Problem Set 6 for Econ 422: Probability and Expected Utility in Finance - Prof. Eric Zivot, Assignments of Economics

Download Problem Set 6 for Econ 422: Probability and Expected Utility in Finance - Prof. Eric Zivot and more Assignments Economics in PDF only on Docsity! Econ 422 E. Zivot Spring 2008 Problem Set 6 Due Monday 8/14/08 Hand in the following textbook problems. BM Chapter 8 Quiz Questions: 8.1, 8.2 BM Chapter 8 Practice Questions 8.10, 8.12, 8.15, 8.16, 8.18 Using the Normal Distribution 1. Suppose the returns on long-term government bonds are normally distributed. Based on the historical record from 1926 – 2004 the annual mean return was estimated to be 5.8 percent and the annual standard deviation was estimated to be 9.3 percent. What is the approximate probability that the return on bonds will be less than -3.5 percent in any given year? What range of returns would you expect to see 95 percent of the time? What range would you expect to see 99 percent of the time? 2. Using probability distributions. Suppose the returns on large-company stocks are normally distributed. Based on the historical record from 1926 – 2004 the annual mean return was estimated to be 12.4 percent and the annual standard deviation was estimated to be 20.3 percent. Determine the probability that in any given year you will lose money by investing in a diversified portfolio of large company stocks. Hint: use the normdist function in EXCEL. Expected Utility Hypothesis Problems 1. A consumer has the utility function u = U(W) = ln(W) where u represents the level of utility, U(W) represents the general form of a utility function, W is the consumer's wealth, and ln(W) represents a specific utility function that specifies that the value of the utility index associated with a given level of wealth, W, is given by the natural log (base e) of W. The consumer's wealth is not fixed, but is a random variable given by the probability distribution: Possible value of W: 100,000 150,000 200,000 250,000 Probability of W: 0.2 0.5 0.2 0.1 a. Compute the consumer's expected wealth i.e. E(W). b. Extend the table to show the possible values of U(W). c. Compute the expected utility of wealth, i.e. .E[U(W)] d. Show that E[U(W)]≠U(E(W)). e. Find the certainty equivalent wealth, that is the wealth, which if held for certain, would be regarded as equivalent in the utility sense to the random wealth distribution given above. i.e. find Wc such that U(Wc)=E[U(W)]. 2. A person's utility of wealth function is U(W)=ln(W), i.e. the person's utility associated with any given level of wealth is equal to the natural log of the wealth level. If the person chooses occupation A his or her wealth is given by the following wealth distribution: Wealth: 1,400,000 2,400,000 Probability: 0.8 0.2 If the person chooses occupation B his or her wealth is given by the following wealth distribution: Wealth: 1,500,000 1,700,000 Probability: 0.5 0.5 a. Which occupation will the person choose and why? Or will the person be indifferent to the alternatives? Explain. b. Compute the certainty equivalent wealth for occupation B. Risk-Return Questions 1. The probability distribution for next year’s price of Google stock is given by Price P1 $300 $400 $500 $600 Probability 0.1 0.2 0.5 0.2 a. Compute the expected value and variance of next year’s price. b. Compute the expected annual rate of return and standard deviation of the annual rate of return if the current price is $500 and you do not expect to receive a dividend at time 1. 2. Which one of the following is a correct ranking of securities based on their volatility over the period of 1926 to 2004? Rank from highest to lowest. (2.5 points) a. large company stocks, U.S. Treasury bills, long-term government bonds b. small company stocks, long-term corporate bonds, large company stocks c. small company stocks, long-term government bonds, long-term corporate bonds d. small company stocks, large company stocks, long-term corporate bonds e. long-term corporate bonds, large company stocks, U.S. Treasury bills