1.1: Preferences 1.2: Risk Premia 1.3: Portfolio Choice 1.4: Conclusions Expected Utility and Risk Aversion George Pennacchi University of Illinois George Pennacchi University of Illinois Expected utility and risk aversion 1/ 58
1.1: Preferences 1.2: Risk Premia 1.3: Portfolio Choice 1.4: Conclusions
Expected Utility and Risk Aversion
George Pennacchi
University of Illinois
George Pennacchi University of Illinois
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Introduction
Expected utility is the standard framework for modeling investorchoices. The following topics will be covered:
1 Analyze conditions on individual preferences that lead to anexpected utility function.
2 Consider the link between utility, risk aversion, and risk premiafor particular assets.
3 Examine how risk aversion a¤ects an individual�s portfoliochoice between a risky and riskfree asset.
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Preferences when Returns are Uncertain
Economists typically analyze the price of a good using supplyand demand. We can do the same for assets.
The main distinction between assets is their future payo¤s:Risky assets have uncertain payo¤s, so a theory of assetdemands must specify investor preferences over di¤erent,uncertain payo¤s.
Consider relevant criteria for ranking preferences. Onepossible measure is the asset�s average payo¤.
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Criterion: Expected Payo¤
Suppose an asset o¤ers a single random payo¤ at a particularfuture date, and this payo¤ has a discrete distribution with npossible outcomes (x1; :::; xn) and corresponding probabilities(p1; :::; pn), where
Pni=1 pi = 1 and pi � 0.
Then the expected value of the payo¤ (or, more simply, theexpected payo¤) is �x � E [ex ] =Pn
i=1 pixi .
Is an asset�s expected value a suitable criterion fordetermining an individual�s demand for the asset?
Consider how much Paul would pay Peter to play thefollowing coin �ipping game.
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St. Petersburg Paradox, Nicholas Bernoulli, 1713
Peter continues to toss a coin until it lands �heads.�Heagrees to give Paul one ducat if he gets heads on the very �rstthrow, two ducats if he gets it on the second, four if on thethird, eight if on the fourth, and so on.
If the number of coin �ips taken to �rst obtain heads is i , thenpi =
� 12
�iand xi = 2i�1: Thus, Paul�s expected payo¤ equals
�x =P1i=1 pixi =
121+
142+
184+
1168+ ::: (1)
= 12 (1+
122+
144+
188+ :::
= 12 (1+ 1+ 1+ 1+ ::: =1
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St. Petersburg Paradox
What is the paradox?
Daniel Bernoulli (1738) explained it using expected utility.
His insight was that an individual�s utility from receiving apayo¤ di¤ered from the size of the payo¤.
Instead of valuing an asset as x =Pni=1 pixi , its value, V ,
would beV � E [U (ex)] =Xn
i=1piUi
where Ui is the utility associated with payo¤ xi .
He hypothesized that Ui is diminishingly increasing in wealth.
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Criterion: Expected Utility
Von Neumann and Morgenstern (1944) derived conditions onan individual�s preferences that, if satis�ed, would make themconsistent with an expected utility function.
De�ne a lottery as an asset that has a risky payo¤ andconsider an individual�s optimal choice of a lottery from agiven set of di¤erent lotteries. The possible payo¤s of alllotteries are contained in the set fx1; :::; xng.A lottery is characterized by an ordered set of probabilities
P = fp1; :::; png, where of course,nPi=1pi = 1 and pi � 0. Let a
di¤erent lottery be P� = fp�1 ; :::; p�ng. Let �, �, and �denote preference and indi¤erence between lotteries.
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Preferences Over Di¤erent Random Payo¤s
Speci�cally, if an individual prefers lottery P� to lottery P,this can be denoted as P� � P or P � P�.
When the individual is indi¤erent between the two lotteries,this is written as P� � P.
If an individual prefers lottery P� to lottery P or she isindi¤erent between lotteries P� and P, this is written asP� � P or P � P�.
N.B.: all lotteries have the same payo¤ set fx1; :::; xng, so wefocus on the (di¤erent) probability sets P and P�.
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Expected Utility Axioms 1-3
Theorem: There exists an expected utility functionV (p1; :::; pn) if the following axioms hold:
Axioms:1) CompletenessFor any two lotteries P� and P, either P� � P, or P� � P, orP� � P.2) TransitivityIf P�� � P�and P� � P, then P�� � P.3) ContinuityIf P�� � P� � P, there exists some � 2 [0; 1] such thatP� � �P�� + (1� �)P, where �P�� + (1� �)P denotes a�compound lottery�; namely, with probability � one receives thelottery P�� and with probability (1� �) one receives the lottery P.
