{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info icon This preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
- 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
Image of page 2
- 6 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % r r Asset 2 Asset 1 μ P σ P ² ² ² ² ² ³ ´ ´ ´
Image of page 3

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern