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Unformatted text preview: UNIVERSITY OF TORONTO Joseph L. Rotman School of Management Nov. 14, 2006 Brean/Kan MGT337Y MIDTERM EXAMINATION #1 Pomorski/Xu SOLUTIONS 1. (a) When capital markets do not exist, we have I = 118125 C and C 1 = W 1 , so the utility function can be written as U ( C ,C 1 ) = ln C × 25(118125 C ) 3 4 . To maximize the utility, we take derivative of U with respect to C , and set it equal to zero 1 C 1 (118125 C ) × 3 4 = 0 . Solving this equation, we get C * = 67500, I * = 50625, and C * 1 = W 1 = 84375. (b) When capital markets exist, the optimal investment is the point at which the marginal rate of return of the investment equals to the interest rate from the capital market. That is, d W 1 d I 1 = 25 × 3 4 × I 1 4 1 = 0 . 25 , and we found the optimal investment is I * = 50625 and W 1 = 84375. The budget constraint is then C + C 1 1 . 25 = 118125 50625 + 84375 1 . 25 = 135000 , which implies C 1 = 1 . 25(135000 C ). The utility function now can be written as U = ln( C C 1 ) = ln( C × 1 . 25(135000 C )) . To maximize the utility, we take the derivative of U with respect to C , and set it equal to zero, we get C * = 67500, and C * 1 = 84375. 2. (a) Let S be the price of the security, we have S = 1 1 + r + 2 (1 + r ) 2 + ··· + 100 (1 + r ) 100 . (1) 1 Multiply both sides of (1) by (1 + r ), we obtain (1 + r ) S = 1 + 2 1 + r + ··· + 100 (1 + r ) 99 . (2) Subtract (1) from (2), we have rS = 1 + 1 1 + r + 1 (1 + r ) 2 + ··· + 1 (1 + r ) 99 100 (1 + r ) 100 ⇒ rS = 1 + A 99 r 100 (1 + r ) 100 ⇒ S = 1 + A 99 r r 100 r (1 + r ) 100 . The second equality follows because the terms in the middle on the right hand side are the present value of an annuity of $1 for 99 years. Putting r = 0 . 08, we have S = $168 . 11....
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