332final08a - UNIVERSITY OF TORONTO Joseph L Rotman School...

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UNIVERSITY OF TORONTO Joseph L. Rotman School of Management Dec. 21, 2008 Ezer/Kan/Florence RSM332 FINAL EXAMINATION Pomorski/Zhou SOLUTIONS 1. (a) In a year, bond A will have one year to maturity and bond B will have two years to maturity. If the economy is strong, bond A ’s YTM will be 11.1% and its price will be P strong A, 1 = $1000 / (1 + 0 . 111) = $900 . 09. If the economy is weak, bond B ’s YTM will be 6.3% and its price will be P weak B, 1 = $1000 / (1 + 0 . 063) 2 = $884 . 98. (b) The expected next year’s price of bond A is E [ P A, 1 ] = 0 . 4 × 1000 1 + 0 . 111 + 0 . 6 × 1000 1 + 0 . 037 = 938 . 628 . The expected holding period return on bond A is E [ R A ] = E [ P A, 1 ] - P A, 0 P A, 0 = 938 . 628 - 856 . 80 856 . 80 9 . 55% . (c) The current value of your portfolio is $856.80+$800.00=$1,656.80. The value of this portfolio next year will depend on the economic conditions: Portfolio value Portfolio return Strong economy 900.09+713.34=1613.43 1613 . 43 1656 . 80 - 1 ≈ - 2 . 618% Weak economy 964.32+884.98=1849.30 1849 . 30 1656 . 80 - 1 11 . 619% Thus, the expected return of the portfolio is μ p = 0 . 4 × - 0 . 02618 + 0 . 6 × 0 . 11619 0 . 05924 . The variance of portfolio returns is σ 2 p = 0 . 4 × ( - 0 . 02618 - 0 . 05924) 2 + 0 . 6 × (0 . 11619 - 0 . 05924) 2 = 0 . 004865 . 1
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The standard deviation is the square root of the variance, or about 6.97%. Note: you can also solve this question by computing portfolio weights of bonds A and B and then using the formulas for portfolio mean and standard deviation. Keep in mind that the portfolio is not equal-weighted: the weight of bond A is about 52% and the weight of bond B is about 48%. (d) Bond C will deliver exactly $1,000 next year. We know the payoff up front, so this bond is risk-free in a one-year portfolio: the standard deviation of returns on bond C is 0. Denote the weight of bond C in your new portfolio by w (the weight of your portfolio from part (c) is 1 - w ). We want w to satisfy: σ p = q w 2 × 0 + (1 - w ) 2 × 0 . 004865 = | 1 - w | × 0 . 0697 = 0 . 03227 . While this equation has two solutions ( w = 0 . 537 and w = 1 . 463), we need to choose the solution with w < 1. This is because we are already holding bonds A and B in our portfolio, so we have 1 - w > 0 which implies w < 1. Thus, your holdings of bond C should account for 53.7% of the overall value of your portfolio. (e) Denote the number of units of bond C by x . The value of your position in bond C is 915 x . The total value of your portfolio is then 1656 . 8 + 915 x and the weight of bond C is w = 915 x/ (1656 . 8 + 915 x ). We need to solve: w = 915 x 1656 . 8 + 915 x = 0 . 537 x = 2 . 1 . 2. (a) The covariance between market returns and returns on stock A is σ AM = ρ AM σ A σ M = - 0 . 3 × 0 . 5 × 0 . 2 = - 0 . 03 , and the market beta of stock A is β A = σ AM σ 2 M = - 0 . 03 (0 . 2) 2 = - 0 . 75 .
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