Question

Name your files with your student number first: e.g., ‘1234567Ass1c.doc0 . Typed, or handwritten and scanned answers are acceptable. This tutorial problem set is the third part of your individual assignment. Four tutorials will be collected and marked and the best three marks will count for the assignment. 1. Expected utility Question 1 Each part is worth 1 mark Suppose Xanthia’s utility function for wealth is given by: U(W) = 1 1 − γ W1−γ Assume that Xanthia’s risk aversion parameter is γ = 2. Xanthia starts with wealth of $100. (a) Xanthia is offered a gamble where he has a 50% chance of gaining $50, but a 50% chance of losing $20. What is the expected value of the gamble? (b) What is the expected utility of this gamble to Xanthia? Will Xanthia take the gamble? (c) What is Xanthia’s risk premium for this gamble? (d) If Xanthia had risk-neutral preferences over wealth, would she be willing to purchase insurance at fair prices (premiums)? Briefly explain why or why not. 2. Portfolio Variance Q2. 1 mark Recall from your earlier stats course the following laws about variance: var(X + Y ) = varX + varY + 2 ∗ cov(X, Y ) var(c ∗ Y ) = c 2 ∗ varY covar(cX, cY ) = c 2 ∗ covarXY 1 2 25005 ECONOMICS AND FINANCE OF THE LIFECYCLE: PROBLEM SET 8 Assume that an investor is choosing from a number of stocks with identical portfolio returns. Each stock has variance V and the covariance between any two stocks has the value C. Calculate the variance of a portfolio equally divided between two of these stocks (i.e., weights equal to 0.5). Calculate the variance of a portfolio equally divided between three of these stocks (i.e., weights equal to 0.33). What about the variance of a portfolio equally divided over 1000 stocks? 3. Optimal portfolio choice Q3. Each part is worth 1 mark Table 1. Inflation-adjusted financial asset returns and volatility (annual). Risky asset Risk free asset Expected simple return 10.5% 5.13% Standard deviation simple return 23.0% 0 Expected log return 10.0% 5.0% Standard deviation log return 25.0% 0 Refer to the information in Table 1. An investor has a fixed amount of wealth to invest over several years and he has to choose how to allocate his wealth between a risky asset and a risk free asset. The investor aims to maximize the utility of his final real (inflation-adjusted) wealth. Real returns to the risky asset are log-normally distributed and the distribution is the same from year to year (independent and identically distributed). The investor has constant relative risk aversion (power) utility. The risk-free asset is a savings account that has a guaranteed real rate of return as shown in Table 1. The risky asset is a well-diversified share portfolio (like an index fund) with real return and variance as shown in Table 1. (a) If ωt is the fraction of wealth invested in the risky asset, and the investor has a risk aversion parameter of γ = 2, use the data from Table 1 and the formula below to calculate the optimal value of ωt . ωt = E(r) − rf + 1/2σ 2 γσ2 (b) How does the fraction ωt change as the variance of the risky asset increases? Briefly explain why. (c) Will the investor change the fraction of total wealth invested in the risky asset from year to year? Briefly explain why or why not. 25005 ECONOMICS AND FINANCE OF THE LIFECYCLE: PROBLEM SET 8 3 (d) Assume now that the investor earns a regular, risk-free salary as well as investment returns. How might this change his portfolio allocation strategy? (e) How is the investment plan given by the formula different from a ‘buy and hold’ strategy?