{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

hw5soln

# hw5soln - ORIE 3500/5500 Fall 10 HW 5 Solutions Assignment...

This preview shows pages 1–3. Sign up to view the full content.

ORIE 3500/5500, Fall ’10 HW 5 Solutions Assignment 5 Solutions Problem 1 Suppose the investor buys m shares of company A and K - m shares of company B. So, the value of the resulting portfolio is mX A + ( K - m ) X B . Now, V ar ( mX A + ( K - m ) X B ) = m 2 σ 2 A + ( K - m ) 2 σ 2 B + 2 m ( K - m )Cov( X A , X B ) . Differentiating with respect to m and setting that equal to 0 , we get δV ar ( mX A + ( K - m ) X B ) δm = 0 2 m ( σ 2 A - Cov( X A , X B )) + 2( K - m )( - σ 2 B + Cov( X A , X B )) = 0 m = K ( σ 2 B - Cov( X A , X B )) σ 2 A + σ 2 B - 2Cov( X A , X B ) . Differentiating twice with respect to m, we get δ 2 V ar ( mX A + ( K - m ) X B ) δm 2 = 2( σ 2 A + σ 2 B - 2Cov( X A , X B )) = 2 V ar ( X A - X B ) 0 . Hence, the variance is minimized for m = K ( σ 2 B - Cov( X A , X B )) σ 2 A + σ 2 B - 2Cov( X A , X B ) . Problem 2 Since the possible values of X are 1 , 2 , and 4, p 1 + p 2 + p 4 = 1. We also have Var( X ) = E ( X - EX ) 2 = (1 - 2) 2 p 1 + (2 - 2) 2 p 2 + (4 - 2) 2 p 4 = p 1 + 4 p 4 = 2 - 2 p 2 . (a) To maximize Var( X ), we should choose p 2 = 0. Solving p 1 + p 4 = 1 , EX = p 1 + 4 p 4 = 2 for p 1 and p 4 , we get p 1 = 2 / 3 and p 4 = 1 / 3. (b) Var( X ) is minimized when p 2 = 1. Then p 1 + 1 + p 4 = 1 , EX = p 1 + 2 + 4 p 4 = 2 , yields p 1 = p 4 = 0. Problem 3 (a) We have Cov( X, Y ) = E ( XY ) - E ( X ) E ( Y ). Note that E ( XY ) = Z -∞ Z -∞ xyf X,Y ( x, y ) dy dx = Z 1 0 Z 1 - x x - 1 xy · 6 x 2 dy dx 1

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

View Full Document
ORIE 3500/5500, Fall ’10 HW 5 Solutions = Z 1 0 6 x 3 Z 1 - x x - 1 y dy dx = Z 1 0 6
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 6

hw5soln - ORIE 3500/5500 Fall 10 HW 5 Solutions Assignment...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online