Finance-Theory · Unit 9 · Video 3 · Interactive Practice

A Share Is Worth Its Dividends: Deriving the Dividend Discount Model

IKey Formulas

Formula	Name	What it says
$P_t = \dfrac{\mathsf{E}_t[D_{t+1} + P_{t+1}]}{1+r}$	One-year pricing	Today's price is the expected dividend plus the expected ex-dividend resale price one year out, discounted one year at a rate held fixed at $r$
$P_t = \displaystyle\sum_{k=1}^{N} \frac{\mathsf{E}_t[D_{t+k}]}{(1+r)^k} + \frac{\mathsf{E}_t[P_{t+N}]}{(1+r)^N}$	After $N$ substitutions	Substituting each year's price forward uses $\mathsf{E}_t\big[\mathsf{E}_{t+1}[X]\big] = \mathsf{E}_t[X]$ and $(1+r)(1+r) = (1+r)^2$ ; the last piece is the resale term
$\displaystyle\lim_{N\to\infty} \frac{\mathsf{E}_t[P_{t+N}]}{(1+r)^N} = 0$	No-bubble condition	An assumption: the expected price does not keep growing as fast as $r$ forever
$P_t \equiv \displaystyle\sum_{k=1}^{\infty} \frac{\mathsf{E}_t[D_{t+k}]}{(1+r_{t+k})^k}$	Dividend discount model	One risk-adjusted rate per horizon, like a spot rate; dividends satisfy $D_t \ge 0$ and prices are ex-dividend

Key Insight: A claim that never pays anything is worth nothing. With every dividend zero, the price equals the discounted expected resale price for every $N$ , so the expected price would have to grow at exactly $r$ forever; the no-bubble condition rules that out and leaves $P_t = 0$ .

IISubstituting Forward Moves Value, Not Price

Each substitution turns part of the resale term into a discounted dividend and leaves the price unchanged.

Horizon N (years) 0

IIIA Claim That Never Pays

With every dividend zero, today's price can rest only on an expected resale price growing at exactly $r$ .

Price today $5.00 Horizon N (years) 20

IVOne Rate per Horizon

Each expected dividend is discounted at the risk-adjusted rate for its own horizon, raised to that horizon.

r(t+1) % 8.0 r(t+2) % 9.0 r(t+3) % 10.0

VQuiz Questions

Problem 1 · Price One Year at a Time

Given: A share is expected to pay a dividend of $2 one year from now and then to sell, ex-dividend, for an expected $53. The risk-adjusted rate is $r = 10\%$ . Find today's price $P_t$ .

✅ Correct! The dividend and the resale price both arrive in one year, so both are discounted:

(2 + 53)/1.1 = 50

❌ Not discounted. $55 is what you expect to hold one year from now. Today's price divides it by

1 + r

55/1.1 = 50

❌ The dividend is a year away too. Adding $2 undiscounted to

53/1.1 \approx 48.18

gives about $50.18. The dividend is paid at

t+1

, so it sits inside the numerator:

(2 + 53)/1.1 = 50

❌ Nothing to subtract.

P_{t+1}

is already ex-dividend, so it excludes the $2. Holding the share for the year earns the dividend plus the resale price:

(2 + 53)/1.1 = 50

, not

(53 - 2)/1.1 \approx 46.36

❌ Not quite. Use

P_t = \mathsf{E}_t[D_{t+1} + P_{t+1}]/(1+r)

Show solution

Buying today at $P_t$ earns, one year from now, the dividend $D_{t+1}$ and the ex-dividend price $P_{t+1}$ . With the rate held fixed at $r$ :

P_t = \frac{\mathsf{E}_t[D_{t+1} + P_{t+1}]}{1+r} = \frac{2 + 53}{1.10} = \frac{55}{1.10} = 50

Today's price is $50.00.

Problem 2 · Two Years of Substitution

Given: A share's expected dividends are $1.10 in one year and $1.21 in two years, and its expected ex-dividend price two years from now is $60.50. The risk-adjusted rate is fixed at $r = 10\%$ . Find today's price using $P_t = \dfrac{\mathsf{E}_t[D_{t+1}]}{1+r} + \dfrac{\mathsf{E}_t[D_{t+2}]}{(1+r)^2} + \dfrac{\mathsf{E}_t[P_{t+2}]}{(1+r)^2}$ .

✅ Correct!

1.10/1.1 + 1.21/1.21 + 60.50/1.21 = 1 + 1 + 50 = 52

❌ Two-year cash flows get two years of discounting. Dividing everything by

1.1

gives

62.81/1.1 = 57.10

. Substituting next year's price in multiplies the two factors: the second dividend and the resale price are divided by

(1.1)^2 = 1.21

❌ The resale term is still there. The two dividends are worth

1 + 1 = 2

, but after only two substitutions the resale term

60.50/1.21 = 50

remains. It vanishes only as

N \to \infty

, and only under the no-bubble condition.

❌ Check the second dividend. It is paid in two years, so it is divided by

(1.1)^2

1.21/1.21 = 1

, not

1.21/1.1 = 1.10

❌ Not quite. Discount each amount by

(1.1)^k

, where

k

is the year in which it arrives.

