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3330%20PS5%20solution

3330%20PS5%20solution - Cornell University Fall 2009...

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Cornell University Fall 2009 Economics 3330: Problem Set 5 Solutions 1. Question 5 of Chapter 8 of Text A portfolio management organization analyzes 60 stocks and constructs a mean-variance efficient portfolio using only those 60 securities. a. How many estimates of expected return, variances, and covariances are needed to optimize this portfolio? To optimize this portfolio one would need: n = 60 estimates of means n = 60 estimates of variances 770 , 1 2 n n 2 = estimates of covariances Therefore, in total: 890 , 1 2 n 3 n 2 = + estimates b. If one could safely assume that stock market returns closely resemble a single-index structure, how many estimates would be needed? In a single index model: r i r f = α i + β i (r M r f ) + e i Equivalently, using excess returns: R i = α i + β i R M + e i The variance of the rate of return on each stock can be decomposed into the components: (l) The variance due to the common market factor: 2 M 2 i σ β (2) The variance due to firm specific unanticipated events: ) e ( i 2 σ In this model: σ β β = j i j i ) r , r ( Cov The number of parameter estimates is: n = 60 estimates of the mean E(r i ) n = 60 estimates of the sensitivity coefficient β i n = 60 estimates of the firm-specific variance σ 2 (e i ) 1 estimate of the market mean E(r M ) 1 estimate of the market variance 2 M σ Therefore, in total, 182 estimates.

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Thus, the single index model reduces the total number of required parameter estimates from 1,890 to 182. In general, the number of parameter estimates is reduced from: ) 2 n 3 ( to 2 n 3 n 2 + + 2. Question 6 of Chapter 8 of Text The following are estimates for two stocks. Stock Expected Return Beta Firm-Specific Std. Dev. A 13% .8 30% B 18 1.2 40 The market index has a standard deviation of 22% and the risk-free rate is 8%. a. What are the standard deviations of stocks A and B? The standard deviation of each individual stock is given by: 2 / 1 i 2 2 M 2 i i )] e ( [ σ + σ β = σ Since β A = 0.8, β B = 1.2, σ (e A ) = 30%, σ (e B ) = 40%, and σ M = 22%, we get: σ A = (0.8 2 × 22 2 + 30 2 ) 1/2 = 34.78% σ B = (1.2 2 × 22 2 + 40 2 ) 1/2 = 47.93% b. Suppose that we were to construct a portfolio with the following proportions: Stock A 0.30, Stock B 0.45, T-bills 0.25. Compute the expected return, standard deviation, beta, and nonsystematic standard deviation of the portfolio.
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3330%20PS5%20solution - Cornell University Fall 2009...

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