Solving For Cost Of Equity With Constant Growth

Estimating an Implied Cost of Equity

A share price and a dividend forecast can be used to estimate the return that equity investors require, provided dividends are expected to grow at a constant rate indefinitely. The calculation is an inversion of the constant-growth dividend model: rather than using a required return to estimate a share price, it uses the price to solve for the required return implied by the assumptions.

This estimate depends on three inputs: the current share price, the next expected annual dividend, and the annual dividend growth rate. A crucial detail is that the dividend input is the one expected at the end of the next year, not the dividend that has just been paid. All amounts and rates must also refer to the same share and annual time period.

Definition: Let $P_0$ be the current share price, $D_1$ the dividend expected at the end of the next year, $g$ the constant annual dividend growth rate, and $k_e$ the annual cost of equity, or required return on equity. Under the constant-growth dividend model, the implied cost of equity is $$ k_e=\frac{D_1}{P_0}+g. $$ The ratio $D_1/P_0$ is the next expected annual dividend divided by the current share price. The formula estimates $k_e$ only when the model’s assumptions are appropriate.

Model Assumptions and Conditions

The model treats the value of a share as the present value of its expected future dividends. It assumes dividends are paid annually at year-end, the next expected dividend is $D_1$, and each subsequent annual dividend grows at the same rate $g$. The required return $k_e$ is constant across the dividend stream. In the usual positive-dividend valuation setting, $P_0>0$, $D_1>0$, $1+k_e>0$, $1+g>0$, and $k_e>g$. These conditions make the discounted dividend series converge to a finite positive value.

The constant-growth formula is not a general valuation rule for every share. It does not represent a forecast with an explicitly changing growth rate, and it cannot produce a finite positive model value for a positive dividend stream when the required return is not greater than the growth rate. A calculated implied return should therefore be checked against the assumptions, not accepted automatically.

Why the Inversion Works

Under the annual constant-growth assumptions, the dividend expected at the end of year $t+1$ is $D_1(1+g)^t$, where $t=0,1,2,\ldots$. Discounting each dividend to the present at annual rate $k_e$ gives the model price:

P_0=\sum_{t=0}^{\infty}\frac{D_1(1+g)^t}{(1+k_e)^{t+1}} =\frac{D_1}{1+k_e}\sum_{t=0}^{\infty} \left(\frac{1+g}{1+k_e}\right)^t.

The series is geometric. If $D_1\ne0$ and $k_e\ne-1$, its terms have a nonzero first term, and it converges exactly when

\left|\frac{1+g}{1+k_e}\right|<1.

If $D_1=0$, the dividend series is identically zero and converges whenever its terms are defined; in particular, the discounting terms require $k_e\ne-1$. In the usual positive-dividend setting, $1+k_e>0$ and $1+g>0$, so the convergence condition reduces to $k_e>g$. The geometric-series sum is then $1/(1-q)$, with $q=(1+g)/(1+k_e)$. Substitution gives:

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

To solve for the cost of equity, multiply both sides by $k_e-g$, divide by $P_0$, and add $g$:

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

The algebra is simple, but it is valid as a valuation inversion only if the inputs describe a share whose dividends reasonably fit the constant-growth model. For example, if $P_0>0$ and $D_1>0$, the calculated result satisfies $k_e>g$, as required by the usual positive-dividend setting.

Applying the Formula

A reliable calculation has four steps: identify the current price per share, identify or calculate the next expected dividend per share, express the annual growth rate as a decimal, and substitute into the formula. Keep the dividend yield and growth rate in the same units until the final step. For example, $0.06+0.04=0.10$, or 10.0%; adding 6% to 0.04 would mix percentage and decimal forms.

Worked Example: Using a Forecasted Next Dividend

A hypothetical company’s shares trade at $P_0=\$48.00$. Its expected dividend at the end of the next year is $D_1=\$2.40$ per share, and analysts assume dividends will grow at 4.0% annually indefinitely. Estimate the implied annual cost of equity.

Method. Apply the constant-growth inversion using $D_1$, not a dividend from a different year. The inputs are $P_0=48.00$, $D_1=2.40$, and $g=0.040$.

First calculate the next expected dividend divided by the current price:

\frac{D_1}{P_0}=\frac{\$2.40}{\$48.00}=0.050=5.0\%.

Then add the annual growth rate:

k_e=0.050+0.040=0.090=9.0\%.

Check. The estimated return exceeds the 4.0% growth rate, and the model-implied price is $\$2.40/(0.090-0.040)=\$48.00$, the stated share price. The common mistake avoided is to treat the dividend-yield calculation alone as the full implied return.

Worked Example: Converting a Just-Paid Dividend to Next Year’s Dividend

A hypothetical insurer has just paid an annual dividend of $D_0=\$1.80$ per share. Assume the dividend grows at 5.0% per year, and the current share price is $P_0=\$30.00$. Estimate the cost of equity under the constant-growth model.

Method. The given dividend is $D_0$, the dividend just paid. Calculate $D_1$ before using the model:

D_1=D_0(1+g)=\$1.80(1+0.050)=\$1.80(1.050)=\$1.89.

Now divide the next expected dividend by the current price:

\frac{D_1}{P_0}=\frac{\$1.89}{\$30.00}=0.063=6.3\%.

