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Chapter 13: Problem 22

Using the CAPM, show that the ratio of the risk premiums on two assets is equal to the ratio of their betas.

Short Answer

Expert verified
Using the Capital Asset Pricing Model (CAPM), we can find the risk premiums for two assets, Asset A and Asset B, as \(Risk \ Premium_A = \beta_A (R_m - R_f)\) and \(Risk \ Premium_B = \beta_B (R_m - R_f)\), respectively. The ratio of their risk premiums can be expressed as: \[\frac{Risk \ Premium_A}{Risk \ Premium_B} = \frac{\beta_A (R_m - R_f)}{\beta_B (R_m - R_f)}\] Simplifying the ratio, we get: \[\frac{Risk \ Premium_A}{Risk \ Premium_B} = \frac{\beta_A}{\beta_B}\] Hence, the ratio of the risk premiums on two assets is equal to the ratio of their betas.

Step by step solution

01

Write down the risk premiums for two assets using the CAPM formula

First, we need to find the risk premiums for two assets using the CAPM formula. Let's call these assets Asset A and Asset B. Their risk premiums can be calculated as follows: \[Risk \ Premium_A = R_A - R_f = \beta_A (R_m - R_f)\] \[Risk \ Premium_B = R_B - R_f = \beta_B (R_m - R_f)\]
02

Find the ratio of the risk premiums

Now we need to find the ratio of risk premiums for Asset A and Asset B. This can be expressed as: \[\frac{Risk \ Premium_A}{Risk \ Premium_B} = \frac{\beta_A (R_m - R_f)}{\beta_B (R_m - R_f)}\]
03

Simplify the ratio

Now we just need to simplify the ratio. Notice that both the numerator and denominator have the same term \((R_m - R_f)\), so we can cancel it out: \[\frac{Risk \ Premium_A}{Risk \ Premium_B} = \frac{\beta_A (R_m - R_f)}{\beta_B (R_m - R_f)} \cdot \frac{1}{(R_m - R_f)}\] \[\frac{Risk \ Premium_A}{Risk \ Premium_B} = \frac{\beta_A}{\beta_B}\]
04

Conclusion

Using the Capital Asset Pricing Model (CAPM), we have shown that the ratio of the risk premiums on Asset A and Asset B is equal to the ratio of their betas: \[\frac{Risk \ Premium_A}{Risk \ Premium_B} = \frac{\beta_A}{\beta_B}\]

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Most popular questions from this chapter

You have 100,000 dollar to invest in a portfolio containing Stock X Stock Y , and a risk-free asset. You must invest all of your money. Your goal is to create a portfolio that has an expected return of 12.5 percent and that has only 80 percent of the risk of the overall market. If Xhas an expected return of 28 percent and a beta of \(1.6,\) Y has an expected return of 16 percent and a beta of 1.2 ,and the risk-free rate is 7 percent, how much money will you invest in Stock X? How do you interpret your answer?

A stock has an expected return of 14 percent, a beta of \(1.6,\) and the expected return on the market is 11 percent. What must the risk-free rate be?

Ratios Stock Y has a beta of 1.45 and an expected return of 17 percent. Stock \(Z\) has a beta of .85 and an expected return of 12 percent. If the risk-free rate is 6 percent and the market risk premium is 7.5 percent, are these stocks correctly priced?

Using information from the previous chapter on capital market history, determine the return on a portfolio that is equally invested in large-company stocks and long-term government bonds. What is the return on a portfolio that is equally invested in small-company stocks and Treasury bills?

A stock has an expected return of 13 percent, the risk-free rate is 5 percent, and the market risk premium is 7 percent. What must the beta of this stock be?
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