Deriving The Gordon Growth Cost Of Equity

Why Dividend Yield and Growth Determine the Required Return

A common-stock investor may receive a cash dividend and may also benefit if the share price rises. Under the Gordon growth model, the share price reflects a stream of dividends that is expected to grow at a constant rate indefinitely. This model gives a compact estimate of the required return: the yield from the next dividend plus the expected growth rate. The derivation matters because it clarifies which dividend belongs in the calculation, why growth is added, and when the result is meaningful.

The estimate is a model-based cost of equity, not a guarantee of the return that a shareholder will actually earn. It relies on a steady perpetual dividend-growth assumption and a required return high enough to discount that growing stream to a finite value. Each worked example states the annual timing convention and checks the result against the valuation equation.

Definition: Let $P_0$ be the current price per common share, $D_t$ the dividend per share expected at the end of year $t$, $g$ the constant annual dividend growth rate, and $r_e$ the annual required return on common equity. Under the Gordon growth model, dividends satisfy $D_t=D_1(1+g)^{t-1}$, and the current price is the present value of all expected future dividends. The model requires $r_e>g$. Its cost-of-equity estimate is $r_e=\frac{D_1}{P_0}+g$.

Assume that dividends are paid once per year, at each year-end, and that the next dividend is $D_1$. The constant-growth pattern makes the second dividend $D_1(1+g)$, the third dividend $D_1(1+g)^2$, and so on. If the same required return $r_e$ discounts every annual payment, the present value of the dividend stream is a geometric series.

Write the dividend present-value equation.
Each dividend is discounted from its payment date. For example, $D_1$ is discounted one year and $D_2$ is discounted two years.

Substitute constant growth.
Replace each future dividend with its expression in terms of $D_1$ and $g$. The resulting terms form a geometric series with first term $D_1/(1+r_e)$ and common ratio $(1+g)/(1+r_e)$.

Sum the series and rearrange.
Because $r_e>g$, the common ratio is less than one, so the infinite series converges. Rearranging the resulting price equation isolates the required return.

P_0=\frac{D_1}{1+r_e}+\frac{D_1(1+g)}{(1+r_e)^2}+\frac{D_1(1+g)^2}{(1+r_e)^3}+\cdots =\frac{D_1}{r_e-g}

To see why the series converges, its common ratio is $(1+g)/(1+r_e)$. With ordinary positive annual rates and $r_e>g$, this ratio is less than one. The sum of an infinite geometric series with first term $a$ and common ratio $q$ is $a/(1-q)$. Here that gives

P_0 =\frac{\frac{D_1}{1+r_e}}{1-\frac{1+g}{1+r_e}} =\frac{\frac{D_1}{1+r_e}}{\frac{r_e-g}{1+r_e}} =\frac{D_1}{r_e-g}.

Starting from this valuation equation, multiply both sides by $r_e-g$, divide by $P_0$, and add $g$. The result is the required-return formula:

P_0(r_e-g)=D_1 \quad\Longrightarrow\quad r_e-g=\frac{D_1}{P_0} \quad\Longrightarrow\quad r_e=\frac{D_1}{P_0}+g.

The term $D_1/P_0$ is the next-dividend yield: the next expected annual dividend divided by today’s share price. The second term is the constant expected growth rate. Both terms must be expressed on the same annual basis. If $D_1$ and $P_0$ are both in dollars per share, their ratio is dimensionless and is interpreted as an annual rate under the model’s timing assumptions.

Assumptions and Interpretation

The equation is not a general-purpose formula for every share. Its interpretation depends on the assumptions used to obtain it. In particular, the dividend-growth pattern must be suitable for a constant-growth model, and the required return must exceed the growth rate. If $r_e\leq g$, the infinite present-value series does not converge to a finite price under this model, so the Gordon valuation equation cannot be used as stated.

