Reconciling Conflicting Cost Of Equity Estimates

Why Two Cost-of-Equity Estimates Can Disagree

The cost of equity is the return investors require for bearing the risk of a company’s common shares. Two standard estimation methods can give different results because they infer that required return from different evidence. The capital asset pricing model (CAPM) uses a risk-free rate, equity beta, and market risk premium. The constant-growth dividend model uses the share price, the next expected dividend, and a perpetual dividend-growth assumption.

A difference between the estimates is not, by itself, evidence that one calculation is wrong. It is a prompt to investigate what each method assumes. A defensible conclusion explains which inputs are more credible for the company and valuation date, how uncertainty affects the estimates, and whether the available evidence supports a point estimate or a range.

Definition: A reconciliation of cost-of-equity estimates is an assessment of why independently calculated estimates differ and what the difference implies for a reasonable required-return estimate. Reconciliation is not the mechanical averaging of the estimates; it is a reasoned comparison of their inputs, assumptions, and relevance.

What Each Method Is Inferring

Under CAPM, the expected return on equity is estimated as the risk-free rate plus compensation for systematic market risk. The model assumes that beta measures the security’s relevant sensitivity to market returns and that the risk-free rate and market risk premium are measured consistently with the return period and market being analyzed.

k_e = r_f + \beta_E \bigl(E[R_M]-r_f\bigr)

Here, $k_e$ is the cost of equity, $r_f$ is the risk-free rate, $\beta_E$ is equity beta, and $E[R_M]-r_f$ is the expected market risk premium. The formula does not guarantee a precise required return: beta and the expected market risk premium are estimates, and a beta from an inappropriate comparison or period can weaken the result.

The constant-growth dividend model assumes dividends grow at a constant rate $g$ indefinitely, the next dividend is $D_1$, and investors discount those dividends at a constant required return $k_e$ greater than $g$. It also assumes the observed share price $P_0$ and forecast dividend refer to compatible dates and share units.

P_0 = \frac{D_1}{k_e-g} \qquad\Longrightarrow\qquad k_e = \frac{D_1}{P_0}+g

The term $D_1/P_0$ is the forward dividend yield. This estimate is particularly sensitive to the price, the forecast of the next dividend, and the perpetual growth rate. The model is not appropriate when a stable perpetual growth assumption is not credible, or when $k_e \leq g$; in those cases the constant-growth expression does not provide a meaningful finite valuation.

Why the Dividend-Growth Formula Holds

The formula follows from discounting the entire stream of dividends under its stated assumptions. If the first payment is $D_1$ one period from now and each later dividend grows by $g$, the dividend at time $t$ is $D_1(1+g)^{t-1}$. Discounting each payment at $k_e$ gives an infinite geometric series:

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

The first term is the first payment discounted one period. Each successive term is the preceding term multiplied by $(1+g)/(1+k_e)$. Since $k_e>g$, this ratio is less than 1, so the infinite geometric series converges. Its first term is $D_1/(1+k_e)$, and its ratio is $(1+g)/(1+k_e)$. Therefore:

P_0 = \frac{D_1/(1+k_e)}{1-(1+g)/(1+k_e)} = \frac{D_1}{k_e-g}

Rearranging gives $k_e=D_1/P_0+g$. The derivation also shows why the model’s conditions matter: if growth is not constant, the geometric-series argument no longer describes the dividend stream; if $k_e\leq g$, the series does not converge to a finite positive value.

A Practical Reconciliation Process

A useful comparison starts by putting both estimates on the same basis: the same company, valuation date, currency, and annual return convention. Then inspect the inputs instead of treating the outputs as competing facts.

Verify the calculations and timing.
Confirm that CAPM uses a coherent risk-free rate, beta, and market risk premium. For the dividend model, use the next expected dividend $D_1$, not a past dividend $D_0$, and match it to the price $P_0$ at the valuation date.

Identify what drives the gap.
Compare the CAPM risk premium implied by beta with the dividend model’s forward yield and growth assumption. Ask whether a difference in systematic risk, price, dividend forecast, or growth can reasonably explain the result.

Test uncertain assumptions.
Recalculate each estimate over supportable input ranges. Do not adjust an input solely to force the methods to agree; the range should reflect evidence about that input.

State a conclusion with its basis.
Report a range when the evidence supports uncertainty, and explain which estimates or inputs carry more weight. If a point estimate is required, justify the selected value rather than defaulting to the midpoint of two conflicting outputs.

