Business Risk, Beta, and Required Return
A business can be exposed to economic conditions that affect many companies at once. For example, demand for a discretionary product may fall during a broad recession, while demand for a more defensive product may be less sensitive to economic conditions. Investors who hold a diversified portfolio cannot eliminate this market-wide exposure simply by owning more securities. CAPM represents the exposure with beta and uses it to estimate the return equity investors require.
A company can also face substantial risks specific to its own operations, such as a factory outage or the loss of a major customer. These risks may create large fluctuations in that company’s returns, but diversification can reduce their effect on a portfolio. CAPM distinguishes these two kinds of risk: it links required return to market-related risk, not to total volatility by itself.
CAPM and the Required Return
The Capital Asset Pricing Model (CAPM) estimates an asset’s expected return, or required return, from the risk-free rate, the expected market risk premium, and the asset’s beta. The market risk premium is the expected market return in excess of the risk-free rate. State the inputs on a consistent basis: for example, use annual rates throughout when estimating an annual required return.
The formula shows why business exposure matters to cost of equity. If a change in the business makes its returns more sensitive to market movements, beta rises. When the expected market risk premium is positive, that increase raises the CAPM required return. A rise in company-specific volatility, by itself, does not necessarily change beta or the CAPM estimate.
Worked Example: Estimating Beta From Market Outcomes
A hypothetical company’s annual return and the market’s annual return are observed in four equally likely economic outcomes. The returns are shown as percentages. The risk-free rate is 3% per year. Estimate beta using the four outcomes, then calculate the CAPM required return. The mean market return in these outcomes is used as the expected market return.
| Outcome | Market return | Company return |
|---|---|---|
| 1 | 2% | -1% |
| 2 | 5% | 4% |
| 3 | 9% | 12% |
| 4 | 12% | 13% |
Method. Calculate the mean of each return, then calculate the covariance of company and market returns and divide by the market variance. Because the outcomes are equally likely, each mean is the arithmetic average.
The mean market return is \((2+5+9+12)/4=7\%\). The mean company return is \((-1+4+12+13)/4=7\%\). Their deviations from their respective means, in percentage points, are:
Market: \(-5,-2,2,5\). Company: \(-8,-3,5,6\).
The covariance is the average product of paired deviations: \[ \operatorname{Cov}(R_i,R_M)=\frac{(-8)(-5)+(-3)(-2)+(5)(2)+(6)(5)}{4} =\frac{40+6+10+30}{4}=21.5 \] percentage-points squared. The market variance is \[ \operatorname{Var}(R_M)=\frac{(-5)^2+(-2)^2+2^2+5^2}{4} =\frac{25+4+4+25}{4}=14.5 \] percentage-points squared. Therefore, \[ \beta_i=\frac{21.5}{14.5}=1.4827586. \] The expected market risk premium is \(7\%-3\%=4\%\). The CAPM required return is \[ 3\%+1.4827586(4\%)=8.9310344\%. \] Rounded only at the end, the estimated beta is 1.4828 and the required return is 8.93% per year.
Check. The beta exceeds 1, consistent with the company return moving more sharply than the market in these outcomes. The calculation uses paired market and company deviations; comparing the two standard deviations alone would not produce beta.
Why Beta Is a Covariance Ratio
Beta is the slope of the best-fitting linear relationship between asset returns and market returns. This slope measures how much the asset return tends to change when the market return changes. The covariance ratio follows directly by finding the slope that minimizes squared prediction errors.
For a proposed intercept \(a\) and slope \(b\), predict the asset return as \(a+bR_M\). The mean squared prediction error is \(E[(R_i-a-bR_M)^2]\).
For any fixed slope \(b\), the error is minimized by \(a=E[R_i]-bE[R_M]\). Substituting this intercept centers each return around its mean.
The centered error is \(E[((R_i-E[R_i])-b(R_M-E[R_M]))^2]\). Differentiating with respect to \(b\) and setting the result to zero gives \(-2\operatorname{Cov}(R_i,R_M)+2b\operatorname{Var}(R_M)=0\).
When market variance is positive, the minimizing slope is \(b=\operatorname{Cov}(R_i,R_M)/\operatorname{Var}(R_M)\), the covariance definition of beta.
This derivation also clarifies a key distinction. Asset volatility measures how widely its returns vary around their own mean. Beta measures how much of those movements is associated with market movements. Volatility uncorrelated with the market can increase total risk without increasing the covariance in beta’s numerator.
Systematic Business Risk and Company-Specific Risk
Systematic risk is risk related to broad market or economic movements. It cannot be eliminated simply by diversifying across many assets, because those assets may respond to the same economic conditions. A business whose sales and returns are especially sensitive to such conditions may have a higher beta.
Company-specific risk is risk that primarily affects one company or a limited group of companies. An unexpected equipment failure is an example. In a diversified portfolio, gains and losses from different company-specific events can offset one another. CAPM therefore does not add a separate premium for that risk to the required return of a well-diversified investor.
This does not mean company-specific risk is unimportant. It can affect a company’s actual results and can matter greatly to an investor who is not diversified. The point is narrower: in CAPM, the risk rewarded with a higher expected return is the component associated with market covariance. A business may have high total volatility but a moderate beta if much of its volatility is company-specific.
Worked Example: Separating Beta From Company-Specific Volatility
Two hypothetical firms have the following estimated characteristics. The annual risk-free rate is 3.5%, the expected market return is 9.0%, and both beta estimates are based on returns measured over the same annual horizon. Firm Cedar has beta 0.55 and company-specific return standard deviation of 22%. Firm Flint has beta 1.35 and company-specific return standard deviation of 7%. Calculate each CAPM required return and interpret the different volatility figures.
