2. (both 176 & 276 attempt) Topie: Markowitz Portfolio Optimization. Let Rr, for i = 1,2,...,n, be Independent samples of a return R of mean a and variance o'. Define the estimators n 1 n- 1 E(R - A)?. 1 Show that (4) = u and (82) = 0'. (That means, they are unblased estimators of u and o', respec- tively.) 11 Suppose a stock's rate of return has annual mean and variance of a and o'. To estimate these quantitles, we divide 1 year Into n equal periods and record the return for each period. Let A = p and of = 03 be the mean and the variance for the return for each period. (You might think In this way, the returns in these periods are assumed to be Independent and Identically distributed.) Using the formulae In Question 2. (1), If A,, and &, are the estimators of these two random variables, then A = nan and 82 = no2. Let o(4) and o(8?) be the standard deviations of these estimators. 1. Show that o (() Is Independent of n. 2. Assuming that the returns are normally distributed, show that o(82) depends on n