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FNCE100_PS3

# FNCE100_PS3 - University of Pennsylvania The Wharton School...

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University of Pennsylvania The Wharton School FNCE 100 A. Craig MacKinlay PROBLEM SET #3 Fall Term 2005 Diversification, Risk and Return 1. We have three securities with the following possible payoffs. Probability Return on Return on Return on State of Outcome Security #1 Security #2 Security #3 1 . 10 . 25 . 25 . 10 2 . 40 . 20 . 15 . 15 3 . 40 . 15 . 20 . 20 4 . 10 . 10 . 10 . 25 (a) What is the expected return and standard deviation on each security? (b) What is Cov( R 1 , R 2 ), Cov( R 1 , R 3 ), and Cov( R 2 , R 3 )? What is Corr( R 1 , R 2 ), Corr( R 1 , R 3 ), and Corr( R 2 , R 3 )? (Cov( · , · ) denotes covariance and Corr( · , · ) denotes correlation). (c) What is the expected return, E ( R p ), and standard deviation, σ ( R p ), of a portfolio which has half of its funds invested in Security #1 and half in Security #2? (d) What is E ( R p ) and σ ( R p ) of a portfolio which has half of its funds invested in Security #1 and half in Security #3? (e) What is E ( R p ) and σ ( R p ) of a portfolio which has half of its funds in Security #2 and half in #3? (f) Compare your answers in Parts (a), (c), (d), and (e). Comment on the effects of diversification. 2. Assume there are n securities, each having: E ( R i ) = . 01 i = 1 , 2 , . . . , n σ 2 ( R i ) = . 01 i = 1 , 2 , . . . , n cov( R i , R j ) = . 005 i = 1 , 2 , . . . , n ; j = 1 , 2 , . . . , n ; j 6 = i 31

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(a) What is the expected return and variance of an equally weighted portfolio con- taining all n securities? (i.e., x i = 1 /n, i = 1 , . . . , n ). (b) What value will the variance approach as n gets larger? (c) What characteristics of securities are most important in the determination of the variance of a “well diversified” portfolio? 3. A risky security cannot have an expected return that is less than the riskfree rate ( R F ) because no risk-averse investor would be willing to hold this asset, in equilibrium. (True, False & explain). 4. “The risk of a portfolio is the variance of its return; however, the variance of the return on an individual asset is not an appropriate measure of its risk.” Discuss. 5. A neighbor purchased a lottery ticket yesterday but now, owing to an unpredicted crisis, is in desperate need of cash. He offers to sell the ticket to you. You know the payoff and the probability of winning. All of your considerable fortune is invested in a highly diversified portfolio. How would you determine an appropriate price for the ticket? 6. Some relevant data pertaining to three Dow-Jones stocks over the January 1971– December 1975 period are: σ ( R i ) β i A: Aluminum Company of America . 093 . 662 B: Eastman Kodak . 070 . 979 C: Union Carbide Corp. . 085 1 . 231 The pairwise correlations between the returns of these three securities are: 32
ρ AB = . 137 ρ AC = . 476 ρ BC = . 422 . Using the capital asset pricing model and assuming that E ( R m ) = . 010 per month and R F = . 002 per month, calculate the expected return on each stock. Why is E ( R B ) > E ( R A ) when σ ( R A ) > σ ( R B )? 7. Given two random variables z and y , Probability of state State of Variable z Variable y of nature nature . 2 I 18 0 . 2 II 5 - 3 . 2 III 12 15 . 2 IV 4 12 . 2 V 6 1 (a) Calculate the mean and variance for each of these variables, and the covariance and correlation between them.

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FNCE100_PS3 - University of Pennsylvania The Wharton School...

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