Problem 4. Consider the asset allocation problem between the S&P 500 stock index (SP) and the value-weighted stock index VW, which is more broadly diversified. Note that the VW portfolio contains the SP portfolio. Thus, if the VW portfolio is the mean-variance efficient portfolio of all stocks it contains, one should not want to hold more (or less) of the SP portfolio than that already included in the VW portfolio. Let us investigate this using historical data. Suppose you have the following estimates from historical data: E(reSPSP,SP = 0.001834, and SP,VW = 0.001822.

(a) Compute the optimal portfolio which combines the VW portfolio and the SP portfolio, i.e., construct the mean-variance efficient portfolio of these two portfolios. What is the Sharpe ratio of this optimal portfolio?

(b) What is the result in part (a) tell us about the return (and current pricing) of the S&P 500 index (i.e., large stocks)?

(c) Now compute the optimal portfolio which combines the optimal portfolio p from part (a) of problem 3 and the SP portfolio. Assume that b30,SP = 0.000078.