Ch2 Expected utility representation
and risk aversion
2.1 Expected utility representation
Problem: ,
: subjective preference and objective probability
Target:
Subjective preference utility function
Objective probability random variable
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1. Expected utility representation
, if there exists a , such that ,
has an expected utility representation. Problem: the existence of the expectation utility representation.
2. Axiom system: preference relation
(1) Completeness and transitivity
(2) Continuity
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(3) Independence:
3. Expected utility theorem: Suppose a preference relation satisfies the axiom system, then has an expected utility representation.
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4. Discussion of the expected utility theory
Allais paradox
A: (1 million, 100%)
B: (5 million, 10%; 1 million, 89%; 0, 1%)
C: (5 million, 10%; 0, 90%)
D: (1 million, 11%, 0, 89%)
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Result: A B, C D
0.11*(
==C Contradiction
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Explanation:
(1) Traditional finance: framework Assumption Reality
(2) Traditional finance: framework Change the axiom system
(3) Behavioral finance
Traditional finance:
Problem:
We need additional assumption.
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2.2 Risk aversion
1. Risk: uncertain rate of return
(1) Outcomes
(2) Probabilities
2. Measurement of risk:
(1) Distribution
All possible outcomes:
Probabilities:
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(2) Statistics
Mean: , weighted average, the first moment
Risk: deviation from the mean
Mean deviation:
Mean absolute deviation (MAD):
Variance: , second central moment Variance deviation from the mean risk
Note: Mean and variance are not enough to describe the distribution. But the importance of all moments beyond variance is much small than the expected value and variance. So variance risk.
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3. Portfolio risk
(1) Portfolio
: the ratio of wealth invested in security to the total wealth.
(2) The rate of return of security :
(3) The rate of return of portfolio :
Proof:
(4) The expected rate of return of portfolio :
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(5) Portfolio risk when
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(6) Portfolio risk when
risk of security
weight of security
?
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(7)
: tend to move together
: tend to move in opposite direction.
(8) Risk management
Hedging: purchase a security that is negatively correlated with a existing security.reduce risk
Diversification: purchase a wide variety of securities
such that reduce risk
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4. Risk aversion: new assumption on
(1) Definition:
, is risk averse.
, is risk neutral.
, is risk seeking.
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(2) Sufficient and necessary condition: an agent is risk averse if and only if
Proof:
Sufficiency:
,
Necessity:
,
Risk aversion
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(3) Sufficient and necessary condition: an agent is risk averse if and only if , (is a concave function)
Proof:
+
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5. Degree of risk aversion
(1) Certainty equivalent: the amount of wealth that one is ready to pay to escape a zero-mean risk
+
, small risk, subjective risk aversion and objective risk
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(2) Absolute risk aversion
Decreasing absolute risk aversion: Constant absolute risk aversion: Increasing absolute risk aversion: Common case: or
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(3) Relative risk aversion Absolute:
Relative: unit wealth
Relative risk aversion
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6. Classical utility function
(1) Quadratic utility function
Advantage:
, a function of the first two moments. Problem:
,
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(2) Constant absolute risk aversion (CARA) utility function
,
Advantage: when
(3) Constant relative risk aversion (CRRA) utility function
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