Worked Comparisons

Worked Example: Diagnosing a Gap Between CAPM and Dividend Growth

A hypothetical company has an estimated equity beta of 1.20. For an annual estimate, an analyst uses a risk-free rate of 4.00% and an expected market risk premium of 5.50%. The company’s current share price is $40.00, its next expected annual dividend is $1.60, and a stable perpetual growth rate of 5.00% is proposed. Calculate both estimates and identify the main assumption to investigate.

Method and assumptions. Apply CAPM using the stated annual market inputs. For the dividend estimate, assume the $1.60 dividend is the next annual payment, the $40.00 price is observed at the valuation date, and 5.00% growth is sustainable indefinitely. The constant-growth model also requires $k_e>g$.

CAPM calculation. The market risk premium is already stated as 5.50%, so:

k_e = 4.00\% + 1.20(5.50\%) = 4.00\% + 6.60\% = 10.60\%

Dividend-growth calculation. The forward yield is $1.60/40.00=0.0400=4.00\%$. Thus:

k_e = \frac{\$1.60}{\$40.00}+5.00\% =4.00\%+5.00\% =9.00\%

The estimates differ by $10.60\%-9.00\%=1.60$ percentage points. A useful diagnostic is to ask what perpetual growth rate would make the dividend estimate equal the CAPM result, holding the price and dividend fixed:

g = 10.60\%-4.00\%=6.60\%

The proposed 5.00% growth assumption is 1.60 percentage points below that implied rate. This does not prove that growth should be 6.60%; it identifies growth, alongside CAPM risk inputs, as a key assumption behind the difference. The method avoids treating the 9.00% and 10.60% outputs as numbers to average without examining their premises.

Check. The dividend-model result exceeds its 5.00% growth assumption, so $k_e>g$, as required. The forward yield and growth sum to 9.00%.

Worked Example: Correcting a Rounded Dividend Yield

For a hypothetical insurer, a share trades at $28.00 and the next annual dividend is forecast to be $1.60. Assume a sustainable constant growth rate of 3.00%. A CAPM estimate uses a 3.20% risk-free rate, equity beta of 1.25, and a 4.80% market risk premium. Calculate both results and explain why rounding the yield too early can distort a range.

Method and assumptions. Use the forward dividend yield with the stated current price and next dividend. Assume the dividend model’s growth rate is constant indefinitely and that the CAPM inputs describe the same annual valuation basis.

Dividend-growth estimate. First calculate the yield:

\frac{\$1.60}{\$28.00} =0.057142857\ldots =5.7142857\ldots\%

Then add the 3.00% growth assumption:

k_e=5.7142857\ldots\%+3.00\% =8.7142857\ldots\%

CAPM estimate. The market risk premium is supplied directly:

k_e=3.20\%+1.25(4.80\%) =3.20\%+6.00\% =9.20\%

The difference is $9.20\%-8.7142857\ldots\%=0.4857143\ldots$ percentage points. If the assumptions are credible, the two calculations indicate a neighborhood from about 8.714% to 9.20%; that is a provisional estimate range, not proof that every value in it is equally likely.

Common mistake and correction. Rounding the forward yield to 5% before adding growth gives the wrong calculation $5\%+3\%=8\%$. The exact yield is about 5.714%, not 5%, so the corrected estimate is 8.714% to three decimal places. This matters if 8.00% is presented as the lower endpoint: it is not the result of the stated dividend and price. Keep intermediate values unrounded and round only the final result.

Check. The dividend-model estimate, 8.714%, is greater than the 3.00% growth rate, and the CAPM calculation is $3.20\%+6.00\%=9.20\%$.

Worked Example: Using Input Ranges to Form a Supported Estimate Range

A hypothetical firm has a share price of $32.00 and an expected next annual dividend of $1.28. Analysts consider perpetual growth between 4.50% and 5.00% plausible. For CAPM, the risk-free rate is 4.00%, the market risk premium is 5.00%, and the equity beta is considered to lie between 0.95 and 1.05. Find the ranges from each method and assess whether they overlap.

Method and assumptions. Treat the stated beta and growth intervals as plausible input ranges, not probability distributions. Keep the price, next dividend, risk-free rate, and market risk premium fixed. Assume the constant-growth dividend model applies throughout the stated growth range and $k_e>g$.

Dividend-growth range. The forward yield is:

\frac{\$1.28}{\$32.00}=0.0400=4.00\%

At 4.50% growth, the estimate is $4.00\%+4.50\%=8.50\%$. At 5.00% growth, it is $4.00\%+5.00\%=9.00\%$. The dividend-model range is therefore 8.50% to 9.00%. Both endpoints exceed the corresponding growth assumption.

