Constant-Growth Gordon Model

Introduction

A share’s value depends on the cash flows investors expect to receive and the return they require for bearing the risk of those cash flows. When a company is expected to pay dividends that grow at a stable rate indefinitely, the constant-growth Gordon model gives a compact way to connect those expectations to a per-share value.

The model is useful only when its timing and growth conditions fit the forecast. In particular, the dividend used in the numerator is the one expected next, not necessarily the dividend just paid. The model also requires the required return to exceed the perpetual dividend growth rate.

Definition: The constant-growth Gordon model values a share as the present value of dividends that are expected to grow at a constant annual rate forever. Let $P_0$ be the share value now, $D_0$ the dividend just paid, $D_1$ the dividend expected one period from now, $r$ the annual required return on the share, and $g$ the constant annual dividend growth rate. Under the model’s stable-growth assumptions, $$ P_0=\frac{D_1}{r-g}, \qquad r>g. $$

Model Inputs and Timing

The formula is most naturally stated with annual dividends and annual rates. The value $P_0$ is measured immediately after the dividend at time 0 has been paid; $D_1$ is expected at time 1. If the most recent dividend is $D_0$, the constant-growth assumption gives $D_1=D_0(1+g)$. If a problem directly supplies the next expected dividend, use that amount as $D_1$ without growing it again.

The required return $r$ is the annual return used to discount the share’s future dividends. It must be expressed on the same annual basis as $g$. A required return of 9% and growth of 3% are entered as 0.09 and 0.03, respectively. The difference $r-g$ is therefore 0.06, or six percentage points.

The model represents a perpetuity of dividends with stable growth. Its basic conditions are that dividends continue indefinitely, grow at the same rate each period, and are discounted at a constant required return. The condition $r>g$ ensures that the discounted dividend stream converges to a finite value. If $r\leq g$, the Gordon formula does not provide a finite positive valuation for a positive dividend stream. If dividends are expected to grow at different rates over time, a single constant-growth model is not the appropriate description of that forecast.

Worked Example: Converting the Last Dividend to the Next Dividend

A hypothetical company has just paid an annual dividend of $2.40 per share. Its dividends are expected to grow by 4% per year indefinitely, and investors require an annual return of 10%. Estimate the value per share immediately after the dividend has been paid. State the model condition and check the result.

Method. First convert the dividend just paid, $D_0$, to the next expected dividend, $D_1$, and then apply the constant-growth Gordon model.

The annual growth rate is $g=0.04$, so

D_1=D_0(1+g)=\$2.40(1.04)=\$2.496.

The required return is $r=0.10$. The convergence condition holds because $0.10>0.04$. The return-growth spread is

$$ r-g=0.10-0.04=0.06. $$

Therefore,

P_0=\frac{\$2.496}{0.06}=\$41.60.

Check. The next dividend is $2.496, and 6% of the estimated value is $0.06(\$41.60)=\$2.496$, matching $D_1$. The estimated value is $41.60 per share. A common mistake is to use the $2.40 dividend just paid in the numerator; doing so would give $40.00 and would value the stream as though its time-1 dividend were only $2.40.

Why the Formula Has This Form

The formula summarizes the present values of all future dividends. To see why it has a denominator of $r-g$, suppose the model’s conditions hold: the first future dividend is $D_1$, each following dividend grows by $g$, and each dividend is discounted at the annual required return $r$. The time-1 dividend is discounted for one period, the time-2 dividend for two periods, and so on.

Write the dividend present values.
With $D_t=D_1(1+g)^{t-1}$, the value is the infinite sum of each dividend discounted back to time 0.

Identify the geometric ratio.
Each term is the previous term multiplied by $q=(1+g)/(1+r)$. For the usual case of nonnegative growth and a positive required return, $r>g$ gives $0<q<1$.

Sum the convergent series.
The present value is its first term, $D_1/(1+r)$, divided by $1-q$. Simplifying gives $D_1/(r-g)$.

\begin{aligned} P_0 &=\frac{D_1}{1+r}+\frac{D_1(1+g)}{(1+r)^2} +\frac{D_1(1+g)^2}{(1+r)^3}+\cdots\\ &=\frac{D_1}{1+r}\left(1+q+q^2+\cdots\right), \qquad q=\frac{1+g}{1+r}\\ &=\frac{D_1}{1+r}\frac{1}{1-q} =\frac{D_1}{1+r}\frac{1+r}{r-g} =\frac{D_1}{r-g}. \end{aligned}

The geometric-series step requires the ratio’s absolute value to be less than 1. In the standard setting of nonnegative dividend growth and positive required return, this is equivalent to $r>g$. The condition is not merely a convenient check: without convergence, adding the discounted dividends does not produce the finite value represented by the formula.

Applying the Model to a Market Comparison

A model value can be compared with a quoted market price, provided the inputs reflect the same valuation date and the same expectations. Such a comparison is conditional on the model: it does not establish that the forecast dividend, growth rate, or required return will actually occur. A value above the quoted price means only that the share appears undervalued relative to the specified Gordon-model inputs.

