1. Expected Utility Theory
How to deal with uncertainty

2. Introduction
we have talked about individual decision making in the absence of uncertainty
in reality, we usually make decision under uncertainty
example:
1. uncertainty from product quality (second-hand vehicle)
2. uncertainty in dealing with others
-> often the outcome depends on what others do
3. purchase of financial assets (stocks and bonds) whose return is contingent on which state is realized.
This is the essence of Financial Economics

3. 2 Goals 1) Individual maximizes their expected Utility10
0.4
E(W) = 0.4(10) + 0.6(2) = 5.2
Asset i
E[U(W)] = 0.4U(10) + 0.6U(2) = ?
0.6 2
Prefer the one with higher E[U(W)] 9
0.3
Asset j
E(W) = 0.3(9) + 0.7(4) = 5.5
E[U(W)] = 0.3U(9) + 0.7U(4) = ?
0.7 4
2) Individual preferences over risk and return y C2
Return x C1
Risk

4. Probability
probability of an event occurring is the relative frequency with which the event occurs
if αi = the probability of event i occurring and there are n possible events (states) then
1. αi > 0, i = 1…n
2.  αi = 1
Lottery (X) with prizes (outcomes, states, events)
X1, X2, X3,...,Xn with corresponding probabilities
α1, α2, α3,...,αn , respectively (mutually exclusive and exhaustive)
then the expected value of this lottery is
E(X) = α1X1 + α2X2 + α3X αnXn
E(X) =  αiXi n
‘i=1 n
‘i=1

5. Example 1 Gamble (X) flip of a coin if heads, you receive $1 X1 = +1
if tails, you pay $1 X2 = -1
E(X) = (0.5) (1) + (0.5) (-1) = 0
if you play this game many times, it is likely that you break-even

6. Example 2 Gamble (X) flip of a coin if heads, you receive $10 X1 = +10
if tails, you pay $1 X2 = -1
E(X) = (0.5) (10) + (0.5) (-1) = 4.50
if you play this game many times, you will be a big winner
How much would you pay to play this game:
perhaps as much as a $4.50
But of course the answer depends upon your preference to risk

7. Fair Gambles if the cost to play = expected value of
these gambles the outcome
then the gamble is said to be actuarially fair
Common empirical findings:
1. individuals may agree to flip a coin for small amounts of money, but usually refuse to bet large sums of money
2. people will pay small amounts of money to play actuarially
unfair games (Lotto 649, where cost = $1, but E(X) < 1)
- but will avoid paying a lot
Why do these empirical findings occur? Becoz’ it is not about E(W)

8. St. Petersburg Paradox Gamble (X):
A coin is flipped until a head appears You receive $2n where n is the flip on which the head occurred
states: X1 = $2 X2 = $4 X3 = $8 ... Xn = $2n
prob: α1 = 1/2 α2 = 1/4 α3 = 1/ αn = 1/2n
E(X) =
Paradox: no one would pay an “actuarially fair” price to play this game (no one would even pay close to the fair price)

9. Explaining the St. Petersburg Paradox
this paradox arises because individuals do not make decisions based purely on wealth, but rather on the utility of their expected wealth
if we can show that the marginal utility of wealth declines as we get more wealth, then we can show that the expected value of a game is finite
Assume U(X) = ln(X) U'(X)=1/x > 0 MU positive
U"(X)=-1/x2 < 0 Diminishing MU
E(U(W)) = E( αi U(Xi)) = ( αi ln(Xi)) = < 
an individual would pay an amount up to 1.39 units of utility to play this gamble  
‘i=1
‘i=1

10. Expected Utility Theory
Objective: to develop a theory of rational decision-making under uncertainty with the minimum sets of reasonable assumptions possible
the following five axioms of cardinal utility provide the minimum set of conditions for consistent and rational behaviour
What do these axioms of expected utility mean?
  1. all individuals are assumed to make completely rational decisions (reasonable)
2. people are assumed to make these rational decisions among
thousands of alternatives
(hard)

11. 5 Axioms of Choice under uncertainty
A1.Comparability (also known as completeness).
For the entire set of uncertain alternatives, an individual can say either that
either x is preferred to outcome y (x > y)
or y is preferred to x (y > x)
or indifferent between x and y (x ~ y).
A2.Transitivity (also know as consistency).
If an individual prefers x to y and y to z, then x is preferred to z. If (x > y and y > z, then x > z).
Similarly, if an individual is indifferent between x and y and is also indifferent between y and z, then the individual is indifferent between x and z. If (x ~ y and y ~ z, then x ~ z).

