Valuing equities: the dividend discount model · RiskAtoms Academy

By the end you will be able to

1Value a share as the present value of its expected dividends.
2Use the Gordon growth formula to value a share whose dividend grows at a constant rate.
3Explain why the growth rate must be below the required return, and how sensitive the value is to the gap between them.
4Value a share whose dividend grows fast for a few years and then settles down.

Why it matters

A share promises nothing and never matures. So how can anyone say what it is worth? The same way you priced a bond: add up what it should pay you, discounted to today. The twist is that the payments are expected to grow, and they never stop.

Core explanation

A share is worth its future dividends

Think of a share in a steady electricity company. Each year it pays a dividend, and each year the dividend creeps up as the business grows. What should you pay for that share today?

You already know the tool. In L1.3.03 you discounted a future amount to find its present value, and in L1.4.03 you priced a bond as the sum of its discounted cash flows. A share works the same way. Its cash flows are its dividends, so its value is the present value of all of them. This is the dividend discount model.

“But I will sell it in a few years.” Click each card to see why that does not change the answer.

Two numbers drive the model.

The required return, $r$ : the yearly return investors demand for the risk of holding the share. Shares are riskier than bonds (L1.5.01), so $r$ is usually higher than bond yields. Later levels show how to estimate it; here it is given.
The dividend growth rate, $g$ : how fast the dividend is expected to rise each year.

Zero growth is a perpetuity. If the dividend never changes, the share is a perpetuity (L1.3.04). A $5.00 dividend for ever at 10% is worth $5.00 ÷ 0.10 = $50.00.

With growth, there is an equally short answer. If the dividend grows at a steady $g$ for ever, the endless sum collapses to next year's dividend divided by $r - g$ . The formal version below shows why. This is the Gordon growth model, and it carries one strict condition: $g$ must be below $r$ . If dividends grew as fast as the discount rate, every future dividend would be worth as much today as the first one, and the total would be infinite.

Why the gap matters

The value depends on

r - g

, a small number. Northwind Utilities, in the worked example, has

r - g

= 5%. Move

r

up by one percentage point and the gap grows by a fifth, so the value falls by 17%. That sensitivity is the model's greatest weakness, and an important lesson about equity risk.

The formal version

The general model, on a few lines. Write $D_t$ for the dividend in year $t$ . You have seen each line before, applied to bonds:

\begin{aligned} P_0 &= \text{PV of all future dividends} \\[2pt] &= \frac{D_1}{1+r} + \frac{D_2}{(1+r)^2} + \frac{D_3}{(1+r)^3} + \cdots \\[2pt] &= \sum_{t=1}^{\infty} \frac{D_t}{(1+r)^t} \end{aligned}

Unlike a bond, the sum runs to infinity ( $\infty$ ) and has no face value at the end. The Gordon growth formula is new, so let's build it one piece at a time.

Two-stage growth, on a few lines. Young companies often grow fast for a while, then slow down. Forecast each dividend for the first $n$ years, then use Gordon for the rest. The Gordon value at year $n$ is called the terminal value:

\begin{aligned} P_0 &= \text{PV of dividends in years } 1 \text{ to } n + \text{PV of the value at year } n \\[2pt] &= \sum_{t=1}^{n} \frac{D_t}{(1+r)^t} + \frac{P_n}{(1+r)^n}, \qquad P_n = \frac{D_{n+1}}{r - g_{\text{long}}} \end{aligned}

Reading it backwards. Rearrange Gordon and a market price tells you the return the market expects: $r = \dfrac{D_1}{P_0} + g$ , the dividend yield plus growth.

Try it

Start at Northwind's numbers: $2.00 dividend, 8% required return, 3% growth. Slide $g$ slowly towards $r$ and watch the value run away. Then push $g$ past $r$ : the discounted dividends stop shrinking, and there is no value at all.

Try itDividend discount explorer

Zero growth: the preferred share pricer →

Dividend just paid, D₀$2.00

Required return, r8.00%

Dividend growth, g3.00%

Next year's dividend D₁ = D₀ × (1 + g) = $2.06

Value of one share

$41.20

r − g5.00%

Dividend yield5.00%

From first 10 years38%

Each dividend, worth todayYears 1 to 40. Shrinking bars mean a finite total.

Value against growthThe dashed line is r. Value explodes as g closes in on it.

The “from first 10 years” figure is a surprise for most people: even at Northwind's modest growth, only 38% of the value comes from the next decade of dividends. For more on summing a geometric series, try the sequences and series calculator.

Worked example

A utility and a growth company

All rates are annual and effective; dividends are paid once a year. Northwind Utilities, a US power company, has just paid a dividend of $2.00. Dividends are expected to grow 3% a year for ever, and investors require 8%.