Show solution

After substituting $P_{t+1}$ into today's equation, every amount that arrives in year 2 is divided by $(1+r)^2 = 1.21$ :

P_t = \frac{1.10}{1.1} + \frac{1.21}{1.21} + \frac{60.50}{1.21} = 1 + 1 + 50 = 52

Today's price is $52.00. The same price comes from pricing one year at a time: $P_{t+1} = (1.21 + 60.50)/1.1 = 56.10$ , then $P_t = (1.10 + 56.10)/1.1 = 52$ .

Problem 3 · Paying for a Claim That Never Pays

Given: A share will never pay a dividend of any kind, not even a liquidating one. It trades today at $20, and the risk-adjusted rate is held fixed at $r = 8\%$ . Find the expected price ten years from now that today's $20 requires, and the share's value once the no-bubble condition is imposed.

Expected price in ten years that justifies $20 today?

Value under the no-bubble condition?

✅ Correct! With no dividends,

20 = \mathsf{E}_t[P_{t+10}]/(1.08)^{10}

forces

\mathsf{E}_t[P_{t+10}] = 20 \times 1.08^{10} \approx 43.18

. The resale term equals 20 at every horizon, so it never goes to zero; the no-bubble condition then leaves a value of $0.

❌ Check the required price. With every dividend zero,

P_t = \mathsf{E}_t[P_{t+N}]/(1+r)^N

for every

N

; solve for

\mathsf{E}_t[P_{t+10}]

❌ That is simple interest.

20 \times (1 + 10 \times 0.08) = 36

. Substituting forward compounds the rate:

20 \times 1.08^{10} \approx 43.18

❌ A flat price loses money. A share that pays nothing and is expected to stay at $20 earns 0%, not 8%. Its whole return must come from resale, so the expected price grows at exactly

r

20 \times 1.08^{10} \approx 43.18

❌ Wrong direction.

20/1.08^{10} \approx 9.26

discounts today's price. The future price must be larger:

\mathsf{E}_t[P_{t+10}] = 20 \times 1.08^{10} \approx 43.18

❌ Check the value. With every dividend zero, the sum of discounted expected dividends is zero, and that sum is all the no-bubble condition leaves.

❌ That is the resale term, not a value. At $20 the resale term is

\mathsf{E}_t[P_{t+N}]/(1.08)^N = 20

for every

N

, so its limit is 20, not 0. That is exactly what the no-bubble condition rules out.

❌ Discounting the price again.

20/1.08^{10} \approx 9.26

is not a value for anything the share pays. The share pays nothing, so the sum of its discounted expected dividends is $0.

❌ The condition pins it down. Once the resale term must vanish,

P_t

equals the sum of discounted expected dividends, and every one of those is zero, so

P_t = 0

whatever anyone forecasts for prices.

Show solution

With $D_{t+k} = 0$ for all $k$ , the $N$ -year formula keeps only the resale term:

P_t = \frac{\mathsf{E}_t[P_{t+N}]}{(1+r)^N} \quad\text{for every } N

At $N = 10$ :

\mathsf{E}_t[P_{t+10}] = 20 \times 1.08^{10} \approx 20 \times 2.1589 \approx 43.18

The expected price must grow at exactly 8% a year forever, and the resale term equals 20 at every horizon. Each buyer pays only because a later buyer is expected to pay more, for a share that will never pay any of them: the bigger-fool chain. The no-bubble condition requires

\lim_{N\to\infty} \frac{\mathsf{E}_t[P_{t+N}]}{(1+r)^N} = 0,

and the only price for which the resale term goes to zero is $P_t = 0$ . The share is worth $0; a buyer at $20 hands the seller a free lunch.

Problem 4 · Two Horizons, Two Rates

Given: A company is expected to pay a dividend of $5.40 in one year and then, in two years, to liquidate and pay out an expected $60.50 per share as a liquidating dividend; nothing is paid after that. The risk-adjusted rates are $r_{t+1} = 8\%$ for the one-year horizon and $r_{t+2} = 10\%$ for the two-year horizon. Find $P_t$ from the dividend discount model.

✅ Correct!

5.40/1.08 + 60.50/1.10^2 = 5 + 50 = 55

. Each dividend is discounted at its own horizon's rate, raised to its own horizon.

❌ One rate for both horizons. Discounting the year-2 payment at 8% gives

60.50/1.08^2 \approx 51.87

and a total of about $56.87. The model gives each horizon its own rate:

60.50/1.10^2 = 50

❌ Missing the power. The year-2 denominator is

(1 + r_{t+2})^2

, not

1 + r_{t+2}

60.50/1.21 = 50

, not

60.50/1.10 = 55

❌ The rate is already a two-year rate.

r_{t+2}

is the rate for the two-year horizon, like a spot rate, so it is squared:

(1.10)^2 = 1.21

. Multiplying

(1.08)(1.10) = 1.188

treats 10% as a rate for the second year alone and gives about $55.93.

❌ Not quite. Use

P_t = \sum_k \mathsf{E}_t[D_{t+k}]/(1 + r_{t+k})^k

with one term per payment.

Show solution

The liquidating dividend is a payment like any other, and every later dividend is zero, so the sum has two nonzero terms:

P_t = \frac{\mathsf{E}_t[D_{t+1}]}{1 + r_{t+1}} + \frac{\mathsf{E}_t[D_{t+2}]}{(1 + r_{t+2})^2} = \frac{5.40}{1.08} + \frac{60.50}{(1.10)^2} = 5 + \frac{60.50}{1.21} = 5 + 50 = 55

Today's price is $55.00.

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