PPT 9 - 1 9 UTILITY FUNCTIONS Reading Luenberger Chapter 9(Many of the slides in this lecture are modifications from last year’s course by

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Unformatted text preview: 10/29/11 1 9. UTILITY FUNCTIONS Reading: Luenberger, Chapter 9 (Many of the slides in this lecture are modifications from last year’s course by Professor Martin) 2 9.1 EXPECTED UTILITY FRAMEWORK W O Initial Wealth (fixed) W End of Period Wealth (random variable) U(w) A Utility Function E[U(W)] Expected Utility Key Assumption: Investors maximize expected utility for utility functions Key Assumption: Investors maximize expected utility for utility functions Key Assumption: Investors maximize expected utility for utility functions Key Assumption: Investors maximize expected utility for utility functions having certain properties having certain properties having certain properties having certain properties This key assumption will be true if investors behavior satisfies certain preference (behavioral) properties that have been stated in the form of axioms developed in connection with a theory for expected utility maximization by von Neumann and Morgenstern (1944). See for example the following textbook: Pennachi, G. (2008). Theory of Asset Pricing , Pearson/Addison-Wesley 10/29/11 3 Basic Properties Used only to rank investments Used only to rank investments Used only to rank investments Used only to rank investments Utility functions are invariant under positive affine transformations, namely: If U 1 and U 2 are related by U 2 = a U 1 + b, a>0, then U 1 and U 2 are equivalent Key Assumptions Key Assumptions Key Assumptions Key Assumptions (1) Investors prefer more to less (2) Investors are unsatiated, never satisfied U(w) is a strictly increasing functions of wealth w. ⇒ ⇒ 10/29/11 4 9.2 RISK AVERSION A gamble: gamble: gamble: gamble: H a random variable with payoffs h 1 , h 2 : Positive Return (Payoff): h 1 with P(h 1 ) = p Negative Return (Payoff): h 2 with P(h 2 ) = 1 – p Expected Payoff: p h 1 + (1 – p) h 2 H is a fair game fair game fair game fair game if: p h 1 + (1 – p) h 2 = 0 Definition Definition Definition Definition A risk averse risk averse risk averse risk averse person is unwilling to accept or is indifferent to any fair game, while a strictly risk averse strictly risk averse strictly risk averse strictly risk averse individual is unwilling to accept a fair game. 10/29/11 5 End of period wealth: W = W o +H where H = h 1 or h 2 Expected Utility: E(U(W)) = E[U (W + H)] = p U (W + h 1 ) + (1 – p) U (W + h 2 ) This traces out a straight line as p varies from 0 to 1 in the next figure....
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This note was uploaded on 03/23/2012 for the course AMATH 541 taught by Professor Kk.t during the Winter '11 term at University of Washington.

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PPT 9 - 1 9 UTILITY FUNCTIONS Reading Luenberger Chapter 9(Many of the slides in this lecture are modifications from last year’s course by

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