Utility
The satisfaction that an individual obtains from a particular course of action
Utility Theory
- A framework (Normative Model) to explain how financial decisions can be made rationally, in light of uncertainty
- Can explain why rational investors won’t necessarily maximise expected wealth (eg insurance/gambling)
The Expected Utility Theorem
- States that a function, U(w), can be constructed representing an investor’s utility of wealth
- States that investors faced with uncertainty makes decisions based on the decision on the basis of maximising the expected value of utility
4 Axioms of utility theory
- Comparability
- Transitivity
- Independence
- Certainty equivalence
Comparability
An investor can state a preference between all available certain outcomes
- eg U(A)>U(B) or U(A)=U(B)
Transitivity
If A is preferred to B and B is preferred to C
- U(A)>U(B) & U(B)>U(C) ==> U(A)>U(C)
- U(A)=U(B) & U(B)=U(C) ==> U(A)=U(C)
Independence
If an investor is indifferent between two uncertain outcomes, A and B, then they are also indifferent between the following two gambles:
- A with prob p and C with prob (1-p)
- B with prob p and C with prob (1-p)
Certainty equivalence
Suppose 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 prob p and C with prob (1-p)
Non-satiation
- People prefer more wealth to less
Risk-Averse Investor
- A risk-averse investor values an incremental increase in wealth less highly than an incremental decrease and will reject a fair gamble (Where E[W] remain unchanged)
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
Risk-neutral Investor
- A risk-neutral investor is indifferent between taking a fair gamble or not
Absolute Risk-Aversion
If the absolute value of the certainty equivalent decreases with increasing wealth the investor is said to exhibit declining ARA
ie. Inc/Dec ARA <==> Inc/Dec |Cx|
This assumes Cx<0 for fair gamble and risk-averse
Relative Risk-Aversion
If the absolute value of the certainty equivalent decreases as a proportion of total wealth increases the investor is said to exhibit declining RRA
ie. Inc/Dec RRA <==> Inc/Dec |Cx/W|
This assumes Cx<0 for fair gamble and risk-averse
Limitations of Utility Theory
- We need to know the precise form and shape of the individual’s utility function. Usually, the best we can hope for is to identify a general feature such as risk aversion and to use the rule to identify broad types of choice
- The theorem cannot be applied seperately to each of several sets of risky choices facing an individual
- For corporate risk management, it may not be possible to consider a utility function for the firm as though the firm was an individual since firms are a coalition of different interest groups
