(Expected Utility Theory - EUT).
1
(Expected Utility Theory)
.
•
von NeumannMorgenstern
• Risk Attitudes
W
o
Risk Averse
▪
▪ U(W)
Concave
U′(WU′′(W)
▪
o Risk Neutral
▪
▪ U(W)
LinearU′(W
U′′(W)=0
▪
o Risk Seeking / Risk Lover
▪
▪ U(W)
ConvexU′(W
U′′(W)
▪
(Expected Utility Theory - EUT).
2
Expected Utility Criterion
•
•
Prospect
U(Wipi
E[U(W)]=i=1∑npiU(Wi
o E[U(W)]
o pii
o Wii
o U(Wi)i
•
Constructing Utility Functions
•
•
o Certainty Equivalent - CECE
o
Lotteries
•
o U(W)=aWbW2a,b
o
U(W)=ln(W)
o U(W)=eaWa
o Power Utility / CRRAU(W)=1γW1γγγ =1
Constant Relative Risk Aversion - CRRAγ=1
Axioms of Expected Utility Theory
• Completeness
AB
AB (AB
BA (BA
A ~ B
• TransitivityABBCACABBCA
C
•
ContinuityA
BBCp
Ap
Cp
B
(Expected Utility Theory - EUT).
3
• Independence
AB
ApCpBpC
pAB
Properties of Utility Functions
• Non-satiation
U′(W)
•
o U′′(W)⟹
o U′′(W)=0⟹
o U′′(W)⟹
• Unique up to a Positive Affine Transformation
U(W)V(W)=a+bU(W)ab
b
Risk PremiumPratt-Arrow
• Risk Premium - RP
o
o
E[W]CERP=E[W]CE
o CE
U(CE)=E[U(W)]
o
CEE[W]RPCE=E[W]RP=0CEE[W]RP
• Arrow-Pratt
W
o Absolute Risk Aversion - ARAARA(W)=U′(W)U′′(W)
▪ ARA(W)
W (DARA - Decreasing ARA)
▪ ARA(W)
CARA - Constant ARA
▪ ARA(W)
W (IARA - Increasing ARA)
o Relative Risk Aversion - RRARRA(W)=WU′(W)U′′(W)=W×ARA(W)
(Expected Utility Theory - EUT).
4
▪ RRA(W)
W (DRRA)
▪ RRA(W)
CRRA - Constant RRA
▪ RRA(W)
W (IRRA)
Expected ValueVariance
• Expected ValueE[X]=i=1∑npixi
• Variance
Standard DeviationVar(X)=σ2=E[(XE[X])2]=i=1∑npi(xiE[X])2
SD(X)=σ=Var(X)
• Mean-Variance Analysis
1 Quadratic Utility
2
Normal Distribution
Skewness
Paradoxes
Prospect Theory
•
Allais Paradox
o
Certainty Effect
o
•
Ellsberg Paradox
o
RiskAmbiguity
subjective probabilities
o
(Expected Utility Theory - EUT).
5
Ambiguity Aversion
• Framing Effects
o
Invariance
o
•
EUTOverconfidence
Anchoring
Herding Behavior
Loss Aversion
