Chapter 2: Utility Theory Flashcards
What is Utility
The satisfaction that an individual receives from a particular course of action
Two important economic terms are non-satiation and risk aversion
Define each term
Non-satiation - preference of more to less
Risk aversion - Dislike of risk
State the expected utility theorem
- U(w) can be constructed representing an investor’s utility of wealth at some future date
- Decisions are to maximise expected utility given different probability outcomes
Expected utility is usually not the same as expected wealth, explain why this might be the case.
High expected wealth can come from high risk investments, which decrease utility for a risk averse investor.
True or False, explain
The utility function will determine whether utility increases or decreases with wealth
False, since we assume non-satiation, utility will always increase with an increase wealth, the type of utility function will only determine the rate of this increase.
State the expected utility theory axioms
**1. Comparibility **
Investos can make a choice between:
A>B, prefers A
B<A, prefers B
A=B, indifferent
2. Transivity
If U(A)>U(B) and U(B)>U(C), then U(A)>U(C)
3. Independence
If an invvestor is indifferent between A and B, then she is also indifferent between the gambles
pU(A)+(1-p)U(C) and
pU(B)+(1-p)U(C)
4. Certainty equivalance
If U(A)>U(B)>U(C), then there exists a unique p such that
pU(A) + (1-p)U(C) = U(B)
For non-satiation, risk aversion, risk-seeking and risk-neutral
- State in terms of the utility function
- State what each means about marginal utility of wealth under risk appetite
- State what the investor will do when presented with a fair gamble
- State the shape of the utility function
Non-satiation: U’(w) > 0 and increasing
Risk-aversion: U’’(w) < 0 and concave
Marginal utility decreases
Reject a fair gamble
Risk-neutral: U’’(w) = 0 constant U(w)
Marginal utility constant
Indifferent to a fair gamble
Risk-seeking: U’’(w) >0 Convex U(w)
Marginal utility increases
Accepted a fair gamble
Define the certainty equivalent of the gamble and the certianty equivalent of the portfolio
Cw - the certainty equivalent of the portfolio, consisting of the combination of existing wealth w and gamble x
U(Cw) = E[U(w+x)] for absolute gamble
U(Cw) = E[U(wy)] for propotional gamble
Cx - Certainty equivalent of the gamble x alone, which will depend on the current level of wealth (Cx = Cw -w)
Relate Cx and Cw to risk aversion
For a risk averse investor, Cw<w => Cx<0
Cx is a measure of risk aversion, a higher |Cx| implies higher risk-aversion
Relate cx to absolute risk aversion
If |Cx| decreases with an increase in wealth, then the investor exhibits decline absolute risk-aversion
Relate cx to relative risk aversion
If |Cx/w| decreases with an increase in wealth, then the investor exhibits declining relative risk-aversion
Write down the formula for A(w) and R(w)
state the relationship between them
A(w) = -U’‘(w)/U’(w)
R(w) = -w*U’‘(w)/U’(w)
R(w) = wA(w)
Tabulate the relationship between A’(w), R’(w) with relative and absolute risk aversion
Less than zero - declining
Equal to zero - constant
Greater than zero - increasing
State the quadratic utility function
* its form
* its limitation
* Absolute and relative risk-aversion
- U(w) = w + dw^2
- 0 < w < -1/2d, d < 0
- Both increasing ARA and RRA
State the log utility function
* its form
* its limitation
* Absolute and relative risk-aversion
- U(w) = ln(w)
- w > 0
- Declining ARA and cconstant RRA (iso-elastic, allows for myopic decisions)
State the power utility function
* its form
* its limitation
* Absolute and relative risk-aversion
- U(w) = (w^r-1)/r
~~~
w > 0 , r < 1, r =! 0
~~~
* Decliining ARA and constant RRA
State the negative exponential utility function
* its form
* its limitation
* Absolute and relative risk-aversion
- U(w) = -exp(-aw)
- a > 0
- Constant ARA and increasing RRA
Two ways to construct an individual’s utility function
- Direct questioning
- Indirect questioning
How can we construct utility functions by direct questioning
- Series of questions that allow the shape of an individual’s utility function to be roughly determined.
- A utility curve of a pre-determined functional form can then be fitted by a least squares method to the points determined by the answers to the questions
- Constrain using economic properties - non-satiation, risk avaersion, declining absolute risk aversion, and (maybe) constant relative risk aversion
How can one construct a utility function by indirect questioning
- Fix two values for the utility function for the extreme values of wealth being considered.
- Individual is asked to identify a certain level of wealth such that they would be indifferent between that certain level of wealth and a gamble that yield of the two extemes with particular probabilities
- The process is repeated for various scenarios until a sufficient number of plots is found.
- Can also use premiums that an individual is willing to pay
How is utility theory applied in insurance
A risk averse investor will buy insurance if
E[U(w-L)] < E[U(w-P)]
where L is the random loss and
P is the premium
Two ways to construct an individual’s utility function
- Direct questioning
- Indirect questioning
Limitations of utility theory
- We need to know the precise form and shape of the utility function.
- The theorem cannot be applied seperately to each of the several sets of risky choices facing an individual.
- May not be possible to consider the utility of a firm as though a firm was an individual
