03_Notes_ExpectedUtility

03_Notes_ExpectedUtility - HANDOUT ON RISK AVERSION If you...

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HANDOUT ON RISK AVERSION If you have comment on these notes (mistakes, typos, “unclear” points, please send me an email) Rene Schwengber, June 15 , 2011 The goal for these notes is to complement your notes taken in class 1 . This is required reading material. 1. Portfolio choice problem We talked in class on the importance of the question: “How to best invest resources in a uncertain environment”, or, “How to optimally invest money for future consumption?”. The goal of this documente is to familiarize the student with the general problem for a consumer choosing financial assets. “Create affordable portfolio X from the set of available assets X in order to maximize the agent utility” in symbols: max X U ( X ) (1.1) subject to pX I X X (1.2) the first contraint is affordability (your portfolio costs less than your total wealth invested I ) , the second represent that you can choose only among available securities, i.e., your portfolio has to be created with available assets from start. In other words, this problem reads: Pick X in order to maximize U , however, you must pick X that is (1) affordable (2) and available to be bought. There are several concepts behind this problem. We need to understant the set X , presumably the agent has a preference, % over X , this preference can be represented by the utility function U (i.e, the function U maps X to a real number, and preserves the order of % ). We start it’s description bellow. First a review of some concepts from decision theory. Definition. Given a set of alternative choices, say C , and an ordering % over it 2 we say that a real valued function u : C → R represents the order % if for all pair of alternatives x, y in C we have: x % y if and only if u ( x ) u ( y ) . In words, the agent’s ranking % over the set C , is preserved by the function u . If he prefer x over y , then his utility associated with x is higher than his utility associated with y. 1 examples and other part of these notes can be found in chapter 3 of the book by Danthine, check the syllabus for the reference. For the part of choice you can read Theory of Value chapter 4, or you intermediate micro book. 2 an order is any binary relation R that satisfies (i) reflexivity, xRx (ii) transitivity xRy, yRz implies xRz and (iii) anti-symmetric if xRy and yRx then y = x . 1
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4751 - Spring 2011 - June 15, 2011 Rene Schwenger Fact. If we have a preference % over C , that is represented by a function that maps C into the set real numbers, U : C → R , it must be true then that such preference is complete . That is, for any pair of elements X, Y in C , we must have X % Y or Y % X . This is the case, since associated to X we have a number U ( X ) which in turn is always T than the number U ( Y ) . As observed in class an investment decision scenario, the set of available choices is best described by a set of random variables.
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