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Unformatted text preview: UNIVERSITY OF TORONTO Joseph L. Rotman School of Management Nov. 13, 2007 Bal/Chang/Lenouvel MGT337Y MIDTERM EXAMINATION #1 Pomorski/Rahaman SOLUTIONS 1. a. Since the bond trades at par, the yield to maturity is equal to the coupon rate, 5%. b. The prices of the three securities satisfy the following system of equations: 1000 1 + 0 . 02 = 1000 1 + r 1 1000 = 50 1 + r 1 + 1050 (1 + r 2 ) 2 . 08 = (1 + r 3 ) 3 (1 + r 2 ) 2 1 . The solution is r 1 = 2%, r 2 = 5 . 08%, and r 3 = 6 . 04%. c. f 2 = (1 + r 2 ) 2 1 + r 1 1 = 8 . 25% . d. Thus, P = 100 1 + r 1 + 100 (1 + r 2 ) 2 + 1100 (1 + r 3 ) 3 = 1111 . 14 . e. There are two instruments here: the zerocoupon bond and oneyear interest rate, 5%. Since the face value of the bond is $1,000, its price should be: P = $1000 1 + 0 . 05 = $952 . 38 . However, the price the bond trades at is different: $920. To make arbitrage profit, we need to create a portfolio that replicates the cash flows of the bond. To ensure we get exactly $1,000 next year, we need to invest (deposit) $952.38 today at the interest rate of 5%. The price of this portfolio (the amount of money we need to spend today) is $952.38. This means that either the bond is too cheap or the portfolio (deposit) is too expensive. Thus, we want to buy the bond and “sell” the deposit (that is, borrow at the interest rate of 5%). The cash flows of this strategy are as follows: 1 t 1 buy bond $920 +$1000 borrow $952.38 $952.38 $1000 net 32.38 2. a. Probabilities need to sum up to one, so the missing probability is 10.20.5=0.3. We should then use the formula for the expected value to compute the missing returns: E ( R A ) = 0 . 2 × . 2 + 0 . 5 × . 12 + 0 . 3 × x = 0 . 085 ⇒ x = . 05 E ( R B ) = 0 . 2 × y + 0 . 5 × . 07 + 0 . 3 × . 06 = 0 . 069 ⇒ y = 0 . 08 where x and y are missing returns on A and B, respectively. Probability R A R B r f Expansion 0.2 20% 8% 3% Normal 0.5 12% 7% 3% Recession 0.35% 6% 3% b....
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