final08a - MGT 337 Final Examination 2008 Solutions 1. Term...

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MGT 337 Final Examination 2008 … Solutions 1. Term Structure / Bond Portfolios / Arbitrage ( 20 marks ) a) Spot rates: 95 = 100/(1+r 1 ) r 1 = 5.26% 160 = 200/(1+r 2 ) 2 r 2 = 11.80% Forward rates: f 1 = r 1 = 5.26% f 2 = (1+r 2 ) 2 /(1+r 1 )-1 = 0.187 = 18.7%. b) In a year, bond B will be a zero-coupon bond with the face value of $200 and one year to maturity. We are given its yield to maturity, so we can compute its price. It will be $160.00 with the probability of 0.3, $180.18 with the probability of 0.5, and $198.02 with the probability of 0.2. The bond’s expected return will be equal to the difference between next year’s price and today’s price, divided by today’s price. Thus, it will be 0%, 12.61%, and 23.76% with the probability of 0.3, 0.5, and 0.2, respectively. Altogether, the expected return on the bond is 11.06%. c) Bond A costs $95 and in a year will be worth $100 for sure. Thus, the return on bond A is 5.26% (naturally, this is the first spot interest rate from a). The expected return on bond B is 11.06%. Consider a portfolio that assigns the weight of w A to bond A and the weight w B = 1 - w A to bond B. Portfolio expected return is 0.0526w A + 0.1106(1-w A ) = 0.1, which implies w A = 18.28% and w B = 81.72%. d) Return on A is risk-free, so the variance of portfolio return will only depend on the weight of bond B and the variance of the return on B: Var(portfolio) = w B 2 Var(R B ) We know the probability distribution of the return on B so we can compute its variance: Var(R B ) = 0.3(0-0.1106) 2 + 0.5(0.1261 - 0.1106) 2 + 0.2(0.2376-0.1106) 2 = 0.00702.
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Thus, the variance of the portfolio return is 0.004686 and its standard deviation is 0.068452 (6.84%). e) The rate the bank offers is lower than the rate implied by bond prices. Thus, there is an arbitrage opportunity. To take advantage of this opportunity we sell high and buy low. 15% is too low, so we want to borrow money at that forward rate. If we borrow 1000, we will get that amount in a year and we will need to repay 1150 in two years. The second leg of our arbitrage strategy should offset these cash flows. Suppose we buy x A units of bond A and x B units of bond B. This will offset the cash-flow from the forward transaction when: Year 1: -1000 = 100x A + 0x B Year 2: 1150 = 0x A + 200x B These equations yield x A = -10, x B = 5.75. So our strategy should be to short sell 10 units of A and buy 5.75 units of B. If we do that, our cash-flow today is +10 × $95 - 5.75 × $160 = $30. Our cash-flows in one year and two years are zero, as the cash-flows from our bond portfolio and forward contract offset one another exactly. Thus, we have a free lunch of $30 today.
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2. Capital Asset Pricing Model ( 20 marks ) a) P has zero beta (and zero alpha), so its expected return should be equal to the risk-free rate. Thus, the risk-free rate should be 4%. Q has the beta of 1 (and zero alpha), so its expected return (estimated at 13%) should be equal to the expected market return. Thus,
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This note was uploaded on 04/26/2011 for the course RSM 333 taught by Professor Sabrinabutti during the Spring '11 term at University of Toronto.

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final08a - MGT 337 Final Examination 2008 Solutions 1. Term...

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