Answer Happy

3.3. Let rot be the return of a stock index at time t. Sharpe's single-index model assumes that the log returns of the n stocks in the index are gen- erated by rit = αi+ßirot+Eit, 1 ≤ i ≤ p, where it is uncorrelated with rot and Cov(€it, €jt) = 0²1{i=j}. The model also assumes that (rot,…….‚¯pt), 1 ≤ t ≤n, are i.i.d. vectors. (a) Suppose Var (rot) = o. Show that the covariance matrix F = (fij) of the log return of the n stocks under the single-index model is given by F = 33¹ + o²I, where ß = (B₁,..., ßp)T. (b) Let σij = Cov(rit, rit) and Σ = - (ij)1<i,j<p. Let S = (s¿j) be the sample covariance matrix based on (rit,…..‚¯pt)T, 1 ≤ t ≤ n. Let R(a) = aF + (1 − a)S. Consider the quadratic loss function L(a) = ||·||2, where ||A|| is the Frobenius norm of a square matrix A defined by ||A||2 tr(ATA). Show that the minimizer a* of E[L(a)] is given by = 3 Basic Investment Models and Their Statistical Analysis Σ?=1 Σ;=1 [Var (sij) — Cov (fij, Sij)] ·σij)²]* Σ=1 Σ;=1 |Var (fij − Sij) + (E(fij) — σij 90