Deriving CAPM from Mean-Variance Efficiency

Why Mean-Variance Efficiency Leads to CAPM

The CAPM relation is not an isolated rule for assigning required returns. It follows from a particular equilibrium argument: investors choose portfolios by balancing expected return against variance, a risk-free asset is available, and the market portfolio is efficient among risky portfolios. The risk-free asset matters because it changes the efficient set: investors can combine it with one common efficient risky portfolio rather than choose different risky portfolios along a curved frontier.

The derivation makes a distinction that is easy to miss. An investor’s risk aversion affects how much the investor allocates to the risky portfolio, but it does not change the composition of that portfolio under the model’s shared assumptions. In market equilibrium, the common risky portfolio is the market portfolio. Its efficiency then imposes a relationship between each asset’s expected return and its covariance with the market.

Model setup: Consider a one-period market with a risk-free asset paying return \(r_f\) and risky assets with finite expected returns and variances. Investors can trade divisible assets in frictionless, competitive markets, share the same expectations about returns and covariances, and choose portfolios using mean and variance. Assume unrestricted portfolio weights, including the ability to lend or borrow at \(r_f\). In equilibrium, the value-weighted market portfolio of risky assets is efficient. These assumptions allow the tangency and equilibrium arguments below; if they fail, the derivation need not hold.

The Risk-Free Asset and the Tangent Portfolio

Let \(P\) be any risky portfolio, with expected return \(E[R_P]\) and standard deviation \(\sigma_P\). Combining it with the risk-free asset gives a complete portfolio. If fraction \(y\) is invested in \(P\) and fraction \(1-y\) in the risk-free asset, its expected return and variance are

E[R_C]=r_f+y\bigl(E[R_P]-r_f\bigr), \qquad \operatorname{Var}(R_C)=y^2\operatorname{Var}(R_P).

The first equation follows by taking the weighted average of the two returns. The second follows because a risk-free return has zero variance and zero covariance with every return. For \(y\geq 0\), the complete portfolio’s standard deviation is \(y\sigma_P\), so expected return is linear in standard deviation. The slope is the portfolio’s Sharpe ratio, \(\bigl(E[R_P]-r_f\bigr)/\sigma_P\). Thus, among risky portfolios with positive expected excess return, the portfolio producing the steepest line from \(r_f\) is the tangency portfolio.

For example, under a mean-variance objective with risk-aversion parameter \(A>0\), an investor choosing \(y\) for a fixed risky portfolio maximizes

r_f+y\bigl(E[R_P]-r_f\bigr)-\frac{A}{2}y^2\operatorname{Var}(R_P).

Differentiating with respect to \(y\) and setting the derivative equal to zero gives

y=\frac{E[R_P]-r_f}{A\operatorname{Var}(R_P)}.

This result describes the allocation between the risky portfolio and the risk-free asset, not the risky portfolio’s composition. With shared beliefs and unrestricted choices, investors select the same tangency portfolio but may choose different values of \(y\) because their risk aversion differs. Market clearing means the aggregate risky holdings must equal the market portfolio. Consequently, in this model the market portfolio is the tangency portfolio.

Worked Example: Choosing an Allocation to the Efficient Risky Portfolio

A hypothetical investor has risk aversion \(A=4\). The risk-free return is 3% per year, the efficient risky portfolio has expected return 9% per year, and its annual return variance is \(0.04\). Using the mean-variance objective above, calculate the investor’s risky-portfolio weight, expected complete-portfolio return, and standard deviation.

Method and calculation. The risky portfolio’s expected excess return is \(0.09-0.03=0.06\). Its variance is \(0.04\), so the optimal fraction invested in it is

y=\frac{0.06}{4(0.04)} =\frac{0.06}{0.16} =0.375.

The fraction invested in the risk-free asset is \(1-0.375=0.625\). The complete portfolio’s expected return is

$$ E[R_C]=0.03+0.375(0.09-0.03) =0.03+0.0225 =0.0525, $$

or 5.25% per year. Its variance is

\operatorname{Var}(R_C)=0.375^2(0.04) =0.140625(0.04) =0.005625,

so its standard deviation is \(\sqrt{0.005625}=0.075\), or 7.50% per year. As a check, the risk-free asset contributes no variance, and the risky allocation scales the risky portfolio’s standard deviation by \(0.375\): \(0.375(0.20)=0.075\). The common mistake is to treat \(y\) as a change in the risky portfolio’s composition; it is only the investor’s allocation between that portfolio and the risk-free asset.