Method. First find the market risk premium: \(9.0\%-3.5\%=5.5\%\). Then apply CAPM separately using each firm’s beta.
For Cedar, \(\beta=0.55\), so the return premium for market risk is \(0.55(5.5\%)=3.025\%\). Its required return is \(3.5\%+3.025\%=6.525\%\), or 6.53% per year to two decimal places.
For Flint, \(\beta=1.35\), so the return premium for market risk is \(1.35(5.5\%)=7.425\%\). Its required return is \(3.5\%+7.425\%=10.925\%\), or 10.93% per year to two decimal places.
Cedar has the larger company-specific standard deviation, but that figure is not inserted into the CAPM formula. Flint has the larger beta, so CAPM assigns it the larger required return. Common mistake avoided: using a company’s total or company-specific standard deviation as if it were beta. Standard deviation measures volatility; beta measures market sensitivity.
Check. The difference in required returns is \(10.925\%-6.525\%=4.400\%\). It also equals the beta difference, \(1.35-0.55=0.80\), multiplied by the 5.5% market premium: \(0.80(5.5\%)=4.400\%\).
How a Change in Business Exposure Affects Beta
A company’s beta can change when its business mix changes the relationship between its returns and market returns. For example, adding a line of business that tends to perform poorly in broad downturns can increase market sensitivity. Adding a more defensive line may reduce it. The direction depends on the market covariance of the activity being added, not merely on whether it is volatile or profitable.
For a portfolio of assets, portfolio beta is the market-value-weighted average of the individual betas, provided the weights represent the portfolio’s shares of market value and the portfolio return is the corresponding weighted return. This relationship can help explain the effect of combining business exposures. Revenue shares are not automatically appropriate weights: a revenue percentage does not necessarily equal the market-value share of the underlying assets or investment.
Worked Example: Assessing a More Market-Sensitive Business Mix
A hypothetical investment portfolio holds two operating businesses. Business A has beta 0.60 and represents 40% of the combined market value; Business B has beta 1.40 and represents 60%. Assume these are the only two holdings, their portfolio weights are based on market values, and their returns combine in those same weights. The risk-free rate is 4% and the expected market return is 9%. Calculate the portfolio beta and its CAPM required return, then compare it with a portfolio consisting only of Business A.
Method. Apply the portfolio beta formula, then use CAPM. The market risk premium is \(9\%-4\%=5\%\).
The combined beta is \[ \beta_P=0.40(0.60)+0.60(1.40)=0.24+0.84=1.08. \] The portfolio required return is \(4\%+1.08(5\%)=4\%+5.40\%=9.40\%\).
For Business A alone, beta is 0.60 and its CAPM required return is \(4\%+0.60(5\%)=7.00\%\). Under these assumptions, combining the businesses raises beta from 0.60 to 1.08 and raises the CAPM required return from 7.00% to 9.40%.
Check. The combined beta, 1.08, lies between the two component betas, 0.60 and 1.40, as a positive weighted average should. Common mistake avoided: weighting betas by revenue shares without establishing that those shares represent the portfolio’s market-value weights.
Interpretation and Common Exam Traps
Beta is an estimate of systematic sensitivity, not a complete description of business risk or a promise about future returns. Estimates may vary with the observation period, the market index, the business mix, and other estimation choices. A historical beta is evidence about past co-movement; using it in CAPM assumes it is a reasonable estimate of the future exposure relevant to the valuation.
- Confusing volatility with beta. A high standard deviation does not necessarily mean a high beta. Only the portion of return variation that covaries with the market contributes to beta.
- Adding a premium for every risk. CAPM does not add a separate return premium for diversifiable company-specific risk under its standard diversified-investor assumptions.
- Reading beta as a probability or percentage. A beta of 1.2 does not mean a 1.2% chance of loss. It describes sensitivity relative to market returns.
- Ignoring the market risk premium. The risk premium is \(E[R_M]-r_f\), not the expected market return alone. CAPM adds the risk-free rate once, then multiplies beta by the premium.
- Assuming all business volatility raises required return. A business change affects CAPM return to the extent it changes systematic exposure, represented by beta, when the market risk premium is positive.
Check Your Understanding
Use the stated assumptions and keep rates on an annual basis.
- A stock has beta 0.90, the risk-free rate is 4%, and the expected market return is 9%. Find its CAPM required return. (Answer: 8.50% per year)
- For a hypothetical asset, \(\operatorname{Cov}(R_i,R_M)=0.0036\) and \(\operatorname{Var}(R_M)=0.0064\), with returns expressed as decimals. The risk-free rate is 2.5% and the expected market risk premium is 6%. Find beta and the required return. (Answer: beta 0.5625; required return 5.875% per year)
- A portfolio invests 25% of its market value in an asset with beta 0.40 and 75% in an asset with beta 1.20. The risk-free rate is 3% and the expected market return is 8%. Find the portfolio beta and CAPM required return. (Answer: beta 1.00; required return 8.00% per year)
- A firm’s factory outage risk rises, but the estimated covariance of its return with the market and the market variance remain unchanged. Under CAPM, does the estimated required return change solely because of this company-specific risk? (Answer: No; with beta and the market risk premium unchanged, the CAPM required return is unchanged.)
- The market return has variance 0.0225, and an asset’s return has covariance 0.0180 with the market. If the risk-free rate is 3.2% and the expected market return is 8.2%, calculate beta and the CAPM required return. (Answer: beta 0.80; required return 7.20% per year)