Chapter 2 - Utility Theory

Question 1

Expected Utility Theorem (EUT)

Answer 1

states that:
- a function, U(w) can be constructed representing an investor’s utility of wealth, w
- the investor faced with uncertainty makes decisions on the basis of maximising the E(utility) *
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*i.e highest E(U) not highest E(w)

Question 2

4 axioms which EUT can be derived from

Answer 2

CTIC (CT is cool)
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1) COMPARABILITY
2) TRANSITIVITY
3) INDEPENDENCE
4) CERTAINTY EQUIVALENCE

Question 3

Comparability

Answer 3

An investor can state a preference between all available certain outcomes.
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U(A)>U(B), U(B)>U(A) or U(A) = U(B)

Question 4

Transitivity

Answer 4

If A is preferred to B & B is preferred to C then A is preferred to C.
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U(A)>U(B) and U(B)>(C) –> U(A)>U(C)
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This implies that investors are consistent in their rankings of outcomes

Question 5

Independence

Answer 5

If an investor is indifferent between 2 certain outcomes, A&B, then they are also indifferent between the 2 ffg gambles:
i) A with probability p & C with prob (1-p)
ii) B with probability p & C with prob (1-p)
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if U(A)=U(B) and U(C)=U(C then
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pU(A) + (1-p)U(C) = pU(B) + (1-p)U(C)
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The choice between any 2 certain outcomes is independent of all other certain outcomes

Question 6

Certainty equivalence

Answer 6

Suppose that A is preferred to B & B is preferred to C, then there exists a unique probability, p, s.t the investor is indifferent between B & a gamble giving A with prob p, and C w prob (1-p).
**********Thus if:
U(A)>U(B)>U(C) then there exists a unique p (0<p<1) s.t:
pU(A) +(1-p)U(C) = U(B) …. B is the ‘certainty equivalent’
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It represents the certain outcome or level of wealth that yields the same certain utility as the e(utility) yielded by the gamble or lottery involving outcomes A & C.

Question 7

What are the 2 characteristics of utility functions?

Answer 7

Investor’s risk-return preferences are described by the form of their utility function. It’s assumed that they are:
1) NON-SATIATED (prefer more to less) — U’(w) > 0
2) RISK-AVERSE — U’‘(W) < 0
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U’(W) = marginal utility of wealth
R.A -> investor values an incremental increase in wealth less highly than an incremental decrease & will reject a fair gamble.
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fair gamble is one where the e(wealth)= 0 or unchanged

Question 8

Risk-seeking investor

Answer 8

def: values an incremental increase in wealth more highly than an incremental decrease & will seek a fair gamble.
U’‘(W) > 0
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convex utility function
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for RISK-NEUTRAL, U’‘(w) =0

Question 9

Additive or absolute gamble

Answer 9

U(cw) = E[U(w+x)] where cx = cw - w = certainty equivalent of the gamble

Question 10

Multiplicative or proportional gamble

Answer 10

U(cw) = E[U(w*y)]

Question 11

What’s the relationship between ARA (absolute risk aversion) & cx (certainty equivalent)?

Answer 11

increasing / decreasing ARA <–> increasing / decreasing |cx|
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directly proportional

Question 12

What’s the relationship between RRA (absolute risk aversion) & cx/w (certainty equivalent)?

Answer 12

increasing / decreasing RRA <–> increasing / decreasing |cx/w|
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directly proportional

Question 13

ARA (absolute risk aversion)

Answer 13

A(w) = -U’‘(w) / U’(w)

Question 14

RRA (relative risk aversion)

Answer 14

R(w) = -w(U’‘(w) / U’(w)) = wA(w) `

Question 15

Construction of utility functions

Answer 15

One approach involves questioning individuals about their preferences. Questioning may be direct/indirect, it’s done so that the shape of an individual’s utility function can be roughly determined. -> via fit of least squares method (framed in terms of how much an individual is prepared to pay for insurance against various risk)

Question 15

State-dependent utility functions

Answer 15

def: used to model an investor’s behaviour over all possible levels of wealth with a single utility function. This can be overcome by using:
i) utility functions with the same functional form but different parameters over different ranges of wealth, or
ii) state-dependent utility functions, which model the situation where there is a discontinuous change in state of the investor at a certain level of wealth.
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depending on different states for instance, insolvency or HSD model

Question 16

Maximum premium calculation

Answer 16

E[U(a-X)] = U(a-P)
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The maximum premium, P, which an individual will be prepared to pay in order to insure themselves against a random loss X is given by the solution of the above equation where a is the initial level of wealth.

Question 17

Minimum premium calculation

Answer 17

E[U(a+Q-Y)] = U(a)
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The minimum premium, Q, which an insurer should be prepared to charge for insurance against a risk with potential loss Y is given by solution to the above equation where a is the initial wealth of the insurer.

Question 18

Limitations of utility theory

Answer 18

FSC
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1) need to know the precise FORM & shape of the individual’s utility function (this info is normally not available)
2) EUT cannot be applied SEPARATELY to each several sets of risk choices facing an individual
3) for CORPORATE RM, it may not be possible to consider a utility function for the firm as though the firm was an individual.