth phase explicitly and apply the Gordon formula only to the stable phase that follows. Value Northline's shares assuming its $0.60 dividend grows 15% for four years and then 4% forever, with a required return of 11%.

Year	Dividend	Discount factor at 11%	Present value
1	$0.6900	0.900901	$0.6216
2	$0.7935	0.811622	$0.6440
3	$0.9125	0.731191	$0.6672
4	$1.0494	0.658731	$0.6913
Subtotal	-	-	$2.6241

Next, the terminal value. The year-5 dividend is $1.0494 x 1.04 = $1.0914, so the value of everything from year 5 onward, measured at the end of year 4, is $1.0914 / (0.11 - 0.04) = $15.5911. Discount that back four years: $15.5911 x 0.658731 = $10.2704.

Total value per share = $2.6241 + $10.2704 = $12.89.

Two observations. First, the terminal value is 79.6% of the total - which is typical, and is why the terminal growth assumption deserves more scrutiny than the detailed forecast years. Second, $12.89 sits far below Northline's $30.00 market price, so either the market expects much more than four years of 15% growth, or a lower required return, or the valuation's assumptions are wrong. A model that disagrees with the market is a question, not a verdict.

Key idea: A two-stage model forecasts the high-growth years explicitly and applies Gordon to the stable phase, with the terminal value usually dominating the answer.

Valuing the cash flow instead of the dividend

Most firms do not pay out everything they could, and many pay nothing, so analysts often discount free cash flow instead. Two versions, both using Northline's Lesson 2 figures.

Free cash flow to equity (FCFE) is what is left for shareholders after debt is serviced: FCFF - after-tax interest + net new borrowing. Northline's FCFF was $370,000, its interest $120,000 at a 25% tax rate gives after-tax interest of $90,000, and suppose it borrowed a net $50,000. Then FCFE = $370,000 - $90,000 + $50,000 = $330,000, or $1.10 per share on 300,000 shares. Growing at 4% and discounted at an 11% cost of equity: $1.10 x 1.04 / (0.11 - 0.04) = $16.34 per share.

Free cash flow to the firm (FCFF) values the whole enterprise, so it is discounted at the weighted average cost of capital and the debt is subtracted afterward. With FCFF of $370,000 growing 4% and a WACC of 9%: enterprise value = $370,000 x 1.04 / (0.09 - 0.04) = $7,696,000. Subtracting $1,380,000 of interest-bearing debt leaves $6,316,000 of equity, or $21.05 per share.

Three methods, three answers: $12.89, $16.34, and $21.05. They differ because they rest on different assumptions about payout, borrowing, and discount rate, not because two of them are arithmetic errors. Presenting a single valuation number without its assumptions attached conceals exactly this spread, and the honest output of a valuation is a range with the reasons for its width.

Key idea: Discount dividends at the cost of equity, FCFE at the cost of equity, or FCFF at the WACC and subtract debt - and expect a range, not a number.

What a price-earnings ratio implies

Divide the Gordon model by next year's earnings and a useful identity appears: P0 / E1 = payout ratio / (r - g). Northline's payout is 40%, so at r = 11% and g = 4% the model implies a P/E of 0.40 / 0.07 = 5.71. Its actual P/E is 20.0.

Run the identity backward to see what the market believes. If the P/E is 20 and the payout 40% at an 11% required return, then 20 = 0.40 / (0.11 - g), so 0.11 - g = 0.02 and g = 9%. The market is pricing Northline for 9% perpetual dividend growth. Whether that is plausible is now a concrete question about the business - can a firm with these margins and this asset turnover sustain 9% growth indefinitely? - rather than a vague debate about whether the stock is expensive.

This is the most useful thing multiples do. A P/E is not a valuation; it is a compressed statement of assumptions, and reverse-engineering it turns a number into an argument you can examine.

Key idea: P/E = payout / (r - g), so any multiple can be inverted to reveal the growth rate the market is assuming.

A word on market efficiency

If markets process public information quickly, prices already reflect what is publicly knowable, and a valuation model built from public data should usually land near the market price. When it does not, the more likely explanations are that your growth or discount-rate assumption differs from the consensus, or that your model omits something, rather than that you have found a mispricing. The evidence on how efficient markets are is genuinely mixed and is a live research area, but the practical stance for a student is humility about disagreements with the market price.

As stated in Lesson 1, this course is education, not investment advice, and nothing here recommends buying or selling anything. The models are tools for understanding what a price implies, not instructions for acting on it.

Key idea: A model that disagrees with the market price is usually revealing your assumptions rather than a mispricing.

Common wrong turns

• "A stock's value is just its current earnings or price-to-earnings ratio." The model says value is the present value of future dividends (cash to owners), not a single accounting number.
• "The growth rate can be anything." In the Gordon model g must be below r; otherwise the denominator is zero or negative and the price is meaningless.
• "A company that pays no dividends is worthless." No. It can be valued on the dividends or free cash flow it is expected to pay eventually.
• "Use this year's dividend D0 in the formula." Use next year's expected dividend D1; if given D0, grow it by (1 + g) first.
• "Our firm grows 15%, so use g = 15%." The sustainable growth rate applies to the near term. A perpetual g must sit below long-run economy-wide growth.
• "The terminal value is a rounding detail." It was 79.6% of the two-stage valuation here.
• "Discount FCFF at the cost of equity." FCFF belongs to all capital providers, so it is discounted at the WACC, and debt is subtracted from the result.
• "A P/E of 20 means overvalued." It means the market assumes 9% growth at these inputs. Argue with the assumption, not the number.

Try it

A company just paid a $1.20 dividend. Analysts expect 12% dividend growth for three years, then 3% forever. The required return is 9%. (a) Compute the dividends for years 1 to 3 and their present values. (b) Compute the terminal value at the end of year 3 and its present value. (c) Compute the share price and the terminal value's share of it. (d) If instead a naive single-stage model were applied with g = 3% from the start, what price results, and why is it lower?

Answer: (a) D1 = $1.3440, D2 = $1.5053, D3 = $1.6859. Present values at 9%: $1.2330 + $1.2670 + $1.3018 = $3.8018. (b) D4 = $1.6859 x 1.03 = $1.7365, so TV = $1.7365 / (0.09 - 0.03) = $28.9415, and its present value is $28.9415 / 1.09^3 = $22.3482. (c) P0 = $3.8018 + $22.3482 = $26.15, of which the terminal value is $22.3482 / $26.15 = 85.5%. (d) Single-stage: $1.20 x 1.03 / (0.09 - 0.03) = $20.60. It is $5.55 lower because it discards the three years of 12% growth, which permanently raise the base from which the 3% perpetuity grows.

Recap

• A share is worth the present value of all its expected future dividends.
• The Gordon growth model gives P0 = D1 / (r minus g); with D1 = 2.00, r = 10 percent, g = 4 percent, the price is 33.33.
• With zero growth, P0 = D / r; a level 3-dollar dividend at 12 percent is worth 25.
• Required return splits into dividend yield plus growth: r = D1 / P0 + g.
• The model must be adapted for firms with no dividends or irregular growth.
• Sustainable growth is ROE times the retention ratio, which for Northline was 15%.
• The two-stage model gave $12.89 a share, of which 79.6% was terminal value.
• P/E = payout / (r - g), so Northline's P/E of 20 implies 9% perpetual growth at an 11% required return.

Sources

1. Dahlquist, J., & Knight, R. (2022). Multiple approaches to stock valuation. In Principles of Finance. OpenStax, Rice University. openstax.org
2. Dahlquist, J., & Knight, R. (2022). Dividend discount models (DDMs). In Principles of Finance. OpenStax, Rice University. openstax.org
3. Dahlquist, J., & Knight, R. (2022). Discounted cash flow (DCF) model. In Principles of Finance. OpenStax, Rice University. openstax.org
4. Dahlquist, J., & Knight, R. (2022). Preferred stock. In Principles of Finance. OpenStax, Rice University. openstax.org
5. Dahlquist, J., & Knight, R. (2022). Efficient markets. In Principles of Finance. OpenStax, Rice University. openstax.org
6. Damodaran, A. (n.d.). Implied equity risk premiums for the S&P 500. Useful Data Sets. NYU Stern School of Business. pages.stern.nyu.edu
7. U.S. Securities and Exchange Commission. (n.d.). Dividend. Investor.gov ↗ Glossary. SEC Office of Investor Education and Advocacy. investor.gov

The big picture

Every valuation so far assumed we knew the discount rate, the required return. Where does that rate come from? It comes from risk. This lesson makes "return" and "risk" precise: expected return as a probability-weighted average, and risk as the standard deviation of returns.

