Decisions under Uncertainty
Our Main Question
• How individuals behave when they are
dealing with uncertainty?
2
The Approach
• Goal: Develop the theory of investor choice with a
few ingredients:
– A) 5 axioms of rational choice under uncertainty
– By-Product: Expected Utility Theorem to measure utility in the
presence of risk.
– B) Assumption of non-satiation (i.e., greed)
– C) Risk-aversion
– D) Measuring the objects of choice (i.e., the assets that
investors invest) using Mean and Variance of asset
return.
– D) Mapping trade-offs between Mean and Variance that
provides indifference curves of investors.
– such trade-off reveals an investor’s degree of risk-aversion.
3
The End-product
• By the end of this lecture, we will be able to formulate
the following diagram of individual investor’s
indifference curves.
Returns
EU2
EU1
Where EU2 >EU1
Message:
Risk
[i] For the same level of risk, everyone prefers higher returns.
[ii] For the same level of returns, everyone prefers lower risk.
4
Why care about uncertainty?
Simple answer: Because in reality, almost every decision
we make involves uncertainty.
• Example:
– Uncertainty from product quality. (e.g., used vehicle, order
food before eating, any durable goods consumption)
– Uncertainty in dealing with others. (e.g., marriage, firm’s
cutting price may or may not induce price war)
– Purchase of financial assets (stocks, bonds, exchange rate
changes, etc.)
• its return depends on the future.
This is the essence of Financial Economics
5
An example to motivate
Expected utility theory says:
• When payoffs in the next period are uncertain, every individual has his unique
utility function to describe his own preference.
• With such utility function & GREED, individuals make the following decision:
– Max [Expected utility of wealth] subject to his constraints.
• Question: Suppose in this period, Robert can decide to hold either Asset i or
Asset j which both pays off in next period. This is his constraint. In the next
period, Asset i pays him $10 with probability of 0.4 and $2 with probability of
0.6. Asset j pays him $8 with probability of 0.3 and $3 with probability of 0.7.
• Answer: He chooses to hold the asset that gives him the highest expected utility
of wealth, but NOT the highest expected wealth.
0.4
$10
0.6
$2
$8
E(W) = 0.4(10) + 0.6(2) = 5.2
E[U(W)] = 0.4U(10) + 0.6U(2) = ?
Asset i
0.3
Asset j
0.7
$4
He would choose the asset that gives him highest E[U(W)]
E(W) = 0.3(8) + 0.7(3) = 4.5
E[U(W)] = 0.3U(8) + 0.7U(3) = ?
6
St. Petersburg Paradox
•
What measure happiness? To convince you that is NOT E(W), let’s consider the
following famous paradox.
“Suppose someone offers to toss a fair coin repeatedly until it comes up heads, and
to pay you $1 if this happens on the first toss, $2 if it takes two tosses to land a head,
$4 if it takes three tosses, $8 if it takes four tosses, etc. What is the largest sure gain
you would be willing to forgo in order to undertake a single play of this game?”
•
•
•
State (The number of toss a head first comes up)
Payoff
Probability
1st toss
20=$1
½
2nd toss
21=$2
(½)2 = ¼
3rd toss
22=$4
(½)3 = 1/8
The table illustrates only 3 possible states, whereas theoretically, you can construct
this table infinitely. The point is, the Expected wealth (i.e, E(W)) if we play this
gamble is infinite if the cost of playing it is finite. Again, the expected payoff of this
gamble is ∞ given finite playing cost.
You may probably pay $2, or even $10 to play this gamble, depending on your riskpreference. But you would probably not pay $1,000 to play. The point is, no one on
earth would pay any amount close to the expected payoff of this gamble to play.
Why? Because we are not maximizing our E(W), but E(U(W)). It is really the
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E(U(W)) that measures our level of happiness.
Rational decision theory
•
To develop a theory of rational decision making under uncertainty, we impose some precise yet
reasonable axioms about an individual’s behavior.
•
We assume 5 axioms of cardinal utility.
– Axiom 1: Comparability (a.k.a., completeness)
– Axiom 2: Transitivity (a.k.a., consistency)
– Axiom 3: Strong independence
– Axiom 4: Measurability
– Axiom 5: Ranking
•
What do these axioms of generally mean?
