Portfolio Diversification Gauss file(s) prop_diversify.g, apple.csv, ford.csv Matlab file(s) prop_diversify.m, diversify.mat The data files contain daily share prices of Apple and Ford from 2 January 2001 to 6 August 2010, a total of T = 2413 observations.

(a) Compute the daily percentage returns on Apple, y1,t, and Ford, y2,t. Draw a scatter plot of the returns and interpret the graph.

(b) Assume that the returns are iid from a bivariate normal distribution with means µ1 and µ2, variances σ 2 1 and σ 2 2 , and correlation ρ. Plot the bivariate normal distribution for ρ = {−0.8, −0.6, −0.4, −0.2, 0.0, 0.2, 0.4, 0.6, 0.8}.

(c) Derive the maximum likelihood estimators.

(d) Use the data on returns to compute the maximum likelihood estimates.

(e) Let the return on a portfolio containing Apple and Ford be pt = w1y1,t + w2y2,t, where w1 and w2 are the respective weights. (i) Derive an expression of the risk of the portfolio var(pt). (ii) Derive expressions of the weights, w1 and w2, that minimize var(pt). (iii) Use the sample moments in part

(d) to estimate the optimal weights and the risk of the portfolio. Compare the estimate of var(pt) with the individual sample variances.