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#4 Solutions - Portfolio Theory

#4 Solutions - Portfolio Theory - UNIVERSITY OF NORTH...

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U NIVERSITY OF N ORTH C AROLINA A T C HAPEL H ILL K ENAN -F LAGLER B USINESS S CHOOL B USI 408: C ORPORATE F INANCE S OLUTIONS TO P RACTICE P ROBLEM SET #4: P ORTFOLIO T HEORY P ROF . A RZU O ZOGUZ 1. Consider the following distribution of returns, with ρ AM = 0.38, ρ BM = 0.92: Probability R A R B R M 30% -20% -5% 40% 5% 10% 30% 40% 15% E(R) ? ? 12% σ ? ? 15% a. Fill in the question marks. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) % 12 . 8 08124 . 0 0066 . 0 07 . 0 15 . 0 30 . 0 07 . 0 10 . 0 40 . 0 07 . 0 05 . 0 30 . 0 % 7 07 . 0 15 . 0 30 . 0 10 . 0 40 . 0 05 . 0 30 . 0 % 37 . 23 6 3 23 . 0 0546 . 0 08 . 0 40 . 0 30 . 0 08 . 0 05 . 0 40 . 0 08 . 0 20 . 0 30 . 0 % 8 08 . 0 40 . 0 30 . 0 05 . 0 40 . 0 20 . 0 30 . 0 2 2 2 2 2 2 2 2 = = = + + = = = + + = = = = + + = = = + + = B B B A A A R E R E σ σ σ σ b. Find the covariance and the correlation between the returns of A and B. ( ) ( ) ( ) ( ) ( ) [ ] ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( ) 916 . 0 0812 . 0 2337 . 0 0174 . 0 , cov , 0174 . 0 07 . 0 15 . 0 08 . 0 4 . 0 30 . 0 07 . 0 10 . 0 08 . 0 05 . 0 40 . 0 07 . 0 05 . 0 08 . 0 20 . 0 30 . 0 , cov = = = = + + = = B A B A B A B B A A B A R R R R corr R E R R E R E R R σ σ c. What is the expected return and standard deviation of a portfolio with 40% in A, 40% in B, and 20% in M?
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( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )( ) ( )( )( )( )( ) ( )( ) ( ) ( ) ( )( )( )( )( ) ( )( )( )( )( ) ( )( )( )( )( ) ( ) ( ) 142 . 0 02018 . 0 15 . 0 20 . 0 15 . 0 0812 . 0 92 . 0 20 . 0 40 . 0 15 . 0 2336 . 0 38 . 0 20 . 0 40 . 0 15 . 0 0812 . 0 92 . 0 20 . 0 40 . 0 0066 . 0 40 . 0 0174 . 0 40 . 0 40 . 0 15 . 0 2336 . 0 38 . 0 20 . 0 40 . 0 0174 . 0 40 . 0 40 . 0 0546 . 0 40 . 0 , cov , cov , cov , cov , cov , cov , cov , cov , cov , cov % 4 . 8 084 . 0 12 . 0 20 . 0 07 . 0 40 . 0 08 . 0 40 . 0 20 . 0 40 . 0 40 . 0 2 2 2 2 3 1 3 1 2 = = + + + + + + + + = + + + + + + + + = = = = × + × + × = × + × + × = ∑∑ = = P M M M M B M B M A M A M M B M B B B B B A B A B M A M A B A B A A A A A i j j i j i P M B A P R R R R R R R R R R R R R R R R R R R R R E R E R E R E σ ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω σ 2. Briefly explain why the covariance of a security with the rest of a well-diversified portfolio is a more appropriate measure of the risk of the security than the security’s variance. The covariance is a more appropriate measure of a security’s risk in a well- diversified portfolio because the covariance reflects the effect of the security on the variance of the portfolio. Investors are concerned with the variance of their portfolios and not the variance of the individual securities. Since covariance measures the impact of an individual security on the variance of the portfolio, covariance is the appropriate measure of risk. 3. A portfolio consists of 120 shares of Atlas stock, which sells for $50 per share, and 150 shares of Babcock stock, which sells for $20 per share. What are the weights of the two stocks in this portfolio?
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