A discounted cash flow, by hand

Grow the base one year at a time: 5.00 x 1.05 = 5.25 in year 1, 5.25 x 1.05 = 5.51 in year 2, and so on to 8.14 in year 10. Then ask what each year's cash is worth today. Year 1's cash arrives one year from now, so divide it by 1.09 once: 5.25 / 1.09 = 4.82. Year 2's arrives two years out, so divide by 1.09 twice: 5.51 / 1.09² = 4.64. Year 10's is divided by 1.09 ten times, which is 2.37, so 8.14 shrinks to 3.44.

The right-hand column adds to 40.93: everything the explicit decade is worth today. Notice the shape. The cash column rises, the present-value column falls, because the discount rate is larger than the growth rate. A business growing faster than its discount rate for a decade would show a rising right-hand column, and that should make you suspicious of the growth rate before it makes you excited about the value.

Nothing in P&G's filings says 5 percent. Its owner earnings per share rose from a five-year mean of 3.84 at the start of its run to 5.90 at the end, about 3.1 percent a year over 14 years, in nominal dollars. So 5 percent is above P&G's own record: it is a round default the bench uses for every filer, so that the companies can be compared on the same sliders. The 9 percent is a judgment about the return you require for waiting and for risk, which is Part 2.

The honest way to write a DCF is to put the choices first, in a sentence: "base 5.00, ten-year mean owner earnings, all-capex approximation; 5 percent for ten years, then 2.5 percent; 9 percent discount rate". Anyone who disagrees with a choice can see it and change it. A value presented without its sentence is a number with its assumptions hidden.

Read the rate as the return you demand for handing over money today against a stream of cash you cannot be sure of. A safer, steadier stream earns a lower rate; a cyclical, indebted or poorly understood one earns a higher rate. That is a judgment about the business and about you, and two careful people can land a point apart.

One point is not a rounding error. Holding growth at 5 percent and the terminal at 2.5 percent, P&G is worth 113.24 at 8 percent, 95.16 at 9 percent and 81.95 at 10 percent. The swing is lopsided because discounting compounds: every year's divisor changes, and the terminal value's divisor changes most of all, since it sits ten years out and also depends on the gap between the rate and terminal growth. Analyst session 6 sets this row beside the growth and terminal levers and asks which one moves the answer most.

Textbook finance derives a discount rate from a model of market returns, usually the capital asset pricing model, where a riskfree rate is topped up by a company's measured sensitivity to the market (its beta) times an equity risk premium. That derivation is not covered here. It needs market prices to estimate the beta, these pages use no market prices, and the derivation does not remove the judgment: it relocates it into the choice of premium and the period the beta is measured over.

If you want the full machinery, Aswath Damodaran publishes his cost-of-capital lectures, datasets and spreadsheets free at pages.stern.nyu.edu/~adamodar ↗. For every exercise here, the rate is a stated choice, the range in Analyst session 6 brackets it by a point either way, and the sentence you write with the value says which rate you used.

Year 10's owner earnings are 8.14. Year 11's, the first year of stage two, are 2.5 percent higher. Divide year 11's cash by the gap between the discount rate and terminal growth, 0.09 - 0.025 = 0.065, and you get the value at the end of year 10 of everything that follows: 128.39. That value sits ten years in the future, so it is divided by 1.09 ten times, like year 10's cash, giving 54.24 today.

Add the two stages: 40.93 + 54.24. On a calculator that is 95.17; the model prints 95.16. Neither is wrong. Each part is rounded to cents before you add them; the model adds the unrounded parts and rounds once. Say so whenever you show the split, because a reader who adds your parts and gets a different total stops trusting the rest.

The terminal share, 54.24 of 95.16, is 57.0 percent, and at these defaults it is the same for every base, because the base scales both parts equally. Print it beside every value. A DCF where most of the answer comes from beyond the forecast is a statement about the long run more than about the next decade.

The divisor in the Gordon formula is r - g. At 9 and 2.5 percent it is 0.065. Raise terminal growth to 3.5 percent and P&G's value rises to 105.65; lower it to 1.5 percent and it falls to 87.47. Keep raising it and the divisor heads toward zero while the terminal value heads toward infinity. At g equal to r the formula divides by zero, and above it the sum of the stream does not converge at all: each later year is worth as much or more today than the one before.

The intrinsic value bench in Analyst session 10 has a break button that sets terminal growth equal to the discount rate. It does not print an enormous number; it prints no value, and the reason. A model that returned a figure there would be lying.

Damodaran's terminal value lecture gives the economic reason for the cap, not just the arithmetic one: a stable growth rate cannot exceed the growth rate of the economy, though it can be lower, because a company growing faster than its economy for ever would eventually become the economy. He uses the riskfree rate as a simple proxy for nominal economic growth, which keeps terminal growth below the discount rate whenever the discount rate carries any premium for risk.