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Unformatted text preview: lOMoARcPSD|1381793 Problem Set 4 Solutions Portfolio Management (University of New South Wales) Distributing prohibited | Downloaded by adam negile ([email protected]) lOMoARcPSD|1381793 FINS2624 PROBLEM SET 4 SOLUTIONS Question 1. A reasonable utility function should satisfy the following two key properties of actual investor preferences: 1. The higher the wealth, the better: I.e. U(W2) > U(W1) if W2 > W1 2. Risk aversion: the more certain the wealth, the better. I.e. if the wealth could take the following two possible outcomes: , 50% , 50% We know that the expected wealth 2 , But the utility of certain outcome with the same level of wealth than the expected utility from two possible outcomes E[U( . is higher In the given example above: 2 2 We will use these two relationships to verify various utility functions. , Criterion 1: Criterion 2: More formally, criterion 1 means: 0 and criterion 2 means: 0 Distributing prohibited | Downloaded by adam negile ([email protected]) lOMoARcPSD|1381793 5 4 Utility Function ‐ U(W) 3 2 F(W)=W^2 1 F(W)=1/W 0 ‐1 0 1 2 3 4 5 6 F(W)=‐W ln(W) ‐2 W W^(1‐g)/(1‐g) ‐3 ‐4 ‐5 W a) U(W) = W2 2 2 0→ 0→ 1 2 b) U(W) = 1 2 0→ 1 0→ 2 c) U(W) = -W 1 0→ 0→ 1 2 d) U(W) = W 1 0→ 0→ 1 2 Distributing prohibited | Downloaded by adam negile ([email protected]) lOMoARcPSD|1381793 e) U(W) = ln(W) 1 0→ 1 f) , 1 0→ 2 2 0→ 0→ 1 2 Distributing prohibited | Downloaded by adam negile ([email protected]) lOMoARcPSD|1381793 Question 2. a) Investors will pick up the portfolio yielding highest utility per utility function: 1 2 , 0 So you can put numbers in and find out which portfolio will give highest U. But without any calculation, we should be able to tell it should be portfolio A (highest E(r), but lowest ) b) A = 1 1 2 0.2 1 1 0.2 2 0.18 1 2 0.12 1 1 0.22 2 0.0958 1 2 0.15 1 1 0.28 2 0.1108 c) Set 0.12 0.15 1 2 0.22 0.12 0.15 1 2 0.28 0.03 2 0.28 0.22 1 2 0.28 0.22 2 Distributing prohibited | Downloaded by adam negile ([email protected]) lOMoARcPSD|1381793 Question 3. 2 , a) w1 = w2 = w3 = 1 0.12 3 , 0.07 0.17 0.12 1 3 1 3 1 1 , 3 3 1 3 1 3 You can use the formulae above to do the calculation. But, it is easier to use the covariance matrix. 1 9 1 1 , 3 3 1 9 1 9 1 3 , 1 9 1 3 1 1 , 3 3 1 9 , 1 9 1 9 1 1 , 3 3 1 9 , 1 9 1 1 , 3 3 1 9 , 1 9 1 1 , 3 3 1 1 , 3 3 , 1 9 1 1 , 3 3 , 1 9 1 1 , 3 3 1 9 , 1 9 19.28% b) - Portfolio with asset 1 only has E(rP) = 12% and Equally weighted portfolio has E(rP) = 12% and 25% 19.28% Diversification strives to smooth out unsystematic risk events in a portfolio so that the positive performance of some investments will neutralize the negative performance of Distributing prohibited | Downloaded by adam negile ([email protected]) lOMoARcPSD|1381793 others. Therefore, the benefits of diversification will hold only if the securities in the portfolio are not perfectly correlated. The diversification benefit is the reduction in the volatility of portfolio returns of 5.72% c) 3 The investor will choose the one yielding highest utility per 0.17 If choosing asset 2: 3 0.3 0.035 0.12 If choosing equally weighted portfolio: 3 0.1928 0.0642 An investor with a risk aversion parameter of 3 will prefer the equally weighted portfolio. d) Please replicate the calculation in c) with A = 1 An investor with a risk aversion parameter of 1 will prefer the portfolio consisting of only asset 2. e) 1 3 1 2 1 3 , 1 6 1 1 , 2 3 1 6 , 1 2 , 1 3 , 1 6 1 3 1 6 1 6 1 1 , 3 2 1 1 , 2 3 , 1 6 1 1 , 2 3 1 2 1 6 1 1 , 2 3 , 1 6 1 1 , 2 3 , 1 6 = sum of all items in the covariance matrix above. Distributing prohibited | Downloaded by adam negile ([email protected]) 1 6 lOMoARcPSD|1381793 Question 4. 