4.3Lesson 4.3 · Module 4

The Gordon Growth Model

Turn the DDM into usable constant-dividend and constant-growth formulas. The model's usefulness depends on realistic long-run assumptions, especially the gap between r and g.

Constant-dividend stock = perpetuity
Gordon Growth: P₀ = D₁/(r−g)
D₀ was just paid; D₁ is next
Why r > g is required
Expected return = dividend yield + growth

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Learning objectives

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

1Value a constant-dividend stock as a perpetuity: P₀ = D / r.
2Value a constant-growth stock using the Gordon Growth Model: P₀ = D₁ / (r − g).
3Distinguish D₀ (just paid) from D₁ (the next dividend) and use the correct one.
4Explain why the model requires r > g, both mathematically and economically.
5Explain why small changes in r − g produce large changes in value.
6Decompose expected return into dividend yield plus growth.
7Solve the Gordon model for value, return, or growth depending on the unknown.
8Identify when the Gordon model is appropriate and when it is not.

From Lesson 4.2

In Lesson 4.2 we derived the Dividend Discount Model by pushing the resale price infinitely far into the future: $P_{0} = \sum_{t = 1}^{\infty} E [D_{t}] / (1 + r)^{t}$ . That formula is correct but impractical — forecasting every individual dividend forever is impossible. This lesson introduces two simplifying assumptions that turn the DDM into closed-form formulas you can actually compute: constant dividends, and constant growth.

3.1Section 1

Begin from the general DDM

General Dividend Discount Model

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

The stock's value is the present value of all expected future dividends.

This is exact in principle but unusable in practice: forecasting E[ $D_{t}$ ] for every year into the indefinite future is not feasible. We need simplifying assumptions.

3.2Section 2

A constant-dividend stock is a perpetuity

Suppose a company pays a constant $$5$ dividend every year, forever, and the required return is $r = 10%$ . The DDM becomes a familiar geometric series:

Constant-dividend series

P_{0} = \frac{5}{1.10} + \frac{5}{1.1 0 ^{2}} + \frac{5}{1.1 0 ^{3}} + \dots

Each $5 dividend is discounted back by one more power of (1+r). This is a level perpetuity.

Perpetuity formula

P_{0} = \frac{D}{r}

A level perpetuity paying D each period, discounted at r, is worth D divided by r.

P_{0} = \frac{5}{0.10}

P₀ = $50

At $50 the dividend yield is D/P₀ = 5/50 = 10%, which equals r. Growth is 0%, so the total expected return is 10%.

3.3Section 3

Dividend and discount-rate effects

From $P_{0} = D / r$ the comparative statics are immediate:

Higher dividend D

D ↑ → P₀ ↑. More cash to shareholders means more value.

Higher required return r

r ↑ → P₀ ↓. A higher discount rate shrinks the present value of the same dividends.

3.4Section 4

Constant dividend growth

Now relax the assumption slightly: instead of a flat dividend, assume the dividend grows at a constant rate $g$ forever. So $D_{2} = D_{1} (1 + g)$ , $D_{3} = D_{1} (1 + g)^{2}$ , and so on. The same geometric-series math collapses the infinite sum into a single, elegant formula.

The Gordon Growth Model

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

A dividend growing at a constant rate g forever, discounted at r, is worth next year's dividend divided by (r − g). This is a growing perpetuity.

$P_{0}$: stock value today
$D_{1}$: next dividend (one period from now)
$r$: required return (cost of equity)
$g$: constant dividend growth rate

This is the Gordon Growth Model (also called the constant-growth DDM). It is the workhorse formula for valuing mature, stable dividend payers.

3.5Section 5

A full worked example

Worked example

P_{0} = \frac{D _{1}}{r - g} = \frac{D _{0} ( 1 + g )}{r - g}

Given D₀ = $2, g = 4%, r = 10%. First compute D₁, then plug into the model.

D_{1} = $2 \times (1.04) = $2.08, P_{0} = \frac{$2.08}{0.10 - 0.04} = \frac{$2.08}{0.06}

P₀ = $34.67

A stock that just paid a $2 dividend, growing at 4% per year, with a 10% required return, is worth $34.67 per share.

3.6Section 6

D₀ vs. D₁: which dividend do you use?

A common and costly mistake is using the wrong dividend. The two look similar but are separated by one full period.

D₀ — the dividend just paid. If you buy today, you will not receive D₀; it already went to the previous owner.
D₁ — the next dividend, the first one available to a buyer today. This is what enters the Gordon formula.

If you are given D₀, grow it one period first.

Gordon model using D₀

P_{0} = \frac{D _{0} ( 1 + g )}{r - g}

When the most recent dividend D₀ is what you know, grow it by one period to obtain D₁ = D₀(1+g), then divide by (r − g).

The numerator must always be the next dividend the buyer will receive — never the one already paid.

Payment timeline

D₀ (just paid)You buy todayD₁ (first dividend you receive)

3.7Section 7

Why the model requires r > g

The Gordon formula only makes sense when $r > g$ . There are two reasons.

Mathematical

The infinite series converges only when $(1 + g) / (1 + r) < 1$ , i.e. $g < r$ . Otherwise the sum diverges and no finite price exists.

Economic

No company can grow faster than the overall economy forever. A perpetual growth rate above the economy's long-run rate is not credible.

What when g ≥ r?

The model is invalid. It does not imply negative value or infinite value — it simply cannot be used. You would need a different model, such as multi-stage growth.