12. 5 Axioms of Choice under uncertainty
A3.Strong Independence.
Suppose we construct a gamble where the individual has a probability α of receiving outcome x and a probability (1-α) of receiving outcome z. This gamble is written as:
G(x,z:α)
Strong independence says that if the individual is indifferent to x and y, then he will also be indifferent as to a first gamble set up between x with probability α and a mutually exclusive outcome z, and a second gamble set up between y with probability α and the same mutually exclusive outcome z.
  If x ~ y, then G(x,z:α) ~ G(y,z:α)
NOTE: The mutual exclusiveness of the third outcome z is critical to the axiom of strong independence.

13. 5 Axioms of Choice under uncertainty
A4.Measurability. (CARDINAL UTILITY)
If outcome y is less preferred than x (y < x) but more than z (y > z),
then there is a unique probability α such that:
the individual will be indifferent between
[1] y and
[2] A gamble between x with probability α
z with probability (1-α).
In Maths,
if x > y > z or x > y > z ,
then there exists a unique α such that y ~ G(x,z:α) 

14. 5 Axioms of Choice under uncertainty
A5.Ranking. (CARDINAL UTILITY)
If alternatives y and u both lie somewhere between x and
z and we can establish gambles such that an individual is
indifferent between y and a gamble between x (with probability α1)
and z, while also indifferent between u and a second gamble, this
time between x (with probability α2) and z, then if α1 is greater
than α2, y is preferred to u.
If x > y > z and x > u > z
then if y ~ G(x,z:α1) and u ~ G(x,z:α2),
  then it follows that if α1 > α2 then y > u,
or if α1 = α2, then y ~ u

15. One more assumption People are greedy, prefer more wealth than less.
The 5 axioms and this assumption is all we need in order to develop a expected utility theorem and actually apply the rule of
max E[U(W)] = max ∑iαiU(Wi)

16. Utility Functions Utility functions must have 2 properties
1. order preserving: if U(x) > U(y) => x > y
2. Expected utility can be used to rank combinations of risky
alternatives:
U[G(x,y:α)] = αU(x) + (1-α) U(y)
Deriving Expected utility theorem, one of the most elegant derivations in Economics, is tough. Don’t worry about a formal derivation. Just apply it.
Remark:
  Utility functions are unique to individuals
- there is no way to compare one individual's utility function with
another individual's utility
- interpersonal comparisons of utility are impossible
if we give 2 people $1,000 there is no way to determine who is happier

17. One more element: Risk Aversion
Consider the following gamble:
Prospect a prob = α G(a,b:α)
prospect b prob = 1-α
Question: Will we prefer the expected value of the gamble with
certainty, or will we prefer the gamble itself?
ie. consider the gamble with
10% chance of winning $100
90% chance of winning $0 E(gamble) = $10
would you prefer the $10 for sure or would you prefer the gamble?   
if prefer the gamble, you are risk loving
if indifferent to the options, risk neutral
if prefer the expected value over the gamble, risk averse

18. Preferences to Risk U(W) U(W) U(W) a b W a b W a b W Risk Preferring
U(b)
U(b)
U(b)
U(a)
U(a)
U(a) a b W a b W a b W
Risk Preferring
Risk Neutral
Risk Aversion
U'(W) > 0
U''(W) = 0
U'(W) > 0
U''(W) > 0
U'(W) > 0
U''(W) < 0

