Question

A mutual fund manager expects his portfolio to earn a rate of return of 11 percent this year. The beta of the portfolio is 0.8. If the market risk premium is 10 percent and the risk-free rate is 4 percent, is it a good idea to invest? How could you use a stock index fund (S&P 500 index) and risk-free fund (invested in T-bills) to create a portfolio with the same risk as the manager, but a higher return?

Answer 1

Solution:-

Part 1:

As per CAPM, the required rate of return is as follows:

Required rate of return= Risk free rate + beta*Market risk premium= 4% + 0.8*10% = 12%

Thus, the expected return from mutual fund of 11% is lower than the required return of 12%. Thus, it is not a good idea to invest in this portfolio.

Part 2:

Now, we know that the market risk premium is 10% which means that the expected return of market (i.e. S&P) is 14% (Risk premium 10% + risk free rate 4%).

The desired beta of a portfolio with S&P and risk free assets is 0.8. The beta of S&P is 1 and beta of risk free asset is 0. Now, Let the % weightage of S&P in total portfolio be X. Therefore, the share of risk free asset will be (100-X). The portfolio beta is as follows:

Beta of S&P*weightage of S&P + beta of risk free asset*weightage of risk free asset= Portfolio beta

X*1 + (100-X)*0 = 0.8

X= 0.8 or 80%

Thus, the S&P should have 80% weightage in portfolio and risk free asset should have remaining 20%.

Expected return= 14%*0.8 + 4%*0.2= 12%

Hence, as we can see this portfolio of S&P and risk free asset has the same portfolio beta of 0.8 that the mutual fund had but has a higher expected return of 12% (compared to 11% of mutual fund)