投资学6单元题目及答案.docx

Problem1: Consider a risky portfolio. The end-of-year cash flow derived from the portfolio will be either $70,000 or $200,000 with equal probabilities of .5. The alternative risk-freeinvestment in T-bills pays 6% per year. a. If you require a risk premium of 8%, how much will you be willing to pay for theportfolio? b. Suppose that the portfolio can be purchased for the amount you found in (a). Whatwill be the expected rate of return on the portfolio? answers: a.The expected cash flow is: (0.5 * $70,000) + (0.5 *200,000) = $135,000With 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 = $118,421b. 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 derived as follows:$118,421*1 + E(r) = $135,000Therefore, E(r) = 14%. The portfolio price is set to equate the expected rate or return with the required rate of return.Problem2: 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 ofrisk aversion for which the risky portfolio is still preferred to bills?（Hint: use this utility function U = E(r) 0.005As 2） answers: When we specify utility by U = E(r) 0.005As 2, the utility level for T-bills is 7%. The utility level for the risky portfolio is: U = 12 0.005A 182 = 12 1.62AIn order for the risky portfolio to be preferred to bills, the following inequality must hold:12 1.62A 7 A 5/1.62 = 3.09A must be less than 3.09 for the risky portfolio to be preferred to bills.Problem3: Draw the indifference curve in the expected returnstandard deviation plane corresponding to a utility level of 5% for an investor with a risk aversion coefficient of 3.(Hint: Choose several possible standard deviations, ranging from 5% to 25%, and find the expected rates of return providing a utility level of 5%. Then plot the expected returnstandard deviation points so derived. answers: Points on the curve are derived by solving for E(r) in the following equation:U = 5 = E(r) 0.005As 2 = E(r) 0.015s 2 - E(r) =U+ 0.015s 2 The values of E(r), given the values of s 2, are therefore:s(%)s 2R(%)0 0 5.00 5 25 5.38 10 100 6.50 15 225 8.38 20 400 11.00 25 625 14.38 The bold line in the following graph (labeled Q3, for Question 3) depicts the indifference curve. Use the following data in answering next 3 questions: Problem4(*):Based on the utility formula above, which investment would you select if you were risk averse with A=4?a. 1. b. 2. c. 3. d. 4.answers: c Utility for each portfolio = E(r) 0.005 4 s 2, We choose the portfolio with the highest utility value. In 3,U=21-0.02162=15.88; In 4, U=24-0.02212=15.18Problem5(*): Based on the utility formula above, which investment would you select if you were risk neutral?a. 1. b. 2. c. 3. d. 4. answers: d When investors are risk neutral, then A = 0; the portfolio with the highest utility is the one with the highest expected return.Problem6(*): The variable (A) in the utility formula represents the:a. investors return requirement.b. investors aversion to risk.c. certainty equivalent rate of the portfolio.d. preference for one unit of return per four units of risk.answers: bConsider historical data showing that the average annual rate of return on the S&P 500 portfolio over the past 70 years has averaged about 8.5% more than the Treasury bill return and that the S&P 500 standard deviation has been about 20% per year. Assume these values are representative of investors expectations for future performance and that the current T-bill rate is 5%. Use these values to solve problems 7 to 8.Problem7: Calculate the expected return and variance of portfolios invested in T-bills and the S&P500 index with weights as follows: answers: The portfolio expected return and variance are computed as follows:(1)(2)(3)(4)rPortfoliosPortfolio(%)s2Portfolio(%)WBillsrBillsWIndexrIndex(1)(2)+(3)(4)(3) 20%05%113.50%13.50%20.00 400.00 0.25%0.813.50%11.80%16.00 256.00 0.45%0.613.50%10.10%12.00 144.00 0.65%0.413.50%8.40%8.00 64.00 0.85%0.213.50%6.70%4.00 16.00 15%013.50%5.00%0.00 0.00 Problem8: Calculate the utility levels of each portfolio for an investor with A=3 and A=5.What do you conclude? answers: Computing utility from U = E(r) 0.005 As 2 = E(r) 0.015s 2 , we arrive at the values in the column labeled U(A = 3) in the following table:rPortfolio s2Portfolio(%)U(A = 3)U(A = 5)13.50%4007.50 3.50 11.80%2567.96 5.40 10.10%1447.94 6.50 8.40%647.44 6.80 6.70%166.46 6.30 5.00%05.00 5.00 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.The colu