Chapter5 Solutions - T HE E CONOMICS OF F INANCIAL M ARKETS...

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THE ECONOMICS OF FINANCIAL MARKETS R. E. BAILEY Solution Guide to Exercises for Chapter 5 Portfolio selection: the mean-variance model 1. An investor uses the mean-variance criterion for selecting a portfolio of two risky assets. As- set 1 has an expected return of 20% and a variance of 4. Asset 2 has an expected return of 60% and a variance of 36. There is no risk-free asset available. (a) Explain how to construct the efficient portfolio frontier for the cases in which the cor- relation coefficient between the returns, ρ 12 , is equal to +1 and also when it is equal to - 1 . Answer : The expected rate of return on the portfolio is given by: μ P = 1 5 a + 3 5 (1 - a ) , where a is the proportion of the portfolio invested in asset 1. The variance of the rate of return on the portfolio is: σ 2 P = 4 a 2 + 36(1 - a ) 2 + 2 a (1 - a ) × 2 × 6 × ρ 12 , where ρ 12 is the correlation coefficient between the rates of return on the two assets. Case: ρ 12 = +1 : σ 2 P = 4 a 2 + 36(1 - a ) 2 + 24 a (1 - a ) (1) = (2 a + 6(1 - a )) 2 (2) σ P = 2 a + 6(1 - a ) (3) Case: ρ 12 = - 1 : σ 2 P = 4 a 2 + 36(1 - a ) 2 - 24 a (1 - a ) (4) = (2 a - 6(1 - a )) 2 (5) σ P = ± (2 a - 6(1 - a )) (6) σ P = +(2 a - 6(1 - a )) for a 3 4 (7) σ P = - (2 a - 6(1 - a )) for a < 3 4 (8) Case: ρ 12 = +1 : σ P = 6 - 4 a Hence: a = 6 - σ P 4 a = - 2 + σ P 4 μ P = 1 5 ± 6 - σ P 4 ² + 3 5 ± - 2 + σ P 4 ² (9) = 6 - σ P - 6 + 3 σ P 20 = 2 σ P 20 = σ P 10 . (10) 1
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- 6 aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l, , , , , , , , , , , , , , , , , , , , , , , , , , , , σ P a ρ 12 = - 1 ± ± ± ±² ρ 12 = - 1 H H H Y ρ 12 = +1 ³ ³ ³ ³ ³ + 6 2 3 4 1 Figure 1: Standard deviation and a with ρ 12 = ± 1 . Case: ρ 12 = - 1 and a 3 / 4 : σ P = 2 a - 6(1 - a ) = - 6 + 8 a Hence: a = 6 + σ P 8 a = 2 - σ P 8 μ P = 1 5 ± 6 + σ P 8 ² + 3 5 ± 2 - σ P 8 ² (11) = 6 + σ P + 6 - 3 σ P 40 = 12 - 2 σ P 40 = 3 10 - σ P 20 . (12) Case: ρ 12 = - 1 and a < 3 / 4 : σ P = - 2 a + 6(1 - a ) = 6 - 8 a Hence: a = 6 - σ P 8 a = 2 + σ P 8 μ P = 1 5 ± 6 - σ P 8 ² + 3 5 ± 2 + σ P 8 ² (13) = 6 - σ P + 6 + 3 σ P 40 = 12 + 2 σ P 40 = 3 10 + σ P 20 . (14) (b) Describe, in general terms, how to construct the portfolio frontier when - 1 < ρ < +1 . Answer : When - 1 < ρ < +1 , the portfolio proportions a and 1 - a are chosen to minimize σ P (or σ 2 P ) for each level of μ P . For each level of μ P , the solution provides one point on the portfolio frontier. As μ P is chosen at different levels, so the frontier is traced out. The multiple asset case is similar, except that now there are n portfolio proportions to choose: a 1 , a 2 , . .., a n (such that the proportions sum to 1). For - 1 < ρ < +1 , the frontier is a hyperbola in the space of ( μ P , σ P ) : 2
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- 6 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % r r Asset 2 Asset 1 μ P σ P ² ² ² ² ² ³ ´ ´ ´
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This note was uploaded on 03/21/2011 for the course ECON 6120 taught by Professor Crabbe during the Spring '11 term at University of Ottawa.

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Chapter5 Solutions - T HE E CONOMICS OF F INANCIAL M ARKETS...

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