Finance-Theory · Unit 9 · Video 7 · Interactive Practice

The Dividend Yield Is Half the Answer: Cost of Capital as r = D/P + g

IKey Formulas

Formula	Name	What it needs
$P_t = \dfrac{D}{r - g}, \quad r > g$	Gordon growth model (the starting point)	A dividend $D$ growing at rate $g$ forever, discounted at $r$ ; $P_t$ is the ex-dividend price, so $D$ is next period's dividend
$r = \dfrac{D}{P_t} + g$	Cost of capital: dividend yield plus growth	Multiply the Gordon model by $r - g$ , divide by $P_t$ , add $g$ . $D/P_t$ is the dividend-price ratio, the stock version of coupon over price
$r = \dfrac{D_0(1+g)}{P_t} + g$	The same formula in observable inputs	The dividend just paid $D_0$ and today's price (observed) plus $g$ (a judgment), with $g$ measured on the dividend cycle: quarterly dividends, quarterly $g$
$r = (0.052)(1.049) + 0.049 \approx 10.35\%$	Duke Power, September 1992	Yield $D_0/P_0 = 5.2\%$ with Value Line's $g = 4.9\%$ ; I/B/E/S's $g = 4.1\%$ gives $(0.052)(1.041) + 0.041 \approx 9.51\%$

Key Insight: The dividend yield measures what current operations pay out; $g$ is the one place a company's future growth opportunities enter its cost of capital. The formula ties $r$ and $g$ together: assume one and it lets you infer the other, but it hands you neither on its own.

IITwo Pieces of the Required Return

Duke Power's 5.2% dividend yield was only about half of its cost of capital; growth supplied the rest.

Dividend yield D₀/P₀ (%) 5.2% Growth g (%) 4.9%

IIIThe Price the Yield Alone Cannot Explain

Priced as a flat perpetuity, Duke Power was worth about half its market price; each assumed $r$ implies a $g$ .

Assumed cost of capital r (%) 10.35%

IVNon-Payers and Negative Cash Flows

Inside its range a negative $D$ gives a negative price; outside it, deeper losses mean a higher “value”.

Steady-state D ($ per share) −1.50

💡 These are steady-state expressions: years in which a firm pays nothing or loses money while investing to grow are valued one at a time, and a steady-state formula applies only after them.

VQuiz Questions

Problem 1 · Cost of Capital from the Dividend Just Paid

Given: A stock trades ex-dividend at $40. It has just paid a dividend of $2.00 per share, and dividends are expected to grow at $g = 5\%$ a year forever. Find the cost of capital $r$ .

✅ Correct! Next period's dividend is

2.00 \times 1.05 = 2.10

, so the yield term is

2.10/40 = 5.25\%

and

r = 5.25\% + 5\% = 10.25\%

❌ Close, but which dividend? The price is ex-dividend, so the

D

D/P_t

is next period's dividend,

D_0(1+g)

, not the one just paid.

❌ That is only half the answer.

5.25\%

is the dividend yield on next period's dividend; the cost of capital adds the growth rate

g

❌ Not quite. Only the dividend grows by

(1+g)

; the

g

term is added once, unscaled:

r = D_0(1+g)/P_t + g

❌ Not quite. Use

r = D_0(1+g)/P_t + g

Show solution

The price is ex-dividend, so the dividend in the formula is next period's: $D = D_0(1+g) = 2.00 \times 1.05 = 2.10$ dollars.

r = \frac{D_0(1+g)}{P_t} + g = \frac{2.10}{40} + 0.05 = 0.0525 + 0.05 = 0.1025 = 10.25\%

Using the dividend just paid instead gives $2.00/40 + 0.05 = 10.00\%$ , a quarter of a percentage point too low: the missing piece is $D_0 g/P_t = 0.05 \times 0.05 = 0.25\%$ .

Problem 2 · Match the Dividend Cycle

Given: A company pays dividends quarterly. It has just paid $0.50 per share, its ex-dividend price is $50, and its dividends are expected to grow $1\%$ per quarter forever. Find its cost of capital per quarter.

✅ Correct! Quarterly dividend, quarterly growth, quarterly cost of capital:

0.505/50 + 0.01 = 2.01\%

per quarter.

❌ The cycles are mixed. Multiplying the dividend by 4 puts an annual yield next to a quarterly

g

. The formula has no unit of time built in, so every input must be on the same cycle.

❌ Close, but which dividend? At the ex-dividend price the next dividend is

0.50 \times 1.01 = 0.505

, not the $0.50 just paid.

❌ That is only the dividend yield. Add the quarterly growth rate

g = 1\%

❌ Not quite. Keep the dividend, the growth rate and the answer all per quarter.