19. The Utility Function U(W) 3.40 3.00 2.30 U'(W) > 0 U''(W) < 0
Let U(W) = ln(W)
U'(W) > 0
U''(W) < 0
U'(W) = 1/w
U''(W) = - 1/W2
MU positive
But diminishing
1.61 W 1 5 10 20 30

20. U[E(W)] and E[U(W)]
U[(E(W)] is the utility associated with the known level of expected wealth (although there is uncertainty around what the level of wealth will be, there is no such uncertainty about its expected value)
E[U(W)] is the expected utility of wealth is utility associated with level of wealth that may obtain
The relationship between U[E(W)] and E[U(W)] is very important

21. Expected Utility
Assume that the utility function is natural logs: U(W) = ln(W)
Then MU(W) is decreasing
U(W) = ln(W)
U'(W)=1/W
MU>0
MU''(W) < => MU diminishing
Consider the following example:
80% change of winning $5
20% chance of winning $30
E(W) = (.80)*(5) + (0.2)*(30) = $10
U[E(W)] = U(10) = 2.30
E[U(W)] = (0.8)*[U(5)] + (0.2)*[U(30)]
= (0.8)*(1.61) + (0.2)*(3.40)
= 1.97
Therefore, U[(E(W)] > E[U(W)] -- uncertainty reduces utility

22. The Markowitz Premium U(W) 3.40 1.61 W 1 5 10 30 U(W) = ln(W)
U[E(W)] = U(10) = 2.30
E[U(W)]
= 0.8*U(5) + 0.2*U(30)
= 0.8* *3.40
= 1.97
Therefore, U[E(W)] > E[U(W)]
Uncertainty reduces utility
Certainty equivalent: 7.17
That is, this individual will take
7.17 with certainty rather than
the uncertainty around the gamble
U[E(W)] = 2.30
E[U(W)] = 1.97
1.61
2.83 W 1 5 CE
= 7.17 10 30

23. The Certainty Equivalent
The Expected wealth is 10
The E[U(W)] = 1.97
How much would this individual take with certainty and be indifferent the gamble
Ln(CE) = 1.97
Exp(Ln(CE)) = CE = 7.17
This individual would take 7.17 with certainty rather than the gamble with expected payoff of 10
The difference, (10 – 7.17 ) = 2.83, can be viewed as a risk premium – an amount that would be paid to avoid risk

24. The Risk Premium Risk Premium:
the amount that the individual is willing to give up in order to avoid the gamble
Recall the gamble
80% change of winning $5
20% chance of winning $30
E(W) = (.80)*(5) + (0.2)*(30) = $10
Suppose the individual has the choice now between the gamble and the expected value of the gamble
E[U[W)] = 1.97
Certainty equivalent = $7.17
Investor would be willing to pay a maximum of $2.83 to avoid the gamble ($10 - $7.17) ie will pay an insurance premium of $2.83.
THIS IS CALLED THE MARKOWITZ PREMIUM
Ln(CE)=1.97, i.e U(CE)=E[U(W)], CE=7.17, RP= =$2.83

25. The Risk Premium Risk Premium In general,
Risk
Premium =
an individual's expected wealth,given he plays the gamble -
level of wealth the individual would accept with certainty if the gamble were removed (ie the certainty equivalent)
In general,
if U[E(W)] > E[U(W)] then risk averse individual (RP > 0)
if U[E(W)] = E[U(W)] then risk neutral individual (RP = 0)
if U[E(W)] < E[U(W)] then risk loving individual (RP < 0)
risk aversion occurs when the utility function is strictly concave
risk neutrality occurs when the utility function is linear
risk loving occurs when the utility function is convex

26. The Arrow-Pratt Premium
Risk Averse Investors
assume that utility functions are strictly concave and increasing
Individuals always prefer more to less (MU > 0)
Marginal utility of wealth decreases as wealth increases
A More Specific Definition of Risk Aversion
W = current wealth
 gamble Z and the gamble has a zero expected value
E(Z) = 0
what risk premium (W,Z) must be added to the gamble to make the individual indifferent between the gamble and the expected value of the gamble?

