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
I.e. is an increasing function
Define a risk-averse investor.
- A risk-averse investor values an incremental increase in wealth less than than an incremental decrease and will reject a fair gamble.
- They exhibit diminishing marginal utility of wealth and therefore have a concave utility function 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
What does it mean by having increasing/decreasing/constant absolute risk aversion?
Increasing - the wealthier you get the less you are willing to invest in risky assets
Decreasing - the wealthier you get the more you are willing to invest in risky assets
Constant - the wealthier you get the amount you are willing to invest in risky assets stays the same
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 of the gamble and wealth (Cw) is the value that would give you the same level of utility that taking the gamble would.
Cw= w+Cx
What is the formula for the certainty equivalence of an additive gamble.
U(C_w) = E(U(w+x))
We require U(w+c_x) = U(c_w)
> w+c_x = c_w
> c_x = c_w - w
C_w is the certainty equivalent of the wealth and the gamble
C_x is the certainty equivalent of the gamble only.
What is the formula for the quadratic utility function and what are its attributes?
U(w) = w+dw^2
Increasing ARA
Increasing RRA
What is the formula for the quadratic utility function and the condition for risk aversion and non-satiation?
U(w) = w+dw^2
For risk aversion, U’‘(w) <0, therefore
U’’(w) = 2d and therefore d<0
For Non-satiation, U’(w) >0
U’(w) = 1+2d but we know d must be negative
1+2dw >0
w < -1/2d (as d<0 so sign changes)
What is the formula for the log utility function and what are its attributes?
U(w) = log(w) (w>0)
Declining ARA
Constant RRA (iso-elastic)
What is the formula for the power utility function and what are its attributes?
U(W)=(w^gamma -1)/gamma (w>0)
Declining ARA
Constant RRA (iso-elastic)
What attributes do the utility function U(w)=sqrt(w) have?
U(w) = sqrt(w)
Decreasing ARA
Constant RRA (the proportion in risk assets will stay the same)
What is the rational behind the maximum premium formula?
The maximum premium you would pay is the utility of your initial wealth minus you premium that is equal to the expected utility of taking on the risk. Hence, the formula is:
U(a-P) = E(U(a-X))
X is the random loss
a is initial wealth
P is premium
What is the rational behind the minimum premium an Insurance company is prepared to charge formula?
The minimum premium is the utility the insurer would have without taking on the risk which equates to the insurers expected utility of the insurers initial wealth, plus the minimum premium (Q) minus the random loss incurred (Y).
U(a) = E(U(a+P-Y))
E(U(a-P+y)) = ProbU(a+p-y) + ProbU(a+p)
