Expected Utility Criterion (EU)
We now want to consider both the amount of final wealth as well
as its risk, though not only in terms of variance but in a more
general form, to evaluate lotteries!
The Expected Utility (EU) criterion allows us to do such an
evaluation!
For better intuition consider the following psychological principle:
„Changes in stimulus do not necessarily induce similar changes in
sensation or perception!“
Accordingly, an additional monetary unit may not have the same
value (utility) for each DM => subjective view (perception)!
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Expected Utility Criterion (EU)
In case of EU the [pure] wealth is replaced by [its] utility!
The transformation of wealth to utility is performed by each DM
based on his subjective utility function!
Thereby the DM transforms the payoffs of a lottery resp. the
values of his final wealth to corresponding utility values!
Then, intuitively, these utility values are summed up weighted by
the corresponding probabilities, i.e. the so called expected utility
is calculated as the fundamental criterion to make decisions!
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Expected Utility Criterion (EU)
Thus the corresponding value function is given as follows:
( ) [ ( )]
~ = ΕU w
~ = Ε[U (w + ~
V w
x )]
f
f
0
Conditioned on whether x is a discrete or a continuous lottery, the
following expressions hold accordingly:
Ε[U ( ~
x )] = ∑ piU (w0 + xi ) resp.
i
∞
Ε[U ( ~
x )] = ∫ U (w0 + t ) f x (t )dt
−∞
Since U(.) can take various forms, the initial wealth w0 and the
lottery x cannot be simply separated from each other!
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Expected Utility Criterion (EU)
Example:
Let us consider two DMs (A and B) each endowed with an initial
wealth of € 200 with the following utility functions:
( )
( )
U A w f = w2f and U B w f = w f
Which one of the two lotteries x and y depicted on slide 12 in slide
set 1 would A and B prefer respectively?
In other words: Which one of the two lotteries x and y does have a
greater expected utility for A and B respectively?
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Expected Utility Criterion (EU)
Example (cont.):
The calculation of EU of the two lotteries for both A and B is
straightforward:
Ε[U A ( w0 + ~
x )] = 0,5 * (200 + 50) 2 + 0,5 * (200 + 150) 2 = 92500
Ε[U ( w + ~
y )] = 1* (200 + 100) 2 = 90000
A
0
Ε[U B ( w0 + ~
x )] = 0,5 * 200 + 50 + 0,5 * 200 + 150 ≈ 17,26
Ε[U ( w + ~
y )] = 1* 200 + 100 ≈ 17,32
B
0
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Expected Utility Criterion (EU)
Obviously, A and B will make different decisions; while A prefers
lottery x over y, for B lottery y has a higher expected utility than
lottery x!
Since both lotteries have the same expected value and lottery x is
more risky than lottery y, one could argue that for A risk has a
positive utility value while for B the opposite is true!
In this respect, A seems to be risk seeking and B risk averse.
None of the two is risk neutral, as none of them evaluates the
lotteries based on their expected values alone (otherwise at least
one of them would have to be indifferent between x and y)!
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Expected Utility Criterion (EU)
In consequence, it seems that different forms of utility functions
may represent different risk attitudes.
However:
Is there truly a relationship between the risk attitude of a DM
and the form of her/his utility function?
How can this relationship be defined (described)?
How can risk seeking (aversion) be measured?
What does this imply for the actual decision making of a DM?
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Expected Utility Criterion (EU)
It is not plausible to assume that every mathematical function could
also be the utility function of an arbitrary DM.
In order to account for some of the most basic and natural principles
of the decision process, the utility function must in fact have certain
properties.
At this stage we only impose two such properties, in particular:
The utility function is assumed to be monotonicaly increasing on
its entire domain u` ≥ 0. In other words, each additional unit of
wealth increases utility, or at least does not decrease it!
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Expected Utility Criterion (EU)
U is assumed to be continuous, and hence differentiable, on its
entire domain.
To be as general as possible we will, at the moment, not impose
any additional restrictions on the form of the utility function!
Rather we will analyze how the various properties of a utility
function may influence (characterize) the behavior of a DM,
especially her/his risk attitude; more generally, the relation
between the behavior of a DM and the form of her/his utility
function is analyzed!
During our expertise, additional assumptions about the form of the
utility function may be applied.
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Certainty Equivalent (CEQ)
Let us assume that a DM is endowed with a [certain] initial wealth
w0 and a risky lottery x.
What about the following question:
How much certain wealth would yield the DM the same level of
satisfaction (utility) as his actual wealth holdings, i.e. the original
endowment w0 and the lottery x?
In other words: What is the certain amount of wealth, between
which and the current endowment (w0 + x) the DM would be
indifferent?
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Certainty Equivalent (CEQ)
Let us call this [certain] amount of wealth the certainty
equivalent and denote it by w*.
