Chapter 2 - Utility Theory Flashcards
Expected Utility Theorem (EUT)
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)
4 axioms which EUT can be derived from
CTIC (CT is cool)
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1) COMPARABILITY
2) TRANSITIVITY
3) INDEPENDENCE
4) CERTAINTY EQUIVALENCE
Comparability
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)
Transitivity
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
Independence
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
Certainty equivalence
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.
What are the 2 characteristics of utility functions?
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
Risk-seeking investor
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
Additive or absolute gamble
U(cw) = E[U(w+x)] where cx = cw - w = certainty equivalent of the gamble
Multiplicative or proportional gamble
U(cw) = E[U(w*y)]
What’s the relationship between ARA (absolute risk aversion) & cx (certainty equivalent)?
increasing / decreasing ARA <–> increasing / decreasing |cx|
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directly proportional
What’s the relationship between RRA (absolute risk aversion) & cx/w (certainty equivalent)?
increasing / decreasing RRA <–> increasing / decreasing |cx/w|
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directly proportional
ARA (absolute risk aversion)
A(w) = -U’‘(w) / U’(w)
RRA (relative risk aversion)
R(w) = -w(U’‘(w) / U’(w)) = wA(w) `
Construction of utility functions
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)
State-dependent utility functions
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
Maximum premium calculation
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.
Minimum premium calculation
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.
Limitations of utility theory
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.