Add the growth rate:

k_e=0.063+0.050=0.113=11.3\%.

Check. The model-implied price is $\$1.89/(0.113-0.050)=\$1.89/0.063=\$30.00$. A common mistake is to use $D_0=\$1.80$ directly, which would give $1.80/30+0.05=0.110$, or 11.0%. That is not the stated model calculation because it uses the dividend already paid instead of the next expected dividend.

Worked Example: Comparing Two Implied Cost Estimates

An analyst compares two hypothetical dividend-paying companies. Company A has a current share price of $\$40.00$, a next expected annual dividend of $\$2.20$, and expected constant annual dividend growth of 3.0%. Company B has a current share price of $\$55.00$, a next expected annual dividend of $\$3.30$, and expected constant annual growth of 4.0%. Estimate both implied costs of equity and determine which estimate is higher.

Method for Company A. Calculate the dividend-to-price ratio, then add the growth assumption:

\frac{D_{1,A}}{P_{0,A}}=\frac{\$2.20}{\$40.00}=0.055=5.5\%, \qquad k_{e,A}=0.055+0.030=0.085=8.5\%.

Method for Company B. Use its own dividend, price, and growth assumption:

\frac{D_{1,B}}{P_{0,B}}=\frac{\$3.30}{\$55.00}=0.060=6.0\%, \qquad k_{e,B}=0.060+0.040=0.100=10.0\%.

The difference is $10.0\%-8.5\%=1.5$ percentage points, so Company B has the higher implied estimate by 1.5 percentage points under the stated assumptions.

Check. The implied prices are $\$2.20/(0.085-0.030)=\$40.00$ for A and $\$3.30/(0.100-0.040)=\$55.00$ for B. The common mistake avoided is comparing only the dividend amounts: the price and growth assumption both affect each implied cost.

Common Errors and Interpretation Checks

A mathematically correct substitution can still answer the wrong question if the inputs are misread. Before reporting an estimate, check the dividend date, the units, and whether the assumed growth pattern is suitable for the model.

Potential error	Why it is wrong	Correction
Using $D_0$ when the model requires $D_1$	The dividend already paid is not the next expected dividend.	If given $D_0$, calculate $D_1=D_0(1+g)$ first.
Adding rates in inconsistent forms	A decimal and a percentage do not represent the same written quantity.	Convert both inputs to decimals or both to percentages before adding.
Using a total company dividend with a price per share	The dividend and price are then on different measurement bases.	Use dividend per share with price per share, or consistently convert both to total-equity amounts.
Ignoring the model conditions	The rearranged equation does not make an unsuitable dividend forecast appropriate.	Check that the constant-growth assumptions and convergence conditions are reasonable.

The result is an implied estimate, not a guarantee of the return an investor will actually earn. It is the cost of equity that makes the constant-growth dividend model’s present value equal the observed price, conditional on the dividend and growth assumptions. If dividends are expected to follow changing growth rates, or the inputs are uncertain, this single constant-growth estimate may not be a suitable description of the equity valuation.

Key takeaway: For a share priced at $P_0$, use the next expected dividend per share $D_1$ and the annual constant growth rate $g$: $k_e=D_1/P_0+g$. Verify dividend timing, consistent units, and the model’s convergence assumptions before interpreting the estimate.

Check Your Understanding

For each hypothetical share, use the constant-growth model and give the implied annual cost of equity.

A share is priced at $\$25.00$, its next expected annual dividend is $\$1.25$, and constant annual growth is 4.0%. (Answer: 9.0%)
A company has just paid an annual dividend of $\$2.00$. Its dividends are expected to grow by 3.0% annually, and its share price is $\$40.00$. First determine the next expected dividend. (Answer: $D_1=\$2.06$; $k_e=8.15\%$)
A share is priced at $\$64.00$, with a next expected annual dividend of $\$3.20$ and constant growth of 2.5%. (Answer: 7.5%)
Company X has $P_0=\$36.00$, $D_1=\$2.70$, and $g=4.0\%$. Company Y has $P_0=\$50.00$, $D_1=\$3.00$, and $g=5.0\%$. Which company has the higher implied cost, and by how many percentage points? (Answer: X is 11.5%, Y is 11.0%; X is higher by 0.5 percentage points.)
A share is priced at $\$20.00$, its next expected annual dividend is $\$1.10$, and its constant annual growth assumption is 5.0%. Calculate the implied cost and check the usual positive-dividend condition $k_e>g$. (Answer: $k_e=10.5\%$, which is greater than $g=5.0\%$.)

Potential error	Why it is wrong	Correction
Using \(D_0\) when the model requires \(D_1\)	The dividend already paid is not the next expected dividend.	If given \(D_0\), calculate \(D_1=D_0(1+g)\) first.
Adding rates in inconsistent forms	A decimal and a percentage do not represent the same written quantity.	Convert both inputs to decimals or both to percentages before adding.
Using a total company dividend with a price per share	The dividend and price are then on different measurement bases.	Use dividend per share with price per share, or consistently convert both to total-equity amounts.
Ignoring the model conditions	The rearranged equation does not make an unsuitable dividend forecast appropriate.	Check that the constant-growth assumptions and convergence conditions are reasonable.

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