Worked Cost Of Equity Estimation Case

Estimating Cost of Equity in Context

A cost-of-equity estimate is an input to decisions about a company’s financing and investment, but no formula can make uncertain inputs certain. A useful estimate therefore requires both calculation and judgment: identify the model, check whether its assumptions fit the company, and explain what a difference between methods may indicate.

This case uses two standard approaches: the Capital Asset Pricing Model (CAPM) and the constant-growth dividend model. CAPM estimates the return investors require for bearing systematic risk. The dividend model infers a return from the current share price and expected dividends. They answer the same broad question from different inputs, so agreement is informative and disagreement calls for investigation.

Definition: The cost of equity, $r_e$, is the return required by the company’s common shareholders for investing in its equity at the relevant level of risk. Under CAPM, $r_e=r_f+\beta_E(R_M-r_f)$, where $r_f$ is the risk-free rate, $\beta_E$ is the equity beta, and $R_M-r_f$ is the expected market risk premium. Under the constant-growth dividend model, $r_e=D_1/P_0+g$, where $P_0$ is the current share price, $D_1$ is the next expected dividend per share, and $g$ is the constant expected dividend growth rate.

What Each Estimate Requires

CAPM separates the return into a risk-free component and compensation for systematic risk. Beta measures how the equity’s returns move with returns on the market portfolio. A beta above 1 indicates greater systematic sensitivity than the market under the measurement model; a beta below 1 indicates lower sensitivity. CAPM does not add compensation for risks that can be diversified away.

The standard CAPM calculation assumes investors can diversify, evaluate risk and return in a common single-period framework, and form sufficiently consistent expectations about the market portfolio. Its practical estimate also depends on proxies: a government security may stand in for the risk-free asset, a historical or forward-looking beta may stand in for true beta, and an estimated market risk premium may stand in for the market’s expected excess return. The result is sensitive to these choices.

The constant-growth dividend model starts from the assumption that dividends per share grow at a constant rate $g$ indefinitely and that the required return $r_e$ is greater than $g$. It also assumes the current price is consistent with the present value of that growing dividend stream. It is most useful when a company pays dividends and its long-term payout and growth expectations are reasonably stable. A recent change in dividend policy, irregular payments, or an implausible perpetual growth forecast can make the result unreliable.

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}, \qquad r_e>g

The last equality is the sum of a convergent geometric series: its first term is $D_1/(1+r_e)$, and its common ratio is $(1+g)/(1+r_e)$. The series converges when $(1+g)/(1+r_e)<1$, equivalent to $r_e>g$. Rearranging $P_0=D_1/(r_e-g)$ gives $r_e=D_1/P_0+g$. This derivation also shows why using a growth rate at least as large as the required return is not a valid constant-growth valuation.

Worked Case: Two Methods, Different Evidence

Worked Example: Northline Metering’s Cost of Equity

Northline Metering Inc. is a hypothetical publicly traded company. Its equity beta is estimated at 1.15 from a liquid share history during which its business mix remained broadly stable. The annual risk-free rate is 4.10%, and the expected market return is 9.60%. Northline’s latest annual dividend was $1.60 per share; the current share price is $34.00. A long-term dividend growth forecast is 3.20%. Estimate its cost of equity using both methods and recommend how to use the results. Treat all rates as annual and assume the dividend forecast is a perpetual growth rate only when applying the dividend model.

Method 1: CAPM. First calculate the market risk premium, rather than using the expected market return itself:

R_M-r_f=9.60\%-4.10\%=5.50\%

Then apply CAPM:

r_e=4.10\%+1.15(5.50\%) =4.10\%+6.325\% =10.425\%\approx 10.43\%

Method 2: constant-growth dividend model. The formula requires the next expected dividend, so grow the latest dividend by 3.20%:

D_1=\$1.60(1.032)=\$1.6512

The expected dividend yield is:

