3330%20PS4%20solution

3330%20PS4%20solution - Cornell University Fall 2009...

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Cornell University Fall 2009 Economics 3330: Problem Set 4 Solutions 1. Questions 10 and 11 of Chapter 6 of Text a. Calculate the expected return and variance of portfolios invested in T-bills and the S&P 500 index with weights in the following table. Assume that the S&P return has been 8.5% more than the T-bill return of 5%. Assume that the standard deviation has been 20% for the S&P and zero for the T-bills. W bills W index 0 1 0.2 0.8 0.4 0.6 0.6 0.4 0.8 0.2 1 0 The portfolio expected return and variance are computed as follows: (1) W Bills (2) r Bills (3) W Index (4) r Index r Portfolio (1) × (2)+(3) × (4) σ Portfolio (3) × 20% σ 2 Portfolio 0.0 5% 1.0 13.5% 13.5% = 0.135 20% = 0.20 0.0400 0.2 5% 0.8 13.5% 11.8% = 0.118 16% = 0.16 0.0256 0.4 5% 0.6 13.5% 10.1% = 0.101 12% = 0.12 0.0144 0.6 5% 0.4 13.5% 8.4% = 0.084 8% = 0.08 0.0064 0.8 5% 0.2 13.5% 6.7% = 0.067 4% = 0.04 0.0016 1.0 5% 0.0 13.5% 5.0% = 0.050 0% = 0.00 0.0000 b. Calculate the utility levels of each portfolio for an investor with A=3. What is the optimal portfolio? Computing utility from U = E(r) – 0.5 × A σ 2 = E(r) – 1.5 σ 2 , we arrive at the values in the column labeled U(A = 3) in the following table: W Bills W Index r Portfolio σ Portfolio σ 2 Portfolio U(A = 3) U(A = 5) 0.0 1.0 0.135 0.20 0.0400 0.0750 0.0350 0.2 0.8 0.118 0.16 0.0256 0.0796 0.0540 0.4 0.6 0.101 0.12 0.0144 0.0794 0.0650 0.6 0.4 0.084 0.08 0.0064 0.0744 0.0680 0.8 0.2 0.067 0.04 0.0016 0.0646 0.0630 1.0 0.0 0.050 0.00 0.0000 0.0500 0.0500 The column labeled U(A = 3) implies that investors with A = 3 prefer a portfolio that is invested 80% in the market index and 20% in T-bills to any of the other portfolios in the table.
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c. Calculate the utility levels of each portfolio for an investor with A=5. What is the optimal portfolio? The column labeled U(A = 5) in the table above is computed from: U = E(r) – 0.5A σ 2 = E(r) – 2.5 σ 2 The more risk averse investors prefer the portfolio that is invested 40% in the market index, rather than the 80% market weight preferred by investors with A = 3. 2.
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This note was uploaded on 12/22/2009 for the course ECON 3330 taught by Professor Mbiekop during the Fall '08 term at Cornell.

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3330%20PS4%20solution - Cornell University Fall 2009...

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