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Chapter5 Solutions

# 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 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % r r Asset 2 Asset 1 μ P σ P ² ² ² ² ² ³ ´ ´ ´

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