Why it matters: risk is the reason different investments carry different required returns, and those returns are the discount rates that drive every valuation. Measuring risk is the bridge from the mechanics of discounting to where the discount rate actually comes from.

Expected return

The return on an investment over a period is its gain (price change plus any dividend) divided by the starting price. When the future is uncertain, we work with the expected return. The expected return is the probability-weighted average of the possible returns. If the economy might boom, stay normal, or fall into recession, we weight each scenario's return by its probability.

Consider a stock with these forecasts. Multiply each return by its probability, then add.

Scenario	Probability	Return	Probability times Return
Boom	0.30	positive 25 percent	positive 7.5 percent
Normal	0.40	positive 12 percent	positive 4.8 percent
Recession	0.30	negative 5 percent	negative 1.5 percent
Expected return			positive 10.8 percent

The expected return is 0.30 times 25 percent plus 0.40 times 12 percent plus 0.30 times negative 5 percent, which equals 7.5 plus 4.8 minus 1.5, or positive 10.8 percent. It is not any single outcome; it is the average you would expect over many repetitions.

Key idea: Expected return weights each possible return by its probability, giving the long-run average outcome rather than any one scenario.

Risk as standard deviation

Two investments can share an expected return yet differ wildly in how spread out their possible returns are. Finance measures that spread with the variance and its square root, the standard deviation. Variance is the probability-weighted average of squared deviations from the expected return; standard deviation is its square root. The bigger the standard deviation, the more the actual return is likely to stray from the expected return, and the riskier the investment.

Worked example, continuing the stock above (mean 10.8 percent). Square each scenario's deviation from the mean, weight by probability, and sum to get variance, then take the square root.

Scenario	Deviation from 10.8 percent	Squared	Times probability
Boom	14.2 percent	0.020164	0.0060492
Normal	1.2 percent	0.000144	0.0000576
Recession	negative 15.8 percent	0.024964	0.0074892
Variance			0.013596

Variance is about 0.0136, and the standard deviation is the square root, about 0.1166, or 11.66 percent. Roughly speaking, returns will often land within a band of about one standard deviation around the 10.8 percent mean, that is, between about negative 0.9 percent and positive 22.5 percent.

Key idea: Standard deviation summarizes how far returns typically stray from the mean; larger standard deviation means more risk.

The risk-return trade-off

The foundational pattern of investing is that higher expected returns come only with higher risk. The risk-return trade-off is the principle that greater expected return requires accepting greater risk. History bears this out: over the long run, stocks have delivered higher average returns than bonds, and bonds more than Treasury bills, and in exactly that order, riskier assets have swung more violently along the way.

The extra return a risky asset offers above the safe rate is its risk premium. The risk premium is the expected return of a risky asset above the risk-free rate, the reward for bearing risk. No investment reliably offers high return with low risk; if one seems to, look harder.

Key idea: Return and risk rise together; the risk premium is the compensation investors demand for accepting uncertainty.

Same mean, different risk

The claim that two assets can share an expected return and differ in risk is easy to state and worth doing. Here is a second stock, B, under the identical three scenarios.

Scenario	Probability	Stock A return	Stock B return
Boom	0.30	25.0%	16.0%
Normal	0.40	12.0%	10.5%
Recession	0.30	-5.0%	6.0%

Stock B's expected return is 0.30 x 16.0 + 0.40 x 10.5 + 0.30 x 6.0 = 4.80 + 4.20 + 1.80 = 10.8% - identical to A. Its variance is 0.30 x (0.0520)^2 + 0.40 x (-0.0030)^2 + 0.30 x (-0.0480)^2 = 0.0008112 + 0.0000036 + 0.0006912 = 0.001506, so its standard deviation is 3.88% against A's 11.66%.

An investor choosing between them faces no trade-off at all: B offers the same expected return with a third of the volatility, so no risk-averse investor would hold A on its own. That is a strong statement, and the next lesson qualifies it heavily - once you can combine assets in a portfolio, an individually volatile stock can still be worth holding if it moves against the others.

A convenient way to compare risk per unit of return is the coefficient of variation, standard deviation divided by expected return. Here A scores 11.66 / 10.8 = 1.08 and B scores 3.88 / 10.8 = 0.36. Unlike raw standard deviation, this comparison stays meaningful when the expected returns differ.

Key idea: Equal expected returns with unequal standard deviations make the choice obvious; the coefficient of variation compares risk per unit of return when the means differ.

Averaging returns: two different averages

A subtle trap sits in the word "average." Consider a stock that rises 50% in year 1 and falls 50% in year 2. The arithmetic mean return is (50% - 50%) / 2 = 0%. But $100 becomes $150 and then $75, so the investor has lost a quarter of their money.

The geometric mean captures what actually happened: it is the constant rate that produces the same ending wealth. Here it is (1.50 x 0.50)^(1/2) - 1 = 0.750^(1/2) - 1 = -13.397% a year. Two years at -13.397% turns $100 into $75.00, which is the truth.

The gap appears in every real return series. Suppose a fund returns +22%, -12%, +18%, and +5% over four years. The arithmetic mean is 33 / 4 = 8.25%. Cumulative wealth is 1.22 x 0.88 x 1.18 x 1.05 = 1.33019, so the geometric mean is 1.33019^(1/4) - 1 = 7.39%. The 0.86-point gap is not an error; it is volatility drag, and a close approximation is geometric mean = arithmetic mean - variance / 2. With a variance of 0.017619 here, that predicts 8.25% - 0.88% = 7.37%, within two basis points of the exact 7.39%.

The rule for which to use: arithmetic for forecasting a single future period, geometric for describing realized performance over multiple periods. A fund advertising its arithmetic mean is quoting a number no investor experienced.

Key idea: Geometric mean = arithmetic mean - variance/2 approximately; use arithmetic to forecast one period and geometric to report what happened.

Measuring risk from history rather than scenarios

Scenario tables are for teaching; in practice risk is estimated from realized returns. The procedure differs in one detail: divide by n - 1 rather than n, because using the sample mean costs a degree of freedom.

Take the same four returns. Deviations from the 8.25% mean are 13.75, -20.25, 9.75, and -3.25 percentage points; their squares are 189.06, 410.06, 95.06, and 10.56, summing to 704.75. The sample variance is 704.75 / 3 = 234.92, so the sample standard deviation is 15.33%.

Two warnings come with any historical estimate. First, four observations is far too few - the standard error of a volatility estimate from n observations falls roughly with the square root of n, so short samples give wildly unstable numbers. Second, using history assumes the future resembles the past, which is a modelling choice rather than a fact, and it fails exactly when it matters most, during regime changes.

Key idea: Sample variance divides by n - 1, and short samples produce volatility estimates too unstable to rely on.

What the normal distribution buys, and what it hides

If returns were normally distributed, the standard deviation would translate directly into probabilities: about 68% of outcomes within one standard deviation of the mean, 95% within two, and 99.7% within three. For stock A with a mean of 10.8% and a standard deviation of 11.66%, that gives a one-standard-deviation band of -0.86% to 22.46% and a two-standard-deviation band of -12.52% to 34.12%.

You can also compute a loss probability. The distance from the mean to zero is (0 - 10.8) / 11.66 = -0.926 standard deviations, and the normal distribution puts about 17.7% of outcomes below that point. So under this model roughly one year in six loses money.

Now the caveat that professionals take seriously. Real asset returns have fat tails: extreme moves happen far more often than a normal distribution predicts. The single-day fall in US stocks in October 1987 was more than twenty standard deviations from the mean by the volatility estimates of the time - an event a normal model assigns a probability so small that it should not have occurred in the entire history of the universe. It occurred, and comparable outliers have occurred repeatedly since.

Standard deviation remains the workhorse because it is tractable and because portfolio mathematics is built on it. Treat it as a measure of typical variation, not as a reliable guide to disaster. Practitioners supplement it with downside measures such as semi-deviation, which counts only shortfalls below the mean, and with stress tests that ask what happens in scenarios no distribution predicted.

Key idea: The normal distribution turns standard deviation into probabilities but badly understates extreme events, so it is a good model of ordinary variation and a poor model of crises.

What history says about the trade-off

The claim that riskier assets have paid more is testable, and long-run United States data support it in the expected order: over multi-decade horizons, stocks have delivered higher average annual returns than long-term government bonds, which in turn have beaten Treasury bills - and the standard deviations rank in exactly the same order. Published series maintained for teaching and valuation work, such as those on annual returns to stocks, bonds and bills since 1928, let you check this rather than take it on trust.

