T8 - UNIVERSITY OF TORONTO Joseph L Rotman School of Management RSM332 Tutorial#8 Problem Set Nov.10/Nov.11 2011 1 There are two stocks A and B The

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UNIVERSITY OF TORONTO Joseph L. Rotman School of Management RSM332 Tutorial #8 Problem Set Nov.10/Nov.11 2011 1. There are two stocks, A and B . The table below gives next year’s returns of the two stocks, R A and R B , depending on the state of the world, and the probability of each state of the world. You also know that E ( R A ) = 8 . 5% and E ( R B ) = 6 . 9%. Finally, you can borrow and lend at the risk-free interest rate r f = 3%. Probability R A R B r f Expansion 0.2 20% 3% Normal 0.5 12% 7% 3% Recession 6% 3% (a) Fill in the blanks in the table. (5 marks) (b) Compute the variances of R A and R B and the correlation between them. (5 marks) (c) What is the standard deviation of a portfolio that combines A and B (but not the risk-free asset) and has the expected return of 7.3%? (5 marks) (d) What is the correlation between R A and returns on an equally-weighted portfolio of A and the risk-free asset (i.e., portfolio weights are 0.5 for each of the two)? (5 marks) Solution: (a) Probabilities need to sum up to one, so the missing probability is 1-0.2-0.5=0.3. We should then use the formula for the expected value to compute the missing returns: E ( R A ) = 0 . 2 × 0 . 2 + 0 . 5 × 0 . 12 + 0 . 3 × x = 0 . 085 x = - 0 . 05 E ( R B ) = 0 . 2 × y + 0 . 5 × 0 . 07 + 0 . 3 × 0 . 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 0.3 -5% 6% 3% 1
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(b) σ 2 A = 0 . 2 × (0 . 2 - 0 . 085) 2 + 0 . 5 × (0 . 12 - 0 . 085) 2 + 0 . 3 × ( - 0 . 05 - 0 . 085) 2 = 0 . 008725 σ 2 B = 0 . 2 × (0 . 08 - 0 . 069) 2 + 0 . 5 × (0 . 07 - 0 . 069) 2 + 0 . 3 × (0 . 06 - 0 . 069) 2 = 0 . 000049 Cov[ R A ,R B ] = 0 . 2 × (0 . 2 - 0 . 085)(0 . 08 - 0 . 069) + 0 . 5 × (0 . 12 - 0 . 085)(0 . 07 - 0 . 069) + + 0 . 3 × ( - 0 . 05 - 0 . 085)(0 . 06 - 0 . 069) = 0 . 000635 ρ A,B = Cov[ R A ,R B ] σ A σ b = 0 . 97 . (c) We first need to calculate the weights of the portfolio that has expected return of 7.3%: 0 . 073 = 0 . 085 w A + 0 . 069 w B 1 = w A + w B , which yields w A = 0 . 25, w B = 0 . 75. Now we can use the portfolio variance formula: σ 2 P = w 2 A σ 2 A + w B σ 2 B + 2 w A w B cov ( R A ,R B ) = 0 . 000811 , σ P = 0 . 0285 . (d) Since the variance of the risk-free rate is zero and the covariance between the risk- free rate and anything else is zero, the variance of the equally-weighted portfolio and the covariance of the portfolio with R A are σ 2 P = 0 . 5 2 σ 2 A + 0 . 5 2 × 0 + 2 × 0 . 5 × 0 . 5 × 0 = 0 . 5 2 σ 2 A
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This note was uploaded on 12/20/2011 for the course RSM 332 taught by Professor Raymondkan during the Fall '08 term at University of Toronto- Toronto.

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T8 - UNIVERSITY OF TORONTO Joseph L Rotman School of Management RSM332 Tutorial#8 Problem Set Nov.10/Nov.11 2011 1 There are two stocks A and B The

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