Chapter 2 - Utility Theory Flashcards
Define ‘Utility’.
Utility is the satisfaction obtained by an individual from a particular course of action.
What is ‘The Utility Theorem’
- A utility function, U(w), can be constructed as representing an investor’s utility of wealth, w, at some future date.
- Investors should base decisions on what will achieve the highest expected utility, given their beliefs of the different outcomes.
How would you find the proportion of wealth, say a, to invest into a risky share to maximise expected utility?
- Find your equation for expected utility with the parameter a
- Differentiate E(U(W)) with respect to a and set equal to 0 to calculate a
- Differentiate E(U(w)) twice and substitute a found in part 2 and in order for this to be a maximum then the second derivate must be <0.
4.Substitute your value of a back into E(U(w)) to calculate your expected utility.
What are the 4 axioms of Utility Theorem?
- Comparability
- Transitivity
- Independence
- Certainty Equivalence
What do the 4 axioms of Utility Theorem infer?
An investor whose behaviour is consistent with these axioms will always make decisions in accordance with the expected utility theorem.
What is the ‘Comparability’ axiom of the utility theorem?
The investor is able to state a preference of all certain outcomes.
i.e.
U(A) > U(B) - A is preferred to B
U(B) > U(A) - B is preferred to A
U(A) = U(B) - indifferent between A & B
What is the ‘Transitivity’ axiom of the utility theorem?
If A is preferred to B and B preferred to C, then A if preferred to B. i.e.
U(A) > U(B) and U(B)>U(C) then U(A)>U(C)
What is the ‘Independence’ axiom of the utility theorem?
If an investor is indifferent between two certain outcomes, A and B, then they are also indifferent between the following two gambles:
(i) A with probability p and C with probability (1-p) and
(ii) B with probability p and
C with probability (1 - p).
Hence, if U(A) = U(B) (and of course U(C) is equal to itself), then: p U(A) + (1–p) U(C) = p U(B) + (1–p) U(C)
Define Non-satiation.
Preferring more to less i.e. U’(w)>0
Define a risk-averse investor.
Risk averse investors derive less additional utility from the prospect of a possible gain than they lose from the prospect of an identical loss with the same probability of occurrence. Therefore a risk averse investor will reject a fair gamble.
U’‘(w)<0
What shape is utility curve i.e. y=U(w), x=w for a
1. Risk averse investor
2. Risk seeking investor
- Concave - U’‘(w) <0
- Convex - U’‘(w) >0
Define a risk-seeking investor.
A risk-seeking investor values an incremental increase in wealth more highly than an incremental decrease and will seek a fair gamble.
They exhibit increasing marginal utility of wealth.
i.e. U’‘(w)>0
What is the Absolute Risk Aversion formula?
A(w) = -U’‘(w)/U’(w)
What is the Relative Risk Aversion formula?
R(w) = -w*U’‘(w)/U’(w)
Explain certainty equivalence.
The certainty equivalence is the value that give you the same level of utility that taking the gamble would.