Two honest qualifications. The realized premium varies enormously by period and by country, so a single historical average is a weak forecast of the future premium. And survivorship matters: the United States is one of the more successful markets of the last century, so extrapolating its record to all markets overstates the general case. The direction of the risk-return relationship is well supported; its magnitude is genuinely uncertain, which is why Module 6 treats the equity risk premium as an assumption to be tested rather than a constant to be looked up.

Key idea: Long-run data support the risk-return ordering, but the size of the equity risk premium is uncertain and varies by period and country.

Common wrong turns

• "Expected return is what you will earn." No. It is a probability-weighted average; the actual result will usually differ, sometimes a lot.
• "A higher expected return is simply better." Only if you can bear the accompanying risk; a higher standard deviation means a wider range of outcomes, including losses.
• "Standard deviation and variance are different measures of risk." They are the same information; standard deviation is just the square root of variance, in the same units as returns.
• "You can get high return with no risk." Reliably, no. Extra return is the reward for extra risk; a too-good-to-be-true safe high return signals a catch.
• "Up 50% then down 50% leaves you even." It leaves you down 25%. The geometric mean is -13.397% a year.
• "Average annual return means what I earned per year." Only the geometric mean does. The arithmetic mean of that four-year series was 8.25% against a realized 7.39%.
• "Three standard deviations covers the worst case." Real returns have fat tails; October 1987 was beyond twenty standard deviations on contemporary estimates.
• "Stocks return about 10%, so use 10%." Realized premiums vary hugely by period and country, and the US record is among the best available.

Try it

A stock has a 20% chance of returning 30%, a 50% chance of returning 10%, and a 30% chance of returning -15%. (a) Compute its expected return. (b) Compute its variance and standard deviation. (c) Compute its coefficient of variation. (d) Separately, a fund returns +30%, -20%, +25%, +10%, and -5% over five years. Compute the arithmetic and geometric mean annual returns and say which one an investor actually experienced.

Answer: (a) 0.20 x 30 + 0.50 x 10 + 0.30 x (-15) = 6.0 + 5.0 - 4.5 = 6.50%. (b) Deviations are 23.50, 3.50, and -21.50 points; squares 0.055225, 0.001225, 0.046225; weighted 0.011045 + 0.000613 + 0.013868 = 0.025525, so the standard deviation is 15.98%. (c) 15.98 / 6.50 = 2.46 - very high risk per unit of expected return. (d) Arithmetic mean = (30 - 20 + 25 + 10 - 5) / 5 = 8.00%. Cumulative wealth = 1.30 x 0.80 x 1.25 x 1.10 x 0.95 = 1.3585, so geometric mean = 1.3585^(1/5) - 1 = 6.32%. The investor experienced the geometric 6.32%; the 1.68-point gap is volatility drag.

Recap

• Expected return is the probability-weighted average of possible returns; the example stock's is 10.8 percent.
• Risk is measured by variance and its square root, standard deviation; here about 0.0136 and 11.66 percent.
• Two assets with the same mean can differ sharply in standard deviation, hence in risk.
• Higher expected return requires higher risk; the risk premium is the reward above the risk-free rate.
• Stock B matched A's 10.8% expected return with a standard deviation of 3.88% against 11.66%.
• Geometric mean is approximately arithmetic mean minus variance/2, and it is what an investor actually earns.
• Sample variance divides by n - 1; short samples give unstable volatility estimates.
• Normal-distribution probabilities understate extreme moves, so standard deviation describes ordinary variation only.

Sources

1. Dahlquist, J., & Knight, R. (2022). Risk and return to an individual asset. In Principles of Finance. OpenStax, Rice University. openstax.org
2. Dahlquist, J., & Knight, R. (2022). Measures of spread. In Principles of Finance. OpenStax, Rice University. openstax.org
3. Dahlquist, J., & Knight, R. (2022). Probability distributions. In Principles of Finance. OpenStax, Rice University. openstax.org
4. Dahlquist, J., & Knight, R. (2022). Historical picture of returns to stocks. In Principles of Finance. OpenStax, Rice University. openstax.org
5. Dahlquist, J., & Knight, R. (2022). Historical picture of returns to bonds. In Principles of Finance. OpenStax, Rice University. openstax.org
6. Damodaran, A. (n.d.). Annual returns on stocks, T-bonds and T-bills: 1928 to the present. Useful Data Sets. NYU Stern School of Business. pages.stern.nyu.edu
7. U.S. Securities and Exchange Commission. (n.d.). What is risk? Investor.gov ↗ Investing Basics. SEC Office of Investor Education and Advocacy. investor.gov

The big picture

The most important free lunch in finance is diversification: holding many different assets reduces risk without necessarily reducing expected return. This lesson explains why diversification works, splits risk into the part you can remove and the part you cannot, and uses the Capital Asset Pricing Model to turn risk into a required return.

Why it matters: the CAPM is where the discount rates used throughout this course actually come from. It also underlies the cost of equity in Module 6, so this lesson connects risk directly to valuation.

Why diversification works

Diversification means spreading investments across many assets to reduce risk without necessarily lowering expected return. When you combine assets whose returns do not move in lockstep, their ups and downs partly cancel, and the portfolio's swings shrink below the average swing of its parts.

Imagine two stocks with return standard deviations of 20 percent and 30 percent, held in equal amounts. If they moved perfectly together (correlation positive 1), the portfolio's standard deviation would be the simple weighted average, 25 percent. But if they are uncorrelated, the portfolio's standard deviation falls to about 18 percent; if they moved perfectly opposite (correlation negative 1), risk could shrink to as low as 5 percent. The less correlated the assets, the more risk diversification removes.

Correlation of the two stocks	Portfolio standard deviation
positive 1 (move together)	25 percent
0 (unrelated)	about 18 percent
negative 1 (move opposite)	5 percent

Key idea: Combining imperfectly correlated assets lowers portfolio risk below the average of the individual risks; the lower the correlation, the greater the benefit.

Two kinds of risk

Diversification splits risk into two types.

• Unsystematic risk (also firm-specific or diversifiable risk) is unique to one company: a lawsuit, a failed product, a factory fire. Unsystematic risk is firm-specific risk that a diversified portfolio can eliminate. Across a large portfolio these events are independent and wash out.
• Systematic risk (also market risk) affects nearly all assets at once: recessions, interest-rate shifts, wars. Systematic risk is market-wide risk that cannot be diversified away. It is the only risk investors are ultimately rewarded for bearing.

The key insight: because unsystematic risk is free to eliminate, the market pays a risk premium only for systematic risk. An asset's contribution to the systematic risk of a diversified portfolio is measured by its beta. Beta measures an asset's systematic risk relative to the overall market, whose beta is 1.0. A beta of 1.0 means the asset moves with the market; a beta of 1.5 means it swings 50 percent more than the market; a beta of 0.5 means it is half as volatile.

Key idea: Only systematic (market) risk earns a premium, because unsystematic risk can be diversified away for free; beta measures an asset's systematic risk.

The Capital Asset Pricing Model

The Capital Asset Pricing Model (CAPM) gives an asset's required return as the risk-free rate plus beta times the market risk premium. It says expected return rises in proportion to beta:

E(R) = Rf + beta times (E(Rm) minus Rf)

Here Rf is the risk-free rate, E(Rm) is the expected market return, and (E(Rm) minus Rf) is the market risk premium. The market risk premium is the expected return of the market above the risk-free rate, the reward per unit of beta.

Key idea: The CAPM turns beta into a required return by adding a beta-scaled market risk premium to the risk-free rate.

Worked example

Suppose the risk-free rate is 3 percent, a stock's beta is 1.2, and the market risk premium is 5 percent. Plug in.

E(R) = 3 percent + 1.2 times 5 percent = 3 percent + 6 percent = 9 percent

Investors should require 9 percent to hold this stock. That 9 percent is exactly the kind of number we plugged in as the required return when valuing stocks and projects; the CAPM is where discount rates come from. A higher-beta stock would demand a higher return: at beta 1.4 the required return is 3 + 1.4 times 5 = 10 percent. A safer, low-beta stock demands less: at beta 0.6 it is 3 + 0.6 times 5 = 6 percent.

Beta	CAPM required return (Rf 3 percent, premium 5 percent)
0.6	6.0 percent
1.0	8.0 percent
1.2	9.0 percent
1.4	10.0 percent

Key idea: A stock's CAPM return rises with its beta, so riskier (higher-beta) stocks require higher expected returns.

