Chapter 2 - Utility Theory

Question 1

Utility

Answer 1

Utility is the satisfaction that an individual obtains from a particular course of action

Question 2

Utility function (Preference function)

Answer 2

In the application of utility theory to finance and investment choice, it is assumed that a numerical value called the utility can be assigned to each possible value of the investor’s wealth by what is known as a preference function or utility function.

This is the key assumption underlying the application of utility theory to finance.

Question 3

The expected utility theorem

Answer 3

1. The expected utility theorem states that a function, U(w), can be constructed representing an investor’s utility of wealth, w, at some future date
2. Desicions are made on the basis of maximising the expected value of utility under the investor’s particular beliefs about the probability of different outcomes

Question 4

4 axioms underlying utility theory

Answer 4

The expected utility theorem can be derived formally from the following four axioms.

1. Comparability
2. Transitivity
3. Independence
4. Certainty equivalence

Question 5

Comparability

Answer 5

An investors can state a preference between all available certain outcomes.

Question 6

Transitivity

Answer 6

If A is preferred to B and B is preferred to C, then A is preferred to C.

Question 7

Independence

Answer 7

If an investor is indifferent between two certain outcomes, A and B, then he is also indifferent between the following two gambles

(i) A with probability p and C with probability (1-p)
(ii) B with probability p and C with probability (1-p)

Question 8

Certainty equivalence

Answer 8

Suppose that A is preferred to B and B is preferred to C. Then there is a unique probability, p, such that the investor is indifferent between B and a gamble giving A with probability p and C with probability (1-p).
B is known as the certainty equivalent of the above gamble.

Question 9

Non-satiation

Answer 9

It is usually assumed that people prefer more wealth to less. This is known as the principle of non-satiation and can be expressed as:
U’(w) > 0
(marginal utility of wealth)

Question 10

Risk-averse investor

Answer 10

A risk-averse investor values an incremental increase in wealth less highly that an incremental decrease and will reject a fair gamble.
The utility function condition is:
U’‘(w) < 0.
Concave utility function.
Risk aversion = diminishing marginal utility of wealth

Question 11

Fair gamble

Answer 11

A fair gamble is one that leaves the expected wealth of the individual unchanged. Equivalently it can be defined as a gamble that has an overall expected value of 0.

Question 12

Risk-seeking investor

Answer 12

A risk-seeking investor values an incremental increase in wealth more highly that an incremental decrease and will seek a fair gamble. The utility function condition us U’‘(w) > 0.
Convex utility function.

Question 13

Risk-neutral investor

Answer 13

A risk-neutral investor is indifferent between fair a fair gamble and the status quo. In this case U’‘(w) = 0.

Question 14

Certainty equivalent of a fair gamble for a risk-averse investor

Answer 14

For a risk-averse investor this is negative, ie the investor would have to be paid to accept the gamble.

Question 15

Absolute risk aversion

Answer 15

If the absolute value of the certainty equivalent decreases with increasing wealth, the investor is said to exhibit declining absolute risk aversion.
If the absolute value of the certainty equivalent increases, the investor exhibits increasing absolute risk aversion.

Investors who hold an increasing absolute amount of wealth in risky assets as they get wealthier exhibit declining absolute risk aversion.

Question 16

Relative risk aversion

Answer 16

If the absolute value of the certainty equivalent decreases as a proportion of the total wealth as wealth increases the investor is said to exhibit declining relative risk aversion.
If the absolute value of the certainty equivalent increases as a proportion of total weight as wealth increases the investor is said to exhibit increasing relative risk aversion.

Investors who hold an increasing proportion of their wealth in risky assets as they get wealthier are exhibiting declining relative risk aversion.

Question 17

Relationships between the first derivatives of the A(w) and R(w) functions and declining or increasing absolute and relative risk aversion

Answer 17

Absolute risk aversion Relative risk aversion
Increasing A’(w) > 0 R’(w) > 0
Constant A’(w) = 0 R’(w) = 0
Decreasing A’(w) < 0 R’(w) < 0

Question 18

3 commonly used utility functions

Answer 18

1. Quadratic utility function
2. Log utility function
3. Power utility function (part of a wider class of functions known as HARA - hyperbolic absolute risk aversion)

Question 19

Iso-elastic utility function

Answer 19

A utility function exhibiting constant relative risk aversion.
The use of iso-elastic utility functions simplifies the determination of an optimal strategy for a multi-period investment decision, because it is possible to make a series of so-called “myopic” decisions.

Question 20

“Myopic” decisions

Answer 20

The decision at the start of each period only has to consider the possible outcomes at the end of that period and can ignore subsequent periods.

Question 21

State-dependent utility functions

Answer 21

Can be used to model the situation where there is a discontinuous change in the state of the investor at a certain level of wealth.

Question 22

Discuss how a utility function may depend on current wealth / Describe a situation that requires the use of state-dependent utility functions

Answer 22

Such a situation arises when we consider an insurance company that will become insolvent if the value of its assets falls below a certain level. At asset levels just above the insolvency position, the company will be highly risk-averse and this can be modelled by a utility function that has a discontinuity at the insolvency point. However, the consequence of applying the same utility function when the company has just become insolvent would be that the company would be prepared to accept a high probability of losing its remaining assets for a chance or regaining insolvency.

Question 23

3 limitations of utility theory

Answer 23

1. To calculate expected utility we need to know the precise form and shape of the individual’s utility function.
2. The expected utility theorem cannot be applied separately to each of several sets of risky choices facing an individual.
3. For corporate risk management, it may not be possible to consider a utility function for the firm as though the firm was an individual. The firm is a coalition of interest groups, each having claims on the firm.

Question 24

State two alternative decision rules that can be used for risky choices

Answer 24

Alternative decision rules that can be used for risky choices include those under mean-variance portfolio theory and stochastic dominance.