You are trying to decide how to allocate your money over thenext year between one-year T-Bills and a stock index. Suppose thatT-Bills are going to return 2.5% for sure over the next year, whileyou forecast that the expected stock index return is 4.0% and thestandard deviation of the stock index return is 15.0%.
(a) Suppose that w is the fraction of your wealth that youallocate to the stock index. For w = 0%, 1%, 2%, . . . , 99%, 100%,(i.e. for 101 different values of w), calculate the expected return(E[rw]) and standard deviation of return (Ïw) for the strategy thatinvests a fraction w in the stock market and (1âw) in T-Bills. (Youdonât need to present any of these numbers). Plot out first E[rw]against w, and then Ïw against w. (Again, you donât need to presentthese plots). What shape do these graphs have? Why?
(b) Now, for each of these 101 different strategies, calculatethe utility of the strategy (Uw) for an investor with the utilityfunction U(E[r], Ï) = E[r]â2Ï 2 . Plot out Uw against w. (Please doinclude this graph in your write-up). What does your graph suggestis the right allocation to stocks for this investor? Does thisagree with the formula for the optimal allocation?