The portfolio variance formula, worked

The table above came from a formula worth stating and using. For two assets with weights w1 and w2, standard deviations s1 and s2, and correlation rho:

Portfolio variance = w1^2 x s1^2 + w2^2 x s2^2 + 2 x w1 x w2 x rho x s1 x s2

Give the two stocks expected returns as well: stock 1 has s1 = 20% and an expected return of 12%, stock 2 has s2 = 30% and an expected return of 16%, held 50/50. Substituting the fixed terms: variance = 0.25 x 0.04 + 0.25 x 0.09 + 2 x 0.25 x rho x 0.06 = 0.01 + 0.0225 + 0.03 x rho = 0.0325 + 0.03 x rho.

Correlation	Portfolio variance	Portfolio standard deviation	Expected return	Risk reduction vs 25%
+1.0	0.062500	25.00%	14.0%	0.00 points
+0.5	0.047500	21.79%	14.0%	3.21 points
0.0	0.032500	18.03%	14.0%	6.97 points
-0.5	0.017500	13.23%	14.0%	11.77 points
-1.0	0.002500	5.00%	14.0%	20.00 points

The column that makes the point is the fourth one. Expected return is 14.0% in every row, because portfolio expected return is a simple weighted average, 0.5 x 12% + 0.5 x 16%, and correlation has nothing to do with it. Risk, by contrast, falls from 25.00% to 5.00% as correlation moves from +1 to -1. That asymmetry - return averages, risk does not - is precisely why diversification is called the only free lunch in finance.

Key idea: Portfolio expected return is a weighted average regardless of correlation, but portfolio risk falls as correlation falls, so risk can be cut without giving up return.

The best mix, not just an equal one

Fifty-fifty is convenient, not optimal. With uncorrelated assets, the weights that minimize variance are w1 = s2^2 / (s1^2 + s2^2) = 0.09 / (0.04 + 0.09) = 0.6923 in stock 1 and 0.3077 in stock 2.

Check the result: variance = 0.6923^2 x 0.04 + 0.3077^2 x 0.09 = 0.019172 + 0.008521 = 0.027692, so the standard deviation is 16.64% against 18.03% for the equal-weighted mix. The expected return is 0.6923 x 12% + 0.3077 x 16% = 13.23%.

Note the trade-off honestly: the minimum-variance portfolio gives up 0.77 points of expected return to save 1.39 points of standard deviation. Whether that is worth doing depends on the investor, which is exactly what the efficient frontier formalizes - every point on it is the lowest risk available for its level of return, and choosing among them is a preference, not a calculation.

Key idea: The minimum-variance mix is rarely equal weights, and moving toward it trades expected return for lower risk.

How many stocks is enough?

For n equally weighted stocks each with variance s^2 and average pairwise covariance c, the portfolio variance is s^2 / n + (1 - 1/n) x c. The first term vanishes as n grows; the second does not. Suppose each stock has a 40% standard deviation and the average pairwise correlation is 0.30, so c = 0.30 x 0.16 = 0.048.

Number of stocks	Portfolio standard deviation
1	40.00%
2	32.25%
5	26.53%
10	24.33%
20	23.15%
50	22.41%
Infinite	21.91%

Going from 1 stock to 10 removes 15.67 points of risk. Going from 10 to 50 removes 1.92 more. Going from 50 to infinity removes 0.50. The floor of 21.91% is the square root of the average covariance, and it is the systematic risk no amount of diversifying can touch. This is the arithmetic behind the common guidance that most diversification benefit is captured within a few dozen holdings - and behind the sharper point that the last portion of risk is not removable at any number of stocks.

Key idea: Diversification benefits decline steeply with the number of holdings and stop at a floor equal to the square root of average covariance - the systematic risk.

Computing a beta

Beta has a definition, not just an interpretation: it is the covariance of the asset with the market divided by the variance of the market, which is equivalent to the correlation times the ratio of standard deviations.

beta = Cov(i, m) / Var(m) = rho x s_i / s_m

Suppose a stock has a 28% standard deviation, the market has 16%, and their correlation is 0.62. Then the covariance is 0.62 x 0.28 x 0.16 = 0.027776 and the market variance is 0.16^2 = 0.025600, so beta = 0.027776 / 0.025600 = 1.085. Equivalently, 0.62 x (0.28 / 0.16) = 0.62 x 1.75 = 1.085.

The second form is instructive. Beta rises with volatility and with correlation. A stock can be extremely volatile and still have a low beta if it moves independently of the market - gold miners and some biotech firms behave this way. Volatility is total risk; beta is only the part that moves with everything else, which is why they are not interchangeable.

Betas also add up. A portfolio's beta is the weighted average of its holdings' betas: 40% at 1.30, 35% at 0.85, and 25% at 0.40 gives 0.520 + 0.298 + 0.100 = 0.9175. Applying the CAPM with a 3% risk-free rate and a 5% market risk premium, the portfolio's required return is 3% + 0.9175 x 5% = 7.59%.

Key idea: Beta = correlation x (asset volatility / market volatility), so a volatile but uncorrelated asset can have a low beta, and portfolio betas are weighted averages.

The security market line and alpha

Plot required return against beta and the CAPM becomes a straight line, the security market line, running from the risk-free rate at beta 0 with a slope equal to the market risk premium. Every fairly priced asset lies on it.

The vertical distance between an asset's expected return and the line is its alpha. The stock above with a beta of 1.2 requires 9%. If an analyst expects it to return 11%, its alpha is +2.0% and it plots above the line, which in CAPM terms means it is underpriced. If the expectation were 7.5%, alpha would be -1.5% and the stock would plot below the line.

Read that carefully, because it is the most misused idea in the lesson. Alpha is defined relative to a model. A positive alpha means the asset's expected return exceeds what this particular model says it should - which is evidence of mispricing only if the model is right. If the model is missing a risk factor that the asset is exposed to, the apparent alpha is really compensation for a risk the model failed to measure.

Key idea: Alpha is the gap between expected return and the security market line, and it measures mispricing only to the extent the model is correct.

How well does the CAPM hold up?

Intellectual honesty requires saying this plainly: the CAPM's empirical record is weak. Testing it since the 1970s has repeatedly found that the relationship between beta and average return is much flatter than the model predicts - low-beta assets have earned more than the CAPM says, and high-beta assets less. Researchers have also documented return patterns related to firm size and to the ratio of book value to market value that beta does not explain, which motivated multi-factor models.

A widely cited survey of this evidence by Fama and French concluded that the model's empirical problems are serious enough to invalidate most of the ways it is used in practice. That is a strong statement from two authors central to the field.

So why is the CAPM still taught first and still used? Because its logic is sound and durable: diversifiable risk should not be rewarded, only exposure to common risk should be priced, and required return should rise with that exposure. Those propositions survive the empirical failures of the specific one-factor implementation. The professional stance is to use the CAPM as a disciplined starting point for a cost of equity, to check the answer against alternatives, and never to present it as a measurement.

Key idea: The CAPM's core logic is sound and its empirical performance is poor, so treat its output as a reasoned estimate rather than a measured required return.

Common wrong turns

• "Diversification eliminates all risk." No. It removes firm-specific risk but leaves systematic market risk, which no amount of diversifying can erase.
• "Investors are paid for taking any risk." Only systematic risk earns a premium; bearing avoidable firm-specific risk is not rewarded.
• "A high-beta stock is always a bad investment." Not so. It carries more systematic risk and therefore a higher required (and expected) return; whether it is a good buy depends on price.
• "Beta measures total risk." No. Beta measures only systematic risk relative to the market, not the full standard deviation of returns.
• "Diversifying costs some expected return." It does not. Expected return was 14.0% at every correlation while risk ranged from 5% to 25%.
• "Portfolio standard deviation is the weighted average of the parts." Only when correlation is exactly +1. At zero correlation it was 18.03% against a 25% weighted average.
• "A volatile stock must have a high beta." Beta = correlation x volatility ratio, so an uncorrelated volatile stock can have a low beta.
• "The CAPM tells you the required return." It gives an estimate from a model whose empirical record is poor. Use it, and check it.

Try it

Asset X has a 24% standard deviation, asset Y has 34%, and they are held 60/40. (a) Compute portfolio standard deviation at correlations of +1, +0.35, and 0, and compare each with the weighted average. (b) A stock has a 30% standard deviation against a market standard deviation of 18%, with a correlation of 0.55. Compute its beta and its CAPM required return if the risk-free rate is 4% and the market risk premium is 5.5%. (c) A portfolio holds 30% at beta 1.45, 45% at beta 1.00, and 25% at beta 0.55. Compute the portfolio beta and required return. (d) If that portfolio is expected to return 11%, what is its alpha?

