Someone offers you $1,000 today or $1,100 in a year. Almost nobody needs a course to say "it depends": on whether you trust them, on what else you could do with the money, on whether you need it now. Those three words, trust, alternatives and need, are the whole subject. Corporate finance is what happens when you ask the question about a factory instead of a loan, with a hundred million at stake and a board waiting for the answer.
Every company faces the same three decisions. Which projects should we spend on? Where does the money come from, borrowed or from owners? And what is the whole business worth? A graduate has one tool for all three: a way of turning cash that arrives at different times, with different risks, into a single number today.
If a bank pays 6% a year, $1,000 today becomes $1,060 next year and $1,123.60 the year after, because the interest earns interest. That is compounding. Run it backwards and you get its twin, present value: what a future sum is worth now. At 6%, a promise of $1,000 in 20 years is worth only about $312 today, because $312 invested at 6% would grow to $1,000 by then.
At the default, $1,000 grows to $3,207 in 20 years, and $1,000 due in 20 years is worth $312 now. The rule of 72 gives a shortcut: money doubles in roughly 72 divided by the percentage rate, so about 12 years at 6% (the exact answer is 11.9). The same arithmetic is why waiting costs: a rate is the price of time.
A project is a stream of cash: an outlay now, receipts later. To judge it, discount each receipt back to today and add them up. The sum, minus the outlay, is the net present value (NPV). If NPV is positive, the project earns more than the discount rate demands, and doing it makes the owners richer by exactly that amount. If it is negative, they would do better lending the money at the discount rate.
Worked: spend $1,000 now for $300 a year over five years. At an 8% discount rate the five receipts are worth $1,198 today, so NPV is +$198. At 15% the receipts are worth $1,006, and NPV is nearly zero. The rate at which NPV hits zero is the project's internal rate of return (IRR), here 15.2%. Slide the rate and watch the line cross zero.
IRR is popular because it is one percentage, easy to compare with a hurdle rate. It is also the part of finance where the textbooks warn you most. Two classic failures.
Several IRRs. If the cash flow changes sign more than once, for example a mine that needs a big cleanup bill at the end, the NPV curve can cross zero twice. A project with flows of −100, +230 and −132 (in millions) has IRRs of both 10% and 20%, and it is worth doing only when the discount rate lies between them. A single IRR number hides that.
Scale. Project A turns 100 into 150 (IRR 50%). Project B turns 1,000 into 1,300 (IRR 30%). If you can do only one and the cost of capital is 10%, A adds 36 and B adds 182. Higher IRR, much lower value. The two NPV curves cross at about 27.8%, and below that rate B wins. The rule graduates keep: when IRR and NPV disagree, trust NPV, because it measures money, not percent.
Illustrative cash flows (millions), invented to show the traps.
So far the future was certain. It is not. A share can fall 40% in a year, and investors want to be paid for taking that chance. The central observation of modern finance, made by Harry Markowitz in 1952, is that you are paid for the risk you cannot avoid, and that you can avoid a lot of it for nothing. Hold many assets that do not move in lockstep and their ups and downs partly cancel.
Risk here is measured as volatility: the standard deviation of returns. For a portfolio of N assets, each with volatility σ and an average pairwise correlation ρ, the portfolio's volatility is:
Take stocks with 30% volatility each and a correlation of 0.3. One stock: 30%. Ten stocks: 18.2%. Fifty: 16.8%. The floor is 30% × √0.3 = 16.4%, no matter how many you hold. That floor is market risk, which cannot be diversified away; the rest is specific risk, which can.
Assumes equal volatility and equal pairwise correlation, a simplification. Real correlations vary and rise in crises.
If investors diversify, the only risk worth a reward is the market risk left over. The Capital Asset Pricing Model, developed in the early 1960s by Jack Treynor, William Sharpe, John Lintner and Jan Mossin (Sharpe's version appeared in 1964), turns that into a price. A stock's beta measures how strongly it moves with the market: beta 1 moves with it, 0.5 half as much, 1.5 one and a half times. The model says the required return is the safe rate plus beta times the market's extra reward:
Inputs are illustrative, not forecasts. Estimates of the premium range widely.
With a 3% safe rate, a 5% premium and a beta of 1.2, the required return is 9%. That number is the discount rate you use in NPV. The model is elegant, which is why it is taught everywhere, and why it is argued about. Critics: Fama and French (1992) found that beta alone did a poor job of explaining average returns across US stocks, while company size and the ratio of book value to price did better, and Fama and French later added those as extra factors. Richard Roll (1977) argued the model cannot truly be tested, since the "market" should hold every asset, including property and human skills. Many practitioners use CAPM anyway as a rough, transparent starting point. Treat the number as an estimate with a wide error bar.
A company's cost of capital is the return its funders demand: lenders want interest, owners want a CAPM-style return. Blend the two, weighted by how much each provides, and adjust the debt for tax, and you get the weighted average cost of capital (WACC). It is the discount rate for the firm's cash flows.