Show solution

Dividends are paid quarterly, so $g$ and $r$ are both per quarter. Next quarter's dividend is $D_0(1+g) = 0.50 \times 1.01 = 0.505$ dollars.

r = \frac{0.505}{50} + 0.01 = 0.0101 + 0.01 = 0.0201 = 2.01\% \text{ per quarter}

The mixed-cycle error annualizes the dividend but keeps the quarterly $g$ : $4 \times 0.505/50 + 0.01 = 0.0404 + 0.01 = 5.04\%$ , which is neither a quarterly nor an annual cost of capital.

Problem 3 · What the No-Growth Formula Misses

Given: A stock's dividend yield $D_0/P_0$ is $4\%$ , and its dividends are expected to grow at $6\%$ a year forever. Find its cost of capital $r$ , and then the value of the share as a no-growth perpetuity $D_0/r$ at that $r$ .

What is the cost of capital?

What is the no-growth value $D_0/r$ ?

✅ Correct! The market price is

D_0/0.04 = 25\,D_0

, which is

25 \times 0.1024 = 2.56

times the no-growth value. The gap is what

g

pays for.

❌ Close, but which dividend?

4\% + 6\%

uses the dividend just paid. Next period's dividend is

D_0(1.06)

, so the yield term is

0.04 \times 1.06 = 4.24\%

❌ That is only the dividend yield. Add the growth rate:

r = 4.24\% + 6\%

❌ Check your cost of capital. Only the dividend grows by

(1+g)

r = (0.04)(1.06) + 0.06

❌ That is the market price.

D_0/0.04 = 25\,D_0

is what the market pays. The no-growth value divides

D_0

by the cost of capital

r

, not by the yield.

❌ Not quite.

1/0.06

discounts at the growth rate. A perpetuity is discounted at the cost of capital,

r = 10.24\%

❌ Close, but which cost of capital?

D_0/0.10 = 10\,D_0

uses the

10\%

that comes from adding

g

to the yield on the dividend just paid. The cost of capital here is

r = 10.24\%

❌ Check the no-growth value. Divide

D_0

by the cost of capital you found,

r = 0.1024

Show solution

Step 1: cost of capital.

r = \frac{D_0}{P_0}(1+g) + g = (0.04)(1.06) + 0.06 = 0.0424 + 0.06 = 0.1024 = 10.24\%

Step 2: no-growth value. A flat perpetuity of $D_0$ at $r$ :

\frac{D_0}{r} = \frac{D_0}{0.1024} = 9.765625\,D_0 \approx 9.77\,D_0

Compare: the market price is $P_0 = D_0/0.04 = 25\,D_0$ , and $25/9.765625 = 2.56$ . Leave out $g$ and the stock looks overvalued by a factor of more than two. Duke Power showed the same pattern: $19.23\,D_0$ against $9.66\,D_0$ .

Problem 4 · A Loss-Making Firm and g Above r

Given: An internet-era firm is estimated to hand shareholders a steady-state cash flow of $D = -2$ dollars per share a year (a loss). An analyst uses $r = 10\%$ and $g = 12\%$ in $P = D/(r - g)$ .

What number does the formula return?

What should the analyst conclude?

✅ Correct! A positive number from a negative cash flow, larger the deeper the loss, is the perverse result the video warns of: the model is being used outside its range.

❌ Check the sign of the denominator.

r - g = 0.10 - 0.12 = -0.02

, and a negative divided by a negative is positive.

❌ Not quite.

-2/0.10

is the no-growth perpetuity; the formula here divides by

r - g

❌ Not quite.

-2/0.12

divides by the growth rate alone. The denominator is

r - g = 0.10 - 0.12 = -0.02

❌ Check the arithmetic. Compute

r - g

first, then divide

D

by it.

❌ That is the perverse result itself. The arithmetic is right,

-4/(-0.02) = 200

, but losses raising value is the sign that the model is misused, not a valuation.

❌ Not quite. Using the steady-state dividend does not rescue a formula whose condition

r > g

fails.

❌ Not quite.

-2 \times 1.12/(-0.02) = 112

: still positive. The trouble is

g > r

, not the timing of

D

❌ Not quite. Ask whether the formula's condition

r > g

holds here.

Show solution

P = \frac{D}{r - g} = \frac{-2}{0.10 - 0.12} = \frac{-2}{-0.02} = 100

The Gordon model requires $r > g$ . Inside that range a negative $D$ gives a negative price, which already makes no sense. Outside it, as here, the formula gives a positive number, and a deeper loss gives a larger one: $-4/(-0.02) = 200$ . That is the sign of misuse. $P = D/(r-g)$ is a steady-state expression, so the years of zero or negative cash flow, while the firm invests to grow, have to be modelled explicitly, one period at a time.

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