Worked Example: Comparing Model Value with a Quoted Price

A hypothetical firm is expected to pay a dividend of $3.15 per share one year from now. Dividends are forecast to grow by 3% annually forever, and the required annual return is 9%. The current quoted share price is $48. Estimate the model value, compare it with the quoted price, and calculate the model value’s premium over that price as a percentage of the quoted price.

Method. Since the problem gives the next dividend directly, use it as $D_1$. First verify that the required return exceeds growth:

$$ r-g=0.09-0.03=0.06>0. $$

The Gordon value is

P_0=\frac{\$3.15}{0.06}=\$52.50.

The model value exceeds the quoted price by

\$52.50-\$48.00=\$4.50.

As a percentage of the quoted price, the premium is

\frac{\$4.50}{\$48.00}\times 100\%=9.375\%.

Check. Six percent of $52.50 is $3.15, exactly the next dividend. To two decimal places for the percentage, the model value is $4.50, or 9.38%, above the quoted price. The common mistake avoided here is growing $D_1$ a second time; the supplied $3.15 already refers to the next dividend.

From Per-Share Value to Total Equity Value

The Gordon formula produces a value per share. If a problem asks for the total equity value represented by all common shares, multiply the per-share value by the number of shares outstanding, using consistent units. This step changes the scale of the answer, not the valuation model.

Worked Example: Scaling a Per-Share Estimate to Equity Value

A hypothetical company has just paid an annual dividend of $1.80 per share. Its dividend is expected to grow by 5% per year indefinitely, and its required annual return is 11%. Estimate the value per share immediately after the dividend, then estimate the total equity value if 2.4 million shares are outstanding. Also calculate the total next-year dividend for those shares.

Method. Convert the last dividend to the next dividend, value one share, and then scale by the share count. The required-return condition is satisfied because $0.11>0.05$.

The next dividend per share is

D_1=\$1.80(1.05)=\$1.89.

The return-growth spread is

$$ r-g=0.11-0.05=0.06. $$

Thus the value per share is

P_0=\frac{\$1.89}{0.06}=\$31.50.

With 2.4 million shares outstanding, the total equity value is

\$31.50(2{,}400{,}000)=\$75{,}600{,}000.

The total next-year dividend is

\$1.89(2{,}400{,}000)=\$4{,}536{,}000.

Check. Six percent of the total equity value is $0.06(\$75{,}600{,}000)=\$4{,}536{,}000$, which matches the total next-year dividend. The estimates are $31.50 per share and $75.6 million of total equity value. The calculation assumes the stated per-share dividend and share count are consistent; multiplying the per-share value by 2.4 rather than 2.4 million would confuse the share count’s units.

Common Traps and Interpretation

Using the wrong dividend date. $D_0$ is the dividend just paid; $D_1$ is the dividend expected one period ahead. If only $D_0$ is supplied, calculate $D_1=D_0(1+g)$. If $D_1$ is supplied, do not grow it again.
Subtracting rates in inconsistent units. Use decimal rates in the formula. For example, 8% minus 3% is $0.08-0.03=0.05$, not 8 minus 3 in a formula where the dividend is in dollars.
Ignoring $r>g$. A result from dividing by zero or by a negative spread is not a valid finite positive Gordon valuation for a positive dividend stream. Check the condition before interpreting the result.
Calling the estimate an unconditional market price. The model value follows from its dividend, growth, and required-return assumptions. Different expectations or a growth pattern that changes over time require a different analysis.
Confusing per-share value with total equity value. The formula gives $P_0$ for one share. Multiply by the number of shares only if the requested quantity is total equity value.

Key takeaway: Under stable perpetual dividend growth, value a share using the next expected dividend divided by the required return minus the growth rate. Confirm the dividend’s timing, keep rates on a consistent annual basis, and verify $r>g$ before relying on the result.

Check Your Understanding

For each problem, use annual rates and state any relevant model condition.

A hypothetical utility has just paid a dividend of $1.25 per share. Dividends are expected to grow by 4% annually forever, and the required return is 9%. Find the value immediately after the dividend. (Answer: $26.00 per share.)
A hypothetical company’s next annual dividend is forecast to be $2.70 per share. The expected constant growth rate is 2%, and the required return is 8%. Its quoted price is $42. Find the model value and the difference between model value and quoted price. (Answer: $45.00 per share; $3.00 above the quoted price.)
A hypothetical firm has just paid a $0.96 dividend per share. Its dividend is expected to grow at 6% annually forever, and the required return is 10%. There are 5 million shares outstanding. Find the next dividend per share, the Gordon value per share, and the total equity value. (Answer: $1.0176; $25.44 per share; $127.2 million.)
A hypothetical share’s next dividend is $4.20, its constant annual growth rate is 3%, and its required return is 7%. Calculate the model value, then calculate the total next-year dividend for 800,000 shares. (Answer: $105.00 per share; $3,360,000 total next-year dividend.)
A hypothetical company just paid a dividend of $2.00. Its forecast is constant growth of 5% forever, with a required annual return of 5%. Can the Gordon model give a finite positive value under these inputs? Explain using the model condition. (Answer: No. $r=g$, so the denominator is zero and the discounted dividend series does not converge to a finite value.)

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