4.5Lesson 4.5 · Module 4

From Earnings to Dividend Growth

The DDM needs dividend forecasts, but firms report earnings. Understand how payout, retention, book equity, and ROE produce sustainable growth.

Earnings = dividends + retained earnings
g = b × ROE
Growth creates value only when ROE > r
Distinguish ROE from shareholder return

Start

Learning objectives

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

1State the earnings identity: earnings = dividends + retained earnings.
2Define and compute the payout ratio p and the retention ratio b.
3Explain how retained earnings increase book value per share.
4Define ROE = EPS/BVPS and distinguish it from shareholder return.
5Derive the sustainable growth formula g = b × ROE.
6Identify when growth creates value (ROE > r) and when it does not.
7Explain why ROE = r produces value-neutral (NPV = 0) growth.

From Lessons 4.3 and 4.4

The DDM and Gordon model value a stock from its dividends, and the multi-stage model needs a dividend growth rate for each stage. But where does that growth rate come from? Companies report earnings, not dividends, and analysts forecast growth from the bottom up. This lesson traces the chain from earnings to dividends to growth, and shows exactly when growth creates value for shareholders.

5.1Section 1

The earnings identity

Every dollar of earnings goes one of two places: out the door as a dividend, or back into the business as retained earnings.

Earnings identity

Earnings = Dividends + Retained Earnings

Earnings per share split into the dividend per share plus the retained earnings per share.

$5 = $2 + $3

If EPS is $5 and the firm pays a $2 dividend, it keeps $3 inside the business. Earnings are not automatically cash in your pocket — only the dividend portion is.

5.2Section 2

The payout ratio

Payout ratio

p = \frac{D P S}{E P S}

The fraction of earnings paid out as dividends.

p = \frac{$2}{$5} = 0.40

p = 40%

A 40% payout means 40 cents of every dollar earned is distributed. The dividend per share is DPS = p × EPS.

5.3Section 3

The retention (plowback) ratio

Retention ratio

b = \frac{Retained Earnings}{Earnings} = 1 - p

The fraction of earnings kept inside the business to reinvest.

b = \frac{$3}{$5} = 0.60 = 1 - 0.40

b = 60%

Whatever is not paid out is retained. So p + b = 1 always.

5.4Section 4

Book value per share

B V P S = \frac{Book value of common equity}{Shares outstanding}

The accounting value of shareholders' equity, per share.

$B V P S$: book value per share
$Retained/share$: added to BVPS each year

B V P S_{0} = $30, retain $3 \Rightarrow B V P S_{1} = $33

Retained earnings flow directly into book equity. If BVPS₀ is $30 and the firm retains $3 per share, then BVPS₁ = $33.

Definition · Book value is accounting, not market

Book value is what the accounting statements say equity is worth, based on historical cost and accounting rules. It is not the same as market value — the price at which the stock actually trades. A firm can trade far above or below its book value.

5.5Section 5

Return on equity (ROE)

ROE

R O E = \frac{E P S _{1}}{B V P S _{0}}

The accounting return the firm earns on its book equity.

R O E = \frac{$6}{$30} = 0.20

ROE = 20%

A $6 profit on $30 of book equity is a 20% accounting return.

ROE is one of the most misunderstood numbers in finance. It is an accounting measure of how profitable the existing book of business is. It is not any of these:

Concept	Definition	Same as ROE?
ROE	EPS / BVPS	— (itself)
Shareholder return	(D₁ + P₁ − P₀) / P₀	No
Cost of equity r	required return investors demand	No
Dividend yield	D₁ / P₀	No

5.6Section 6

Deriving sustainable growth: g = b × ROE

Growth in earnings comes from a simple chain: the firm retains part of its earnings, that retention increases book equity, and the firm earns ROE on the larger book. Let's trace it with numbers.

Start

$B V P S_{0} = $30$ , $R O E = 20%$ , so $E P S_{1} = 0.20 \times 30 = $6$ . Retention $b = 50%$ .

