Risk Aversion and Certainty Equivalent Wealth
FNCE 3P96: Financial Theory
4. Utility Theory
Text book: Chapter 3
Dr. Walid Ben Omrane
1
3-A-Five axioms of choice under uncertainty
• Rational behavior should be characterized by 5 assumptions known as the axioms of cardinal utility (by Von Neumann and Morgenstern )
If all the individuals are assumed to always make rational decisions, it implies that all of the 5 axioms of cardinal utility are satisfied:
•
Then, their utility function will have 2 properties:
1.
It will be order preserving: if U(X)>U(Y), then X better than Y
2.
The expected utility can be used to rank combinations of risky alternatives. 2
3-B-Developing Utility Functions
• In general we can write the expected utility of the wealth as follows:
E U (W ) pU (Wi ) i i
1.
2.
Given the rational behavior of investors.
And the assumption that all investors prefer more wealth to less : U’(W)>0
–
Investors will always seek to maximize their expected utility of wealth. –
All investors will use it as their objective function.
3
3-C-Establishing a definition of risk aversion
•
Utility functions are specific to individuals
•
Every individual has his own subjective utility function (that embodies his preferences) •
It follows that group utility functions, such as the utility function of a firm makes no sense !
•
Suppose that we have the following gamble:
G(16,4,50%) i.e. the gamble pays $16 and $4 with 50% probability, respectively.
Question: will we prefer the actuarial value of the gamble (i.e. its expected outcome = 50%(16+4)=$10) with certainty or the gamble it self ?
1.
The person who prefers the gamble is a risk lover
2.
The person who is indifferent is risk neutral
3.
The person who prefers the expected value with certainty is risk averse
4
3-C-Establishing a definition of risk aversion
Example :
Flipping a (fair) coin (i.e. probability = 50% for each side)
4. Utility Theory
Text book: Chapter 3
Dr. Walid Ben Omrane
1
3-A-Five axioms of choice under uncertainty
• Rational behavior should be characterized by 5 assumptions known as the axioms of cardinal utility (by Von Neumann and Morgenstern )
If all the individuals are assumed to always make rational decisions, it implies that all of the 5 axioms of cardinal utility are satisfied:
•
Then, their utility function will have 2 properties:
1.
It will be order preserving: if U(X)>U(Y), then X better than Y
2.
The expected utility can be used to rank combinations of risky alternatives. 2
3-B-Developing Utility Functions
• In general we can write the expected utility of the wealth as follows:
E U (W ) pU (Wi ) i i
1.
2.
Given the rational behavior of investors.
And the assumption that all investors prefer more wealth to less : U’(W)>0
–
Investors will always seek to maximize their expected utility of wealth. –
All investors will use it as their objective function.
3
3-C-Establishing a definition of risk aversion
•
Utility functions are specific to individuals
•
Every individual has his own subjective utility function (that embodies his preferences) •
It follows that group utility functions, such as the utility function of a firm makes no sense !
•
Suppose that we have the following gamble:
G(16,4,50%) i.e. the gamble pays $16 and $4 with 50% probability, respectively.
Question: will we prefer the actuarial value of the gamble (i.e. its expected outcome = 50%(16+4)=$10) with certainty or the gamble it self ?
1.
The person who prefers the gamble is a risk lover
2.
The person who is indifferent is risk neutral
3.
The person who prefers the expected value with certainty is risk averse
4
3-C-Establishing a definition of risk aversion
Example :
Flipping a (fair) coin (i.e. probability = 50% for each side)