1.1 From Reading Numbers to Valuing Businesses
A business is worth the cash it will produce in the future, measured in today’s money. Every valuation method is a different way to estimate that one number. A discounted cash flow, or DCF, builds it from your own forecast. Comparable companies use what the market pays for similar listed firms, like checking the price of similar houses on the street. Precedent deals use what buyers actually paid to take control. Professionals build several, then investigate where they disagree, because the gaps show which assumptions matter.
Why it matters: No single method is the answer, and the disagreement between them is the useful part.
Summary: A business is worth the present value of the cash it will generate, and every professional method estimates that one number from different evidence: your own cash-flow forecast (intrinsic valuation, the DCF), what the market pays for similar companies (comps) or what buyers paid for control (precedents). Practitioners build several and investigate where they disagree.
- A multiple is a DCF in shorthand: justified P/E = (1 − g ÷ ROE) ÷ (r − g), so 5% growth at a 20% ROE and a 9% cost of equity justifies 18.75x.
- Borrowing a peer’s multiple without its growth and returns can overpay by 50% (18.75x paid versus a justified 12.5x).
- Banks, early-stage, cyclical, distressed and conglomerate businesses each change which method leads and how it is built.
- Treat a gap of more than about 20% between methods as a question to answer, not a number to average.
Volume I taught you that market capitalization is share price multiplied by shares outstanding, and that it reflects the market’s forward-looking expectations. But where does that share price actually come from? Someone, somewhere — an analyst, a fund manager, an investment banker — had to arrive at a view of what the business is worth, and translate that view into a number. Every valuation method in this Part is one way of answering the same underlying question: what is the right price to pay today for a stream of value that will arrive in the future?
There are three broad families of answer to that question, and professionals typically build all three side by side before settling on a final view: intrinsic valuation (build the value from the company’s own projected cash flows — the DCF, covered in 1.3), relative valuation (look at what similar companies are trading for — comparable company analysis, 1.7), and transaction-based valuation (look at what similar companies actually sold for in real deals — precedent transactions, 1.8).
Valuing a business is like pricing a house. You could estimate its worth from the rent it could generate over the next thirty years (intrinsic — the DCF). You could look at what similar houses on the same street sold for last month (relative — comps). Or you could look at what a similar house actually sold for in a recent, comparable sale (transaction-based — precedents). No single method is “correct” — a careful buyer checks all three and triangulates.
All three families answer the same question with different evidence: your own cash-flow forecast, what investors pay today for similar cash flows, and what buyers paid for control. Because a multiple is a compressed cash-flow valuation, the methods should agree when their inputs agree; when they do not, the gap tells you which input to question.
Apply the growing-perpetuity formula from Section 1.2: The Time Value of Money — The Idea Underneath Everything to earnings. A company that earns a return on equity (ROE) on what it reinvests must retain g ÷ ROE of its earnings to grow at rate g and can pay out the rest, so price ÷ next year’s earnings is P/E = (1 − g ÷ ROE) ÷ (r − g), where r is the cost of equity. A peer growing 5% with a 20% ROE and a 9% cost of equity: payout = 1 − 0.05 ÷ 0.20 = 0.75, so P/E = 0.75 ÷ (0.09 − 0.05) = 18.75x. A multiple is a DCF with growth, returns and risk compressed into one number; it transfers between companies only when those inputs match.
The toolkit in this Part assumes a going concern that funds itself with ordinary debt, reports in one currency and pays tax normally. When an assumption fails, the method changes:
| Situation | What changes | Why |
|---|---|---|
| Bank or insurer | Value equity directly (dividends or excess return on equity) at the cost of equity | Deposits and policy reserves are raw material, not financing, so “unlevered” cash flow has no meaning |
| Early-stage, loss-making company | Forecast 10 or more years, until margins stabilize; weight success and failure scenarios | A five-year forecast still in losses puts more than 100% of value in the terminal value |
| Cyclical or commodity business | Build the terminal year on mid-cycle margins | A perpetuity capitalizes whatever year it is given, peak or trough |
| Distressed company | Value default and recovery scenarios; treat equity as an option on the assets | A going-concern DCF ignores the chance that shareholders receive nothing |
| Conglomerate | Value each segment separately (Section 1.9: Sum-of-the-Parts Valuation) | One blended rate misprices segments with different risk |
| Large non-operating assets | Value cash, investments and surplus property separately; add them in the bridge | They produce no operating cash flow, so omitting them undervalues |
| Interest above the IRC §163(j) cap | Use a lower effective tax rate on the excess interest (Section 1.4: The Weighted Average Cost of Capital (WACC)) | Deductions are capped at 30% of adjusted taxable income, computed before depreciation; the excess carries forward |
| Cash flows in another currency, or in real terms | Discount at a rate on the same currency and inflation basis | Mixing bases more than doubled a value in Section 1.2 |
If the business has positive, forecastable cash flows, build the DCF as the anchor and use comps and precedents to test it. If the methods differ by more than about 20%, treat the gap as a question (which input explains it?) rather than averaging it away. If cash flows are negative or unforecastable for more than five years, lead with scenario-weighted DCFs and use multiples as a sanity check; for banks and insurers, switch to equity-level methods. The 20% trigger is a working heuristic, not a standard.
