ec142f08hw3_sol - Kata Bognar [email protected] Economics...

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Kata Bognar [email protected] Economics 142 Probabilistic Microeconomics UCLA Fall 2008 Homework Assignment 3. - Solutions - 1. (Portfolio Problem) Casper Gutman is an expected utility maximizer and has $100 [8] to invest. His vNM utility function over money is u ( x ) = log x. There are two assets available to him to buy, but not to sell: a risk free treasury bill which costs $10 and pays $20 in the next period for sure a risky stock which costs $5 and pays $5 with probability p and $20 with probability 1 - p in the next period. Gutman may buy any amount of these assets, even partial units. (a) For what values of p would Gutman only buy the risk free asset? Answer: To answer this question, you can refer to local risk neutrality. The local risk neutrality tells us that whenever a gamble is favorable, even a risk averse person would bet a small amount on the gamble. In the context of portfolio choice problem, this means that whenever the expected value of the excess return of the risky asset over the risk free asset is positive, a risk averse person would invest some in the risky asset. This implies that whenever the excess return has a negative expected value the person will not invest in the risky asset. If you think about it, this makes perfect sense, if there is no premium on investing in the risky asset then there is no ‘reason’ for this person to take up risk. What is the excess return of the risky asset over the risk free asset? To calculate that, first you need to find the return for a dollar investment for both assets and then you need to find the difference of those values. Using the notation of the book, you want to calculate ˜ x = ˜ R 1 - R 0 . (This is what we denoted by ˜ x in class.) The next table summarizes what we know about the assets. money invested state 1 return ( p ) state 2 return (1 - p ) Y 0 10 20 20 ˜ Y 1 5 5 20 R 0 1 2 2 ˜ R 1 1 1 4 ˜ x 0 - 1 2 Therefore the expected value of the excess return is E x ) = p ( - 1) + (1 - p )2 . This is non-positive whenever p 2 / 3, hence for all such values Gutman only invests in the risk free asset. 1
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Alternatively, you may actually solve part (c) first and interpret the result you get there. In part (c) you calculate the optimal amount of money to spend on investing in the risky asset (the optimal number of the risky asset to buy) as the function of the probability parameter. Then you should pick probability values such that the optimal investment is not positive. 1 (b) Assume that p = 1 / 2 . Write down the maximization problem of Gutman. How many units of the risky assets would he buy? How much money would
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This note was uploaded on 01/25/2010 for the course ECON ECON 142 taught by Professor Bognar during the Fall '09 term at UCLA.

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ec142f08hw3_sol - Kata Bognar [email protected] Economics...

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