1.

Finish the proofs of Theorems 7.2.2 and 7.2.4 by filling in and deriving the relevant covariance relations (7.15). Use and verify the linearity properties of covariances: for all random variables X, Y and Z, we have Cov[ X, Y + Z] = Cov[ X, Y] + Cov[ X, Z], while for all real ?, Cov[ X, ?Y] = ?Cov[ X, Y].

2.

Let X ₁, , X _T be uncorrelated random variables with mean m and variance s ². Consider the weighted average X _w = ? _t w _tX _t, where the weights w _t ? 0, t = 1, T satisfy ? _t w _t = 1. Show that E[ X _w] = m, Cov[ X _t, X _w] = w _ts ² and Var[ X _w] = ? _t s ². [If especially , we get X _w = X and E[ X] = m; Cov[ X _t, X] = Var[ X] = .]

3.

Show that the sample variance is an unbiased estimator of s ².

4.

Show that the best predictor of X _{j,T + 1} is at the same time the best estimator of the risk premium m + ?