RMSC-1003 3 8 14 3 21 1:2: 3 at 3 2. (5%) Suppose that the returns of an asset i satisfy the two-factor model Mizuitbar theFe. i=L.... . . where ba and ba are factor loadings of the risk factors Fi and Fa respectively. Consider a portfolio (1) where the assets i and j are chosen such that ba ba and big * bye. (a) For the portfolio given in (1). what is Er? (b) For the portfolio r in (1) to have no exposure to any factor risk. show that the factor loadings ba . b. ba. and big must satisfy w = ; bj 12 for all i and j. bit - bil biz - biz (c) Under the condition in part (b) and using the no arbitrage argument, show that there exists a constant do such that " 20 = ci for all i, where ci is a constant. (d) Show that there exist constants , and As such that the expected return ; for any asset i satisfies Hi = do + Albil + 12biz . 3. (25%) Consider a portfolio with return up = Xir = I diri where A = ()1, ... . da), r = (71, . .. . ra) with mean a = (/, .... /a) and variance- covariance matrix S. (a.) Find the risk op for the portfolio. (b) Show that Cov (ri,"p) OP (c) Suppose we define the risk measure of this portfolio in terms of op and suppose that op = do P (2) Consider the portfolio with A = (1, ...,1) = 1. Can you provide an interpretation for (2)? 4. (25%) Let W(t) be a standard Brownian motion process. (a) Evaluate ( (e BFV(s) - 2e - W'(s) + 2 ) dWV (s ) . (b) Let Y(t) = F(W(t)) = log [W(t)|. Evaluate d) (t). (c) Evaluate / is W (s) 2 + 3 WV (s ) + 2 dW (s ) . -END