Dividends are the relevant shareholder cash flows. The model values a share by discounting expected dividends. It does not separately add a terminal sale price; in the infinite-horizon model, future value is represented through the continuing dividend stream.
Growth is constant in every future year. A single $g$ applies from the next dividend onward. A temporary high-growth phase followed by a different stable phase does not fit this assumption without a separate valuation treatment.
The required return is stable. The same annual $r_e$ discounts each dividend. A changing risk profile or changing required return is not represented by this simple series.
Timing is annual and consistent. $D_1$ is the dividend expected one year from now, while $P_0$ is the price today. A dividend already paid is not the numerator in the formula unless it happens to equal the next expected dividend.

A useful interpretation follows directly from the valuation equation. If the model holds, the share price one year from now is $P_1=D_2/(r_e-g)$. Since $D_2=D_1(1+g)$, it follows that $P_1=P_0(1+g)$. Thus the model’s one-year price appreciation rate is $g$, while the dividend received over that year is $D_1$. The corresponding total return is

\frac{D_1+P_1-P_0}{P_0} =\frac{D_1}{P_0}+\frac{P_1-P_0}{P_0} =\frac{D_1}{P_0}+g =r_e.

This identity provides an additional check on the interpretation: the formula combines the dividend return with the model-implied price appreciation. It does not imply that the realized price will grow at exactly $g$ in any particular year. Actual outcomes can differ from the model’s constant expectations.

Worked Example: Estimating the Required Return From a Quoted Share

A hypothetical utility share trades at $P_0=\$40.00$. Its next year-end dividend is expected to be $D_1=\$2.40$ per share, and dividends are expected to grow at a constant annual rate of $g=4.00\%$ indefinitely. Estimate the annual required return using the Gordon growth model and verify the price.

Method. Apply the derived formula using the next expected dividend, the current price, and the annual constant growth assumption. The model’s convergence condition will be checked after calculating the estimate.

First calculate the next-dividend yield:

\frac{D_1}{P_0}=\frac{\$2.40}{\$40.00}=0.0600=6.00\%.

Add the expected growth rate:

r_e=6.00\%+4.00\%=10.00\%.

The convergence condition is satisfied because $10.00\%>4.00\%$. Verify by substituting into the Gordon valuation equation:

\frac{D_1}{r_e-g} =\frac{\$2.40}{0.1000-0.0400} =\frac{\$2.40}{0.0600} =\$40.00.

The recalculated value equals the quoted price. The estimate is therefore 10.0% per year under the stated assumptions. This check also confirms that the dividend, price, and rates were combined consistently.

Worked Example: Using the Next Dividend, Not the Dividend Just Paid

A hypothetical food distributor’s share is priced at $P_0=\$35.00$. The dividend just paid was $D_0=\$2.00$, and the next year-end dividend is already estimated at $D_1=\$2.10$. Assume that dividends grow at a constant annual rate of $g=5.00\%$ from the next dividend onward. Estimate the required return and explain which dividend enters the calculation.

Method. The valuation equation starts with the first dividend that has not yet been paid, $D_1$. Use the supplied $D_1$ directly; do not substitute $D_0$ merely because it is also given.

The next-dividend yield is

\frac{D_1}{P_0}=\frac{\$2.10}{\$35.00}=0.0600=6.00\%.

Therefore,

r_e=6.00\%+5.00\%=11.00\%.

Since $11.00\%>5.00\%$, the series converges. The price check is

\frac{\$2.10}{0.1100-0.0500} =\frac{\$2.10}{0.0600} =\$35.00.

The answer is 11.0% per year. The amount $D_0=\$2.00$ does not enter this calculation because it is the dividend already paid, not the next dividend in the present-value stream.

Worked Example: Checking a Required-Return Estimate Against a Price

An analyst evaluates a hypothetical equipment manufacturer using an annual required return of $r_e=8.50\%$. The next annual dividend is expected to be $D_1=\$1.75$ per share, with constant annual growth of $g=5.00\%$. Find the share price implied by the model, then check the return decomposition at that price.

