CAPM for Portfolios Using Weighted Betas

Why Portfolio Beta Is a Weighted Beta

A portfolio combines securities with different exposures to market risk. Its beta summarizes how its return moves with the market, but it is not generally the simple average of the securities’ betas. Each holding contributes in proportion to its share of the portfolio’s value. Once the portfolio beta is found, CAPM converts that beta into a required return using the same risk-free rate and market risk premium used for individual securities.

The method depends on using consistent weights and a consistent return period. For a portfolio of long positions, the weights are usually each holding’s market value divided by the total portfolio value. Cash or a risk-free asset has beta zero. A short position has a negative weight when weights are measured relative to the investor’s net portfolio value.

Definition: For holdings $j=1,\ldots,n$, let $w_j$ be holding $j$'s signed beginning market value divided by the portfolio’s beginning net value, and let $\beta_j$ be its beta. If the portfolio’s return is the weighted sum of the holdings’ returns over the period, its beta is $\beta_p=\sum_{j=1}^{n}w_j\beta_j$. For a fully invested portfolio, $\sum_{j=1}^{n}w_j=1$. Under CAPM, its required return is $r_p=r_f+\beta_p(E[R_m]-r_f)$, where $r_f$ is the risk-free rate and $E[R_m]-r_f$ is the market risk premium.

Here, $E[R_m]$ is the expected return on the market portfolio, and $r_p$ is the return required by CAPM for the portfolio’s systematic risk. The formula calculates a required return; it does not predict the portfolio’s realized return in a particular period. As in CAPM for an individual security, the calculation assumes the model’s inputs refer to the same period and that the market risk premium is defined as $E[R_m]-r_f$.

Why the Weighted-Beta Formula Holds

The result follows from the definition of beta and the linearity of covariance. Assume the holdings’ returns $R_j$ and the market return $R_m$ are measured over the same period, and that portfolio weights $w_j$ are set at the start of that period. If the holdings are maintained over the period, the portfolio return is $R_p=\sum_j w_jR_j$. Beta is covariance with the market return divided by the variance of the market return.

\begin{aligned} \beta_p &=\frac{\operatorname{Cov}(R_p,R_m)}{\operatorname{Var}(R_m)}\\ &=\frac{\operatorname{Cov}\left(\sum_j w_jR_j,R_m\right)}{\operatorname{Var}(R_m)}\\ &=\sum_j w_j\frac{\operatorname{Cov}(R_j,R_m)}{\operatorname{Var}(R_m)}\\ &=\sum_j w_j\beta_j. \end{aligned}

The second equality uses $R_p=\sum_jw_jR_j$; the third uses covariance’s linearity in its first argument, with the beginning-of-period weights treated as fixed. Each ratio in the sum is the beta of one holding. This derivation explains why weights multiply betas: beta measures market co-movement, and the portfolio’s co-movement is the weighted combination of its holdings’ co-movements.

CAPM then applies to the portfolio in the same form as to an individual security: $r_p=r_f+\beta_p(E[R_m]-r_f)$. The portfolio formula also follows by adding the CAPM required returns of the holdings using weights that sum to one:

\sum_j w_j\left[r_f+\beta_j(E[R_m]-r_f)\right] =r_f+\left(\sum_j w_j\beta_j\right)(E[R_m]-r_f)

This form requires $\sum_j w_j=1$. In an ordinary long-only portfolio, that condition holds when all invested value, including any cash, is included. For a leveraged or short portfolio, weights should be signed and measured against net portfolio value; their sum is still one when all assets and liabilities are represented consistently.

A Reliable Calculation Sequence

Keep the beta calculation separate from the CAPM calculation. First establish what each holding is worth and what denominator defines the portfolio. Next calculate signed weights, combine the betas, and check that the weights add to one. Only then use the portfolio beta in CAPM. This sequence catches errors caused by averaging betas before identifying the actual value exposure.

Set the portfolio base.
For a long-only portfolio, add the market values of all holdings. For a portfolio with shorts, use net portfolio value and represent short positions with negative signed values.

Calculate signed weights.
Divide each holding’s signed market value by the selected portfolio base. Include cash or other risk-free holdings if they are part of the portfolio.

Combine the betas.
Multiply each holding’s beta by its weight and add the results. Check that the weights sum to one.

Apply CAPM.
Calculate the market risk premium as $E[R_m]-r_f$, then find $r_p=r_f+\beta_p(E[R_m]-r_f)$.