3.8Section 8

Sensitivity to the r − g gap

Hold $D_{1} = $2$ and $r = 10%$ fixed, and vary $g$ from 2% to 9%. Watch how $P_{0}$ explodes as $g$ approaches $r$ .

g	r − g	P₀ = 2 / (r − g)
2%	8%	$25.00
4%	6%	$33.33
6%	4%	$50.00
8%	2%	$100.00
9%	1%	$200.00

Moving $g$ from 2% to 9% — a seven-point change in an assumption about the indefinite future — multiplies the value by eight times. This is why realistic, defensible long-run growth assumptions matter enormously when using the Gordon model.

3.9Section 9

Expected return = dividend yield + growth

Rearrange the Gordon model

r = \frac{D _{1}}{P _{0}} + g

Solving the Gordon model for r splits the expected total return into two pieces: the dividend yield (D₁/P₀) plus the growth rate g.

Under the Gordon assumptions, the price grows at g, so P₁ = P₀(1 + g) and the capital-gains yield (P₁ − P₀)/P₀ equals g. Total expected return is therefore dividend yield plus growth.

3.10Section 10

A return example

Return decomposition example

r = \frac{D _{1}}{P _{0}} + g

P₀ = $40, D₁ = $2, g = 5%. The dividend yield is D₁/P₀, and the capital-gains yield is g.

r = \frac{$2}{$40} + 5% = 5% + 5%

r = 10%

Next year the price should be P₁ = $40 × (1.05) = $42. Total gain = $2 dividend + $2 price increase = $4, which is 10% of the $40 paid.

3.11Section 11

Solve for value, return, or growth

Solve for value

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

Given D₁, r, and g, compute the price.

Solve for return

r = \frac{D _{1}}{P _{0}} + g

Given the market price P₀, D₁, and g, infer the implied required return.

Solve for growth

g = r - \frac{D _{1}}{P _{0}}

Given r, P₀, and D₁, infer the growth rate the market is pricing in.

These “implied” values are model-dependent estimates. Solving for an implied growth rate tells you what the market must be assuming for the Gordon model to hold — it does not tell you whether that assumption is realistic.

3.12Section 12

When the Gordon model fits — and when it does not

The Gordon model is a precision tool, not a universal one. It fits companies whose dividends plausibly grow at a steady rate forever.

Appropriate for

Stable, mature companies
Sustainable, predictable payout ratios
Constant, credible long-run growth
Stable required return r
Indefinite dividend stream

Inappropriate for

Early-stage growth companies
No foreseeable dividend or payout
Unstable or cyclical dividends
Restructurings and turnarounds
Temporarily very high growth (g would approach or exceed r)

Preview: multi-stage growth

Real companies often grow fast for a while, then settle. A future lesson introduces multi-stage models: value an initial high-growth period explicitly, then attach a Gordon terminal value once growth normalizes.

Try itGordon Growth lab

P₀ = D₁ / (r − g)

Value a constant-growth stock

Set the most recent dividend D₀, the constant growth rate g, and the required return r. The model requires r > g — when that fails, the constant-growth perpetuity formula cannot be used.

Most recent dividend D₀

Growth rate g4.00%

Required return r10.00%

Next dividend D₁

$2.08

$2.00 × (1 + 4.00%)

Estimated value P₀

$34.67

$2.08 ÷ (10.00% − 4.00%)

Dividend yield

6.00%

$2.08 ÷ $34.67

Expected total return

10.00%

6.00% + 4.00%

Return decomposition

Expected total return = dividend yield + growth = 6.00% + 4.00% = 10.00%. Under Gordon, price grows at g, so the capital-gains yield equals g.

Sensitivity: P₀ as g approaches r (D₀ = $2.00, r = 10.00%)

g	r − g	P₀
0%	10%	$20.00
1%	9%	$22.44
2%	8%	$25.50
3%	7%	$29.43
4%	6%	$34.67
5%	5%	$42.00
6%	4%	$53.00
7%	3%	$71.33
8%	2%	$108.00
9%	1%	$218.00

Notice how P₀ grows rapidly as g approaches r. A small change in the r − g gap produces a large change in value. That is why realistic long-run growth assumptions matter so much.

Compare two assumption sets

Scenario A

D₀

g4%

r10%

P₀ = $34.67

Scenario B

D₀

g6%

r10%

P₀ = $53.00

Scenario A values the stock at $34.67 on a 10.00% required return and 4.00% growth. Scenario B values it at $53.00 — the difference comes entirely from the assumptions, not the formula.

3.13Concept check

Two quick checks

Try itConcept checks

Check 1 — Gordon valuation

D₀ = $3, g = 4%, r = 12%. Find D₁ and P₀.

Check 2 — Implied return

P₀ = $60, D₁ = $2.40, g = 5%. What is the implied r?

3.14Common questions

Questions on growth, value, and the r > g condition

04Mastery

Summary and mastery check

Try itLesson 4.3 mastery check

Pass with 4 of 6 correct

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

01
A constant dividend of $5/year forever, r = 10%. What is P₀?
02
D₀ = $2, g = 4%. What is D₁?
03
D₁ = $2.08, r = 10%, g = 4%. What is P₀?
04
P₀ = $60, D₁ = $2.40, g = 5%. What is the implied r?
05
What happens when g ≥ r in the Gordon Growth Model?
06
Under the Gordon model, the expected total return equals:

Lesson summary

1A constant-dividend stock is valued as P₀ = D/r.
2The Gordon Growth Model values a growing perpetuity: P₀ = D₁/(r−g).
3D₀ was just paid; D₁ is the next dividend.
4The model requires r > g.
5Small changes in r−g cause large valuation changes.
6Expected return = dividend yield + growth rate.
7The Gordon model suits stable, mature companies with sustainable payout.

Replay this lesson Continue to Multi-Stage Growth Valuation