Question

Suppose the market index has a standard deviation of 0.40 and the riskless rate is 5%. You are given the following information about two stocks X and Y:

? = [ 10% 20%], ???(??, ???????) = 0.096, ??? ???(??,???????) = 0.240.

Suppose firm-specific errors are independent and identically distributed with a mean of zero and standard deviation of 0.5.

a) What are the standard deviations of stocks X and Y?

b) You were to construct a portfolio with the following proportions: 20% in Stock X, 50% in Stock Y, and 30% in T-bills. Find the expected return, beta, standard deviation, and nonsystematic standard deviation of the portfolio.

Answer 1

(a) We represent the standard deviations of stocks X and Y as and respectively. Using CAPM, we've,

On simpifying,

Henceforth, for stock X,

On solving,

On further simplifying,

On solving,

Similarly,

On solving,

Now, from the CAPM model,

we can obtain using the concept that variance of the sum of two random variables is the sum of the variance of individual random variables in case they are independent: -

On solving,

Similarly,

On solving,

(b) Calculating the expected return,

On putting in the values,

Calculating the beta,

We've: -

On solving,

Please note that looks like there is a discrepancy in the data because using stock Y, the market index return calculations give different values.

On solving, in this case,

Calculating the standard deviation, we know that: -

Hence, using the variance of sum of random variables concept again,

On solving,

Calculating the non-systematic standard deviation of the portfolio, we need to separate out systematic risk of stocks X and stocks Y of the portfolio. If is the required standard deviation, we can write: -

On putting in the values,

On solving,