Worked Example: Value Weights for Three Long Holdings

A hypothetical investment account holds $40,000 of Fund A, with beta 1.20; $35,000 of Fund B, with beta 0.80; and $25,000 of Fund C, with beta $-0.40$. The annual risk-free rate is 3.00%, and the expected annual market return is 9.00%. Find the portfolio beta and its CAPM required return. Use the value-weighted beta method, which avoids treating holdings with different market values as equally important.

Step 1: Find total value. The portfolio value is $40{,}000+35{,}000+25{,}000=\$100{,}000$. The three weights are $40{,}000/100{,}000=0.40$, $35{,}000/100{,}000=0.35$, and $25{,}000/100{,}000=0.25$. They sum to $0.40+0.35+0.25=1.00$.

Step 2: Find portfolio beta.

\beta_p=(0.40)(1.20)+(0.35)(0.80)+(0.25)(-0.40) =0.48+0.28-0.10=0.66

The negative-beta holding reduces the portfolio beta because its weighted contribution is negative. Its weight is still positive: the holding is owned, but the asset itself has negative beta.

Step 3: Apply CAPM. The market risk premium is $9.00\%-3.00\%=6.00\%$. Thus $r_p=3.00\%+(0.66)(6.00\%)=3.00\%+3.96\%=6.96\%$.

Check. A beta of 0.66 gives a required return below the market’s 9.00% when the market risk premium is positive, and above the risk-free rate of 3.00%. Both comparisons are consistent with the result. The portfolio’s CAPM required return is 6.96% per year.

Worked Example: Share Counts and Current Market Values

A hypothetical fund owns 120 shares of Asset A, currently priced at $50 per share, with beta 1.40; 80 shares of Asset B, priced at $75, with beta 0.50; and 400 shares of Asset C, priced at $20, with beta $-0.25$. Its annual risk-free rate is 2.50%, and the expected annual market return is 8.00%. Find its beta and required return. The method is to convert the share counts to market values before weighting; counting shares directly would give weights unrelated to invested dollars.

Step 1: Calculate each market value. Asset A is worth $120\times \$50=\$6{,}000$. Asset B is worth $80\times \$75=\$6{,}000$. Asset C is worth $400\times \$20=\$8{,}000$. Total portfolio value is $\$6{,}000+\$6{,}000+\$8{,}000=\$20{,}000$.

Step 2: Calculate the weights. The weights are $6{,}000/20{,}000=0.30$, $6{,}000/20{,}000=0.30$, and $8{,}000/20{,}000=0.40$. They add to 1.00.

Step 3: Calculate the weighted beta.

\beta_p=(0.30)(1.40)+(0.30)(0.50)+(0.40)(-0.25) =0.42+0.15-0.10=0.47

Step 4: Calculate the required return. The market risk premium is $8.00\%-2.50\%=5.50\%$. Therefore $r_p=2.50\%+(0.47)(5.50\%)=2.50\%+2.585\%=5.085\%$, or 5.09% when rounded to two decimal places.

Check. The dollar exposures used in the beta calculation contribute $6{,}000(1.40)+6{,}000(0.50)+8{,}000(-0.25)=\$9{,}400$ of beta-weighted value. Dividing by the total $\$20{,}000$ gives $9{,}400/20{,}000=0.47$. This confirms the weighted-beta result.

Cash, Shorts, and the Choice of Denominator

A risk-free asset has beta zero under the CAPM model because its return has zero covariance with the market return. Adding cash to a portfolio therefore adds a zero beta contribution, but it can still reduce the portfolio beta by increasing the denominator used to calculate the weights. A short position works differently: its signed market value is negative, so its contribution to portfolio beta has the opposite sign from the beta of the security that was shorted.

For shorts, the denominator matters. If the investor has long holdings of $140,000, a short position worth $20,000, and cash of $10,000, net portfolio value is $140{,}000-20{,}000+10{,}000=\$130{,}000$. Weights relative to this net value sum to one: the long positions and cash have positive weights, while the short has a negative weight. Using total long exposure or gross exposure as the denominator without adjusting the interpretation of the weights does not produce the standard portfolio beta for return on net portfolio value.

Worked Example: A Short Position and Cash

A hypothetical investor has $90,000 in a long position with beta 1.20, $50,000 in another long position with beta 0.70, a short position worth $20,000 in a security with beta 1.50, and $10,000 in a risk-free account. The annual risk-free rate is 3.50%, and the expected annual market return is 8.50%. Find the net portfolio beta and CAPM required return. Use signed values over net value; treating the short as a positive holding would reverse its risk contribution.

Step 1: Find net portfolio value. The long positions total $\$90{,}000+\$50{,}000=\$140{,}000$. Subtract the short liability and add cash: $\$140{,}000-\$20{,}000+\$10{,}000=\$130{,}000$.