Referring to the definition of w*, formally it can be expressed in
the following way:
∞
( )
U w* = Ε[U (w0 + ~
x )] = ∫ U (w0 + t ) f x (t )dt
−∞
Expressed explicitly for w*
∞


*
−1
−1
~
w = U {Ε[U (w0 + x )]} = U  ∫ U (w0 + t ) f x (t )dt 

−∞
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Certainty Equivalent (CEQ)
Let us now once again consider the lottery x from slide 12 in slide
set 1 and the DMs A and B mentioned above. As we already know
( )
( )
U A w f = w 2f and U B w f = w f
From our earlier calculations we also know that
Ε[U A (w0 + ~
x )] = 92500 and Ε[U B (w0 + ~
x )] = 17,26
Thus, the following must hold
( ) ( )
U A w*A = w*A
2
( )
= 92500 and U B w*B = w*B = 17,26
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Certainty Equivalent (CEQ)
This finally yields the following certainty equivalents:
w*A ≈ 304,14 and w*B ≈ 297,9
It is also possible to derive first the inverse of each utility function
and then calculate w* directly:
( )
U −A1 w f = w f
( )
and U B−1 w f = w2f
which equivalently yields
w*A = 92500 ≈ 304,14 and w*B = 17,26 2 ≈ 297,9
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Certainty Equivalent (CEQ)
Obviously, A prices lottery x higher than B. If we now subtract the
initial wealth amount from the corresponding certainty equivalent
we obtain
w*A − w0 ≈ 104,14 and w*B − w0 ≈ 97,9
This value is obviously the certain amount which brings A (B) the
same satisfaction (utility) as lottery x, being endowed with a
particular initial wealth amount w0.
In other words: If A (B) sold lottery x for € 104,14 (€ 97,9), he/she
then (after the sale) would be equally satisfied (provided with the
same utility) as before the sale; thus being indifferent between
keeping or selling the lottery!
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Asking Price
Note that if A (B) is offered an amount higher than € 104,14 (€
97,9) for lottery x, he/she can increase his/her utility through the
sale. Hence he/she would surely sell the lottery for any such price!
If on the other hand A (B) is offered an amount lower than € 104,14
(€ 97,9) for lottery x, the sale of the lottery would obviously
decrease his/her utility. Hence he/she would surely not sell the
lottery for any such price!
Obviously, this particular value represents the [minimum] asking
price for the lottery [demanded by a particular DM], denoted by pa
pa = w* − w0
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Asking Price
The expected value of lottery x is E(x) = 100.
While A would only sell lottery x for an amount higher than E(x), B
on the other hand would sell it even for an amount below E(x).
It is obvious that for A (B) the risk involved in lottery x has an
additional positive (negative) utility.
Expressed in monetary units the risk of lottery x has for A (B) a
value equal to a certain amount of € 4,14 (€ -2,1).
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Asking Price
Now, which of the two DMs is [obviously] risk seeking/averse?
Well, in general one can say:
pa > E(x) … Risk seeking
pa < E(x) … Risk aversion
pa = E(x) … Risk neutrality
A risk averse DM is indifferent between a lower certain amount of
wealth and a higher risky amount of wealth, as certainty has a
positive utility for him/her. A risk seeking individual on the other
hand is indifferent between a lower risky wealth amount and a
higher certain wealth amount, as risk has a positive utility for
him/her!
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Risk Premium
The difference between pa and E(x) represents a suitable monetary
measure for the value of risk involved in a lottery as perceived by a
particular DM.
This measure is called the risk premium and is defined as follows:
π = Ε( ~x ) − pa
Thus one can say (equivalenty to the propositions based on pa)
π < 0 … Risk seeking
π > 0 … Risk aversion
π = 0 … Risk neutrality
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Risk Premium and Jensen‘s
Inequality
If π > 0 (risk aversion) the DM is willing to sacrifice a particular
positive wealth amount in order to get rid off risk!
We will now discover that there is an explicit relation between the
positivity (negativity) of the risk premium and the second derivative
of the utility function!
This can be shown using the so called Jensen‘s Inequality which
states:
„For each concave function f(.) it holds
f [Ε( ~
x )] > Ε[ f ( ~
x )]
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Risk Premium and Concavity of
the Utility Function
Proof on page 29
(E&G)!
Therefore if U is concave (U``< 0), then [according to Jensen‘s
Inequality]:
Ε[U (w0 + ~
x )] < U [Ε(w0 + ~
x )]
From the definition of the certainty equivalent one obtains
( )
U w* < U [w0 + Ε(~
x )]
Since U is monotonicaly increasing
w* < w0 + Ε(~
x ) ⇔ w0 + pa < w0 + Ε(~
x)⇔
p < Ε (~
x)⇔π > 0
a
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Graphical Demonstration
2,5
2
1,5
1
0,5
wo
w*
w o + E(x)
0
0
1
2
3
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5
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Risk Premium and Concavity of
the Utility Function
Accordingly it follows:
π > 0 U`` < 0 … Risk aversion
π < 0 U`` > 0 … Risk seeking
π = 0 U`` = 0 … Risk neutrality
This is also consistent with the so called Arrow-Pratt
Approximation, according to which the risk premium of a particular
lottery x can be approx. calculated in the following way:
x )) 
Var (~
x )  U ′′(w0 + Ε(~
π≅
− ′

x )) 
2  U (w0 + Ε(~
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Risk Premium and Concavity of
the Utility Function
It is obvious that U`` must be negative (i.e. U concave) for π to have a
positive value.