– all individuals are assumed to make completely rational decisions (reasonable)
– people are assumed to make these rational decisions among thousands of alternatives
(hard)
•
CONCERNING C49:
– The following 4 pages of slides explain each axiom in full details. You may skip them
because I would not test you more than remembering their names. However, they don’t
seem as hard as they sounds. I will go through each of them with an example in class.
– Believe it or not, it is just these 5 rules, we establish the expected utility theory. The
derivation is mathematical and I will neither go through it nor test it. If you are interested,
Chapter 3 section B is what you should read.
– With one more assumption, i.e, people are greedy, we get what we want, i.e., individual
maximizing their expected utility of wealth (i.e., Max E(U(W)) subject to constraints)8
5 Axioms of Choice under uncertainty
A1.Comparability (also known as completeness).
For the entire set, S, of uncertain alternatives, an individual can
say that
either outcome 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 known 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).
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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.
10
5 Axioms of Choice under uncertainty
A4.Measurability. (concerning about 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 Math:
if x › y > z or x > y › z ,
then there exists a unique probability α such that y ~ G(x,z:α)
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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 αy) and z, while also indifferent
between u and a second gamble, this time between x (with probability
αu) and z, then if αy is greater than αz, y is preferred to u.
If x > y > z and x > u > z
then by axiom 4, we have y ~ G(x,z:αy) and u ~ G(x,z:αz),
then it follows that if αy > αz then y › u,
or if αy = αz , then y ~ u
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Derivation of Expected utility theory
Question: How do individuals rank various combinations of risky alternatives?
•
The derivation, one of the most elegant inductive proofs of human knowledge,
uses the axioms to show how preferences can be mapped into measurable
utility.
•
That means, I can in principle tell exactly that risky asset i gives me EUi unit
of expected utility, and EUj for risky asset j. By comparing EUi and EUj, I can
claim whether I prefer i to j, or j to i, or indifferent between them.
•
In the end, expected utility theory shows that the correct ranking function for
risky alternatives is the expected utility.
•
Example again: If risky asset j gives random payoff in the next period: $10
with probability 0.5, $5 with probability 0.2, and $-10 with probability 0.3.
Then I evaluate j as
E(U(W)) = ∑i [(Prob. of state i) x (payoffs in state i)]
= 0.5x(U(10)) + 0.2x(U(5)) + 0.3x(U(-10))
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Your preference dictates U(W)
E(U(W)) = ∑i [(Prob. of state i) x (payoffs in state i)]
•
With this in mind, we do an exercise to show how your preference constructs
your unique utility function. Suppose I arbitrarily assign a utility of -10 to a
loss of $1000 and ask the following question:
– If you are faced with a gamble with prob. p of winning $1000, and prob. (1-p) of
losing $1000. What is this precise p that makes you indifferent between:
[i] taking the gamble or
[ii] getting $0 with certainty?
•
•
•
•
In math, we have:
U(0) = pU(1000) + (1-p)U(-1000)
= pU(1000) + (1-p)(-10)
Assume U(0) = 0 for yourself, and if your answer is that p = 0.6, then,
U(1000) = 6.66667.
Repeat this procedure for different payoffs, and you can work out your own
utility function.
MESSAGE: The AXIOMS of preference is convertible to a UTILITY fn.
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Explaining the St. Petersburg Paradox
• Going back to the St. Petersburg Paradox, we explain it with the help of
another big big concept: i.e., RISK-AVERSION.
• If the utility function that an individual has is marginal utility being
always positive (i.e., the greed assumption), and that it has diminishing
MU as W increases (i.e., risk-aversion), such as the following utility
function:
e.g., U(W) = ln(W) U'(W)=1/W > 0
MU positive
U"(W)=-1/W2 < 0
Diminishing MU
Then,
E(U(W)) = ∑i [(Prob. of state i) x (payoffs in state i)]
=  αi ln(Xi) = 1.39 < 
• Thus, an individual would pay an amount up to 1.39 units of expected
utility to play this gamble.
