You have 5 million baths to invest in a portfolio containing Stock X, Stock Y, a

Question

You have 5 million baths 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 13.5 percent and that has only 80 percent of the risk of

the overall market.

If X has an expected return of 31 percent and a beta of 1.8, Y has an expected return of 20 percent and a beta of 1.3, and the risk-free rate is 7 percent, how much money will you invest in Stock X? How do you interpret your answer?

Explanation / Answer

Let proportion invested in stock X be X% and proportion invested in stock Y be Y%. So proportion invested in risk free asset = 1-X-Y

Total return = X*31% + Y*20% + (1-X-Y)*7% = 7% + 24%*X + 13%*Y

This is equal to 13.5% (return of portfolio)

So 7% + 24%*X + 13%*Y = 13.5%, i.e. 24%*X + 13%*Y = 6.5%

Beta of portfolio = 1.8*X+ 1.3*Y + 0*(1-X-Y) = 0.8, i.e. 0.18*X+ 0.13*Y = 0.08

So the 2 equations we have are:

24%*X + 13%*Y = 6.5%

0.18*X + 0.13*Y = 0.08

Solving, we get X = -25%, i.e. negative 25%

This means that we should invest 25% of 5 million = 1,250,000 in shorting stock X, i.e. we should go short on stock X to the extent of 1,250,000.

Hope this helped ! Let me know in case of any queries.

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