Investment Analysis HW Week 12: Solutions
1. You are given some information about two mutual funds: Fidelity Magellan (FMAGX and T Rowe Price Health Sciences Fund (PRHSX)
Fund Average return Standard deviation Beta FMAGX 1% 2% 0.5 PRHSX 1.5% 2.5% 0.75 The S&P500 made an average of 2% per month over the same period. The risk free rate was zero throughout.
(a) Calculate the Sharpe ratio and alpha of each fund Sharpe ratio of a risky asset X is
|E(Rx)-Rf| SDx
Here we don't know the expected return, but we know that the average return is a good approximation for the expected return, so we can use that instead. For FMAGX, we have |1 - 01/2 = 0.5
For PRHSX,we have
|1.5 - 0|/2.5 = 0.6
The alpha is the difference between the actual expected return and the should. be expected return. Again, we use the average returns as the actual expected returns of the two funds. For the should-be expected returns, we can use the CAPM, which says an asset with a known beta should have an expected return of Rf + 3E(RM - Rf). We don't know what E(RM) is, but once again we can use the average market return (of the S&P) as a proxy. So for FMAGX, the should-be expected return was Rf + 3E(RM - Rf) or 0 + 0.5 * (2 - 0) = 1% per month. FMAGX in fact made an average return of 1% per month. So its alpha was zero. For PRHSX, the should-be expected return was Rf + 3E(RM - Rf) or 0 + 0.8 * (2 - 0) = 1.6% per month. PRHSX in fact made an average return of 1.5% per
1
month. So its alpha was : actual minus should-be, or 1.6 -- 1.5 = --0.1% per month.
(b) Assume that each fund will continue to have the same Sharpe ratio and alpha going forward. Which of the two funds will you buy if you intend to hold ONLY the fund and the risk free asset? When you're thinking about holding a risky asset by itself together with Rf, you only look at the Sharpe ratio. Here PRHSX has the higher Sharpe ratio.
(c) Assume that each fund will continue to have the same Sharpe ratio and alpha going forward. Which fund will you buy if you intend to hold the fund as a small part of a large portfolio of other risky assets? When you're thinking about holding an asset as a small part of a large portfolio of other risky assets, you look at the alpha. A positive alpha is a good thing the fund did better/will do better than it should have, a negative alpha is a bad thing. Here one asset has a zero alpha, and the other has a negative alpha. FMAGX did exactly as well as it should have, given its beta, while PRHSX did worse than it should have. We definitely don't want to buy PRHSX, but we're indifferent abou