\frac{D_1}{P_0}=\frac{\$1.6512}{\$34.00}=0.0485647\ldots=4.8565\%

Adding the assumed growth rate gives:

r_e=4.8565\%+3.20\%=8.0565\%\approx 8.06\%

The estimates differ by $10.425\%-8.0565\%=2.3685$ percentage points, or about 2.37 percentage points. Northline’s stated beta evidence supports using CAPM as the primary estimate, while the dividend estimate is a useful cross-check. The difference is material, so it should not be hidden by simply averaging the figures. In particular, verify that 3.20% is a defensible perpetual growth rate and that the share price and dividend inputs refer to a consistent date. If the dividend forecast is uncertain, the dividend estimate deserves less weight; if the beta or market premium is uncertain, the CAPM estimate deserves less weight. The facts provided favor CAPM, but neither estimate is exact.

Common mistake avoided: using the latest $1.60 dividend in place of $D_1$ would produce $\$1.60/\$34.00+3.20\%=7.91\%$. That is not the stated model, because the price at time 0 is valued using the next dividend. Growing the dividend first gives the correct estimate of about 8.06%.

Check the Inputs Before Comparing Results

The calculations are only as useful as their assumptions. For CAPM, check that the beta represents the company’s current equity risk and that the risk-free rate and market risk premium are measured over a compatible horizon and in the same currency. A beta estimated during a major business transformation may not describe the company going forward. The expected market return is not itself the market risk premium: the risk-free rate must be subtracted first.

For the dividend model, distinguish the most recent dividend $D_0$ from the next dividend $D_1$. Confirm that the growth rate is expected to continue over the long term, not merely that dividends grew quickly in a recent year. The condition $r_e>g$ is essential to the constant-growth model. If the company does not pay dividends or its dividend policy is changing substantially, this method may not provide a meaningful estimate even if the arithmetic is straightforward.

A discrepancy between the estimates is a diagnostic signal, not proof that one formula has failed. The two methods may reflect different uncertainty: CAPM relies on beta and the market risk premium, whereas the dividend method relies heavily on price, expected dividends, and perpetual growth. Use company circumstances to decide which inputs are better supported. Where both methods are credible, report the estimates and explain their range; use a combined point only if a decision requires one and the chosen weighting is justified.

Worked Example: A Stable Dividend Payer With Converging Estimates

A hypothetical regional payments firm has a stable dividend policy. Its latest annual dividend was $2.40 per share, its current price is $47.50, and a long-term dividend growth forecast of 3.50% is supported by its expected reinvestment and payout policy. Its beta is 0.90, the annual risk-free rate is 3.80%, and the expected market return is 9.30%. Estimate both costs of equity and state a reasonable conclusion. Assume all inputs are annual.

Method 1: CAPM. The market risk premium is $9.30\%-3.80\%=5.50\%$. Thus:

r_e=3.80\%+0.90(5.50\%)=3.80\%+4.95\%=8.75\%

Method 2: constant-growth dividend model. Calculate the next dividend:

D_1=\$2.40(1.035)=\$2.484

Its expected yield is $\$2.484/\$47.50=0.0522947\ldots=5.2295\%$. Therefore:

r_e=5.2295\%+3.50\%=8.7295\%\approx 8.73\%

The difference is only $8.75\%-8.7295\%=0.0205$ percentage points, approximately 0.02 percentage points. The assumed growth rate is less than either estimated required return, so the convergence condition is satisfied. The stable payout policy and supported long-term growth forecast make the dividend method relevant, while the CAPM inputs also produce a consistent result. A reasonable conclusion is that the evidence supports a cost of equity near 8.7% to 8.8%. If a single illustrative point is required, the simple midpoint is $(8.75\%+8.7295\%)/2=8.73975\%$, or 8.74% rounded to two decimal places; that midpoint is a reporting choice, not a new model.

Common mistake avoided: treating near agreement as proof that the inputs are certain would overstate precision. Both methods still depend on forecasts and estimates. The close results strengthen the conclusion, but the estimate should still be reported at an appropriate precision.