1 2 To draw utility indifference curves, you: 1. Set a target utility level (like U = 10%) 2. Then select the value of (or E[r]). Solve the equation above to get E[r] (or ) 3. Plot all these combinations of E[r] and on the E[r]- space, and connect them Utility Indifference Curves 80% 70% 60% E[r] 50% 4a, A = 3, U = 10% 40% 4b, A = 3, U = 15% 30% 4c, A = 5, U = 10% 20% 10% 0% 0% 10% 20% 30% 40% 50% 60% You can see that the indifference curve in 4c should be steeper than those in 4a and 4b, because of higher A. I.e. more risk averse investors will require a higher increase in E[r] for the increase in . Distributing prohibited | Downloaded by adam negile ([email protected]) lOMoARcPSD|1381793 Question 5. 1 2 1 3 a) 1 2 1 3 1 2 15.42% 1 2 1 3 21.39% b) 1 2 , , 1 2 1 3 1 3 = 355.83 , , 1 1 2 3 1 1 2 3 1 2 , , 1 3 1 2 1 3 1 3 1 3 , 1 2 1 3 , 1 2 , Based on the given covariance matrix, it is simple to do the calculation. If you are not given the covariance between two assets directly but instead the correlation between two assets, then you should be able to calculate: , Distributing prohibited | Downloaded by adam negile ([email protected]) lOMoARcPSD|1381793 BKM Chapter 6 selected end-of-chapter questions 4. a. The expected cash flow is: (0.5 × $70,000) + (0.5 × 200,000) = $135,000. With a risk premium of 8% over the risk-free rate of 6%, the required rate of return is 14%. Therefore, the present value of the portfolio is: $135,000/1.14 = $118,421 b. If the portfolio is purchased for $118,421 and provides an expected cash inflow of $135,000, then the expected rate of return [E(r)] is as follows: $118,421 × [1 + E(r)] = $135,000 Therefore, E(r) = 14%. The portfolio price is set to equate the expected rate of return with the required rate of return. c. If the risk premium over T-bills is now 12%, then the required return is: 6% + 12% = 18% The present value of the portfolio is now: $135,000/1.18 = $114,407 d. 6. For a given expected cash flow, portfolios that command greater risk premiums must sell at lower prices. The extra discount from expected value is a penalty for risk. Points on the curve are derived by solving for E(r) in the following equation: U = 0.05 = E(r) – 0.5Aσ2 = E(r) – 1.5σ2 The values of E(r), given the values of σ2, are therefore: 2 0.00 0.05 0.10 0.15 0.20 0.25 0.0000 0.0025 0.0100 0.0225 0.0400 0.0625 E(r) 0.05000 0.05375 0.06500 0.08375 0.11000 0.14375 The bold line in the graph on the next page (labeled Q6, for Question 6) depicts the indifference curve. 7. Repeating the analysis in Problem 6, utility is now: U = E(r) – 0.5Aσ2 = E(r) – 2.0σ2 = 0.05 The equal-utility combinations of expected return and standard deviation are presented in the table below. The indifference curve is the upward sloping line in the graph on the next Distributing prohibited | Downloaded by adam negile ([email protected]) lOMoARcPSD|1381793 page, labeled Q7 (for Question 7). 2 0.00 0.05 0.10 0.15 0.20 0.25 0.0000 0.0025 0.0100 0.0225 0.0400 0.0625 E(r) 0.0500 0.0550 0.0700 0.0950 0.1300 0.1750 The indifference curve in Problem 7 differs from that in Problem 6 in slope. When A increases from 3 to 4, the increased risk aversion results in a greater slope for the indifference curve since more expected return is needed in order to compensate for additional σ. E(r) U(Q7,A=4) 5 U(Q6,A=3) U(Q8,A=0) U(Q9,A<0) 13. Expected return = (0.7 × 18%) + (0.3 × 8%) = 15% Standard deviation = 0.7 × 28% = 19.6% Why? Because the standard deviation of risk-free rate is 0 – use the normal variance formulae for portfolio returns and you can verify the above numbers. Distributing prohibited | Downloaded by adam negile ([email protected]) ...
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