The corresponding indifference curve is downward

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The corresponding indifference curve is downward sloping in the graph above (see Problem 3), and is labeled Q6. 7. Utility for each investment = E(r) – 0.5 × 4 × σ 2 We choose the investment with the highest utility value. Investment Expected return E(r) Standard deviation σ Utility U 1 0.12 0.30 -0.0600 2 0.15 0.50 -0.3500 3 0.21 0.16 0.1588 4 0.24 0.21 0.1518 8. When investors are risk neutral, then A = 0; the investment with the highest utility is Investment 4 because it has the highest expected return. 9. b
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6-4 10. 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 11. 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. 12. 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. 13. Expected return = (0.7 × 18%) + (0.3 × 8%) = 15% Standard deviation = 0.7 × 28% = 19.6% 14. Investment proportions: 30.0% in T-bills 0.7 × 25% = 17.5% in Stock A 0.7 × 32% = 22.4% in Stock B 0.7 × 43% = 30.1% in Stock C
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15. Your reward-to-variability ratio: 3571 . 0 28 8 18 S = = Client's reward-to-variability ratio: 3571 . 0 6 . 19 8 15 S = = 16. Client P 0 5 10 15 20 25 30 0 10 20 30 40 σ (%) E(r) % CAL (Slope = 0.3571) 17. a. E(r C ) = r f + y[E(r P ) – r f ] = 8 + y(18 8) If the expected return for the portfolio is 16%, then: 16 = 8 + 10 y 8 . 0 10 8 16 y = = Therefore, in order to have a portfolio with expected rate of return equal to 16%, the client must invest 80% of total funds in the risky portfolio and 20% in T-bills. b. Client’s investment proportions: 20.0% in T-bills 0.8 × 25% = 20.0% in Stock A 0.8 × 32% = 25.6% in Stock B 0.8 × 43% = 34.4% in Stock C c. σ C = 0.8 × σ P = 0.8 × 28% = 22.4% 6-5
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