• Later on, we will define this U(CE) = ln(CE) = 1.39, => CE = 4.01485 as
the certainty equivalent.
15
Preferences to Risk: Intro
U(W)
U(W)
U(W)
U(b)
U(b)
U(a)
U(b)
U(a)
U(a)
a b
Risk-loving
U'(W) > 0
U''(W) > 0
W
a
b
Risk Neutral
U'(W) > 0
U''(W) = 0
W
a b
Risk Averse
W
U'(W) > 0
U''(W) < 0
16
Preferences to Risk: Intro
U(W)
U(W)
U(W)
U(b)
U(b)
U(a)
U(b)
U(a)
U(a)
a b
Risk-loving
W
a
b
Risk Neutral
W
a b
Risk Averse
W
Goals:
(a) Formally define what is risk-aversion.
(b) Establish the concept of risk-premium
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Risk Aversion
• Consider the following gamble:
• Outcome a
prob = α
• Outcome b
prob = 1-α
=> G(a,b: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
Question: Would you prefer the $10 for sure or would you prefer the
gamble?
[i] if prefer the gamble, you are risk loving
[ii] if indifferent to the options, risk neutral
[iii] if prefer the expected value over the gamble, risk averse
18
Risk-aversion as shown in Utility Fn
U(W)=lnW
3.40=U(30)
Risk-averse U:
Let U(W) = ln(W)
2.30=U[E(W)]
U'(W) > 0
U''(W) < 0
1.61=U(5)
U'(W) = 1/w
U''(W) = - 1/W2
MU positive
But diminishing
0
1
5
10
20
30
W
19
Risk-aversion as shown in Utility Fn
U(W)=lnW
3.40=U(30)
Risk-averse U:
Let U(W) = ln(W)
2.30=U[E(W)]
E[U(W)]
=0.8U(5) + 0.2U(30)
=0.8(1.61)+0.2(3.4)
=1.97
1.97=E[U(W)]
1.61=U(5)
Certainty Equivalent:
U(CE) = 1.97 = ln(CE)
CE=7.17
0
1
5 7.17 10
20
30
W
20
U[E(W)] VS E[U(W)]
In general,
if
U[E(W)] >
if
U[E(W)] =
if
U[E(W)] <
E[U(W)]
E[U(W)]
E[U(W)]
then risk averse individual
then risk neutral individual
then risk loving individual
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
Certainty Equivalent
Definition: The amount of money that the individual needs to hold for certainty in order to
be indifferent from playing the gamble.
As the example: This person is indifferent between
[i] holding $7.17 for certain
[ii] playing the gamble that has 80% chance with $5 and 20% with $30.
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Markowitz Risk Premium
CE’s Definition: The amount of money that the individual needs to hold for certainty in
order to be indifferent from playing the gamble.
As the example: This person is indifferent between
[i] holding $7.17 for certain
[ii] playing the gamble that has 80% chance with $5 and 20% with $30.
Risk premium: the difference between an individual’s expected wealth, given the gamble,
and the certainty equivalent wealth.
As the example: This person pays a risk-premium of:
RP = E(Gamble) – CE
= $10 - $7.17 = $2.83
Meaning: Any insurance that costs less than $2.83 that ensures him the level of wealth of
$10 will be attractive to him.
Cost of gamble: The difference between an individual’s current wealth and the certainty
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equivalent wealth.
Risk Premium VS cost of gamble
Risk premium: the difference between an individual’s expected wealth, given the
gamble, and the certainty equivalent wealth.
As the example: This person pays a risk-premium of:
RP = E(Gamble) – CE
= $10 - $7.17 = $2.83
Meaning: Any insurance that costs less than $2.83 that ensures him the level of wealth of
$10 will be attractive to him.
Cost of gamble: The difference between an individual’s current wealth and the certainty
equivalent wealth.
e.g., If the individual’s current wealth is:
(a) $10 = E(gamble)
Cost of gamble = $10 - $7.17 = $2.83 RP = C of Gamble
(b) $11 > E(gamble)
Cost of gamble = $11 - $7.17 = $3.83 RP < C of Gamble
(c) $9.5 < E(gamble)
Cost of gamble = $9.5 - $7.17 = $2.43 RP > C of Gamble
NOTE: Risk premium may or may not be the same as Cost of Gamble, depending on
current wealth. If you are risk-averse, risk premium is always positive!!!