Answer: (a) Variance = 0.36 x 0.0576 + 0.16 x 0.1156 + 2 x 0.24 x rho x 0.0816 = 0.020736 + 0.018496 + 0.039168 x rho. At rho = +1: 0.0784, so 28.00%. At rho = +0.35: 0.052940, so 23.01%. At rho = 0: 0.039232, so 19.81%. The weighted average is 0.6 x 24 + 0.4 x 34 = 28.00%, matched only at rho = +1. (b) Beta = 0.55 x (0.30 / 0.18) = 0.9167; required return = 4% + 0.9167 x 5.5% = 9.04%. (c) Portfolio beta = 0.30 x 1.45 + 0.45 x 1.00 + 0.25 x 0.55 = 0.435 + 0.450 + 0.1375 = 1.0225; required return = 4% + 1.0225 x 5.5% = 9.62%. (d) Alpha = 11.00% - 9.62% = +1.38%, so it plots above the security market line - if the model is right.

Recap

• Diversification lowers risk by combining assets that do not move in lockstep; less correlation removes more risk.
• Risk splits into unsystematic (diversifiable, firm-specific) and systematic (market, undiversifiable) risk.
• Only systematic risk is rewarded, and beta measures it relative to the market (beta 1.0).
• The CAPM gives required return = Rf + beta times market risk premium; with Rf 3 percent, beta 1.2, premium 5 percent, it is 9 percent.
• Portfolio variance = w1^2 s1^2 + w2^2 s2^2 + 2 w1 w2 rho s1 s2, while expected return stays a weighted average.
• Risk fell from 25.00% to 5.00% as correlation moved from +1 to -1, with expected return fixed at 14.0%.
• Beta = correlation x (asset volatility / market volatility), and portfolio betas are weighted averages.
• The CAPM's empirical record is poor, so its output is a reasoned estimate rather than a measurement.

Sources

1. Dahlquist, J., & Knight, R. (2022). Risk and return to multiple assets. In Principles of Finance. OpenStax, Rice University. openstax.org
2. Dahlquist, J., & Knight, R. (2022). The capital asset pricing model (CAPM). In Principles of Finance. OpenStax, Rice University. openstax.org
3. Dahlquist, J., & Knight, R. (2022). Correlation analysis. In Principles of Finance. OpenStax, Rice University. openstax.org
4. Fama, E. F., & French, K. R. (2004). The capital asset pricing model: Theory and evidence. Journal of Economic Perspectives, 18(3), 25-46. American Economic Association. aeaweb.org
5. French, K. R. (n.d.). Data library. Tuck School of Business. Dartmouth College. mba.tuck.dartmouth.edu
6. Damodaran, A. (n.d.). Betas by sector. Useful Data Sets. NYU Stern School of Business. pages.stern.nyu.edu
7. U.S. Securities and Exchange Commission. (n.d.). Diversification. Investor.gov ↗ Glossary. SEC Office of Investor Education and Advocacy. investor.gov

The big picture

Every valuation technique in this course needed a discount rate. For a whole company, that rate is its cost of capital: the return the firm must earn on its investments to keep everyone who financed it satisfied. Because a firm raises money from two groups, lenders and shareholders, its overall cost is a blend of the two, weighted by how much of each it uses. That blend is the weighted average cost of capital, or WACC.

Why it matters: the WACC is the hurdle rate a firm uses to decide which projects to accept. It pulls the whole course together, combining bond pricing (the cost of debt) and the CAPM (the cost of equity) into the single discount rate you plug into an NPV calculation.

The cost of debt

The cost of debt is the effective interest rate a firm pays on its borrowing, essentially the yield investors demand on its bonds. It carries one important twist: interest is tax-deductible, so borrowing shelters some income from tax. The figure that matters for the WACC is therefore the after-tax cost of debt, which is the stated rate reduced by the tax saving.

After-tax cost of debt = rd times (1 minus Tax rate)

Here rd is the pre-tax interest rate on the debt. A firm borrowing at 8 percent with a 25 percent tax rate has an after-tax cost of debt of 8 percent times (1 minus 0.25) = 8 percent times 0.75 = 6 percent. The tax deduction makes debt cheaper than its stated rate, which is one reason firms use some borrowing.

Key idea: Because interest is tax-deductible, the relevant cost of debt is the after-tax rate, rd times (1 minus the tax rate), which is always below the stated rate.

The cost of equity

The cost of equity is the return shareholders require for holding the firm's stock. We already know how to estimate it: the Capital Asset Pricing Model (CAPM) from Module 5, which sets the required return equal to the risk-free rate plus beta times the market risk premium. If the risk-free rate is 3 percent, the firm's beta is 1.2, and the market risk premium is 5 percent, the cost of equity is 3 percent + 1.2 times 5 percent = 3 percent + 6 percent = 9 percent.

Equity is not free. Shareholders demand a return for the risk they bear, and the cost of equity is almost always higher than the cost of debt, because equity holders are paid last, after lenders, and so face more risk. This ordering matters: a firm that treats retained earnings as costless is fooling itself.

Key idea: The cost of equity is the shareholders' required return, usually found with the CAPM, and it normally exceeds the cost of debt because equity is riskier.

Weighting them: the WACC

The weighted average cost of capital is the firm's overall required return, found by weighting the cost of each source by its share of total financing at market values. The formula is:

WACC = (E/V) times re + (D/V) times rd times (1 minus Tax)

where E is the market value of equity, D is the market value of debt, V = E + D is the total value of the firm, re is the cost of equity, and rd is the pre-tax cost of debt. The fractions E/V and D/V are the capital structure weights: the proportions of equity and debt in the financing mix. Market values, not book values, are the correct weights because they reflect what investors could sell their claims for today.

Key idea: WACC blends the after-tax cost of debt and the cost of equity, each weighted by its market-value share of the firm's financing.

Worked example

A firm is financed with 6,000,000 dollars of equity and 4,000,000 dollars of debt, so V = 10,000,000 dollars. Its cost of equity is 12 percent, its pre-tax cost of debt is 6 percent, and the tax rate is 25 percent. The weights are E/V = 6,000,000 / 10,000,000 = 0.60 and D/V = 4,000,000 / 10,000,000 = 0.40. The after-tax cost of debt is 6 percent times (1 minus 0.25) = 4.5 percent. Now multiply each weight by its cost and add.

Source	Weight	Cost	Weight times cost
Equity	0.60	12 percent	7.2 percent
Debt (after tax)	0.40	4.5 percent	1.8 percent
WACC			9.0 percent

The firm's WACC is 0.60 times 12 percent + 0.40 times 4.5 percent = 7.2 percent + 1.8 percent = 9.0 percent. This is the hurdle rate the firm should use to discount its projects: an investment must earn more than 9 percent to create value for the mix of investors who financed it. Notice how the pieces connect: the 4.5 percent came from bond pricing and the tax shield, the 12 percent came from the CAPM, and the resulting 9 percent becomes the discount rate for NPV in the next lesson.

Key idea: With 60 percent equity at 12 percent and 40 percent debt at an after-tax 4.5 percent, the WACC is 9.0 percent, the firm's project hurdle rate.

A full WACC for Northline Instruments

The two-source example is the teaching version. A real capital structure usually has three components, and the inputs must be dug out rather than assumed. Take Northline Instruments from Lesson 2, with a tax rate of 25%.

Debt. Northline has 1,500 bonds outstanding with $1,000 face value, a 5.5% coupon paid semiannually, and 8 years to maturity, currently trading at $920 each. The market value of debt is 1,500 x $920 = $1,380,000. The cost of debt is not the 5.5% coupon - it is the yield the market currently demands. Solving $920 = $27.50 x [1 - (1+y)^-16] / y + $1,000 / (1+y)^16 gives a semiannual yield of 3.4069%, so the quoted annual yield is 6.8137%. Using the coupon instead would understate the cost of debt by 1.31 percentage points.

Preferred stock. Northline has 20,000 preferred shares paying a fixed $2.40 dividend, trading at $30.00. Preferred stock is a perpetuity, so its cost is simply the dividend over the price: $2.40 / $30.00 = 8.00%. Market value = 20,000 x $30.00 = $600,000. Note that preferred dividends are not tax-deductible for the issuer, so there is no (1 - T) adjustment.

Common equity. 300,000 shares at $30.00 gives a market value of $9,000,000. The cost is estimated below.

