as372w10a4soln

as372w10a4soln - Winter 10 Actsc 372 Assignment 4 Solution...

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Winter 10 Actsc 372 Assignment 4 Solution 5 Marks for each part. (total 60 marks) 1 a. Let: X 1 = the proportion of Security 1 in the portfolio and X 2 = the proportion of Security 2 in the portfolio and note that since the weights must sum to 1.0, X 1 = 1 – X 2 Recall that the beta for a portfolio (or in this case the beta for a factor) is the weighted average of the security betas, so β P1 = X 1 β 11 + X 2 β 21 β P1 = X 1 β 11 + (1–X 1 ) β 21 Now, apply the condition given in the hint that the return of the portfolio does not depend on F 1 . This means that the portfolio beta for that factor will be 0, so: β P1 = X 1 β 11 + X 2 β 21 β P1 = 0 = X 1 (1.0) + (1 – X 1 )(0.5) and solving for X 1 and X 2 : X 1 = – 1, X 2 = 2 Thus, sell short Security 1 and buy Security 2. To find the expected return on that portfolio, use R P = X 1 R 1 + X 2 R 2 so applying the above: E(R P ) = –1(20%) + 2(20%) = 20% β P1 = –1(1) + 2(0.5) = 0 b. Following the same logic as in part a, we have β P2 = 0 = β P1 = X 3 β 31 + (1–X 3 ) β 41 β P2 = 0 = X 3 (1) + (1 – X 3 )(1.5) and X 3 = 3, X 4 = –2 Thus, sell short Security 4 and buy Security 3. Then,
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E(R P2 ) = 3(10%) + (–2)(10%) = 10% β P2 = 3(0.5) – 2(0.75) = 0 Note that since both β P1 and β P2 are 0, this is a risk free portfolio! c. The portfolio in part b provides a risk free return of 10%, which is higher than the 5% return provided by the risk free security. To take advantage of this opportunity, borrow at the risk free rate of 5% and invest the funds in a portfolio
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as372w10a4soln - Winter 10 Actsc 372 Assignment 4 Solution...

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