CAPM Beta and the Share of Stock Variance Attributed to Market Risk

Summary

The problem asks how to calculate a stock’s beta from its covariance with the market, use CAPM to estimate expected return, and determine what share of the stock’s variance is attributable to market risk. The response gives the beta and expected-return formulas and reports numerical answers using the stated market variance and risk-free rate.

Its variance explanation is flawed. Market-related variance in a single-factor CAPM model is β² times market variance; the market share is that amount divided by the stock’s total variance. With the stated inputs, the response’s beta calculation also misreads the covariance: 165 percent-squared is 0.0165, while the market variance is 0.0121, giving beta about 1.36. These calculations imply market-related variance above the stated total variance, which signals inconsistent inputs or assumptions, not a 55.55 percent share. The post raises a useful distinction between beta and variance contribution, but its proposed result should not be relied upon.

Key ideas

• Beta is the covariance of a stock with the market divided by market variance.
• CAPM estimates expected return by adding beta times the market risk premium to the risk-free rate.
• In a single-factor model, market-related variance is beta squared times market variance.
• The stated covariance and total variance produce an inconsistency that the answer’s variance calculation does not resolve.

Tags

# CAPM (SML) Problem


# CAPM (SML) Problem












I got 1.3636 for beta for the problem below(165/121). But I became so unsure about the answer when I solved (c) because then the market risk becomes larger than the variance of Stock A. Beta^2*σ(M)^2=224.98> σ(A)^2=220

Am I making a mistake? Or if my solution is correct,how do I interpret this result?(all the variance of the Stock AS is all due to the Market?)

Thanks in advance for your help!!

*By 165%^2 and 220 %^2, I mean percent-squared. A variance of 165%^2 equals 165/10,000. Thanks.

Suppose that the riskless rate of return is 4% and the expected market return is 12%. The standard deviation of the market return is 11%. Sup- pose as well that the covariance of the return on Stock A with the market return is 165%^2. (a) What is the beta of Stock A? (b) What is the expected return on Stock A? (c) If the variance of the return on Stock A is 220%^2, what percentage of this variance is due to market risk?

## Answer by emcor (score 1)

https://quant.stackexchange.com/a/14948

a) The formula for Beta is:

$$\beta_i=\frac{\sigma_{i,M}^2}{\sigma_M^2}=\frac{0.165^2}{0.11^2}=2.25$$

b) So by the CAPM equation, the expected return for the asset is:

$$E(R_i)=r_f+\beta(R_M-r_f)=0.04+2.25(0.12-0.04)=0.22=22\%$$

c) If the variance of the stock is $0.22^2$, since this variance was multiplied by $\beta=2.25$, we get:

$$1-(0.22^2/2.25)/(0.22^2)=55.55\%$$ of asset variance explained by market variance.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.