Integrated CAPM Workshop: Required Returns and Alpha

Using CAPM as a Consistent Decision Framework

CAPM brings together three quantities—risk-free return, market risk premium, and beta—to estimate the return investors require for bearing an asset’s systematic risk. In an applied analysis, the calculation is only one part of the task: the inputs must refer to a consistent period and return basis, and a forecast return must be distinguished from the CAPM required return. Comparing the two produces an estimated alpha, not a guarantee about future performance.

The applications here use annual returns in the same currency and on the same nominal or real basis. They treat each asset’s beta as an estimate for the period being analyzed. The market risk premium is the expected market return less the risk-free rate, not the market return itself.

Definition: Under CAPM, the required return for asset $i$ is $r_i^{\mathrm{CAPM}}=r_f+\beta_i(E[R_m]-r_f)$, where $r_f$ is the risk-free rate, $E[R_m]$ is the expected market return, and $\beta_i$ measures the asset’s systematic sensitivity to the market. The market risk premium is $E[R_m]-r_f$. For a forecast expected return $E[R_i]$, define estimated alpha as $\alpha_i=E[R_i]-r_i^{\mathrm{CAPM}}$.

A beta of 1 means the asset has the market’s beta by definition; it does not mean the asset has the same total risk as the market. A beta between 0 and 1 indicates lower positive market sensitivity than the market. A negative beta indicates sensitivity in the opposite direction. When describing sensitivity magnitude, compare absolute betas: for example, $|-1.2|>1$, so a beta of $-1.2$ has greater absolute market sensitivity than a beta of 1, but in the opposite direction.

Why the CAPM Required-Return Formula Has This Form

Beta is defined using covariance with the market return:

\beta_i=\frac{\operatorname{Cov}(R_i,R_m)}{\operatorname{Var}(R_m)}

In CAPM equilibrium, expected excess return is proportional to the asset’s covariance with the market. This reflects that market-related risk, rather than risk that can be diversified away, determines the return premium in the model. Write that proportionality as $E[R_i]-r_f=\lambda\operatorname{Cov}(R_i,R_m)$, where $\lambda$ is the price per unit of covariance risk. For the market itself, $\operatorname{Cov}(R_m,R_m)=\operatorname{Var}(R_m)$, so:

E[R_m]-r_f=\lambda\operatorname{Var}(R_m) \quad\Longrightarrow\quad \lambda=\frac{E[R_m]-r_f}{\operatorname{Var}(R_m)}

Substitute this expression for $\lambda$ into the asset’s excess-return relation, then use the definition of beta:

E[R_i]-r_f = \frac{E[R_m]-r_f}{\operatorname{Var}(R_m)} \operatorname{Cov}(R_i,R_m) = \beta_i(E[R_m]-r_f)

Adding $r_f$ to both sides gives the CAPM required-return formula. This justification depends on the CAPM equilibrium model; it is not a claim that real-world returns always follow the relation exactly.

Worked Applications

Worked Example: Calculate Required Return and Alpha

A hypothetical asset has beta 1.25. The annual risk-free rate is 3.00%, and the expected annual market return is 9.00%. An analyst forecasts a year-end price of $54 and a dividend of $2.00 per share. The current price is $50. Calculate the CAPM required return, the forecast expected return, and alpha.

Method and assumptions. Use annual returns on one consistent basis. Treat the forecast price and dividend as the only one-year shareholder cash flows, with no taxes or transaction costs. First calculate the market risk premium:

E[R_m]-r_f=9.00\%-3.00\%=6.00\%

The asset’s CAPM required return is its risk-free rate plus beta times that premium:

r_i^{\mathrm{CAPM}}=3.00\%+1.25(6.00\%)=3.00\%+7.50\%=10.50\%

The one-year forecast return includes both the dividend and the price change. The forecast price gain is $54-50=\$4.00$, so total forecast gain is $\$2.00+\$4.00=\$6.00$. Therefore:

E[R_i]=\frac{D_1+P_1-P_0}{P_0} =\frac{\$2.00+\$54.00-\$50.00}{\$50.00} =\frac{\$6.00}{\$50.00} =12.00\%

Subtract the required return from the forecast expected return:

\alpha_i=12.00\%-10.50\%=+1.50\text{ percentage points}

The forecast lies 1.50 percentage points above the CAPM required return. Check: the market premium is 6.00%, and beta 1.25 implies a 7.50% premium above the 3.00% risk-free rate, giving 10.50%. The common mistake this method avoids is omitting the dividend when calculating the forecast return.

Worked Example: Infer Beta and Assess a Forecast

For a hypothetical investment, the risk-free rate is 4.20%, the expected market return is 9.20%, and a committee’s CAPM required-return estimate is 6.70%. Infer the beta implicit in that estimate. Then compare the estimate with an analyst’s forecast based on a current price of $80, an expected dividend of $2.40, and a year-end price of $83.52.

