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UCD ECN 134 - HW6s-S10

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Nominal RatesProblem Set 6 (Answer Key)ECN 134Financial Economics Prof. Farshid MojaverPart A: Risk Aversion Consider a portfolio that offers an expected rate of return of 12% and a standard deviation of 18%.T-bills offer a risk-free 7% rate of return. What is the maximum level of risk aversion for which therisky portfolio is still preferred to bills?Answer)When we specify utility by U = E(r) – 0.5Aσ2, the utility level for T-bills is: 0.07The utility level for the risky portfolio is: U = 0.12 – 0.5A(0.18)2 = 0.12 – 0.0162AIn order for the risky portfolio to be preferred to bills, the following inequality must hold:0.12 – 0.0162A > 0.07  A < 0.05/0.0162 = 3.091. When an investment advisor attempts to determine an investor's risk tolerance, which factor wouldthey be least likely to assess? A. the investor's prior investing experienceB. the investor's degree of financial securityC. the investor's tendency to make risky or conservative choicesD. the level of return the investor prefersE. the investor's feeling about lossAssume an investor with the following utility function: U = E(r) - 3/2(s2).2. To maximize her expected utility, she would choose the asset with an expected rate of return of _______ and a standard deviation of ________, respectively. A. 12%; 20% B. 10%; 15% C. 10%; 10% D. 8%; 10% E. none of the aboveU = 0.10 - 3/2(0.10)2 = 8.5%; highest utility of choices. U = E(r) - (A/2)s2, where A = 4.0. 3. Based on the utility function above, which investment would you select? A. 1 B. 2 C. 3 D. 4 E. cannot tell from the information givenU(c) = 0.21 - 4/2(0.16)2 = 15.88 (highest utility of choices).4. Which investment would you select if you were risk neutral? A. 1 B. 2 C. 3 D. 4 E. cannot tell from the information givenIf you are risk neutral, your only concern is with return, not risk.5. An investor invests 30 percent of his wealth in a risky asset with an expected rate of return of 0.13 and a variance of 0.03 and 70 percent in a T-bill that pays 6 percent. His portfolio's expected return and standard deviation are __________ and __________, respectively. A. 0.114; 0.128 B. 0.087;0.063 C. 0.295; 0.125 D. 0.081; 0.052 E. none of the aboveE(rP) = 0.3(13%) + 0.7(6%) = 8.1%; sP = 0.3(0.03)1/2 = 5.19%.6. What percentages of your money must be invested in the risky asset and the risk-free asset, respectively, to form a portfolio with an expected return of 0.09? A. 85% and 15% B. 75% and 25% C. 67% and 33% D. 57% and 43%E. cannot be determined9% = w1(12%) + (1 - w1)(5%); 9% = 12%w1 + 5% - 5%w1; 4% = 7%w1; w1 = 0.57; 1 - w1 = 0.43; 0.57(12%) + 0.43(5%) = 8.99%.Your client, Bo Regard, holds a complete portfolio that consists of a portfolio of risky assets (P) and T-Bills. The information below refers to these assets. 7. What is the equation of Bo's Capital Allocation Line? A. E(rC) = 7.2 + 3.6 * Standard Deviation of CB. E(rC) = 3.6 + 1.167 * Standard Deviation of CC. E(rC) = 3.6 + 12.0 * Standard Deviation of CD. E(rC) = 0.2 + 1.167 * Standard Deviation of CE. E(rC) = 3.6 + 0.857 * Standard Deviation of CThe intercept is the risk-free rate (3.60%) and the slope is (12.00%-3.60%)/7.20% = 1.167.8. What are the proportions of Stocks A, B, and C, respectively in Bo's complete portfolio? A. 40%, 25%, 35%B. 8%, 5%, 7%C. 32%, 20%, 28%D. 16%, 10%, 14%E. 20%, 12.5%, 17.5%Proportion in A = .8 * 40% = 32%; proportion in B = .8 * 25% = 20%; proportion in C = .8 * 35% =28%.Part B: Optimal Risky Portfolio1. a. Even though it seems that gold is dominated by stocks, gold might still be an attractive asset to hold as a part of a portfolio. If the correlation between gold and stocks is sufficiently low, gold will be held as a component in a portfolio, specifically, the optimal tangency portfolio.b. If the correlation between gold and stocks equals +1, then no one would hold gold. The optimal CAL would be comprised of bills and stocks only. Since the set of risk/return combinations of stocks and gold would plot as a straight line with a negative slope (see the following graph), thesecombinations would be dominated by the stock portfolio. Of course, this situation could not persist. If no one desired gold, its price would fall and its expected rate of return would increase until it became sufficiently attractive to include in a portfolio.Standard Deviation(%)05101520250.0010.0020.0030.0040.00GoldStocksrf182. The probability distribution is:Probability Rate of Return0.7 100%0.3 −50%Mean = [0.7  100] + [0.3 (50)] = 55%Variance = [0.7 (100  55)2] + [0.3 (50  55)2] = 4725Standard deviation = 47251/2 = 68.74%3.  P = 30 = yy  y = 0.75E(rP) = 12 + 0.75(30  12) = 25.5%4. Since we do not have any information about expected returns, we focus exclusively on reducing variability. Stocks A and C have equal standard deviations, but the correlation of Stock B with StockC (0.10) is less than that of Stock A with Stock B (0.90). Therefore, a portfolio comprised of Stocks B and C will have lower total risk than a portfolio comprised of Stocks A and B.5. Rearranging the table (converting rows to columns), and computing serial correlation results in thefollowing table:Nominal RatesSmallcompanystocksLargecompanystocksLong-termgovernmentbondsIntermed-termgovernmentbondsTreasurybillsInflation1920s -3.72 18.36 3.98 3.77 3.56 -1.001930s 7.28 -1.25 4.60 3.91 0.30 -2.041940s 20.63 9.11 3.59 1.70 0.37 5.361950s 19.01 19.41 0.25 1.11 1.87 2.221960s 13.72 7.84 1.14 3.41 3.89 2.521970s 8.75 5.90 6.63 6.11 6.29 7.361980s 12.46 17.60 11.50 12.01 9.00 5.101990s 13.84 18.20 8.60 7.74 5.02 2.93Serial Correlation 0.46 -0.22 0.60 0.59 0.63 0.23For example: to compute serial correlation in decade nominal returns for large-company stocks,we set up the following two columns in an Excel spreadsheet. Then, use the Excel function “CORREL” to calculate the correlation for the data.Decade Previous1930s-1.25%18.36%1940s 9.11% -1.25%1950s 19.41% 9.11%1960s 7.84% 19.41%1970s 5.90% 7.84%1980s 17.60% 5.90%1990s 18.20% 17.60%Note that each correlation is based on only seven observations, so we cannot arrive at any statistically significant conclusions. Looking at the results, however, it appears that, with the exception of large-company stocks,


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