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The Arrow-Pratt Premium
We can have a more solid, or mathematical, definition of 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
Z = Random gamble payoffs, where E(Z) = 0, Variance of Z = σ2z
(W,Z) = Arrow-Pratt Premium
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?
24
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:
expected utility of
the current level
of wealth, given the
gamble
RHS:
utility of the current level of wealth
plus
the expected value of
the gamble
less
the risk premium
We use a Taylor series expansion to (*) to derive an expression for the
risk premium (W,Z).
25
Absolute Risk Aversion
• Arrow-Pratt Measure of a Local Risk Premium (derived from (*)
above)
=
1 2
U (W)
)
Z ( 2
U (W)
• Define ARA as a measure of Absolute Risk Aversion
ARA = -
U (W)
U (W)
• This is 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 $1,000 gamble may be trivial to a rich man, but non-trivial to a
poor man
=> ARA will probably decrease as our wealth increases
i.e., ARA ↓ as W ↑
26
Relative Risk Aversion
• Define RRA as a measure of Relative Risk Aversion
RRA = - W *
U (W)
U (W)
• Constant RRA => An individual will have constant risk aversion to a
"proportional loss" of wealth, even though the absolute loss increases
as wealth does.
• That is, with a gamble with 50/50 chance of increasing your wealth by
10% or decreasing it by 10%, you are about as risk-averse to such
gamble regardless of how wealthy or how poor you are. The risk
premium, measured as a percentage of your initial wealth, will stay
constant.
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E.g.: Quadratic Utility
Quadratic Utility - widely used in the academic literature
U(W) = a W - b W2
U'(W) = a - 2bW
ARA =
-U"(W)
2b
--------- = --------U'(W)
a -2bW
d(ARA)
------- > 0
dW
RRA =
U"(W) = -2b
2b
--------a/W - 2b
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
d(RRA)
------- > 0
dW
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An Example
• U=ln(W)
W = $20,000
• G(10,-10: 0.5) 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
29
Arrow-Pratt Measure
•  = -(1/2) 2z U''(W)/U'(W)
• 2z = 0.5*(20,010 – 20,000)2 + 0.5*(19,090 – 20,000)2 = 100
• 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
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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)) = 9.903487428
ln(CE) = 9.903487428  CE = 19,999.9975
• The risk premium RP = $0.0025
• Therefore, the AP and Markowitz premia are the same
31
Markowitz Approach
E(U(W))
= 9.903487
19,990
20,000
CE
20,010
32
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 degree of asymmetry
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E(U) and the end-product
•
Our final step: the bridging of expected utility theory to the indifference curves
Returns
= expected return
= E(R)
EU2
EU1
Risk = standard deviation of return = σR
Derivation is from chapter 3 part F. Interested student may consult the text.
• The transformation is just the following:
Define [i] Wj as the wealth that you will get in next period if you hold asset j.
[ii] W0 your current period wealth.
Thus, return of holding asset j = Rj = (Wj -W0)/W0
Assume Wi is normally distributed, so Rj is normally distributed too. (Big assumption)
NOTE: we simplify by assuming normally distributed returns. So, we can describe a return
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simply by its mean and standard deviation.
Summary
•
•
•
•
•
With 5 axioms, prefer more to less, we have Expected
utility theory, where preferences => Utility function.
With risk-aversion assumption, we solve St.
Petersburg’s paradox.
Assuming we are all greedy, we know every rational
investor will maximize his E[U(W)].
Assuming returns of risky assets being jointly
normally distributed, we leave ourselves with a 2-D Return
diagram with mean of returns and standard deviation
of returns (i.e, mean and s.d.) as our two only
variables of focus
Derive indifference curves on the plane of return and
risk as the right diagram with expected utility theory.
Risk
END-PRODUCT: Essentially, mean and standard
deviation are the choice variables investors concern about
in order to max their E[U(w)], which is given on the
35
indifference curves.