4.2Lesson 4.2 · Module 4

Why does a stock have value today?

A stock's value equals the present value of the economic benefits expected by its owners. The discount rate is the market-required return for bearing the timing and risk of those benefits.

One-period valuation: P₀ = E[D₁+P₁]/(1+r)
The Dividend Discount Model generalizes this
Resale price is not an independent source of value
The discount rate reflects time value and risk

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Learning objectives

By the end of this lesson, you should be able to:

1Value a stock over one period using P₀ = E[D₁ + P₁] / (1 + r).
2Substitute for the future price recursively and derive the Dividend Discount Model.
3Explain why a future sale price is not an independent source of value.
4Explain why a stock paying no current dividend can still have positive value today.
5Explain why future payoffs are discounted: time value and risk.
6Describe the intuitive components of the discount rate r.
7Distinguish a personal required return from the market-required return.
8Explain why equity valuation is hard: both the numerator and the denominator are uncertain.

From Lesson 4.1

In Lesson 4.1 we established that equity is a residual ownership claim and that growth creates value only when the company earns a return above the cost of equity. Now we ask the natural next question: if equity is a claim on future residual cash flows, how do we translate that claim into a price today? The answer is present value — the same tool you used for bonds, applied to a far less certain stream of payoffs.

2.1Section 1

A one-year stock investment

Suppose you buy one share today at price $P_{0}$ , hold it for one year, receive a dividend $D_{1}$ , then sell it for a price $P_{1}$ . What should you pay today?

The total payoff you expect at the end of the year is $D_{1} + P_{1}$ . Because that payoff is uncertain, we work with its expectation, denoted $E [D_{1} + P_{1}]$ , and discount it at the required return $r$ .

One-period stock valuation

P_{0} = \frac{E [ D _{1} + P _{1} ]}{1 + r}

The stock's value today is the present value of the total payoff expected over the next period — the dividend plus the resale price — discounted at the required return.

$P_{0}$: stock price today (value)
$D_{1}$: dividend paid during the next period
$P_{1}$: expected resale price at the end of the next period
$r$: required return (discount rate) for this stock
$E [\cdot]$: expectation operator — accounts for uncertainty

This is the same present-value logic you used for bonds: take expected future cash, discount it back one period.

2.2Section 2

A numerical example

Worked example

P_{0} = \frac{E [ D _{1} + P _{1} ]}{1 + r}

Plug in the expected dividend, expected resale price, and required return.

P_{0} = \frac{$2 + $108}{1.10} = \frac{$110}{1.10}

P₀ = $100

At $100, the expected return exactly equals the 10% required return. You can verify: ($2 + $108 − $100) / $100 = 10%.

The price you actually pay determines whether your expected return is above, below, or equal to the required 10%.

Pay $90: expected return = ($2 + $108 − $90) / $90 = 22.2% — above required, so the stock is attractive at that price.
Pay $100: expected return = ($2 + $108 − $100) / $100 = 10% — exactly the required return; this is fair value.
Pay $105: expected return = ($2 + $108 − $105) / $105 = 4.8% — below required, so at $105 the stock is overvalued relative to these assumptions.

2.3Section 3

Where does P₁ come from?

The future price $P_{1}$ is not a free input. At time 1, the buyer who purchases the share faces the same valuation problem you faced at time 0. So $P_{1}$ must itself equal the PV of payoffs expected from time 1 onward.

Price at time 1

P_{1} = \frac{E _{1} [ D _{2} + P _{2} ]}{1 + r}

The resale price at time 1 is the PV, as of time 1, of the dividend and resale price expected at time 2.

Substitute P₁ into P₀

P_{0} = \frac{E [ D _{1} ]}{1 + r} + \frac{E [ D _{2} + P _{2} ]}{( 1 + r ) ^{2}}

Replace the single-period payoff with two periods of dividends plus a now-more-distant resale price.

Substitute again for P₂

P_{0} = \frac{E [ D _{1} ]}{1 + r} + \frac{E [ D _{2} ]}{( 1 + r ) ^{2}} + \frac{E [ D _{3} + P _{3} ]}{( 1 + r ) ^{3}}

Each substitution pushes the terminal resale price one period further into the future.

Generalize — the Dividend Discount Model

P_{0} = t = 1 \sum \infty \frac{E [ D _{t} ]}{( 1 + r ) ^{t}}

Pushing the resale price infinitely far forward, the stock's value equals the present value of all expected future dividends (broadly defined).

Here Dₜ means any economic benefit shareholders receive — regular dividends, special dividends, buybacks, or liquidation proceeds. This is the Dividend Discount Model (DDM).

2.4Section 4

The future sale price is intermediate, not a new source of value

Selling your share generates a real capital gain in your hands — the cash is real. But the buyer pays you only because the buyer expects later benefits from the share. The chain of value always traces back to the company's future performance.

Causal chain

Future business performanceFuture shareholder benefitsP₁ (resale price)P₀ (price today)

The price today depends on expected future benefits; those future benefits depend on the business. The resale price is just the conduit through which later benefits reach an earlier owner.

2.5Section 5

A stock with no current dividend can still have value

Suppose a company pays no distributions in years 1 through 4, but you expect it to be acquired in year 5 for $200 per share. With $r = 10%$ :

Value with a single distant payoff

P_{0} = \frac{200}{( 1.10 ) ^{5}}

Even with zero dividends for four years, the expected acquisition proceeds in year 5 still produce a positive value today.