Deriving the Market Covariance Condition

Write \(M\) for the market portfolio and define the expected excess return of any risky asset \(i\) as \(\mu_i=E[R_i]-r_f\). Define \(\mu_M=E[R_M]-r_f\). Assume \(\operatorname{Var}(R_M)>0\) and \(\mu_M>0\), so the market has positive risk and a positive slope from the risk-free return.

Because the market portfolio is tangent to the efficient set, a small feasible change from \(M\) toward any risky asset \(i\) cannot improve its Sharpe ratio. Under the model’s unrestricted-weights assumption, consider the portfolio that places fraction \(\varepsilon\) in asset \(i\) and fraction \(1-\varepsilon\) in \(M\). At \(\varepsilon=0\), this is \(M\). Its expected excess return is \(\mu_M+\varepsilon(\mu_i-\mu_M)\). Its variance has derivative \(2\bigl(\operatorname{Cov}(R_i,R_M)-\operatorname{Var}(R_M)\bigr)\) at \(\varepsilon=0\), as follows by expanding the variance of the weighted return.

The Sharpe ratio is expected excess return divided by standard deviation. Differentiating it at \(\varepsilon=0\), setting the derivative to zero at the tangency portfolio, and multiplying by the positive denominator yields

(\mu_i-\mu_M)\operatorname{Var}(R_M) -\mu_M\bigl(\operatorname{Cov}(R_i,R_M)-\operatorname{Var}(R_M)\bigr)=0.

Expanding and cancelling the terms involving \(\mu_M\operatorname{Var}(R_M)\) gives the tangency condition

\mu_i\operatorname{Var}(R_M) =\mu_M\operatorname{Cov}(R_i,R_M), \qquad \mu_i =\frac{\mu_M}{\operatorname{Var}(R_M)} \operatorname{Cov}(R_i,R_M).

This condition says that expected excess return is proportional to covariance with the market. To express the result using beta, define the asset’s market beta by \(\beta_i=\operatorname{Cov}(R_i,R_M)/\operatorname{Var}(R_M)\). Substituting this definition and restoring \(r_f\) gives the CAPM expected-return relation:

E[R_i] =r_f+\beta_i\bigl(E[R_M]-r_f\bigr).

The argument is a first-order condition at the efficient market portfolio. It does not assert that the market portfolio is risk-free or that every asset has the same expected return. It relates an asset’s expected excess return to its covariance with the market, scaled by the market’s variance.

Worked Example: Inferring an Asset’s Expected Return from Covariance

Suppose the risk-free return is 4% per year, the market portfolio’s expected return is 10% per year, and its return variance is \(0.0625\). A hypothetical security has covariance \(0.025\) with the market return. Under the CAPM equilibrium relation, find its beta and expected return.

Method and calculation. First calculate the market risk premium:

$$ E[R_M]-r_f=0.10-0.04=0.06. $$

Then calculate the security’s beta using covariance divided by market variance:

\beta_i=\frac{0.025}{0.0625}=0.4.

The CAPM expected return is

$$ E[R_i]=0.04+0.4(0.06) =0.04+0.024 =0.064. $$

The expected return is 6.40% per year. As a check, the tangency condition gives the same expected excess return directly: \((0.06/0.0625)(0.025)=0.024\), so adding the 4% risk-free return gives 6.40%. The common mistake is to divide covariance by the asset’s variance; beta’s denominator is the market variance.

Checking the Relation Across a Risky Portfolio

The derivation applies to each risky asset that can be considered as a feasible change from the market portfolio. It also implies the same relation for a portfolio of assets: expected returns and covariances combine linearly under fixed portfolio weights. The next example checks tangency using a two-asset market portfolio and shows how the covariance condition reproduces each asset’s expected return.

Worked Example: Verifying Tangency for Two Risky Assets

A hypothetical market contains risky assets A and B, with market weights 0.60 and 0.40. The risk-free return is 4% per year. Their expected returns are 9.6% and 7.0%, and their annual return variances and covariance are \(\operatorname{Var}(R_A)=0.04\), \(\operatorname{Var}(R_B)=0.0225\), and \(\operatorname{Cov}(R_A,R_B)=0.01\). Calculate the market variance, each asset’s covariance with the market, and each beta. Check that both assets satisfy the CAPM relation.