Northwind Utilities with the Gordon model

Step 1 of 5

1
Grow today's dividend into next year's
$D_1 = 2.00 \times 1.03 = 2.06$

Tasman Health, an Australian company, is growing fast. It has just paid A$1.00. Analysts expect 12% growth for 3 years, then 4% for ever. Investors require 9%.

Tasman Health with a two-stage model

Step 1 of 5

1
Forecast the fast-growth dividends
Year Dividend (A$) PV at 9%
1 1.1200 1.0275
2 1.2544 1.0558
3 1.4049 1.0849
Total 3.1682

Year	Dividend (A$)	PV at 9%
1	1.1200	1.0275
2	1.2544	1.0558
3	1.4049	1.0849
Total		3.1682

Regulatory lens · looking ahead

No rules apply yet. When shares have no market price, accountants value them with techniques like this one: IFRS 13 Fair Value Measurement calls it the income approach (L4.7). The extreme sensitivity to $r - g$ is also why regulators stress equity holdings heavily, for example in the Solvency II equity risk charge for insurers (L4.6).

Sparks

SparkOlder than its name

The formula is named after Myron Gordon, who popularised it in the late 1950s. But John Burr Williams had already argued in The Theory of Investment Value (1938) that a share is worth the present value of its future dividends, and worked out the constant-growth case.

SparkMost of the value is far away

At Northwind's 8% required return and 3% growth, the first ten years of dividends make up only 38% of the share's value. A share is a bet on the distant future, which is why news about long-run growth or interest rates can move prices so much in a single day.

Key takeaways

A share is worth the present value of all its expected dividends. Selling the share later does not change this: the buyer pays for the dividends that follow.
If dividends grow at a constant rate g for ever, P₀ = D₁ ÷ (r − g). Grow today's dividend into next year's first.
The formula needs g < r. As g approaches r, the value shoots up; at or above r, there is no finite value.
Value is very sensitive to r − g, and most of it comes from dividends many years away.
For a company growing fast now, forecast the early dividends and add a Gordon terminal value: the two-stage model.

Reaction

Check what stuck

Question 1 · objective 1

You plan to hold a share for one year. You expect a dividend of 0.80 at the end of the year and to sell the share then for 21.00. You require 9% a year. What is the share worth today, to two decimal places?

Question 2 · objective 2

A company has just paid a dividend of 1.50. Dividends are expected to grow 4% a year for ever, and investors require 10%. What is the share worth, to two decimal places?

Question 3 · objective 3

An analyst plugs r = 7% and g = 8% into the Gordon formula and gets a negative price. What has gone wrong?

ANothing: the share is worth less than zeroBThe formula needs g below r; with g ≥ r the sum of dividends has no finite valueCThe dividend should have been discounted twiceDNegative prices mean the share should be sold short

Question 4 · objective 3

A share trades at 40.00. Next year's dividend is expected to be 1.20, growing 5% a year after that. What return does the price imply investors require, in per cent?

%

Question 5 · objective 4

A dividend of 2.00 has just been paid. It will grow 10% a year for two years, then 3% a year for ever. Investors require 8%. What is the share worth, to two decimal places?

Glossary

Dividend discount model: Dividend discount model: values a share as the present value of all the dividends it is expected to pay.
Required return: The annual return investors demand for holding a share, given its risk. Used as the discount rate for its dividends.
Dividend growth rate: The rate at which the dividend per share is expected to grow each year.
Gordon growth model: The dividend discount model for dividends that grow at a constant rate for ever: P₀ = D₁ ÷ (r − g).
Perpetuity: A payment of the same amount every year for ever. Its value today is the payment divided by the rate.
Terminal value: The value, at a future date, of all dividends after that date. Used to close off a forecast.
Dividend yield: Next year's dividend divided by today's share price.

Deep Dives & Origins

Deep Dives

Brealey, Myers and Allen, Principles of Corporate Finance, the chapter on valuing common stocks. Beginner.
Koller, Goedhart and Wessels (McKinsey & Company), Valuation: Measuring and Managing the Value of Companies, Wiley, the chapters on continuing value. Intermediate.

Origins

J. B. Williams, The Theory of Investment Value, Harvard University Press, 1938.
M. J. Gordon, “Dividends, Earnings, and Stock Prices”, Review of Economics and Statistics 41(2), 1959.
Brealey, Myers and Allen, Principles of Corporate Finance, McGraw Hill.
IFRS 13 Fair Value Measurement, IASB.

Watch this lesson

Video version

Prefer to watch? The same lesson as a 14-minute video will sit here, made from this page's script.