The Approximation formula can be simply derived starting with:
U (w0 + pa ) = Ε[U (w0 + ~
x )]
Applaying a Taylor-Expansion of second order one obtains:
U (w0 + pa ) ≅ U [w0 + Ε(~
x )] + [ pa − Ε(~
x )]U ′[w0 + Ε(~
x )]
2
~
~
(
(
)
)
Ε
−
Ε
x
x
Ε[U (w0 + ~
x )] ≅ U [w0 + Ε(~
x )] + Ε[~
x − Ε (~
x )]U ′[w0 + Ε(~
x )] +
U ′′[w0 + Ε(~
x )]
2
[
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Degree of Absolute Risk
Aversion
Plugging in and simplifying the terms finally yields:
x )) 
Var (~
x )  U ′′(w0 + Ε(~
π≅
− ′

x )) 
2  U (w0 + Ε(~
Where the multiplier, depicted in the brackets [.] is usually called
the degree of absolute risk aversion, denoted by Aa.
Obviously, Aa is a local measure being a function of the expected
final wealth:
[
[
]
]
~ )
′
′
(
U
w
Ε
f
~ )= −
Aa (w
f
~ )
U ′ Ε(w
f
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Bid Price
Similarly as in case of the asking price, pa, one can also ask the
question, how much a DM is [at maximum] willing to pay for a
particular lottery i.e. what is the so called bid price, denoted by pb?
By definition it is simply the price at which the DM is indifferent
between buying and not buying a particular lottery x, i.e. where
after the purchase he/she has the same exp. utility as before:
U (w0 ) = Ε[U (w0 − pb + ~
x )]
Depending on the exact form of U and f(x) it may be more or less
difficult to express pb explicitly from the underlying expression!
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Bid Price
If the DM is offered a lottery for a price below pb it is optimal for
him/her to buy the lottery, since he/she can increase his/her utility by
doing so. If on the other hand the purchase price of a lottery
exceeds pb, the DM will optimally not buy the lottery, since doing so
would in fact decrease his/her utility!
This principle comes into play in the field of insurance [among
others], where the insurer buys a lottery whereas the policyholder
(insured) sells one. For the transaction (transfer of risk from the
insured to the insurer) to be realized it is necessary that the price of
the lottery (here the negative of the insurance premium) is higher
than the [minimum] asking price required by the insured and it is
simultaneously lower than the [maximum] bid price of the insurer.
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Note that pa < p < pb must hold
for any transaction at price p!
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Variance vs. Expected Utility
So far we have treated the risk of a lottery first in terms of the
variance of its payoffs and second within the general concept of
expected utitliy (EU).
Under some particular circumstances the variance alone may be a
relatively sufficient measure of risk; but not in general!
It is obvious that in case of an approximation of the risk premium
using a Taylor-Expansion up to a second order and a particular
degree of risk aversion the risk premium only depends on the
variance of a lottery, rendering all of its higher moments as
irrelevant!
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Variance vs. Expected Utility
However, in general this is only an approximation! It is exact (true) if
and only if the third derivative of U and consequently all of its
derivatives of a higher order are equal to zero!
If this is not the case then the risk premium as well as the expected
utility of any lottery is also influenced by all of its non-zero higher
moments, e.g. skewness, curtosis, etc.
To prove this proposition one can simply find lottery pairs x and y,
such that E(x) = E(y) and Var(x) > Var(y), whereas a risk averse DM
(U´´< 0) prefers lottery x despite the higher variance [which can
only be due to the higher order derivatives of U being non-zero thus
letting higher moments of the two lotteries come into play].
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Variance vs. Expected Utility
To put it simply, variance is not a general (universal) measure of
risk! Although it is often the case, a higher (lower) variance alone
may not necessarily imply higher (lower) risk!
A suitable example is offered by the following two lotteries
(according to Ingersoll, 1987):
xi
p(xi)
yi
p(yi)
0
0,5
1
7/8
4
0,5
9
1/8
In this particular case it is true that: E(x) = E(y) und Var(x) < Var(y).
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Variance vs. Expected Utility
However, a particular risk averse DM, e.g. one with the following
utility function U(wf) = (wf)1/2 and w0 = 0 prefers lottery y over x, i.e.
Ε[U ( w0 + ~
x )] < Ε[U ( w0 + ~
y )]
The differences of the higher moments of x and y obviously come
into play, as the corresponding higher order derivatives of U(wf) are
not zero.
Thus, to restrict one‘s attention exclusively to variance as the only
measure of risk may under particular distributional properties of the
underlying lottery be insufficient and even misleading!
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Expected Utility Criterion (EU)
Important Remark:
Note that both EV and MV criteria are just special cases of the EU
criterion, in particular, if U is linear EV, and, if U is quadratic MV.
Try to show this analytically!
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