1

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

),1( IiAi

I

ii

iiiII

p

pIiApApAp

1

11

?

0],,1),,[()],(,),,[(

3

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

Where

),,1( Iiui

I

iiiupcu

1

)(

],,1),,[( IiApc ii

4

A.Von-Neumann-Morgenstern Expected Utility Theory

3. Expected Utility Theory

I

iii

I

iii

upcu

upcuwhere

cciffcucu

12,2,2

11,1,1

2121

)(

)(,

)()( ~

5

A.Von-Neumann-Morgenstern Expected Utility Theory

4. Risk Aversion1) Utility Functions

6

A.Von-Neumann-Morgenstern Expected Utility Theory

2) Risk Premium and Cost of Gamble

7

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 .

0)(~

ZE

),(~

Zw

),(~

Zww

8

A.Von-Neumann-Morgenstern Expected Utility Theory

• RHS

• LHS

0)~

(

)]~

,()~

([)]~

([

ZE

ZwZEwUZwUE

)(')()]~

([ wUwUZwwU

)(2

1)(

])(2

1)(

~)([)]

~([

2

2

wUwU

wUZwUZwUEZwUE

Z

])(

)([

2

1 2

wU

wUZ

9

A.Von-Neumann-Morgenstern Expected Utility Theory

• ARA= Absolute Risk Aversion

• RRA= Relative Risk Aversion

)(

)(

wU

wU

)(

)(

wU

wUw

10

A.Von-Neumann-Morgenstern Expected Utility Theory

• E.g.1. Quadratic Utility function

• E.g.2. Power Utility function

0)(

2)(

2

0)(

2

2

)( 2

dw

RRAd

bwab

RRA

dw

ARAd

bwa

bARA

bwawwU

0)(

2

0)(2

)( 1

dw

RRAdRRA

dw

ARAd

wARA

wwU

11

A.Von-Neumann-Morgenstern Expected Utility Theory

5. Mean-Variance(M-V) Utility function – Assume– Indifference curves of risk averters:

)),~

((~~ 2RENR

dZZfZEUUE

dRERfRUUE

ERUU

w

wwR

j

j

jj

)1,0;()~

()(

),;~

()~

()(

),;~

(

~~

0

0

12

A.Von-Neumann-Morgenstern Expected Utility Theory

0)1,0;()~

(Denum

?)1,0;()

~(

)1,0;(~

)~

(

0)1,0;(~

)~

()1,0;()~

(

0)1,0;()~

)(~

()(

dZZfZEU

dZZfZEU

dZZfZZEU

d

dE

dZZfZZEUdZZfZEUd

dE

dZZfZd

dEZEU

d

UdE

(+) (+)

(+) (+)

?

13

A.Von-Neumann-Morgenstern Expected Utility Theory

0)1,0;(~

)~

(Num dZZfZZEU )(0)(

)(slope

d

dE

14

A.Von-Neumann-Morgenstern Expected Utility Theory

• Convexity: Let be two points on the same

indifference curve.

),(),,( 2211 EBEA

)2

,2

( 2121 EEC

15

A.Von-Neumann-Morgenstern Expected Utility Theory

)]([)]([)]22

([

)]~

([)]~

22([

)~

(2

1)

~(

2

1]

~22

[

)](2

1)(

2

1[)]([

:

22112121

112121

22112121

EUEEUEEE

UE

ZEUEZEE

UE

ZEUZEUZEE

U

BRARUCRU

averterRisk

16

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

)(F wx

iii wwywx

wwywx

allfor)(G)(F

allfor)(G)(F

)(G wy

17

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

w

wwxwy

wwxwyi

somefor)(F)(G

,allfor0)](F)(G[

18

A.Von-Neumann-Morgenstern Expected Utility Theory

3) Mean-Variance Paradox

19

A.Von-Neumann-Morgenstern Expected Utility Theory

a. Mean Variance Analysis

20

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

21

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*

22

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

23

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

24

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