Retain

Retained = $b \times E P S_{1} = 0.50 \times 6 = $3$ . So $B V P S_{1} = 30 + 3 = $33$ .

Next year

$E P S_{2} = R O E \times B V P S_{1} = 0.20 \times 33 = $6.60$ . Growth = $(6.60 - 6) /6 = 10%$ .

Sustainable growth, derived

g = b \times R O E

Earnings grow at the retention ratio times the return earned on the retained capital.

$b$: retention ratio
$R O E$: return on book equity (and on new investment, under the model's assumptions)

E_{1} = R O E \cdot B V_{0}, retained = b E_{1}, B V_{1} = B V_{0} (1 + b R O E) \Rightarrow g = \frac{E _{2} - E _{1}}{E _{1}} = b R O E

The derivation: next period's earnings are ROE times this period's book; retained earnings raise book by b × ROE percent; so earnings themselves grow at b × ROE.

5.7Section 7

Assumptions behind g = b × ROE

The formula is elegant but rests on several assumptions. Treat it as a useful approximation, not a law of nature.

Retention ratio b is stable over time.
ROE is stable, and new investment earns the same ROE as the existing book.
Reinvested capital is productive — it actually generates the assumed return.
Growth is financed internally (no new share issuance, no buybacks).
Share count does not change.

In reality, retention, ROE, and share count all move around. The formula captures the typical relationship, not every firm in every year.

5.8Section 8

Connecting g to the Gordon model

Gordon with sustainable growth

P_{0} = \frac{E P S _{1} ( 1 - b )}{r - b R O E}

Substitute DPS = (1−b)·EPS and g = b·ROE into P₀ = D₁/(r−g).

There is a trade-off built in: a higher retention b lowers today's dividend (1−b)·EPS but raises the growth rate b·ROE. Whether more retention helps depends on whether ROE exceeds r.

5.9Section 9

Three cases: creates, neutral, destroys value

Fix $r = 10%$ and compare what happens when ROE on new investment is above, at, or below r.

ROE = 15% > r

Each retained dollar earns 15% while costing 10%. Reinvestment creates value. Growth is good.

ROE = 10% = r

Each retained dollar earns exactly its cost. NPV = 0. Growth is value-neutral — it neither helps nor hurts.

ROE = 4% < r

Each retained dollar earns 4% while costing 10%. Reinvestment destroys value. The firm would do better to pay the earnings out.

5.10Section 10

The Texas Western example: ROE = r

A classic illustration. Suppose $E P S_{1} = $1$ , $B V P S_{0} = $10$ , book assets grow at $8%$ , and both $R O E = 10%$ and $r = 10%$ . The firm retains $$0.80$ , so $b = 80%$ , $p = 20%$ , and $D_{1} = $0.20$ . With $g = b R O E = 8%$ :

Growth strategy value

P_{0} = \frac{D _{1}}{r - g} = \frac{0.20}{0.10 - 0.08}

P_{0} = \frac{0.20}{0.02} = $10

P₀ = $10

No-growth comparison

P_{0}^{n o - g r o w t h} = \frac{E P S _{1}}{r} = \frac{1}{0.10}

P_{0}^{n o - g r o w t h} = $10

P₀ = $10

Both strategies give $10. Reinvesting at ROE = r produces growth, but no extra value — the NPV of reinvestment is exactly zero.

The one-dollar explanation

Retain $1 of earnings. It earns $R O E = 10%$ , so $$0.10$ per year forever. That stream is worth $0.10/0.10 = $1$ . You put in $1 of capital and got back $1 of value — NPV = 0. This is exactly why ROE = r is the dividing line between value creation and destruction.

