Consider a portfolio with four stocks Google(GOOG), 3M(MMM), Microsoft (MSFT) and IBM(IBM). We are interested in monthly return (log return between 22 business days) of the portfolio elements from 01/01/2007 to 12/31/2016. 1. Find efficient frontier for the benchmark return R* 's between 0.0075 and 0.010, that is {0.0075, 0.0076, 0.0077, ..., 0.0099, 0.010}.

2. Suppose that risk free rate of return rf = 0.0001 (=0.01% = 1bp) per month (22 business days). Find the portfolio weight vector for the tangency portfolio and the Sharpe Ratio.

Weight Vector: Google;0.1666389; 3M:= 0.6063341; Microsoft= 0.1317821 and IBM=0.9524492.

Sharpe Ratio = 0.2432858

0.0 0.2 0.4 0.6 0.8 1.0 IBM MSFT MMM GOOG JUHT 0.00095 0.0085 0.0075 0.036 0.038 10.040 0.042 sigma 0.0 0.2 0.4 0.6 0.8 1.0 IBM MSFT MMM GOOG JUHT 0.00095 0.0085 0.0075 0.036 0.038 10.040 0.042 sigma