Method and assumptions. Assume all rates are annual and expressed on the same basis. Rearrange the CAPM formula to solve for beta. The market risk premium is:

9.20\%-4.20\%=5.00\%

The required return is 2.50 percentage points above the risk-free rate, so:

\beta_i = \frac{r_i^{\mathrm{CAPM}}-r_f}{E[R_m]-r_f} = \frac{6.70\%-4.20\%}{5.00\%} = \frac{2.50\%}{5.00\%} =0.50

This is a positive beta below 1, indicating lower positive market sensitivity than the market under the beta measure. Next, the forecast price gain is $83.52-80=\$3.52$. Adding the dividend gives a total forecast gain of $\$3.52+\$2.40=\$5.92$, and:

E[R_i]=\frac{\$5.92}{\$80}=0.074=7.40\%

The forecast alpha is $7.40\%-6.70\%=+0.70$ percentage points. Check: applying the inferred beta returns $4.20\%+0.50(5.00\%)=6.70\%$, exactly the committee estimate. The common mistake avoided here is dividing the required return itself by the market premium; beta uses the excess of the required return over the risk-free rate.

Worked Example: Interpret Negative Beta Without Reversing the Sign

A hypothetical asset has beta $-1.20$. The annual risk-free rate is 3.00%, and the expected market return is 9.00%. Its current price is $50, and its expected year-end dividend and price are $1.00 and $47.40, respectively. Calculate the required return and alpha, and describe what the beta says about market sensitivity.

Method and assumptions. Use the stated annual rates and a one-year return that includes both the dividend and price change. The market risk premium is $9.00\%-3.00\%=6.00\%$. Retain the negative sign when multiplying by beta:

r_i^{\mathrm{CAPM}} = 3.00\%+(-1.20)(6.00\%) = 3.00\%-7.20\% = -4.20\%

The forecast price change is $47.40-50=-\$2.60$. Adding the $1.00 dividend produces a total forecast loss of $-\$2.60+\$1.00=-\$1.60$. Thus:

E[R_i]=\frac{-\$1.60}{\$50}=-3.20\%

The forecast return is above the CAPM required return, even though both are negative:

\alpha_i=-3.20\%-(-4.20\%)=+1.00\text{ percentage point}

A negative beta indicates inverse market sensitivity. Because $|-1.20|=1.20>1$, its absolute sensitivity is greater than that of a beta-1 asset, but its direction is opposite. Check: an expected return of $-3.20\%$ is one percentage point higher than $-4.20\%$. The common mistake is treating “beta below 1” as necessarily lower sensitivity or dropping the negative sign in the required-return calculation.

Reading Alpha and the Security Market Line

The security market line (SML) plots CAPM required return against beta. For a given risk-free rate and market risk premium, the CAPM-required return is the point on the line at an asset’s beta. An asset’s forecast expected return can be compared with that required return:

Comparison	Alpha sign	Interpretation within the model
Forecast expected return is above required return	Positive	Forecast plots above the SML
Forecast expected return equals required return	Zero	Forecast lies on the SML
Forecast expected return is below required return	Negative	Forecast plots below the SML

Alpha is a difference in returns, so its unit is a percentage point. A positive alpha does not mean the asset is certain to outperform, and it does not by itself establish that a market price is wrong. The forecast may be inaccurate, beta may be estimated with error, or the CAPM assumptions may not describe the relevant setting. Also distinguish forecast alpha from a realized-return residual: a return observed after the period is not the same quantity as a forecast made before it.

Assumptions, Input Discipline, and Common Traps

CAPM is an equilibrium model, not a mechanical law of observed returns. Its standard framework assumes investors make decisions over a common single period using mean and variance, share homogeneous expectations about returns and risk, and can borrow and lend at a common risk-free rate. It also assumes frictionless markets and an efficient market portfolio that includes the relevant risky assets. Departures from these assumptions, or use of an imperfect market proxy, can affect the usefulness of an estimate.

Before calculating, check that the risk-free rate, expected market return, and beta refer to compatible horizons and definitions. In particular, the risk-free rate and expected market return must both be nominal or both be real. Use the expected market return to calculate the premium; do not substitute a realized past market return without recognizing that this changes the estimate’s interpretation. The beta should correspond to the asset and the market proxy being analyzed.

A frequent exam trap is to mistake beta for a probability, a percentage return, or a measure of total volatility. Beta is a sensitivity measure based on covariance with the market. Another trap is to interpret negative alpha as a negative expected return: alpha is the difference between two returns, and its sign alone does not determine whether either return is above or below zero.

Key takeaway: Calculate the market risk premium first, apply beta with its sign intact, and compare the resulting required return with a separately calculated forecast expected return. Interpret alpha as a model-based comparison, not as a certain outcome.

Check Your Understanding

For each problem, keep rates on the same annual basis and show the intermediate calculations.

A hypothetical asset has beta 0.80. The risk-free rate is 3.50%, and the expected market return is 9.00%. Find its CAPM required return. (Answer: 7.90%)
The risk-free rate is 4.00%, the expected market return is 10.00%, and an asset’s CAPM required return is 13.00%. Infer its beta. (Answer: 1.50)
A hypothetical asset has beta $-0.40$, a 2.80% risk-free rate, and a 7.30% expected market return. Its forecast expected return is 1.70%. Calculate its required return and alpha. (Answer: Required return 1.00%; alpha +0.70 percentage points.)
A share sells for $64, with an expected dividend of $1.60 and year-end price of $68.80. Its beta is 1.10, the risk-free rate is 3.20%, and the expected market return is 8.20%. Calculate its forecast return and alpha. (Answer: Forecast return 10.00%; required return 8.70%; alpha +1.30 percentage points.)
Two assets have CAPM required returns of 6.00% and 12.00%, at betas 0.50 and 1.50, respectively. Infer the common risk-free rate and market risk premium. (Answer: Risk-free rate 3.00%; market risk premium 6.00%.)

What You'll Learn