Method. First use the Gordon valuation equation to calculate the price. Then calculate the dividend yield and add growth to confirm that they reproduce the assumed required return.

The difference between the required return and growth is

r_e-g=8.50\%-5.00\%=3.50\%=0.0350.

Because $8.50\%>5.00\%$, the model has a finite value. The implied price is

P_0=\frac{D_1}{r_e-g} =\frac{\$1.75}{0.0350} =\$50.00.

At this price, the next-dividend yield is

\frac{D_1}{P_0}=\frac{\$1.75}{\$50.00}=0.0350=3.50\%.

Adding growth gives $3.50\%+5.00\%=8.50\%$, the assumed required return. The price and return calculations therefore agree. This is a consistency check of the model inputs, not evidence that the hypothetical share will actually trade at that price.

Common Mistakes and Practical Use

A frequent error is to place the dividend just paid in the numerator. In the food distributor example, doing so would produce

\frac{D_0}{P_0}+g =\frac{\$2.00}{\$35.00}+5.00\% =5.714\%+5.00\% =10.714\%.

That is not the Gordon estimate for the stated assumptions. The model’s first cash flow is $D_1$, so the correct calculation is

\frac{D_1}{P_0}+g =\frac{\$2.10}{\$35.00}+5.00\% =6.00\%+5.00\% =11.00\%.

The wrong calculation uses a cash flow from the wrong time. A careful solution labels the dividend’s timing before dividing and checks that the result satisfies the model’s valuation equation. Other errors to avoid include mixing a monthly growth rate with an annual dividend yield, treating a growth rate greater than or equal to the required return as a valid finite Gordon model, or reporting a model estimate as a certain realized return.

For cost-of-equity analysis, the Gordon growth estimate can provide a dividend-based view to consider alongside other evidence. Its usefulness depends on whether a stable, perpetual dividend-growth assumption is credible. When that assumption is not appropriate, the numerical output may be precise without being informative. The derivation makes the model’s central logic transparent: a shareholder’s required return is represented by the next dividend relative to today’s price, plus the price growth implied by the constant-growth dividend stream.

Key takeaway: Under the constant-growth dividend model, $P_0=D_1/(r_e-g)$, with $r_e>g$. Rearranging gives $r_e=D_1/P_0+g$. Use the next expected dividend, keep the timing and annual units consistent, and verify the model’s convergence condition.

Check Your Understanding

For each problem, assume annual year-end dividends and constant growth forever where the Gordon model is specified. Use rates on an annual basis.

A hypothetical share costs $\$30.00$, its next dividend is $\$1.80$, and constant expected growth is 4.00%. Estimate the required return. (Answer: 10.0% per year.)
A hypothetical retailer’s share price is $\$50.00$, its next dividend is $\$2.75$, and constant growth is 3.50%. Find the required return and state whether the convergence condition holds. (Answer: 9.00% per year; yes, because 9.00% exceeds 3.50%.)
A hypothetical share has just paid a dividend of $\$1.20$. Its next dividend is $\$1.26$, its current price is $\$28.00$, and constant growth from the next dividend is 5.00%. Estimate the required return using the correctly timed dividend. (Answer: 9.50% per year.)
A hypothetical share’s next dividend is $\$3.60$, constant growth is 4.50%, and the required return is 9.50%. Calculate the Gordon-model price and verify the dividend-yield-plus-growth decomposition. (Answer: $\$72.00$; the yield is 5.00%, and 5.00% + 4.50% = 9.50%.)
Two hypothetical shares satisfy the Gordon assumptions. Share A has $D_1=\$2.10$, $P_0=\$35.00$, and $g=3.00\%$. Share B has $D_1=\$2.80$, $P_0=\$40.00$, and $g=4.00\%$. Estimate both required returns and identify the higher estimate. (Answer: A = 9.00% per year; B = 11.0% per year; B is higher by 2.00 percentage points.)

Deriving The Gordon Growth Cost Of Equity

What You'll Learn

Why Dividend Yield and Growth Determine the Required Return