Worked Example: Distinguishing the Market Return From the Premium

A hypothetical equipment supplier has an estimated equity beta of 1.30. The annual risk-free rate is 4.00%, and the expected market return is 10.00%. Its latest annual dividend was $0.90, the share price is $15.75, and the long-term dividend growth forecast is 5.00%. Calculate both estimates, test the dividend-model condition, and identify a tempting CAPM error.

CAPM calculation. First find the market risk premium: $10.00\%-4.00\%=6.00\%$. Then:

r_e=4.00\%+1.30(6.00\%)=4.00\%+7.80\%=11.80\%

Dividend-model calculation. The next dividend is $D_1=\$0.90(1.05)=\$0.945$. Its yield is $\$0.945/\$15.75=0.0600=6.00\%$. Therefore:

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

The model’s required condition holds because $11.00\%>5.00\%$. The estimates differ by $0.80$ percentage points. The dividend forecast and price imply an 11.00% return if the 5.00% growth assumption persists, while CAPM implies 11.80% under its risk inputs. A decision-maker should investigate the growth forecast, beta, and market premium before choosing a primary estimate. The data alone do not establish which input set is more reliable.

Common mistake and correction. A tempting but incorrect calculation is $4.00\%+1.30(10.00\%)=17.00\%$. It multiplies beta by the expected market return rather than the market risk premium. CAPM requires subtracting the risk-free rate first; the correct premium is 6.00%, giving 11.80%.

Making a Defensible Recommendation

A recommendation should name the estimate, identify the evidence supporting it, and state the key limitation. The following sequence keeps the comparison focused:

Calculate both methods correctly.
Use the market risk premium in CAPM and the next expected dividend in the constant-growth model.

Test model conditions.
Check whether beta and the premium represent current systematic risk, and whether dividends can reasonably grow at a stable rate below the required return.

Compare and explain.
Quantify the difference, identify which inputs are more credible in the company’s circumstances, and explain why one estimate receives more weight or why both remain informative.

Do not select the larger estimate automatically, assume the two methods must agree, or report several decimal places merely because a calculator produced them. A sound conclusion may favor one method when its inputs are clearly better supported, or give a range when neither method dominates. The justification is part of the estimate.

Check Your Understanding

For each problem, calculate before checking the answer. Rates are annual unless stated otherwise.

A hypothetical retailer has beta 1.10, a risk-free rate of 3.40%, and an expected market return of 8.60%. Estimate its CAPM cost of equity. (Answer: market risk premium $=5.20\%$; cost of equity $=3.40\%+1.10(5.20\%)=9.12\%$.)
A company’s latest dividend is $1.25, its current share price is $25.00, and its expected perpetual growth rate is 4.00%. Find the dividend-model cost of equity, using the next dividend. (Answer: $D_1=\$1.30$; dividend yield $=5.20\%$; cost of equity $=9.20\%$.)
A hypothetical transport company has beta 0.80, a risk-free rate of 4.50%, and an expected market return of 10.50%. Calculate CAPM cost of equity and explain why multiplying beta by 10.50% is incorrect. (Answer: premium $=6.00\%$; cost of equity $=9.30\%$; 10.50% is the market return, not the excess return over the risk-free rate.)
A dividend payer has $D_0=\$2.00$, a 3.00% perpetual growth forecast, and a share price of $40.00. Calculate the dividend-model estimate and check the convergence condition using that estimate. (Answer: $D_1=\$2.06$; estimate $=\$2.06/\$40.00+3.00\%=8.15\%$; $8.15\%>3.00\%$, so the condition holds.)
A hypothetical company has beta 1.25, a risk-free rate of 4.20%, and a market risk premium of 5.00%. Its latest dividend is $1.50, its share price is $30.00, and its supported long-term growth forecast is 4.00%. Calculate both estimates and state how a recently revised payout policy should affect the comparison. (Answer: CAPM $=4.20\%+1.25(5.00\%)=10.45\%$; dividend model $D_1=\$1.56$, so $r_e=\$1.56/\$30.00+4.00\%=9.20\%$; the 1.25-percentage-point difference calls for checking the growth and beta inputs, and a recent payout revision weakens confidence in the dividend-model assumption of stable growth.)

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