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Unformatted text preview: Kata Bognar [email protected] Economics 142 Probabilistic Microeconomics UCLA Fall 2009 Homework Assignment 4. suggested solutions  1. (Arbitrage) Dr. Andrei Smyslov is a risk averse expected utility maximizer. The [6] following three assets are available for him. price payoff in state 1 payoff in state 2 asset A 100 50 200 asset B 40 80 20 asset C 20 30 30 These assets can be bought or sold in any amounts, even partial units. (a) Point out an arbitrage opportunity for Dr. Smyslov. Answer: Suppose that Dr. Smyslov buys a units of asset C , and sells 0.1 units of asset A and 0.25 units of asset B. Since he can buy and sell the above assets in any amounts, even partial units, this position is possible for him. The cost of this portfolio is zero: 20 . 1 × 100 . 25 × 40 = 0 and the net payoffs are: 30 . 1 × 50 . 25 × 80 = 5 and 30 . 1 × 200 . 25 × 20 = 5 in state 1 and state 2, respectively. Thus, this is an arbitrage opportunity. Clearly, there are few other portfolios that would work. To fully charac terize the set of the possible portfolios, think about the conditions that are necessary for arbitrage: (i) the cost of the portfolio should be zero and (ii) the payoff in each state should be positive. If a portfolio allows for arbirtage then any multiple of it does that as well. Thus, we can normalize the amount of one asset. Here, assume that there is a unit of C in the portfolio. Now we can express the conditions above mathematically: 20 100 α 40 β = 0 30 50 α 80 β > 30 200 α 20 β > . Any α ∈ (1 / 15 , 2 / 15) and β = 1 / 2 (5 / 2) α satisfies this system so those are all possible weights in the portfolio. Some of you tried to define a portfolio in which the assets A and B are bought and C are sold. If you try to do that and write down the appropriate system 1 of inequalities to solve, you will get contradicting conditions for the weights. That must be a sign that you do something wrong and at that point you want to think about changing the signs. To avoid this problem you can think in terms of per dollar returns. The idea is the same in this context, you want to put together a portfolio that has zero costs but positive payoffs in both states. But you can prepare a graph as we did in class and decide easily which assets should be bought and which assets should be sold. return in state 1 return in state 2 asset A 1 / 2 2 asset B 2 1 / 2 asset C 3 / 2 3 / 2 6 state 1 state 2 B : A C · · · · · · · · · · · 1/2 ··· 2 You can see from the graph that the decision maker wants to buy C and sell a combination of A and B. An arbitrage opportunity in the per dollar return context would mean that one can divide a dollar investment between the asset A and the asset B such that the returns on that investment in either states will be less than the return on a dollar investment in C in both states. This is possible if the following system has a solution: α 1 2 + (1 α )2 < 3 2 α 2 + (1 α ) 1 2 < 3 2 ....
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 Fall '09
 Bognar
 Utility, Expected utility hypothesis, Dr. Dave Bowman

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