Total market value V = $1,380,000 + $600,000 + $9,000,000 = $10,980,000, so the weights are 12.57% debt, 5.46% preferred, and 81.97% common equity.

Key idea: The cost of debt is the current market yield, not the coupon, and preferred stock costs its dividend divided by its price with no tax adjustment.

Two estimates of the cost of equity

The cost of equity cannot be observed, only estimated, and it is standard to do it two ways and compare.

CAPM. With a risk-free rate of 4.0%, a beta of 1.15, and a market risk premium of 5.0%: re = 4.0% + 1.15 x 5.0% = 9.75%.

Dividend growth model. Rearranging the Gordon model gives re = D1 / P0 + g. Lesson 9 found that Northline's $30.00 price implies about 9% perpetual dividend growth, and its $0.60 dividend grown one year is $0.654, so re = $0.654 / $30.00 + 0.09 = 2.18% + 9.00% = 11.18%.

The two estimates differ by 1.43 percentage points, which is entirely normal and worth confronting rather than hiding. They rest on different assumptions - one on a beta estimated from past returns and a debated market premium, the other on a growth rate implied by the current price. Neither is a measurement. Common practice is to report both, use the midpoint of 10.47%, and carry the range through the sensitivity analysis.

Key idea: Estimate the cost of equity at least two ways; a 1.4-point disagreement is normal and belongs in the sensitivity analysis rather than in a rounding decision.

Assembling the WACC

The three-source formula extends the two-source one directly:

WACC = (E/V) x re + (P/V) x rp + (D/V) x rd x (1 - T)

Using the CAPM cost of equity of 9.75%, and an after-tax cost of debt of 6.8137% x 0.75 = 5.1103%:

Source	Market value	Weight	Cost	Weight x cost
Common equity	$9,000,000	0.81967	9.7500%	7.9918%
Preferred stock	$600,000	0.05464	8.0000%	0.4372%
Debt (after tax)	$1,380,000	0.12568	5.1103%	0.6423%
Total	$10,980,000	1.00000	-	9.0712%

Northline's WACC is 9.07%. Substituting the dividend-model cost of equity of 11.18% instead raises it to 10.24%, and the midpoint estimate gives 9.66%. The honest output is therefore "roughly 9% to 10.25%", and a project whose NPV flips sign inside that band is a project the analysis cannot decide.

Key idea: Report the WACC as a range that reflects the disagreement between cost-of-equity estimates, not as a single figure to four decimal places.

Why book weights give the wrong answer

Northline's book equity from Lesson 2 was $1,800,000, one fifth of its $9,000,000 market value. Using book weights: total book capital = $1,380,000 + $600,000 + $1,800,000 = $3,780,000, giving weights of 36.51% debt, 15.87% preferred, and 47.62% equity. The resulting WACC is 7.78% against the correct 9.07%.

The error is 1.29 percentage points and it is systematically in one direction: book values understate successful firms' equity, so book weights underweight the most expensive source of capital and produce a WACC that is too low. A hurdle rate that is too low approves projects that destroy value, and it does so invisibly, because every one of them shows a positive NPV at the understated rate.

Key idea: Book weights understate equity for successful firms, producing a WACC that is too low and a hurdle rate that quietly approves value-destroying projects.

One firm, one rate, two wrong answers

The WACC reflects the risk of the firm's existing assets. Using it for a project of different risk misprices that project in a predictable direction.

Suppose Northline has two divisions: a stable instruments business with a beta of 0.85 and a software venture with a beta of 1.75. Their appropriate costs of equity are 4.0% + 0.85 x 5.0% = 8.25% and 4.0% + 1.75 x 5.0% = 12.75%, straddling the firm-wide 9.75%.

Now consider a software project with an expected return of 11%. Judged against the firm-wide rate it looks attractive by 1.25 points and gets funded. Judged against its own risk-adjusted rate of 12.75% it falls short by 1.75 points and should be rejected. Meanwhile an instruments project returning 9% is rejected at the firm rate and should have been accepted at 8.25%.

A single hurdle rate therefore systematically overfunds the risky division and underfunds the safe one - and, worse, it does so while the numbers all appear to work. The fix is a divisional or project-specific cost of capital, estimated from the betas of firms operating in that line of business rather than from the parent's own beta.

Key idea: A firm-wide WACC systematically overfunds high-risk divisions and underfunds low-risk ones, so risk-adjusted divisional rates are needed.

Why WACC does not fall forever

Debt looks cheap - 5.11% after tax against 9.75% for equity - which invites the question of why the firm does not fund itself entirely with debt. Two forces push back.

First, more debt makes equity riskier. Leverage magnifies the variability of returns to shareholders, so the equity beta rises with leverage and the cost of equity rises with it. Adding cheap debt therefore raises the price of the remaining equity, offsetting part of the gain.

Second, beyond some point, financial distress becomes costly: lenders demand higher yields, customers and suppliers grow wary, and management attention shifts from operations to refinancing. Those costs are real and grow faster than the tax shield.

The combination gives WACC a shallow U shape against leverage, with an optimum somewhere in the middle that varies enormously by industry - stable, asset-heavy businesses support far more debt than volatile, intangible-heavy ones. Published industry data on capital structure and cost of capital make this pattern visible across sectors, and it is a good check on whether a proposed structure is plausible for the business in question.

Key idea: Debt's tax shield is offset by a rising cost of equity and by distress costs, giving WACC a U shape whose minimum differs by industry.

Common wrong turns

• "Debt is free because interest is small." No. Debt has a real cost, the interest rate, lowered but not eliminated by the tax deduction; the after-tax rate is still positive.
• "Retained earnings cost nothing." Equity financed by retained earnings still carries the shareholders' required return, the cost of equity, because that money could have been paid out and invested elsewhere.
• "Use book values for the weights." The correct weights are market values of debt and equity, which reflect current investor claims, not historical accounting figures.
• "A lower WACC is always better regardless of risk." Piling on cheap debt raises financial risk, which eventually pushes up both the cost of debt and the cost of equity, so WACC does not fall forever.
• "The cost of debt is the coupon rate." It is the current yield to maturity. Northline's 5.5% coupon corresponded to a 6.81% cost of debt.
• "Adjust preferred dividends for tax." Preferred dividends are not deductible for the issuer, so there is no (1 - T) term.
• "Our WACC is 9.0712%." The cost of equity alone ranged over 1.43 points. Report a range, not four decimal places.
• "Use the company WACC for every project." That overfunds the 1.75-beta division and underfunds the 0.85-beta one.

Try it

A firm has $8,000,000 of debt at market value yielding 7.2%, $2,000,000 of preferred stock paying a $3.00 dividend on shares trading at $40.00, and $30,000,000 of common equity. Its beta is 1.30, the risk-free rate is 3.5%, the market risk premium is 5.5%, and the tax rate is 21%. (a) Compute the capital structure weights. (b) Compute the cost of preferred and the cost of equity. (c) Compute the after-tax cost of debt. (d) Compute the WACC, showing each component's contribution.

Answer: (a) V = $8,000,000 + $2,000,000 + $30,000,000 = $40,000,000, so weights are 20.00% debt, 5.00% preferred, and 75.00% equity. (b) Cost of preferred = $3.00 / $40.00 = 7.50%. Cost of equity = 3.5% + 1.30 x 5.5% = 10.65%. (c) After-tax cost of debt = 7.2% x (1 - 0.21) = 5.688%. (d) Contributions: equity 0.75 x 10.65% = 7.9875%; preferred 0.05 x 7.50% = 0.3750%; debt 0.20 x 5.688% = 1.1376%. WACC = 9.50%.

Recap

• The cost of capital is the return a firm must earn to satisfy all its investors, used as the discount rate for its projects.
• The after-tax cost of debt is rd times (1 minus the tax rate); at 8 percent and a 25 percent tax rate it is 6 percent.
• The cost of equity is the shareholders' required return, estimated with the CAPM, and normally higher than the cost of debt.
• WACC = (E/V) times re + (D/V) times rd times (1 minus Tax); with 60 percent equity at 12 percent and 40 percent debt at 4.5 percent after tax, it is 9.0 percent.
• Northline's three-source WACC was 9.07% on a CAPM cost of equity and 10.24% on a dividend-model one.
• Using book rather than market weights cut the WACC to 7.78%, a 1.29-point understatement.
• Preferred stock costs its dividend over its price, with no tax adjustment.
• A single firm-wide hurdle rate overfunds risky divisions and underfunds safe ones.

