Advanced Financial Economics Homework 2 Due on April 14th before class

Transcription

1 Advanced Financial Economics Homework 2 Due on April 14th before class March 30, (20 points) An agent has Y 0 = 1 to invest. On the market two financial assets exist. The first one is riskless. The price is one and its return r f = 2. Short selling on this asset is allowed. The second asset is risky. Its price is 1 and its return r, where r is a random variable with probability distribution 1 with probability π 1, r = 2 with probability π 2, 3 with probability π 3. No short selling is allowed on this asset. a) If the agent invests a in the risky asset, what is the probability distribution of the agent s portfolio return r P? b) A risk averse agent maximizes a von Neumann utility, E{u[Y 0 r f + a( r r f )]}. Show the optimal choice of a is positive if and only if the expectation of r is greater than 2. c) Find the optimal choice of a when u(y ) = 1 exp( by ), b > 0 and when u(y ) = 1 1 γ Y 1 γ, 0 < γ < 1. 1

2 d) Find the absolute risk aversion coefficient in either case of question c). If Y increases, how will the agent react? 2. (10 points) An individual with a well-behaved utility function and an initial wealth of $1 can invest in two assets. Each asset has a price of $1. The first is a riskless asset that pays r f = 1. The second pays amount r 1 and r 2 where r 1 < r 2 with probabilities of π and 1 π, respectively. Denote the units demanded to the riskless asset and the risky asset by x 1 and x 2, respectively, with x 1, x 2 [0, 1]. a) Given a simple necessary condition (involving r 1 and r 2 only) for the demand for the riskless asset to be strictly positive. Given a simple necessary condition (involving r 1, r 2 and π only) so that the demand for the risky asset is strictly positive. b) Assume now that the conditions in item a) are satisfied. Formulate the optimization problem and write down the first order condition. Can you intuitively guess the sign of dx 1 /dr 1? Verify your guess by assuming that x 1 is a function of r 1 written as x 1 (r 1 ), and taking the total differential of the first order condition with respect to r 1. Can you conjecture a sign for dx 1 /dπ? Provide an economic interpretation without verifying it as done previously. 3.(20 points) You are a portfolio manager considering whether or not to allocate some of the money to SP500 index. Denote by r your and r SP the returns of your portfolio and returns of SP500 Index, respectively. Let σ( r) be the standard deviation of r and corr( r your, r SP ) be the correlation between r your and r SP. Your assistant provides you with the following historical return information: 2

3 Year r your (%) r SP (%) a) Show that the addition of the SP500 Index to your portfolio will reduce risk (at no loss in return) provided corr( r your, r SP ) < σ your σ SP assuming as in the case, σ your > σ SP. b) Based on this historical data, could you receive higher returns for the same level of risk (standard deviation) by allocating some of your wealth to the SP500? c) Based on this historical experience, would it be possible to reduce your portfolio s risk below its current level by investing something in the SP500? d) What fraction of the variation in SP500 can be explained by variation in your portfolio s returns? 4.(25 points) Given two random variables X and Y probability state of nature X Y 0.2 I II III IV V 6 1 a) Calculate the mean and variance of each variable, and the covariance between X and Y 3

4 b) Suppose X and Y represent the returns from two assets. Calculate the mean and variance for the following portfolio: portfolio % invested in X % invested in Y c) Find the portfolio that has the minimum variance. d) Let portfolio A have 75% in X and portfolio B have 25% in X. Calculate the covariance between the two portfolios. e) Calculate the covariance between the minimum variance portfolio and portfolio A, and the covariance between the minimum variance portfolio and portfolio B. f) What is the covariance between the minimum variance portfolio and any other portfolio along the efficient set? g) What is the relationship between the covariance of the minimum variance portfolio with other efficient portfolio, and the variance of the minimum variance portfolio? 5.(15 points) Two securities have the following joint distribution of returns, r 1 and r 2 : P (r 1 = 1 and r 2 = 0.15) = 0.1 P (r 1 = 0.5 and r 2 = 0.15) = 0.8 P (r 1 = 0.5 and r 2 = 1.65) = 0.1 a) Compute the means, variances and covariance of returns for the two securities 4

5 b) Plot the feasible mean-standard deviation [ER, σ] combinations, assuming that the two securities are the only investments available. c) Which portfolios belong to the mean-variance efficient set? d) Show that security 2 is mean-variance dominated by security 1, yet enters all efficient portfolios but one. How do you explain this? e) Suppose the possibility of lending, but not borrowing, at 5% (without risk) is added to the previous opportunities. Draw the new set of [ER, σ] combinations. Which portfolios are now efficient? 6.(10 points) Given that assets X and Y are perfectly correlated such that Y = X and the probability distribution for X is probability X(%) What is the percentage of your wealth to put into asset X to achieve zero variance? Graph the minimum variance frontier and the zero variance point. 5