Question

The current market price of a security is $50, the security's expected return is 15%, the riskless rate of interest is 2%, and the market risk premium is 8%.

1. What is the beta of the security?
2. What is the covariance of returns on this security with the returns on the market portfolio?
3. What will be the security's price, if the covariance of its rate of return with the market portfolio doubles?
4. How is your result consistent with our understanding that assets with higher systematic risks must pay higher returns on average?

show formulas and provide brief explanation of findings

Answer 1

Current Market price of the security = $50

Security Expected return = E[R_i] = 15%

Risk-free rate = R_f = 2%

Market Risk premium is the difference between return on market portfolio and the risk free rate.

Market Risk Premium = R_M-R_f = 8%

a. We will use the CAPM equation  

CAPM Equation

E[R_i] = R_f + β_i*(R_M - R_f)

15% = 2% + β_i *(8%)

β_i = 17%/8% = 2.125

Therefore, Beta of the stock = 2.125

b. Below is the formula for beta of a stock i

β_i = Cov(i, M)/σ_M²

where Cov(i,M) is the covariance of return on security i with the return on the market portfolio M.

Cov(i, M) = β_i* σ_M²

Therefore, the covariance between security return and the market is the product of the beta of the security and market variance.

c. It is given that the covariance of its rate of return with the market portfolio doubles. Now, since the variance of the return on market portfolio (σ_M²) doesn’t change and Cov(i, M) = β_i* σ_M², Hence, the beta of the stock will also get double. (From the above answer)

Therefore, β_i,new = 2*2.125 = 4.25

Using CAPM Equation

E[R_i] = R_f + β_i,new*(R_M - R_f) = 2% + 4.25*(8%) = 36%

New Expected return = 36%

Therefore, new security’s price = (1+36%)*50 = $60

d. We know that beta is a measure of Stock’s volatility with respect to the market or systematic risk. As the beta doubled from 2.125 to 4.25 i.e., the systematic risk increased, we can see that the return increased from 15% to 36%. Hence, the result is consistent with the understanding that assets with higher systematic risk must pay higher return on average.