Borrowing a multiple without borrowing the economics. Apply the peer’s 18.75x P/E to a target growing 3% with a 12% ROE at the same 9% cost of equity: its justified P/E is (1 − 0.03 ÷ 0.12) ÷ (0.09 − 0.03) = 0.75 ÷ 0.06 = 12.5x. On $100 million of next-year earnings, paying 18.75x means $1,875 million for a business worth $1,250 million on the same formula: a 50% overpayment, hidden because both pay out 75% of earnings. Avoid it by matching growth and returns on capital, not just the industry label (Section 1.7: Comparable Company Analysis (“Comps”) builds the peer set).
Which valuation method is the most accurate?
None on its own, because each answers a different question: a DCF prices the cash flows on your assumptions, comps show what the market pays today for similar companies, and precedents show what buyers paid for control. Practitioners trust the zone where the three overlap and investigate where they diverge (Section 1.10: Triangulating Valuation — Putting It All Together).
Is a DCF better than valuation multiples?
A DCF is more explicit, not automatically better. Every assumption is visible and can be challenged, but small changes in the discount rate or terminal growth move it a lot (Section 1.3: Discounted Cash Flow (DCF) — Building the Model). Multiples are faster and reflect current prices, yet hide the same growth, return and risk assumptions inside one number.
What is intrinsic value?
Intrinsic value is the present value of the cash a business will generate for its investors, discounted at a rate that reflects that cash’s risk. It is an estimate built from forecasts, not a fact the market reveals, so two careful analysts can reach different intrinsic values and both defend them.
Intrinsic, relative and transaction-based valuation all estimate the present value of future cash, using your forecast, current market prices or past deal prices. A multiple is a compressed DCF, justified by growth, return on equity and the cost of equity, so it transfers only between companies whose economics match; banks, early-stage, cyclical, distressed and conglomerate businesses change which method leads.
Four questions on this chapter. Decide on your answer first, then click “Reveal Answer.”
1. A company grows 4% a year, earns a 16% return on equity on reinvested earnings, and has a 10% cost of equity. Using P/E = (1 − g ÷ ROE) ÷ (r − g), what forward P/E is justified?
- 18.8x
- 16.7x
- 12.5x
- 7.5x
Reveal Answer
Answer: C. Payout = 1 − 0.04 ÷ 0.16 = 0.75, so P/E = 0.75 ÷ (0.10 − 0.04) = 12.5x. Ignoring the reinvestment gives 16.7x; dividing by r or by g alone gives 7.5x or 18.8x.
2. Which company is the clearest case for valuing equity directly instead of building unlevered free cash flow and an EV-to-equity bridge?
- A bank whose deposits fund its lending
- A manufacturer with moderate bank loans
- A software company holding net cash
- A retailer that leases most of its stores
Reveal Answer
Answer: A. For a bank, deposits are raw material rather than financing, so unlevered cash flow has no clear meaning; banks are valued at the equity level with the cost of equity.
3. Two companies in the same industry trade at very different P/E multiples. Under the justified-multiple logic, which difference best explains the gap?
- Their share counts and nominal share prices
- Their growth, return on equity and risk
- Their fiscal year-ends and reporting dates
- Their stock exchange listings and tickers
Reveal Answer
Answer: B. Justified P/E = (1 − g ÷ ROE) ÷ (r − g), so only growth, return on equity and the cost of equity move it; share counts, dates and listings do not.
4. A DCF values a company at $40 a share while trading comps imply $28. What does this chapter’s decision rule recommend first?
