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final07a - UNIVERSITY OF TORONTO Joseph L Rotman School of...

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UNIVERSITY OF TORONTO Joseph L. Rotman School of Management April 30, 2007 Davydenko/Derrien/ MGT337Y FINAL EXAMINATION Florence/Lu/Nevard SOLUTIONS 1. (a) Let S be the price of the security, we have S = 100 1 + r + 101 (1 + r ) 2 + · · · + 199 (1 + r ) 100 . (1) Multiply both sides of (1) by (1 + r ), we obtain (1 + r ) S = 100 + 101 1 + r + · · · + 199 (1 + r ) 99 . (2) Subtract (1) from (2), we have rS = 100 + 1 1 + r + 1 (1 + r ) 2 + · · · + 1 (1 + r ) 99 - 199 (1 + r ) 100 rS = 100 + A 99 r - 199 (1 + r ) 100 S = 100 + A 99 r r - 199 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 = \$1405 . 04. (b) The monthly interest rate is r m = 0 . 12 / 12 = 0 . 01. Let x be the monthly payment, the present value of the twelve payments must be equal to \$10000. This implies xA 12 r m = \$10000 x = \$10000 A 12 0 . 01 = \$10000 11 . 2551 = \$888 . 49 . (3) Therefore, the monthly payment is \$888.49. (c) (i) Since this loan requires you to pay \$900/month whereas the loan in part (b) only requires you to pay \$888.49/month. You obviously would prefer the 12% loan that is compounded on a monthly basis. 1

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(ii) For the last monthly payment, you will pay an interest of \$10.26, so the principal repayment is \$900 - \$10 . 26 = \$889 . 74. On a monthly basis, the interest rate is r m = 10 . 26 889 . 74 = 0 . 01153 . Therefore, the effective annual interest rate is r = (1 + r m ) 12 - 1 = 0 . 1475 , which is much higher than the quoted 8%/year. This explains why you prefer the loan in part in (b). 2. (a) The decomposition of the total risk of Robotoy is σ 2 R = β 2 R σ 2 M + σ 2 , (4) where the first term on the right hand side is the market risk of Robotoy and the second term is the unique risk of Robotoy. It follows that the market risk of Robotoy is (1 . 39) 2 × (0 . 1802) 2 = 0 . 06274 and the unique risk of Robotoy is σ 2 R - β 2 R σ 2 M = (0 . 4021) 2 - 0 . 06274 = 0 . 09895. (b) If you hold just one stock, Robotoy is indeed riskier than Mobitoy. However, if you include Robotoy in a well diversified portfolio (say the market portfolio), then only its systematic risk (beta) matters. In this case, Robotoy is less risky than Mobitoy because it has a lower beta. This explains why the expected return of Robotoy is lower than that of Mobitoy. (c) For a portfolio with 55% in Robotoy and 45% in Mobitoty, its expected return and standard deviation are given by μ p = 0 . 55 × 0 . 1601 + 0 . 45 × 0 . 2092 = 0 . 1822 , σ p = (0 . 55) 2 (0 . 4021) 2 + (0 . 45) 2 (0 . 3422) 2 + 2(0 . 55)(0 . 45)(0 . 6)(0 . 4021)(0 . 3422) 1 2 = 0 . 3369 . (d) In order to find out the expected return of the portfolio, we need to know three things: its beta, the risk-free rate, and the risk premium on the market portfolio. We first figure out the composition of the portfolio. Let x M be the percentage that is invested in the market portfolio, we have σ p = x M σ M Since the portfolio has a standard deviation of 20% and σ M = 18 . 02%, we find that x M = 20 / 18 . 02 = 1 . 11. As the beta of the portfolio is a weighted average of the beta of the individual assets in the portfolio, we have β p = x M β M = 1 . 11 × 1 = 1 .
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final07a - UNIVERSITY OF TORONTO Joseph L Rotman School of...

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