Post on 05-Jan-2016
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Chapter 2
Prospect Theory and
Expected Utility Theory
2
A.Von-Neumann-Morgenstern Expected Utility Theory
1. Von-Neumann-Morgenstern Axioms
:Certain, identifiable Outcomes
C: Choice facing the individual
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A.Von-Neumann-Morgenstern Expected Utility Theory
2. Utility Function Let be the number described in A(σ)
Define u over all possible choices by
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4
A.Von-Neumann-Morgenstern Expected Utility Theory
3. Expected Utility Theory
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5
A.Von-Neumann-Morgenstern Expected Utility Theory
4. Risk Aversion1) Utility Functions
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A.Von-Neumann-Morgenstern Expected Utility Theory
2) Risk Premium and Cost of Gamble
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A.Von-Neumann-Morgenstern Expected Utility Theory
3) Pratt-Arrow Risk Aversion
Def. Assume that an individual faces an “actuarially fair” bet (i.e. ). Let w be the individual’s initial wealth. The risk premium is that amount such that the individual is indifferent between receiving the risk and receiving the nonrandom amount .
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8
A.Von-Neumann-Morgenstern Expected Utility Theory
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• LHS
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A.Von-Neumann-Morgenstern Expected Utility Theory
• ARA= Absolute Risk Aversion
• RRA= Relative Risk Aversion
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A.Von-Neumann-Morgenstern Expected Utility Theory
• E.g.1. Quadratic Utility function
• E.g.2. Power Utility function
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A.Von-Neumann-Morgenstern Expected Utility Theory
5. Mean-Variance(M-V) Utility function – Assume– Indifference curves of risk averters:
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A.Von-Neumann-Morgenstern Expected Utility Theory
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A.Von-Neumann-Morgenstern Expected Utility Theory
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A.Von-Neumann-Morgenstern Expected Utility Theory
• Convexity: Let be two points on the same
indifference curve.
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A.Von-Neumann-Morgenstern Expected Utility Theory
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A.Von-Neumann-Morgenstern Expected Utility Theory
6. Stochastic Dominance1) First Order Stochastic Dominance: An asset is said to be stochastically dominant over another if
an individual receive greater wealth from it in every state of nature.
Asset x, will be stochastically dominant over asset y,
if
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wwywx
allfor)(G)(F
allfor)(G)(F
)(G wy
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A.Von-Neumann-Morgenstern Expected Utility Theory
2) Second Order Stochastic Dominance An asset is said to be second order stochastically dominant over
another if an individual (Risk averter) receives greater accumulated wealth in any given level of wealth.
Asset x, is second order stochastically dominant over asset y,
if
ii
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wwxwy
wwxwyi
somefor)(F)(G
,allfor0)](F)(G[
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A.Von-Neumann-Morgenstern Expected Utility Theory
3) Mean-Variance Paradox
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A.Von-Neumann-Morgenstern Expected Utility Theory
a. Mean Variance Analysis
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a. Stochastic Dominance Analysis
A.Von-Neumann-Morgenstern Expected Utility Theory
EPS Prob(B) Prob(A) F(B) G(A) F-G ∑(F-G)
3.00 0.2 0.2 0.2 0.2 0.0 0.0
4.00 0.0 0.2 0.2 0.4 -0.2 -0.2
5.00 0.2 0.2 0.4 0.6 -0.2 -0.4
6.00 0.0 0.2 0.4 0.8 -0.4 -0.8
7.00 0.2 0.2 0.6 1.0 -0.4 -1.2
8.00 0.0 0.0 0.6 1.0 -0.4 -1.6
9.00 0.2 0.0 0.8 1.0 -0.2 -1.8
10.00 0.0 0.0 0.8 1.0 -0.2 -2.0
11.00 2.0 0.0 1.0 1.0 0.0 -2.0
AdominatesB)EPS(F)EPS(F BA
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B.Kahneman and Tversky Prospect Theory ---Non-expected Utility Theory
1. Three effects1) Certainty effect
State Prob. State Prob.
2,500 0.33 2,400 1.002,400 0.66
0 0.01E(A) 2,409 E(B) 2,400
A B*
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B.Kahneman and Tversky Prospect Theory ---Non-expected Utility Theory
2) Reflection effect
3) Isolation Effect
(4,000, 0.80) (3,000)* (-4,000, 0.80)* (-3,000)
E 3,200 3,000 -3,200 -3,000
Positive Prospect Negative Prospect
(1,000, 0.50) (500)* (-1,000, 0.50)* (-500)
E 1,500 1,500 1,500 1,500
Positive Prospect Negative Prospectw=1,000 w=2,000
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B.Kahneman and Tversky Prospect Theory ---Non-expected Utility Theory
2. Prospect Theory1) Value function
A. Reference pointB. concave for gain convex for lossC. steeper for loss than for gain
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B.Kahneman and Tversky Prospect Theory ---Non-expected Utility Theory
2) Weight function
A. Sharp drop of π at the endpointsB. discontinuities of π at the endpointsC. Non-linearity