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Unformatted text preview: UNIVERSITY OF TORONTO Joseph L. Rotman School of Management Oct. 25, 2011 Kan/Simutin RSM332 MIDTERM EXAMINATION Yang SOLUTIONS 1. Let one period be six months and D be the amount of semiannual deposits. Kathy will deposit D from t = 1 when Larry is five an half years old, till t = 26 when Larry is 18 years old. This forms a 26periods annuity with an effective semiannual rate of r s = 0 . 091 / 2 = 0 . 0455. The future value of this annuity on date t = 26 is (1 + r s ) 26 DA 26 r s = (1 + 0 . 0455) 26 D . 0455 " 1 1 (1 + 0 . 0455) 26 # = 47 . 912 D. The education cost forms a 4year annuity with payment of $24,000 per year. But now the discount rate is the effective annual interest rate: r a = 1 + . 091 2 2 1 = 0 . 09307 . The value of this annuity at t = 26 is 24000 A 4 r a = 24000 r a " 1 1 (1 + r a ) 4 # = 77232 . 17 . Thus, 47 . 912 D = 77232 . 17 D = 1611 . 97 . (b) Let P be the amount of annual installment. At the beginning of the 11th year, we still owe 10 installments. Therefore, the principal outstanding is P A 10 r and the interest proportion of the 11th installment is: rPA 10 r = P " 1 1 (1 + r ) 10 # = 150 . Similarly, at the beginning of 16th year, we still owe 5 installments, and hence the interest portion is rPA 5 r = P " 1 1 (1 + r ) 5 # = 125 . 1 Let x = 1 / (1 + r ) 5 . Then, we have the following equations: P (1 x 2 ) = 150 , (1) P (1 x ) = 125 . (2) Dividing the first equation by the second equation, we obtain 1 + x = 150 125 x = 0 . 2 1 1 + r = (0 . 2) 1 5 . It follows that P = 125 / (1 x ) = 125 / (1 . 2) = 156 . 25. At the beginning of 19th year, we will owe 2 installments, and the interest portion of the 19th installment is rPA 2 r = P " 1 1 (1 + r ) 2 # = 156 . 25 h 1 (0 . 2) 2 5 i = 74 . 17 . 2. (a) Since the yieldtomaturity of a pure discount bond is equal to the spot rate, we have r 1 = 0 . 05 from Bond 1. For bond 2, we have 1000 = 80 1 + r 1 + 1080 (1 + r 2 ) 2 = 80 1 . 05 + 1080 (1 + r 2 ) 2 ....
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 Fall '08
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