1) how to proof above as per the two factor model

2) how to proof the variance-covariance matrix

3) how to proof the variance on the j th return

4) Please proof the covariance is correct

5) why we should use equation 3

The returns r;, of the assets satisfy the two-factor model r = Boi + Blifi+ B2i f2 + ei , i = 1, ...,n, (1) where Bi; and B2i are factor loadings of the risk factors fi and f2 re- spectively. Let r = (ri,...,"n)", Bo = (B01 ..., Bon)", F = (f1, f2)", e = (e1, ..., en)T, and B11 B21 Bin Bzn ... (a) Show that the two-factor model (1) can be expressed as r = Bo + 3"F +e. (2) (b) Show that the variance-covariance matrix for r is given by ,-'F + , (3) where EF and Ee are the variance-covariance matrices of F and e respectively. (c) Let B; = (B1j, B2;)" (j = 1,...,n) be the j-th column of the matrix 3. Show that the variance of the j-th return is Var(r;) = 3}Ef;+ o, . (d) Show further that Cov (ri, ri) F), . (e) Can you give a reason why one would consider estimating Er using equation (3) instead of using the sample variance-covariance matrix of r directly. The returns r;, of the assets satisfy the two-factor model r = Boi + Blifi+ B2i f2 + ei , i = 1, ...,n, (1) where Bi; and B2i are factor loadings of the risk factors fi and f2 re- spectively. Let r = (ri,...,"n)", Bo = (B01 ..., Bon)", F = (f1, f2)", e = (e1, ..., en)T, and B11 B21 Bin Bzn ... (a) Show that the two-factor model (1) can be expressed as r = Bo + 3"F +e. (2) (b) Show that the variance-covariance matrix for r is given by ,-'F + , (3) where EF and Ee are the variance-covariance matrices of F and e respectively. (c) Let B; = (B1j, B2;)" (j = 1,...,n) be the j-th column of the matrix 3. Show that the variance of the j-th return is Var(r;) = 3}Ef;+ o, . (d) Show further that Cov (ri, ri) F), . (e) Can you give a reason why one would consider estimating Er using equation (3) instead of using the sample variance-covariance matrix of r directly