P_{0} = \frac{$200}{1.6105}

P₀ = $124.18

Zero current dividends does not mean zero value. It means the benefits are expected to arrive later.

2.6Section 6

Why discount future payoffs?

Two forces push the value of a future payoff below its face amount.

Time value

A dollar today can be invested and grow. $P V = 100/1.05 = $95.24$ — a certain $100 next year is worth less than $100 today even with no risk.

Risk

An uncertain payoff is worth less than a certain one. The more uncertain, the bigger the discount.

Compare an expected $110 payoff discounted at two different rates:

Lower risk — r = 5%

P V = \frac{110}{1.05} = $104.76

Higher risk — r = 15%

P V = \frac{110}{1.15} = $95.65

The payoff is identical ($110); only the discount rate differs. A riskier stock commands a higher $r$ , and therefore a lower present value for the same expected cash.

2.7Section 7

What determines r?

It is crucial to distinguish two ideas that sound similar:

Personal required return: the return you, individually, would demand to hold the stock.
Market-required return: the return the broad market demands for bearing the timing and risk of this stock's payoffs. This is the $r$ used to price the stock.

Intuitive decomposition of r (not CAPM yet)

r = risk-free rate + risk premium

The required return compensates investors for the pure time value of money (the risk-free rate) plus an additional premium for bearing the stock's specific risk.

The risk premium is larger for stocks whose payoffs are more cyclical, more uncertain, more leveraged, or more sensitive to recession. We are not introducing CAPM here — this is just the intuition that riskier cash flows require a larger discount.

Factors that raise the risk premium

Cyclicality — revenue swings with the economy
Operating uncertainty — unpredictable costs or demand
Leverage — high debt amplifies equity risk
Recession sensitivity — payoffs shrink in downturns
Interest rates — higher risk-free rates raise r directly

Important

Management cannot simply declare a lower cost of equity. The required return is set by investors weighing the risk of the cash flows, not by the company's preference.

2.8Section 8

Personal vs. market-required return

Suppose the expected payoff is $110. The market requires 10%, so the market price is $100. But you personally require 15%.

Market r = 10% → P₀

$100

110 / 1.10

Your personal r = 15% → your valuation

$95.65

110 / 1.15

The market's fair value is around $100. You personally wouldn't buy at $100 because your hurdle is higher. But your personal rejection does not force the market price down to $95.65 — the market price reflects the marginal investor, not you.

2.9Section 9

Numerator vs. denominator

Equity valuation is hard because the DDM asks two difficult questions, and both must be answered:

Numerator — $E [D_{t}]$ : What economic benefits will shareholders actually receive, and when?
Denominator — $r$ : What return does the market require for bearing the timing and risk of those benefits?

For bonds, the numerator (contractual coupons and principal) is largely known. For equities, both the numerator and the denominator are uncertain. That is the fundamental source of difficulty in equity valuation.

Try itOne-period valuation explorer

P₀ = (D₁ + P₁) / (1 + r)

Value a stock over one period

Set what you expect to receive next period (a dividend D₁ and a future price P₁), the return you require, and the price you would actually pay. The lab computes the stock's estimated value and tells you whether the purchase price delivers more or less than your required return.

Expected dividend D₁Expected future price P₁

Required return r10.00%

Purchase price

Estimated current value

$100.00

($2.00 + $108.00) ÷ (1 + 10.00%)

Dividend yield

2.00%

$2.00 ÷ $100.00

Expected capital gain

8.00%

($108.00 − $100.00) ÷ $100.00

Expected total return

10.00%

($2.00 + $108.00 − $100.00) ÷ $100.00

VerdictExpected return matches required

At this purchase price you expect 10.00%, matching the 10.00% required. This is fair value given your assumptions.

Estimated value vs. purchase price

Estimated value

$100.00

Purchase price

$100.00

Estimated value$100.00

Purchase price$100.00

Remember: the estimated current value depends on your assumptions about D₁, P₁, and r. Change r and watch how the same expected payoff maps to a different value today.

2.10Concept check

No-dividend stock and a rising r

Try itConcept check

A stock is expected to pay no dividend and to sell for $55 next year. First with r = 10%, then with r = 15%.

Work out P₀ in each case, then reveal the answer.

r = 10%

P₀ = 55 / 1.10 = $50.00

r = 15%

P₀ = 55 / 1.15 = $47.83

2.11Common questions

Questions on discounting and resale price

03Mastery

Summary and mastery check

Try itLesson 4.2 mastery check

Pass with 3 of 4 correct

Answer all questions, then check your work. You can retry any time — mastery is based on correctness, not speed.

01
E[D₁] = $2, E[P₁] = $108, r = 10%. What is P₀?
02
A stock pays no dividend for 5 years, then pays $200 in year 5. r = 10%. What is P₀?
03
The discount rate r in the DDM represents:
04
If r rises from 10% to 15% on a $110 expected payoff, P₀ moves from $100 to:

Lesson summary

1A stock's value equals the PV of expected economic benefits to shareholders.
2The one-period model is P₀ = E[D₁+P₁]/(1+r).
3The DDM generalizes: P₀ = Σ E[Dₜ]/(1+r)ᵗ.
4Resale price is not independent — it reflects future expected benefits.
5The discount rate is the market-required return for timing and risk.
6Equity valuation is hard because both cash flows and discount rates are uncertain.

Replay this lesson Continue to the Gordon Growth Model