Sources

1. Dahlquist, J., & Knight, R. (2022). The concept of capital structure. In Principles of Finance. OpenStax, Rice University. openstax.org
2. Dahlquist, J., & Knight, R. (2022). The costs of debt and equity capital. In Principles of Finance. OpenStax, Rice University. openstax.org
3. Dahlquist, J., & Knight, R. (2022). Calculating the weighted average cost of capital. In Principles of Finance. OpenStax, Rice University. openstax.org
4. Dahlquist, J., & Knight, R. (2022). Optimal capital structure. In Principles of Finance. OpenStax, Rice University. openstax.org
5. Damodaran, A. (n.d.). Cost of capital by industry sector. Useful Data Sets. NYU Stern School of Business. pages.stern.nyu.edu
6. Damodaran, A. (n.d.). Betas by sector. Useful Data Sets. NYU Stern School of Business. pages.stern.nyu.edu
7. Graham, J. R., & Harvey, C. R. (2001). The theory and practice of corporate finance: Evidence from the field. Journal of Financial Economics, 60(2-3), 187-243. faculty.fuqua.duke.edu

The big picture

We close the course by putting every piece together. Capital budgeting is the process of deciding which long-term projects a firm should fund, and it is where the financial manager creates, or destroys, value. The recipe is now familiar: estimate a project's cash flows, discount them at the firm's cost of capital, and apply the net present value rule.

Why it matters: choosing the right projects is the single most important thing a firm does with its money. Getting the cash flows and the discount rate right is what separates value creation from value destruction, and the tools below are used in real corporate finance every day.

Getting the cash flows right

The hardest part of capital budgeting is not the discounting arithmetic but choosing which cash flows to count. Three rules keep the analysis honest.

• Incremental cash flow: a cash flow that occurs only because the project is undertaken. Count only cash flows that change because you take the project. If a cost happens whether or not you invest, leave it out.
• Sunk cost: money already spent and unrecoverable. Ignore it. A past market study you already paid for is gone and is irrelevant to today's decision, no matter how much it cost.
• Opportunity cost: the value of the best alternative use of a resource the project consumes. Include it. If a project uses a warehouse you could otherwise rent out for 50,000 dollars a year, that forgone rent is a real cost of the project.

Two more habits matter. Use after-tax cash flows, because taxes are a real outflow. And recall from Module 1 that cash flow is not the same as accounting profit: add back non-cash charges such as depreciation, which reduce reported profit but do not actually leave the firm as cash.

Key idea: Analyze projects with incremental, after-tax cash flows; include opportunity costs and exclude sunk costs.

Putting it together with NPV

Recall the project from Module 3: it costs 10,000 dollars today and returns 4,000 dollars, 5,000 dollars, and 6,000 dollars over the next three years. Suppose the firm's WACC, which we computed in the previous lesson as 9 percent, is the right discount rate (rather than the 10 percent we assumed earlier). Discount each inflow at 9 percent, where the present value of a cash flow is the cash flow divided by (1.09) raised to the year number.

Year	Cash flow	PV at 9 percent
1	4,000 dollars	3,669.72 dollars
2	5,000 dollars	4,208.40 dollars
3	6,000 dollars	4,633.10 dollars
Total PV of inflows		12,511.23 dollars

The net present value subtracts the upfront cost from the total present value of the inflows: NPV = minus 10,000 + 12,511.23 = plus 2,511.23 dollars. Because the firm now discounts at its true 9 percent cost of capital rather than 10 percent, the project looks even better than before: a lower hurdle rate raises NPV. Since NPV is positive, the firm creates value by accepting the project.

Key idea: With a 9 percent WACC the project's inflows are worth 12,511.23 dollars today, so NPV = minus 10,000 + 12,511.23 = plus 2,511.23 dollars, and the project is accepted.

Comparing the criteria

Managers use several yardsticks to judge projects. Know their strengths and flaws.

Method	Decision rule	Strength	Weakness
NPV	Accept if NPV is greater than 0	Measures dollar value added; uses all cash flows and the time value of money	Requires a discount rate
IRR	Accept if IRR is greater than the WACC	Intuitive percentage return	Can mislead on project scale and with multiple sign changes in cash flows
Payback period	Accept if cost is repaid within a chosen cutoff	Simple; highlights how quickly cash comes back	Ignores the time value of money and all cash flows after the cutoff

The payback period is the time required for a project's cumulative cash flows to repay its initial cost. Our project recovers 4,000 dollars then 5,000 dollars, a cumulative 9,000 dollars, after two years, and it needs 1,000 dollars more of the year-three 6,000 dollars to break even, so payback is about 2 + 1,000 / 6,000 = 2.17 years. Payback is a rough liquidity check, but because it ignores the time value of money and everything past the cutoff, it should never overrule NPV.

The internal rate of return (IRR) is the discount rate that makes a project's NPV exactly zero; you accept a project when its IRR exceeds the WACC. IRR and NPV usually agree, but IRR can give a misleading ranking when projects differ greatly in size or when cash flows change sign more than once.

The verdict of modern finance is firm: when the methods conflict, follow NPV, because maximizing NPV is the same as maximizing the value of the firm, the goal we began with in Module 1.

Key idea: NPV, IRR, and payback usually agree, but when they conflict follow NPV, because maximizing NPV maximizes the value of the firm.

A complete project, from purchase order to salvage

The three-line example above is a teaching device. Here is the real shape of a capital-budgeting analysis, using Northline Instruments and its 9% weighted average cost of capital from the previous lesson. Assumptions stated in full: equipment costs $2,400,000 and is depreciated straight-line over 6 years to a book value of zero; the project requires $180,000 of net working capital at the outset, recovered at the end; annual revenue is $1,900,000 and cash operating costs are $1,120,000; the tax rate is 25%; the equipment is expected to sell for $300,000 at the end of year 6; all cash flows arrive at year end.

Step 1: the initial outlay. Capital expenditure plus the working-capital investment: $2,400,000 + $180,000 = -$2,580,000 at time zero. Working capital is a genuine outflow even though nothing is consumed - the cash is tied up in inventory and receivables until the project ends.

Step 2: annual operating cash flow. Depreciation is $2,400,000 / 6 = $400,000 a year, so EBIT = $1,900,000 - $1,120,000 - $400,000 = $380,000, and taxes are $95,000. Operating cash flow adds the non-cash depreciation back:

OCF = EBIT x (1 - T) + Depreciation = $380,000 x 0.75 + $400,000 = $685,000

A second route makes the tax effect visible: OCF = (Revenue - Costs) x (1 - T) + T x Depreciation = $780,000 x 0.75 + 0.25 x $400,000 = $585,000 + $100,000 = $685,000. The second term, $100,000 a year, is the depreciation tax shield - cash the firm keeps purely because depreciation reduced taxable income. It is worth $100,000 x 3.889651 for the first five years alone.

Step 3: the terminal year. Two extras land in year 6. The equipment sells for $300,000 against a book value of zero, so the entire proceeds are taxable: after-tax salvage = $300,000 x (1 - 0.25) = $225,000. And the $180,000 of working capital is released. Year 6 therefore brings $685,000 + $225,000 + $180,000 = $1,090,000.

Year	0	1-5 (each)	6
Capital expenditure	-$2,400,000	-	-
Net working capital	-$180,000	-	+$180,000
Operating cash flow	-	+$685,000	+$685,000
After-tax salvage	-	-	+$225,000
Free cash flow	-$2,580,000	+$685,000	+$1,090,000

Key idea: A project's cash flows have three parts - initial outlay including working capital, annual after-tax operating cash flow including the depreciation tax shield, and terminal-year salvage plus working-capital recovery.

Every decision rule on one project

Now apply all four criteria to the same numbers at a 9% cost of capital.

• NPV. The five-year annuity factor is [1 - 1.09^-5] / 0.09 = 3.889651, so years 1-5 are worth $685,000 x 3.889651 = $2,664,411. Year 6 is worth $1,090,000 / 1.09^6 = $1,090,000 / 1.677100 = $649,931. Total present value of inflows = $3,314,343, so NPV = $3,314,343 - $2,580,000 = +$734,343. Accept.
• IRR. The rate that zeroes the NPV is 17.51%, comfortably above the 9% hurdle. Same verdict.
• Profitability index. $3,314,343 / $2,580,000 = 1.285, so each dollar invested returns $1.285 of present value. Above 1, so accept.
• Payback. Three years return $2,055,000, leaving $525,000 of the $2,580,000 outlay to recover from year 4's $685,000: payback = 3 + $525,000 / $685,000 = 3.77 years.