CAPM range. Because the risk-free rate and market risk premium are positive, the CAPM estimate increases with beta. The endpoint calculations are:

\begin{aligned} \beta_E=0.95:\quad & k_e=4.00\%+0.95(5.00\%)=8.75\% \\ \beta_E=1.05:\quad & k_e=4.00\%+1.05(5.00\%)=9.25\% \end{aligned}

The CAPM range is 8.75% to 9.25%. The overlap of the two ranges is 8.75% to 9.00%. That overlap is a useful region of agreement under the stated assumptions, while the full evidence-based span across both methods is 8.50% to 9.25%.

A conclusion should distinguish these two ranges. If the beta evidence is stronger and the dividend growth forecast is especially uncertain, the CAPM range may deserve more emphasis. If both sets of inputs are comparably credible, the overlap may be a reasonable central region to investigate further. Selecting the midpoint of the overlap, 8.875% (about 8.9%), could be a convenient summary only if symmetric uncertainty within that overlap is defensible; it is not automatically the uniquely correct cost of equity.

Check. The yield is 4.00%; adding the growth endpoints gives 8.50% and 9.00%. The CAPM endpoints are 8.75% and 9.25%, so the overlap is 8.75% to 9.00%.

Interpreting the Evidence and Avoiding Traps

The difference between CAPM and dividend-growth estimates is often traceable to a small number of assumptions. A high CAPM result may reflect a high beta, a high market risk premium, or a risk-free rate measured on a different basis. A low dividend-growth result may reflect a high share price, a low next dividend, or conservative growth. Conversely, a low share price or an aggressive perpetual growth assumption can raise the dividend-based result.

Price deserves particular attention. Since the forward yield is $D_1/P_0$, a price from one date should not be combined casually with a dividend forecast prepared for another date. Likewise, substituting the most recent dividend $D_0$ for $D_1$ understates the numerator when the dividend is expected to grow before the next payment. In a simple constant-growth setting, the next dividend is $D_1=D_0(1+g)$, not $D_0$.

Observed disagreement	Diagnostic question
CAPM estimate is higher	Are beta or the market risk premium elevated, and are they appropriate for the company and valuation date?
Dividend-growth estimate is higher	Is the share price low, is the next dividend forecast reliable, or is perpetual growth too optimistic?
Estimates differ sharply	Are dates, annual conventions, dividend timing, and the definition of market risk premium consistent?
Dividend method gives an implausible result	Is constant perpetual growth appropriate, and does the resulting estimate satisfy $k_e>g$?

A common exam trap is to report the arithmetic average of two estimates as a reconciliation. For example, averaging 8.714% and 9.20% produces 8.957%, but that calculation says nothing about why the estimates differ or whether either method has more credible inputs. An average can be reported as a descriptive midpoint only when the analyst explains why equal weighting is appropriate. Otherwise, investigate the inputs and present a range or a justified selected estimate.

Key takeaway: Reconcile the assumptions, not just the outputs. Align dates and conventions, identify which risk, price, dividend, or growth inputs drive the difference, test plausible alternatives, and state a range or point estimate that the evidence supports.

Check Your Understanding

Work each problem using consistent annual inputs. Answers are provided so you can check the calculations.

A hypothetical retailer has beta 1.10, a 3.50% risk-free rate, and a 5.00% market risk premium. Its share price is $25.00, next annual dividend is $1.00, and perpetual growth is estimated at 4.00%. Calculate both cost-of-equity estimates and their difference. (Answer: CAPM 9.00%; dividend-growth 8.00%; difference 1.00 percentage point.)
A company’s next dividend is $2.10, its current price is $35.00, and the proposed constant growth rate is 2.50%. Calculate the dividend-growth estimate and state whether the convergence condition is satisfied. (Answer: forward yield 6.00%; estimate 8.50%; the estimate exceeds growth, so $k_e>g$.)
For a hypothetical manufacturer, the risk-free rate is 4.25%, the market risk premium is 4.00%, and beta is between 0.80 and 1.00. Calculate the CAPM estimate range. (Answer: 7.45% to 8.25%.)
A share trades at $24.00 and the next annual dividend is $1.20. The estimated perpetual growth rate is between 3.00% and 4.00%. Separately, CAPM gives a supported estimate range of 8.20% to 9.10%. Calculate the dividend-model range and its overlap with CAPM. (Answer: dividend-growth range 8.00% to 9.00%; overlap 8.20% to 9.00%.)
Two analysts obtain 7.80% and 9.40% using CAPM and dividend growth, respectively. Give two distinct input checks and explain why simply averaging the values does not reconcile them. (Answer: check whether the CAPM beta and market risk premium are appropriate and consistently measured; check the share-price date, next-dividend forecast, and perpetual growth assumption. The 8.60% average alone does not establish which inputs or estimate are credible.)

What You'll Learn