PS04-solution

.pdf

School

University of California, Los Angeles *

*We aren’t endorsed by this school

Course

433

Subject

Finance

Date

Jan 9, 2024

Type

pdf

Pages

4

Uploaded by oooni

Report
Problem Set 4: Solutions UCLA MQE Core Finance Prof. Pierre-Olivier Weill 1. Consider two risky assets with returns r 1 and r 2 . Assume that the expected returns, standard deviations are ¯ r 1 = 0 . 03, ¯ r 2 = 0 . 08, σ 1 = 0 . 05, σ 2 = 0 . 1. Write a Python program to plot, on the same diagram, the investment opportunity set when the correlation is ρ 12 = 1, ρ 12 = 0 . 1 and ρ 12 = +1. Assume for this calculation that portfolio weights are positive. Answer: Please see the Jupyter notebook on the website. 2. Consider two risky assets with returns r 1 and r 2 . Assume that the expected returns satisfy ¯ r 1 < ¯ r 2 , the standard deviation σ 1 < σ 2 and the correlation is ρ ̸ = 0. (a) Suppose that we combine the two assets in a portfolio with weights w 1 and w 2 with w 1 + w 2 = 1. Find a formula for the portfolio weights, w 1 and w 2 , of the minimum variance portolio. Answer: The formula for the variance of this portfolio is Var( w ) = w 2 1 Var( r 1 ) + w 2 2 Var( r 2 ) + 2 w 1 w 2 Cov( r 1 , r 2 ) . We know that w 1 + w 2 = 1, and thus w 2 = 1 w 1 . Then, we can write the variance of the portfolio as Var( w ) = w 2 1 Var( r 1 ) + (1 w 1 ) 2 Var( r 2 ) + 2 w 1 (1 w 1 )Cov( r 1 , r 2 ) . Now we can take the derivative of the variance with respect to w 1 to find the minimum of this function: 0 = 2 w 1 Var( r 1 ) 2(1 w 1 )Var( r 2 ) + 2(1 2 w 1 )Cov( r 1 , r 2 ) w 1 [2Cov( r 1 , r 2 ) Var( r 1 ) Var( r 2 )] = Cov( r 1 , r 2 ) Var( r 2 ) . Taken together we obtain: w 1 = Cov( r 1 ,r 2 ) Var( r 2 ) 2Cov( r 1 ,r 2 ) Var( r 1 ) Var( r 2 ) w 2 = Cov( r 1 ,r 2 ) Var( r 1 ) 2Cov( r 1 ,r 2 ) Var( r 1 ) Var( r 2 ) .
(b) Derive a condition on ρ for the investment opportunity set to be backward bending. Explain why the investment opportunity set is not always backward bending. Answer: We have already found the minimum. We would like to show that the variance of the portfolio is decreasing in w 2 (the weight of the asset with the large variance) when w 2 = 0 and increasing in w 2 when w 2 = 1. First, we take the partial derivative of the variance of the portfolio Var( w ) with respect to w 2 and evaluate it when w 2 = 0: Var ( w ) ∂w 2 w 2 =0 = [(1 w 2 ) 2 Var( r 1 ) + w 2 2 Var( r 2 ) + 2(1 w 2 ) w 2 ρσ ( r 1 ) σ ( r 2 )] ∂w 2 w 2 =0 = [ 2(1 w 2 )Var( r 1 ) + 2 w 2 Var( r 2 ) + 2 (1 2 w 2 ) ρσ ( r 1 ) σ ( r 2 )] w 2 =0 = 2Var( r 1 ) + 2 ρσ ( r 1 ) σ ( r 2 ) We need that Var( w ) ∂w 2 w 2 =0 < 0, then: 2Var( r 1 ) + 2 ρσ ( r 1 ) σ ( r 2 ) < 0 ρ < Var( r 1 ) σ ( r 1 ) σ ( r 2 ) = σ ( r 1 ) σ ( r 2 ) Second, we take the partial derivative of the variance of the portfolio Var( w ) with respect to w 2 and evaluate it when w 2 = 1: Var ( w ) ∂w 2 w 2 =1 = [(1 w 2 ) 2 Var( r 1 ) + w 2 2 Var( r 2 ) + 2(1 w 2 ) w 2 ρσ ( r 1 ) σ ( r 2 )] ∂w 2 w 2 =1 = [ 2(1 w 2 )Var( r 1 ) + 2 w 2 Var( r 2 ) + 2 (1 2 w 2 ) ρσ ( r 1 ) σ ( r 2 )] w 2 =1 =2Var( r 2 ) 2 ρσ ( r 1 ) σ ( r 2 ) We need that Var( w ) ∂w 2 w 2 =1 > 0, then: 2Var( r 2 ) 2 ρσ ( r 1 ) σ ( r 2 ) > 0 ρ < Var( r 2 ) σ ( r 1 ) σ ( r 2 ) = σ ( r 2 ) σ ( r 1 ) So, we should have that ρ < σ ( r 1 ) σ ( r 2 ) < σ ( r 2 ) σ ( r 1 ) , given σ ( r 2 ) > σ ( r 1 ).
Your preview ends here
Eager to read complete document? Join bartleby learn and gain access to the full version
  • Access to all documents
  • Unlimited textbook solutions
  • 24/7 expert homework help