Explain your answer in depth - show formulas and all relevant stuff
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_{M}^{2}
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_{M}^{2}
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_{M}^{2}) doesn’t change and Cov(i, M) = β_{i}* σ_{M}^{2}, 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.
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