- Discard the DCF because market prices are right
- Discard the comps because intrinsic value wins
- Average the two and report $34 as fair value
- Find which input explains the gap first
Reveal Answer
Answer: D. A gap above about 20% is treated as a question: the methods should agree when their inputs agree, so the gap points to an input (growth, returns or risk) that needs checking.
1.2 The Time Value of Money — The Idea Underneath Everything
A dollar today is worth more than a dollar next year, because today’s dollar can be put to work earning a return. To compare cash at different dates, you shrink future money back to today’s value, which is called discounting. The higher the rate and the longer the wait, the more it shrinks. For example, $100,000 due in three years at 10% is worth $75,131 today. Every price in finance, from bonds to businesses, rests on this one idea.
Why it matters: Once you can move money across time, you can compare any two offers fairly.
Summary: Money has a time value because a dollar today can earn a return. To compare cash at different dates, discount it with PV = CF ÷ (1 + r)ⁿ: $100,000 due in three years at 10% is worth $75,131 today.
- An annuity, a perpetuity and a growing perpetuity have one-line formulas: $10,000 a year for 5 years at 8% is worth $39,927; forever, $125,000; growing 2% a year, $166,667.
- The growing-perpetuity formula only works when the discount rate exceeds the growth rate, and it becomes explosive as the two converge.
- Match nominal cash flows with nominal rates: discounting a nominal stream at the real rate turned a $2,000 value into $4,227.
- Match the compounding period too: 12% compounded monthly is 12.68% a year.
A dollar today is worth more than a dollar a year from now — not because of inflation alone, but because a dollar in hand today can be invested and start earning a return immediately. This single idea, the time value of money, is the foundation every valuation method in this Part rests on. To compare money received at different points in time, you must discount future amounts back to a common point — usually today — using a discount rate that reflects both the time value of money and the risk that the future cash flow might not arrive at all.
The mechanics are a single formula, repeated relentlessly: the present value (PV) of a future cash flow (CF) received n years from now, discounted at rate r, is:
PV = CF ÷ (1 + r)ⁿ
Suppose a company will pay you $100,000 in exactly 3 years, and your required return (the discount rate) is 10% per year. The present value today is: $100,000 ÷ (1.10)³ = $100,000 ÷ 1.331 ≈ $75,131. In other words, you should be willing to pay roughly $75,131 today for the right to receive $100,000 in three years, given a 10% required return. Raise the discount rate and the present value falls further — this single relationship, scaled up across many years of cash flow, is the entire engine of the DCF in the next section.
Three shortcuts turn the single-cash-flow formula into the building blocks of every valuation. An annuity, a fixed payment for n years, is worth CF × [1 − (1 + r)⁻ⁿ] ÷ r: $10,000 a year for 5 years at 8% is worth $10,000 × (1 − 1.08⁻⁵) ÷ 0.08 = $39,927. A perpetuity, a fixed payment forever, is worth CF ÷ r: $10,000 ÷ 0.08 = $125,000. A growing perpetuity, whose first payment arrives in one year and then grows at rate g forever, is worth CF₁ ÷ (r − g): $10,000 ÷ (0.08 − 0.02) = $166,667. The last formula is the Gordon Growth Model that values the long tail of every DCF (Section 1.3: Discounted Cash Flow (DCF) — Building the Model).
Rates must also match their compounding period. A loan quoted at 12% a year compounded monthly costs (1 + 0.12 ÷ 12)¹² − 1 = 12.68% as an effective annual rate, so discounting annual cash flows at the 12% headline would overstate their present value (Section 0.4: The Quant Toolkit covers compounding conventions).
A growing perpetuity is a geometric series: CF₁ ÷ (1 + r) + CF₁(1 + g) ÷ (1 + r)² + CF₁(1 + g)² ÷ (1 + r)³ + … Each term equals the one before multiplied by (1 + g) ÷ (1 + r). While g is below r, that ratio is below 1, the terms shrink, and the sum converges to CF₁ ÷ (r − g). When g reaches r, every term is equal and the sum is infinite; a denominator of zero is the algebra telling you the assumption is impossible. The same algebra explains why growth assumptions near the discount rate are dangerous. At r = 8%, moving g from 2% to 3% raises the value per dollar of next year’s cash from 1 ÷ 0.06 = 16.7x to 1 ÷ 0.05 = 20.0x (+20%); moving it from 5% to 6% raises it from 33.3x to 50.0x (+50%).
Every later section in this Part is these formulas applied with care: the cash flows get more detailed and the discount rate more carefully built, but the arithmetic stays the same.
