Chapter 2: Utility Theory

Question 1

What is Utility

Answer 1

The satisfaction that an individual receives from a particular course of action

Question 2

Two important economic terms are non-satiation and risk aversion
Define each term

Answer 2

Non-satiation - preference of more to less
Risk aversion - Dislike of risk

Question 3

State the expected utility theorem

Answer 3

1. U(w) can be constructed representing an investor’s utility of wealth at some future date
2. Decisions are to maximise expected utility given different probability outcomes

Question 4

Expected utility is usually not the same as expected wealth, explain why this might be the case.

Answer 4

High expected wealth can come from high risk investments, which decrease utility for a risk averse investor.

Question 5

True or False, explain

The utility function will determine whether utility increases or decreases with wealth

Answer 5

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.

Question 6

State the expected utility theory axioms

Answer 6

**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)

Question 7

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

Answer 7

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

Question 8

Define the certainty equivalent of the gamble and the certianty equivalent of the portfolio

Answer 8

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)

Question 9

Relate Cx and Cw to risk aversion

Answer 9

For a risk averse investor, Cw<w => Cx<0
Cx is a measure of risk aversion, a higher |Cx| implies higher risk-aversion

Question 10

Relate cx to absolute risk aversion

Answer 10

If |Cx| decreases with an increase in wealth, then the investor exhibits decline absolute risk-aversion

Question 11

Relate cx to relative risk aversion

Answer 11

If |Cx/w| decreases with an increase in wealth, then the investor exhibits declining relative risk-aversion

Question 12

Write down the formula for A(w) and R(w)

state the relationship between them

Answer 12

A(w) = -U’‘(w)/U’(w)
R(w) = -w*U’‘(w)/U’(w)

R(w) = wA(w)

Question 13

Tabulate the relationship between A’(w), R’(w) with relative and absolute risk aversion

Answer 13

Less than zero - declining
Equal to zero - constant
Greater than zero - increasing

Question 14

State the quadratic utility function
* its form
* its limitation
* Absolute and relative risk-aversion

Answer 14

• U(w) = w + dw^2
• 0 < w < -1/2d, d < 0
• Both increasing ARA and RRA

Question 15

State the log utility function
* its form
* its limitation
* Absolute and relative risk-aversion

Answer 15

• U(w) = ln(w)
• w > 0
• Declining ARA and cconstant RRA (iso-elastic, allows for myopic decisions)

Question 16

State the power utility function
* its form
* its limitation
* Absolute and relative risk-aversion

Answer 16

• U(w) = (w^r-1)/r
  ~~~
  w > 0 , r < 1, r =! 0

~~~
* Decliining ARA and constant RRA

Question 17

State the negative exponential utility function
* its form
* its limitation
* Absolute and relative risk-aversion

Answer 17

• U(w) = -exp(-aw)
• a > 0
• Constant ARA and increasing RRA

Question 18

Two ways to construct an individual’s utility function

Answer 18

• Direct questioning
• Indirect questioning

Question 19

How can we construct utility functions by direct questioning

Answer 19

• 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

Question 20

How can one construct a utility function by indirect questioning

Answer 20

• 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

Question 21

How is utility theory applied in insurance

Answer 21

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

Question 22

Two ways to construct an individual’s utility function

Answer 22

• Direct questioning
• Indirect questioning

Question 23

Limitations of utility theory

Answer 23

• 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