All four agree here, which is the normal case for a single conventional project. The value of computing all four is not that they might disagree but that each answers a different question: how much value (NPV), what return (IRR), how efficiently (PI), and how long is our money exposed (payback).

Key idea: For a single conventional project all four criteria agree, and each still contributes a distinct piece of information.

Finding out what the decision depends on

A single NPV is a point estimate built on a dozen assumptions. Three standard tests probe it.

Sensitivity analysis moves one input at a time. Cut revenue 10%, to $1,710,000, and operating cash flow falls to ($1,710,000 - $1,120,000 - $400,000) x 0.75 + $400,000 = $542,500, taking NPV down to +$95,099. Still positive, but 87% of the value is gone from a 10% revenue miss. Move the discount rate to 12% instead and NPV falls to +$441,500 - a much smaller effect. For this project the revenue forecast is the dangerous input, which is the opposite of the pattern seen in Lesson 6 and is exactly why the test is worth running rather than assuming.

Break-even analysis asks how far an input can move before NPV reaches zero. Write operating cash flow as a function of revenue R: OCF = (R - $1,120,000 - $400,000) x 0.75 + $400,000 = 0.75R - $740,000. The combined discount factor for the operating cash flows is 3.889651 + 1 / 1.677100 = 4.485919, and the terminal extras contribute $405,000 / 1.677100 = $241,488. Setting NPV to zero: OCF x 4.485919 = $2,580,000 - $241,488 = $2,338,512, so OCF = $521,301 and 0.75R = $1,261,301, giving R = $1,681,734. Revenue can fall 11.5% before the project stops creating value.

Scenario analysis moves several inputs together in a coherent story - a recession lowers revenue and raises the discount rate at the same time - because in reality inputs are correlated and one-at-a-time sensitivity understates the downside.

Key idea: Sensitivity finds the input that matters, break-even finds how much room you have, and scenario analysis respects the fact that bad news arrives together.

When the budget runs out first

Everything above assumes a firm can fund every positive-NPV project. Under capital rationing it cannot, and the ranking question returns. Suppose a $2,100,000 budget and three independent projects:

Project	Cost	Present value of inflows	NPV	Profitability index
A	$800,000	$1,040,000	$240,000	1.30
B	$1,500,000	$1,875,000	$375,000	1.25
C	$600,000	$810,000	$210,000	1.35

Ranking by profitability index takes C first, then A, spending $1,400,000 and leaving $700,000 - not enough for B. Total NPV = $210,000 + $240,000 = $450,000.

But B and C together cost exactly $2,100,000 and deliver $375,000 + $210,000 = $585,000. The PI ranking left $135,000 on the table. The reason is that projects are indivisible: you cannot buy 47% of project B. PI ranking is optimal when projects can be scaled continuously and is only a good heuristic when they cannot. Under rationing, enumerate the feasible combinations and pick the highest total NPV.

Key idea: Under capital rationing, rank by profitability index as a starting point but check whole combinations, because indivisible projects can beat the PI ordering.

Where the course lands

Thirteen lessons reduce to one sentence: the value of anything is the present value of the cash it will produce, discounted at a rate that reflects its risk. Every technique here is a way of making that sentence operational. Time value moves cash flows through time. Bond and stock valuation apply it to contractual and residual claims. Risk and return supply the discount rate. Capital budgeting applies the whole apparatus to the decisions a firm actually makes.

Three habits matter more than any formula. State your assumptions - compounding, timing, tax, nominal or real - because the answer is undefined without them. Verify your arithmetic by computing the same quantity a second way, as this lesson did with the two routes to operating cash flow. And test what the answer depends on, because a decision that survives only one set of assumptions is not a decision, it is a hope with a spreadsheet attached.

A closing reminder from Lesson 1: this course is education, not investment advice. It has taught methods for analyzing cash flows and risk, not recommendations about any security, strategy, or company. Decisions about your own money belong with a qualified, licensed professional.

Key idea: Value is discounted expected cash flow; the professional skill is stating assumptions, verifying arithmetic, and testing what the answer rests on.

Common wrong turns

• "Include the sunk cost so the project pays for past spending." No. Sunk costs are unrecoverable and must be excluded; only future incremental cash flows belong in the analysis.
• "A resource we already own is free to use." Not if it has an alternative use. The opportunity cost, such as forgone rent, is a genuine cost of the project.
• "Accounting profit is the cash flow to discount." No. Profit includes non-cash charges like depreciation; capital budgeting uses actual after-tax cash flows, adding depreciation back.
• "A short payback period means a project is a good investment." Not necessarily. Payback ignores the time value of money and any cash flows after the cutoff, so it can favor a worse project over a better one.
• "Working capital is not a real cost." It is cash tied up for the project's life. It belongs in year 0 as an outflow and in the final year as a recovery.
• "Salvage value is a clean inflow." Proceeds above book value are taxable. Selling zero-book equipment for $300,000 delivered $225,000.
• "Depreciation does not matter because it is non-cash." It is non-cash and it reduces taxes. The shield was worth $100,000 a year here.
• "Rank by profitability index under a budget constraint." That ranking left $135,000 of NPV unclaimed because the projects were indivisible.

Try it

A firm evaluates a 5-year project. Equipment costs $1,800,000, depreciated straight-line to zero over 5 years. Net working capital of $140,000 is required at the start and recovered at the end. Annual revenue is $1,450,000 and cash costs are $820,000. The tax rate is 21%, the cost of capital is 10%, and the equipment will sell for $250,000 at the end. (a) Compute annual depreciation, the tax shield, and annual operating cash flow. (b) Compute the initial outlay and the year-5 cash flow. (c) Compute NPV and the profitability index. (d) Compute IRR and payback.

Answer: (a) Depreciation = $1,800,000 / 5 = $360,000; shield = 0.21 x $360,000 = $75,600; OCF = ($1,450,000 - $820,000 - $360,000) x 0.79 + $360,000 = $270,000 x 0.79 + $360,000 = $573,300. (b) Initial outlay = $1,800,000 + $140,000 = -$1,940,000. After-tax salvage = $250,000 x 0.79 = $197,500, so year 5 = $573,300 + $197,500 + $140,000 = $910,800. (c) Years 1-4 at a factor of 3.169865 give $1,817,102; year 5 is $910,800 / 1.61051 = $565,527. Total present value = $2,382,629, so NPV = +$442,819 and PI = $2,382,629 / $1,940,000 = 1.228. (d) IRR = 18.01%. Cumulative cash after 3 years is $1,719,900, leaving $220,100 of the $1,940,000 outlay against year 4's $573,300, so payback = 3 + $220,100 / $573,300 = 3.38 years.

Recap

• Capital budgeting selects long-term projects using incremental, after-tax cash flows.
• Exclude sunk costs, include opportunity costs, and discount actual cash flows rather than accounting profit.
• Discounting the 10,000 dollar project at a 9 percent WACC gives inflows worth 12,511.23 dollars and an NPV of plus 2,511.23 dollars, so it is accepted.
• NPV, IRR, and payback each have uses, but when they conflict follow NPV, because it measures value added and matches the goal of the firm.
• OCF = EBIT x (1 - T) + depreciation, or equivalently (Revenue - Costs) x (1 - T) + T x depreciation.
• The Northline expansion had an NPV of $734,343, an IRR of 17.51%, a PI of 1.285, and a payback of 3.77 years.
• Break-even analysis showed revenue could fall 11.5% before the project stopped creating value.
• Under capital rationing, check whole combinations rather than trusting the profitability-index ranking.

Sources

1. Dahlquist, J., & Knight, R. (2022). Choosing between projects. In Principles of Finance. OpenStax, Rice University. openstax.org
2. Dahlquist, J., & Knight, R. (2022). Using Excel to make company investment decisions. In Principles of Finance. OpenStax, Rice University. openstax.org
3. Dahlquist, J., & Knight, R. (2022). Pro forma financials. In Principles of Finance. OpenStax, Rice University. openstax.org
4. Dahlquist, J., & Knight, R. (2022). Forecasting cash flow and assessing the value of growth. In Principles of Finance. OpenStax, Rice University. openstax.org
5. Dahlquist, J., & Knight, R. (2022). What is working capital? In Principles of Finance. OpenStax, Rice University. openstax.org
6. Graham, J. R., & Harvey, C. R. (2001). The theory and practice of corporate finance: Evidence from the field. Journal of Financial Economics, 60(2-3), 187-243. faculty.fuqua.duke.edu
7. Damodaran, A. (n.d.). Data: Current year. Useful Data Sets. NYU Stern School of Business. pages.stern.nyu.edu