27. The Arrow-Pratt Premium
The risk premium  can be defined as the value that satisfies the following equation:   
E[U(W + Z)] = U[ W + E(Z) - ( W , Z)] (*)
LHS: RHS:
expected utility of utility of the current level of wealth
the current level plus
of wealth, given the the expected value of
gamble the gamble
less
the risk premium
 We want to use a Taylor series expansion to (*) to derive an expression for the risk premium (W,Z)  

28. Absolute Risk Aversion
Arrow-Pratt Measure of a Local Risk Premium (derived from (*) above)
define ARA as a measure of Absolute Risk Aversion
this is defined as a measure of absolute risk aversion because it measures risk aversion for a given level of wealth
ARA > 0 for all risk averse investors (U'>0, U''<0)
How does ARA change with an individual's level of wealth?
ie. a $1000 gamble may be trivial to a rich man, but non-trivial to a poor man
=> ARA will probably decrease as our wealth increases

29. Relative Risk Aversion
Constant RRA => An individual will have constant risk aversion to a "proportional loss" of wealth, even though the absolute loss increases as wealth does
Define RRA as a measure of Relative Risk Aversion

30. Quadratic Utility
Quadratic Utility - widely used in the academic literature
U(W) = a W - b W2
U'(W) = a - 2bW U"(W) = -2b
-U"(W) b
ARA = =
U'(W) a -2bW
d(ARA)
> 0 dW
  b
RRA =
a/W - 2b
d(RRA)
quadratic utility exhibits
increasing ARA
and increasing RRA
ie an individual with increasing RRA would
become more averse to a given percentage
loss in W as W increases
- not intuitive

31. The Empirical Evidence
Empirical evidence (Friend and Blume (1975)) indicates that individuals have decreasing ARA and constant RRA = 2
Power Utility Function
U(W) = -W-1
U'(W) = W-2 > 0
U"(W) = -2W-3 < 0
ARA = 2/W => dARA/dW < 0
RRA = 2W/W = 2 => dRRA/dW = 0
This power utility function is consistent with the empirical evidence of Friend and Blume (1975)
U(W) = -1/W

32. An Example
U=ln(W) W = $20,000
G(10,-10: 50) 50% will win 10, 50% will lose 10
What is the risk premium associated with this gamble?
Calculate this premium using both the Markowitz and Arrow-Pratt Approaches

33. Arrow-Pratt Measure  = -(1/2) 2z U''(W)/U'(W)
U'(W) = (1/W) U''(W) = -1/W2
U''(W)/U'(W) = -1/W = -1/(20,000)
 = -(1/2) 2z U''(W)/U'(W) = -(1/2)(100)(-1/20,000) = $0.0025

34. Markowitz Approach E(U(W)) =  piU(Wi)
E(U(W)) = (0.5)U(20,010) + 0.5*U(19,990)
E(U(W)) = (0.5)ln(20,010) + 0.5*ln(19,990)
E(U(W)) =
ln(CE) =  CE = 19,
The risk premium RP = $0.0025
Therefore, the AP and Markowitz premia are the same

35. Markowitz Approach
E(U(W)) =
19,990
20,000
20,010 CE

36. Differences in two approaches
Markowitz premium is an exact measures whereas the AP measure is approximate
AP assumes symmetry payoffs across good or bad states, as well as relatively small payoff changes.
It is not always easy or even possible to invert a utility function, in which case it is easier to calculate the AP measure
The accuracy of the AP measures decreases in the size of the gamble and its asymmetry

37. Mean & Variance as choice variables
With 5 axioms, prefer more to less, we have Expected utility theorem
With risk-aversion assumption, we solve St. Petersburg’s paradox
With returns of risky assets being jointly normally distributed, we can draw indifference curves on the plane of return (E(r)) and risk (var(r) or standard deviation of r) as the right diagram
Essentially, mean and variance are the choice variables investors concern about in order to max their E[U(w)]
Return
Risk