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# PPT 9 - 9 UTILITY FUNCTIONS Reading Luenberger Chapter...

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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)

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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: 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 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 (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

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4 9.2 RISK AVERSION A 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 if: p h 1 + (1 – p) h 2 = 0 Definition Definition Definition Definition A risk averse risk averse person is unwilling to accept or is indifferent to any fair game, while a 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 0 + H)] = p U (W 0 + h 1 ) + (1 – p) U (W 0 + h 2 ) This traces out a straight line as p varies from 0 to 1 in the next figure. A strictly risk averse investor chooses W o instead of W = W o +H. If the investor is also an expected utility maximizerthen: U(W o ) > E(U(W)) So it is above the straight line. The above holds for all h 1 < 0, h 2 > 0, and p (0,1). 10/29/11

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6 This implies that U is strictly concave , i.e., strictly risk averse investors have strictly concave utility functions. U(W o +h 1 ) U(W o ) W o +h 1 W o +h 2 U(W o +h 2 ) W o w U(w) p U (W 0 + h 1 ) + (1 –p) U (W 0 + h 2 ) If the investor is only risk averse rather than strictly risk averse, then an equality can hold on the previous slide and U must be concave but not strictly concave.
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