Step 2: Calculate signed weights. The weights are $90/130$ for the first long position, $50/130$ for the second, $-20/130$ for the short, and $10/130$ for cash, with values expressed in thousands of dollars. Their sum is $(90+50-20+10)/130=1$.

Step 3: Calculate the portfolio beta. The risk-free account has beta zero.

\begin{aligned} \beta_p &=\left(\frac{90}{130}\right)(1.20) +\left(\frac{50}{130}\right)(0.70) +\left(\frac{-20}{130}\right)(1.50) +\left(\frac{10}{130}\right)(0)\\ &=\frac{108+35-30+0}{130} =\frac{113}{130} \approx 0.8692 \end{aligned}

Step 4: Apply CAPM. The market risk premium is $8.50\%-3.50\%=5.00\%$. Therefore $r_p=3.50\%+(0.8692)(5.00\%)=3.50\%+4.346\%=7.846\%$, or 7.85% when rounded to two decimal places.

Check. The signed dollar-weighted beta is $108+35-30=\$113{,}000$ in beta-weighted value. Dividing by net value of $\$130{,}000$ gives approximately 0.8692. The short security’s positive beta contributes negatively because the position itself is short.

Common Mistakes and Checks

A frequent error is to take an unweighted average of the holdings’ betas. In the first example, that incorrect calculation would be $(1.20+0.80-0.40)/3=0.5333$. It gives each fund equal influence despite their different values. The correct calculation uses weights 0.40, 0.35, and 0.25, giving portfolio beta 0.66. The correction is not to change the CAPM formula; it is to use the correct portfolio beta.

Other errors arise from mixing inputs or misreading the portfolio:

Using share counts as weights. Shares have different prices, so calculate each holding’s market value first.
Using the market return instead of the market risk premium. CAPM uses $E[R_m]-r_f$, not $E[R_m]$ alone.
Giving cash a beta of one. Under CAPM, a risk-free holding has beta zero.
Making a short holding positive. A short position’s signed weight is negative when measured against net portfolio value.
Mixing periods. Do not combine an annual risk-free rate with a market return or beta defined for a different period without first making the inputs consistent.
Calling the required return a forecast. CAPM gives a required return for systematic risk, not the realized return the portfolio will earn.

Before accepting a result, confirm that the signed weights sum to one, that the portfolio beta is the weighted sum using those weights, and that the CAPM premium is the market return minus the risk-free rate. When the market risk premium is positive, a portfolio with beta between zero and one should have a CAPM required return between the risk-free rate and the expected market return. A negative or greater-than-one beta changes that comparison in the direction indicated by the formula.

Key takeaway: Calculate portfolio beta from signed value weights, then apply CAPM to that beta. The beta reflects the portfolio’s combined market exposure; the CAPM required return adds the risk-free rate to that exposure multiplied by the market risk premium.

Check Your Understanding

For each hypothetical portfolio, calculate the requested beta and, where given, its annual CAPM required return. Use the market risk premium as the expected market return minus the risk-free rate.

A portfolio holds $18,000 of an asset with beta 0.90 and $12,000 of an asset with beta 1.40. If the risk-free rate is 3.20% and the expected market return is 8.20%, find the portfolio beta and required return. (Answer: $\beta_p=1.10$; $r_p=8.70\%$.)
A target allocation has 40% in a fund with beta 0.70, 35% in a fund with beta 1.30, and 25% in Treasury bills. The risk-free rate is 4.00% and the expected market return is 9.50%. Find the portfolio beta and required return. (Answer: $\beta_p=0.735$; $r_p=8.04\%$.)
A hypothetical account holds $24,000 of an asset with beta 1.50, $16,000 of an asset with beta $-0.25$, and $10,000 of an asset with beta 0.80. If the risk-free rate is 2.80% and the expected market return is 7.80%, find the portfolio beta and required return. (Answer: $\beta_p=0.80$; $r_p=6.80\%$.)
An investor has $60,000 long in a security with beta 1.10, $25,000 long in a security with beta 0.40, a $15,000 short position in a security with beta 1.60, and $10,000 in cash. The risk-free rate is 3.50% and the expected market return is 9.50%. Use net portfolio value to find beta and required return. (Answer: net value $=\$80{,}000$; $\beta_p=0.65$; $r_p=7.40\%$.)
A portfolio is rebalanced to hold 50% in a fund with beta 1.40, 30% in a fund with beta $-0.20$, and 20% in a risk-free asset. If the risk-free rate is 4.50% and the expected market return is 10.00%, find the portfolio’s beta and CAPM required return. (Answer: $\beta_p=0.64$; $r_p=8.02\%$.)

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