A discounted cash flow (DCF) valuation then does what NPV did for projects, but for the whole firm. Forecast the free cash flow for some years, add a terminal value for everything after, and discount it all. With this firm's cash of $100 million, 8% growth for five years, 2% afterwards and a 9% WACC, the business is worth about $1.88 billion, and 74% of that comes from the terminal value. That is the first lesson of DCF: most of the answer is the distant future, so the answer is only as good as the growth and discount-rate assumptions. Move the terminal growth up to 4% and the value jumps by almost a third.
A company with $100 million of free cash flow now. If growth forever would reach the discount rate, the tool caps it one point below.
The other common method uses multiples: look at what similar companies sell for relative to a measure like earnings or cash flow, and apply the same ratio. The tool above would call this firm's value "18.8 times cash flow". Multiples are quick and market-based, but they only move with the market: if the whole sector is overpriced, so is yours. Careful analysts use both and ask why they disagree.
Should a firm fund itself with debt or equity? In 1958 Franco Modigliani and Merton Miller proved something startling: in a world with no taxes, no bankruptcy costs and no information gaps, it does not matter. The firm's value comes from what its assets earn, not from how the claims on them are sliced. An investor could borrow personally to copy a leveraged firm, so the market will not pay extra for the firm doing it. More debt makes the equity riskier, so shareholders demand a higher return that exactly cancels the cheapness of debt. They received Nobel prizes for this (Modigliani 1985, Miller 1990).
The theorem is a benchmark: it tells you where to look for real effects. The first is tax. In most countries interest is deductible and dividends are not, so debt carries a tax shield and a levered firm's WACC falls. The second is distress: more debt raises the chance of bankruptcy, which is costly. The textbook "trade-off theory" balances the two; pecking order theorists argue managers instead prefer internal cash, then debt, then new shares, because of information gaps. Both are live in the literature.
Assumes an all-equity return of 10% and debt at 5%. Distress costs are not modelled, which is why the tax benefit looks endless here. Real firms hit a limit.
With no tax, the WACC stays at 10% at any debt level while the cost of equity climbs (13.3% at 40% debt). With a 25% tax rate, 40% debt cuts WACC to 9%. Pushed to 90% the tool says 7.75%, which no lender would permit: that is the limit the distress cost supplies.
If prices already reflect what everyone knows, nobody can reliably beat the market, and an NPV calculation is just reading the price. Eugene Fama's 1970 review gave this efficient market hypothesis its structure: weak (past prices carry no news), semi-strong (public information is already in the price) and strong (even private information is). The evidence is mixed in a way that keeps both camps in business.
Fama, Shiller and Lars Peter Hansen shared the 2013 Nobel prize in economics: a prize for one side's theory, the other side's critique and the statistical methods both used. Luck matters more than people think. Try it: give a thousand "fund managers" a coin each and see how many beat the market ten years in a row by chance alone.
Each manager has a 50% chance of beating the market each year, independent of the others. The simulation uses your browser's random numbers.
For 1,000 managers and ten years the expected number with a perfect record is about one (1,000 ÷ 1,024), and there is a 62% chance of at least one. That one looks like a genius. Real streaks are not purely luck, but the arithmetic is why careful readers ask how many managers started.
Corporate finance turns cash that arrives at different times and different risks into a single number today. Discount at a rate that reflects what the money could earn elsewhere at the same risk; trust NPV over IRR; remember that diversification removes risk for free and the rest is what you are paid for. Every valuation is a model, and its biggest input is usually a guess about the far future.
Try this: take any offer with delayed money (a "0% for 12 months" deal, a pension quote, a subscription that saves 20% if you pay a year ahead). Pick a discount rate you believe, and compute the present value of both sides.
Sources. Time value, NPV, IRR and the diversification formula are standard textbook finance (Brealey et al.; Berk & DeMarzo). Markowitz, "Portfolio Selection", Journal of Finance (1952); Sharpe, "Capital Asset Prices", Journal of Finance (1964), with Treynor, Lintner and Mossin independently; Roll, "A Critique of the Asset Pricing Theory's Tests" (1977); Fama and French, "The Cross-Section of Expected Stock Returns", Journal of Finance (1992); Modigliani and Miller, "The Cost of Capital, Corporation Finance and the Theory of Investment", American Economic Review 48(3) (1958); Fama, "Efficient Capital Markets: A Review of Theory and Empirical Work" (1970); Shiller (1981) on excess volatility; Grossman and Stiglitz (1980). Nobel prizes: Markowitz, Miller and Sharpe 1990; Modigliani 1985; Fama, Hansen and Shiller 2013. The SPIVA claim refers to S&P Dow Jones Indices' published scorecards; the share varies by year and market, so no percentage is quoted. Interactive numbers were computed in node: $1,000 at 6% for 20 years = $3,207.14, present value $311.80, doubling 11.90 years; NPV of 5 × $300 on $1,000 at 8% = $197.81, IRR 15.24%; the −100/+230/−132 flows have IRRs of exactly 10% and 20%; A versus B at 10% = $36.36 against $181.82, crossover 27.8%; diversification 30.0%, 18.2%, 16.8% and floor 16.4%; CAPM 3% + 1.2 × 5% = 9%; DCF enterprise value $1,877.9 million with terminal 74.1%, 2,472.7 at 4% terminal growth; WACC 9.0% and cost of equity 12.5% at 40% debt and 25% tax; 1,000/1,024 = 0.98 managers, 62.4% chance of at least one.