5.11Section 11

Growth slowdown: still worth $10

Now let the company grow at 8% through year 5, then slow to 4%. Keep $R O E = 10%$ and $r = 10%$ . Retention is $b = 80%$ during high growth and $b = 40%$ once stable. The forecast looks like this:

Year	BVPS	EPS	b	Retained	Dividend	End BVPS
0	$10.00	$1.000	80%	$0.800	$0.200	$10.80
1	$10.80	$1.080	80%	$0.864	$0.216	$11.66
2	$11.66	$1.166	80%	$0.933	$0.233	$12.60
3	$12.60	$1.260	80%	$1.008	$0.252	$13.60
4	$13.60	$1.360	80%	$1.088	$0.272	$14.69
5	$14.69	$1.469	40%	$0.588	$0.881	$15.28
6	$15.28	$1.528	40%	$0.611	$0.917	$15.89

Despite the rising earnings and dividends, the value remains $$10$ . Every dollar reinvested earns exactly $r = 10%$ , so each reinvestment has NPV = 0. Growth with ROE = r adds nothing.

5.12Section 12

Historical ROE vs incremental ROE

One last, crucial distinction. A firm can report a high historical ROE on its existing book of business — old, profitable investments — while having few good new opportunities. Value creation depends on the return on the next dollar of investment, not on the average return of the past dollars.

Watch the incremental ROE

A mature firm might show 20% ROE on its legacy assets but only earn 6% on new projects. The reported ROE looks great; the growth strategy still destroys value. Always ask: what is the return on the marginal investment?

Try itEarnings & reinvestment simulator

g = b × ROE

From earnings to sustainable growth

Set earnings per share, book value per share, the payout or retention ratio, the ROE earned on retained capital, and the cost of equity. The simulator traces the chain earn → retain → reinvest → grow and shows whether growth creates value.

Earnings per share EPS₁Book value per share BVPS₀

Payout ratio p = DPS/EPS40%

ROE on retained capital15.0%

Cost of equity r10.0%

Payout and retention always sum to 1: p = 40%, b = 60%.

Dividend per share DPS

$2.00

p × EPS = 40% × $5.00

Retained per share

$3.00

b × EPS = 60% × $5.00

Ending BVPS₁

$33.00

$30.00 + $3.00

Next-year EPS₂

$4.95

ROE × BVPS₁ = 15.0% × $33.00

Sustainable growth

9.00%

g = b × ROE = 60% × 15.0% = 9.00%.

ROE vs r

Value-creating

ROE on new investment (15.0%) exceeds the cost of equity (10.0%). Each dollar retained generates more than a dollar of value.

Two different ROEs

Historical ROE = 16.7%

Profitability on the existing book: EPS₁ / BVPS₀ = $5.00 ÷ $30.00.

ROE on new investment = 15.0%

What the next dollar of retained capital earns. Future value creation depends on this number, not the historical one.

5.13Concept check

When g = r, Gordon cannot run forever

Try itWorked check

EPS₁ = $4, payout = 25%, ROE = 16%, r = 12%

Find the retention ratio, the dividend, the growth rate, and decide whether Gordon can value this stock perpetually.

5.14Common questions

Earnings, ROE, and the value of growth

05Mastery

Summary and mastery check

Try itLesson 4.5 mastery check

Pass with 3 of 5 correct

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

01
If EPS = $5 and dividend = $2, what is the payout ratio?
02
BVPS₀ = $30, ROE = 20%, b = 50%. What is g?
03
If ROE = r, does retaining earnings create value?
04
The sustainable growth formula is:
05
Texas Western: ROE = 10%, r = 10%. Growth strategy value vs no-growth value?

Lesson summary

1Earnings = dividends + retained earnings.
2Payout ratio p = DPS/EPS; retention ratio b = 1 − p.
3Book value per share increases by retained earnings per share.
4ROE = EPS/BVPS measures accounting profitability, not stock return.
5Sustainable growth: g = b × ROE.
6Growth creates value only when the return on new investment exceeds the cost of equity.
7When ROE = r, growth is value-neutral (NPV = 0).

Replay this lesson Continue to Growth Opportunities, PVGO, and P/E