Discount every cash flow at a rate built on the same basis as the cash flow: the same currency, the same inflation treatment (nominal with nominal, real with real), the same compounding period, and a risk premium that matches whose cash it is (the whole firm’s in Section 1.4: The Weighted Average Cost of Capital (WACC), shareholders’ in Section 1.5: CAPM — The Cost of Equity). If the cash flow is contractually certain, such as a Treasury coupon, use the Treasury yield of matching maturity; if it is risky, either add risk to the rate or haircut the cash flow, never both. Set the rule aside only for rough comparisons in which every option carries the same error.
Mixing nominal cash flows with a real discount rate. A business will pay $100 next year, growing 3% a year in nominal terms; the nominal required return is 8% and expected inflation is 2.5%. The correct value is $100 ÷ (0.08 − 0.03) = $2,000. Discount the same nominal stream at the real rate, 1.08 ÷ 1.025 − 1 = 5.37%, and you get $100 ÷ (0.0537 − 0.03) = $4,227: more than double, from one inconsistency. Avoid it by labeling every forecast and every rate as nominal or real; company forecasts and market yields are almost always nominal.
Why is a dollar today worth more than a dollar in the future?
Because a dollar today can be invested to earn a return, can be spent now rather than later, and carries no risk of failing to arrive. At a 5% risk-free return, $100 today grows to $105 in a year, so $100 promised a year from now is worth only $100 ÷ 1.05 = $95.24 today, before any allowance for inflation or default risk.
What discount rate should I use to calculate present value?
Use the return you could earn on an alternative with the same risk, currency and timing. For a guaranteed payment, that is the Treasury yield of matching maturity; for a company’s cash flows to all investors, its WACC (Section 1.4: The Weighted Average Cost of Capital (WACC)); for cash flows to shareholders only, its cost of equity (Section 1.5: CAPM — The Cost of Equity). A personal hurdle rate works for personal decisions but is not a market valuation.
What is the difference between present value and net present value?
Present value is what a stream of future cash flows is worth today; net present value (NPV) subtracts what you must pay to get it. If a project costs $70,000 and its cash flows have a present value of $75,131, its NPV is $5,131. A positive NPV means the project earns more than the discount rate, so it creates value.
Present value discounts a future cash flow by (1 + r)ⁿ; annuities, perpetuities and growing perpetuities extend that to streams, with the growing perpetuity requiring the rate to exceed growth. Rates and cash flows must share the same currency, inflation basis and compounding period, or the answer can more than double.
Four questions on this chapter. Decide on your answer first, then click “Reveal Answer.”
1. What is the present value of $50,000 received in 4 years at a 7% annual discount rate?
- $35,649
- $36,000
- $38,145
- $46,729
Reveal Answer
Answer: C. PV = 50,000 ÷ 1.07⁴ = 50,000 ÷ 1.3108 = $38,145. Simple discount, 50,000 × (1 − 4 × 0.07) = 50,000 × 0.72, gives $36,000; discounting one or five years gives the other figures.
2. A payment of $5,000 arrives next year and grows 3% a year forever. At a 9% discount rate, what is it worth today?
- $83,333
- $55,556
- $41,667
- $85,833
Reveal Answer
Answer: A. Value = CF₁ ÷ (r − g) = 5,000 ÷ (0.09 − 0.03) = $83,333. Growing the first payment again gives $85,833; ignoring growth gives $55,556; adding g to r gives $41,667.
3. A loan is quoted at 12% a year compounded monthly. What is its effective annual rate?
- 12.36%
- 12.55%
- 12.00%
- 12.68%
Reveal Answer
Answer: D. EAR = (1 + 0.12 ÷ 12)¹² − 1 = 12.68%. Semiannual compounding would give 12.36% and quarterly 12.55%; the headline 12% ignores compounding.
4. Why does the growing-perpetuity formula CF₁ ÷ (r − g) break down when g equals or exceeds r?
- Accounting rules forbid discount rates below expected growth
- Later payments never shrink, so the sum never converges
- The formula assumes payments stop after a fixed number of years
- Inflation makes later payments worthless once growth matches the rate
Reveal Answer
Answer: B. Each term is the previous one times (1 + g) ÷ (1 + r); when g ≥ r that ratio is at least 1, so the terms never shrink and the sum is infinite.
- 26 U.S.C. §163, Interest (Cornell LII) — §163(j) 30% of adjusted taxable income cap, depreciation add-back, carryforward