Method and calculation. First calculate the market variance from the portfolio weights:

\begin{aligned} \operatorname{Var}(R_M) &=0.60^2(0.04)+0.40^2(0.0225) +2(0.60)(0.40)(0.01)\\ &=0.36(0.04)+0.16(0.0225)+0.48(0.01)\\ &=0.0144+0.0036+0.0048\\ &=0.0228. \end{aligned}

Next, use \(R_M=0.60R_A+0.40R_B\) to calculate each covariance:

\begin{aligned} \operatorname{Cov}(R_A,R_M) &=0.60(0.04)+0.40(0.01)=0.028,\\ \operatorname{Cov}(R_B,R_M) &=0.60(0.01)+0.40(0.0225)=0.015. \end{aligned}

The market expected return is \(0.60(0.096)+0.40(0.07)=0.0856\), or 8.56%. Its premium over the risk-free return is \(0.0856-0.04=0.0456\). The betas are

\beta_A=\frac{0.028}{0.0228}\approx1.2281, \qquad \beta_B=\frac{0.015}{0.0228}\approx0.6579.

Applying the relation gives the following checks:

\begin{aligned} E[R_A]&=0.04+1.2281(0.0456)\approx0.096=9.60\%,\\ E[R_B]&=0.04+0.6579(0.0456)\approx0.070=7.00\%. \end{aligned}

The small rounding in the beta calculations does not change the results: using the unrounded ratios gives 9.60% and 7.00%. Both assets satisfy the tangency condition. The common mistake is to calculate market variance using only the weighted individual variances and omit the covariance term; that would give \(0.018\), not the correct \(0.0228\).

What the Derivation Establishes—and Its Limits

The risk-free asset shapes the result by allowing investors to combine a common tangency portfolio with lending or borrowing at a known return. Individual risk preferences determine the allocation to that portfolio. The market-clearing condition identifies the aggregate risky portfolio as the market portfolio, and its tangency condition gives the covariance relation that becomes CAPM when written in beta form.

This is a model result, not a claim that observed returns always follow the equation exactly. The derivation depends on the stated assumptions, including shared expectations, an available risk-free borrowing and lending rate, and an efficient market portfolio. If investors face different borrowing rates or constraints, or if the chosen market portfolio is not efficient, the straight-line tangency argument does not establish this relation. When applying the result, keep all returns on the same horizon and use the variance and covariance of returns measured on that same basis.

Key takeaway: With a risk-free asset and an efficient market portfolio, the tangency condition is \(E[R_i]-r_f=\bigl(E[R_M]-r_f\bigr)\operatorname{Cov}(R_i,R_M)/\operatorname{Var}(R_M)\). Since the covariance-to-market-variance ratio is beta, this condition is the CAPM expected-return relation.

Check Your Understanding

For each problem, use the one-period CAPM framework and keep returns and variances on the same annual basis.

The annual risk-free return is 3%, the market’s expected return is 11%, its variance is \(0.04\), and a security’s covariance with the market is \(0.012\). Find the security’s beta and CAPM expected return. (Answer: \(\beta=0.30\); expected return \(=5.40\%\).)
An investor has \(A=5\), the market’s expected excess return is 8% per year, and the market variance is \(0.16\). The risk-free return is 4%. Find the investor’s optimal market weight and complete-portfolio expected return. (Answer: market weight \(=0.10\); expected return \(=4.80\%\).)
A market portfolio has annual variance \(0.09\). A security has beta 1.00, and the risk-free return is 4%. Find the security’s covariance with the market and its CAPM expected return if the market risk premium is 6%. (Answer: covariance \(=0.09\); expected return \(=10.00\%\).)
The annual risk-free return is 4%, the market risk premium is 8%, and market variance is \(0.0625\). A security’s covariance with the market is \(0.03125\). Find its beta and expected return, then verify its expected excess return using the covariance condition. (Answer: \(\beta=0.50\); expected return \(=8.00\%\); expected excess return \(=0.04\).)
The annual risk-free return is 2%, the market expected return is 10%, market variance is \(0.04\), and a security’s covariance with the market is \(0.015\). If the security’s observed expected return is 8%, calculate its beta and CAPM-implied expected return, and state whether the observed return satisfies the tangency condition. (Answer: \(\beta=0.375\); CAPM-implied expected return \(=5.00\%\); no, the observed return does not satisfy it.)

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