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Expected Utility Axioms 4-5
4) IndependenceFor any two lotteries P and P�, P� � P if and only if for all � 2(0,1] and all P��:
�P� + (1� �)P�� � �P + (1� �)P��
Moreover, for any two lotteries P and Py, P � Py if and only if forall � 2(0,1] and all P��:
�P + (1� �)P�� � �Py + (1� �)P��
5) DominanceLet P1 be the compound lottery �1Pz + (1� �1)Py and P2 be thecompound lottery �2Pz + (1� �2)Py. If Pz � Py, then P1 � P2 ifand only if �1 > �2.
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Discussion: Machina (1987)
The �rst three axioms are analogous to those used to establisha real-valued utility function in consumer choice theory.
Axiom 4 (Independence) is novel, but its linearity property iscritical for preferences to be consistent with expected utility.
To understand its meaning, suppose an individual chooses P�
� P. By Axiom 4, the choice between �P� + (1� �)P�� and�P + (1� �)P�� is equivalent to tossing a coin that withprobability (1� �) lands �tails,� in which both lotteries payP��, and with probability � lands �heads,� in which case theindividual should prefer P� to P.
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Allais Paradox
But, there is some experimental evidence counter to thisaxiom.
Consider lotteries over fx1; x2; x3g = f$0; $1m; $5mg and twolottery choices:C1: P1 = f0; 1; 0g vs P2 = f:01; :89; :1gC2: P3 = f:9; 0; :1g vs P4 = f:89; :11; 0g
Which do you choose in C1? In C2?
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Allais Paradox
Experimental evidence suggests most people prefer P1 � P2and P3 � P4.
But this violates Axiom 4. Why?
De�ne P5 = f1=11; 0; 10=11g and let � = 0:11. Note that P2is equivalent to the compound lottery:
P2 � �P5 + (1� �)P1
� 0:11f1=11; 0; 10=11g+ 0:89f0; 1; 0g� f:01; :89; :1g
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Allais Paradox
Note also that P1 is trivially the compound lottery�P1 + (1� �)P1. Hence, if P1 � P2, the independenceaxiom implies P1 � P5.Now also de�ne P6 = f1; 0; 0g, and note that P3 equals thefollowing compound lottery:
P3 � �P5 + (1� �)P6
� 0:11f1=11; 0; 10=11g+ 0:89f1; 0; 0g� f:9; 0; :1g
while P4 is equivalent to the compound lottery
P4 � �P1 + (1� �)P6
� 0:11f0; 1; 0g+ 0:89f1; 0; 0g� f:89; 0:11; 0g
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Allais Paradox
But if P3 � P4, the independence axiom implies P5 � P1,which contradicts the choice of P1 � P2 that impliesP1 � P5.
Despite the sometimes contradictory experimental evidence,expected utility is still the dominant paradigm.
However, we will consider di¤erent models of utility at a laterdate, including those that re�ect psychological biases.
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Deriving Expected Utility: Axiom 1
We now prove the theorem by showing that if an individual�spreferences over lotteries satisfy the preceding axioms, thesepreferences can be ranked by the individual�s expected utilityof the lotteries.
De�ne an �elementary�or �primitive� lottery, ei , whichreturns outcome xi with probability 1 and all other outcomeswith probability zero, that is, ei = fp1; :::pi�1;pi ;pi+1:::;png =f0; :::0; 1; 0; :::0g where pi = 1 and pj = 0 8j 6= i .
Without loss of generality, assume that the outcomes areordered such that en � en�1 � ::: � e1. This follows from thecompleteness axiom for this case of n elementary lotteries
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Deriving Expected Utility: Axiom 3, Axiom 4
From the continuity axiom, for each ei , there exists aUi 2 [0; 1] such that
ei � Uien + (1� Ui )e1 (2)
and for i = 1, this implies U1 = 0 and for i = n, this impliesUn = 1.
Now a given arbitrary lottery, P = fp1; :::; png, can be viewedas a compound lottery over the n elementary lotteries, whereelementary lottery ei is obtained with probability pi .
P � p1e1 + :::+ pnen
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Deriving Expected Utility: Axiom 4
By the independence axiom, and equation (2), the individualis indi¤erent between lottery, P, and the following lottery:
p1e1 + :::+ pnen � p1e1 + :::+ pi�1ei�1 + pi [Uien + (1� Ui )e1]+pi+1ei+1 + :::+ pnen (3)
where the indi¤erence relation in equation (2) substitutes forei on the right-hand side of (3).
By repeating this substitution for all i , i = 1; :::; n, theindividual will be indi¤erent between P and
p1e1 + :::+ pnen �
nXi=1
piUi
!en +
1�
nXi=1
piUi
!e1 (4)
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Deriving Expected Utility: Axiom 5
Now de�ne � �nPi=1
piUi . Thus, P � �en + (1� �)e1
Similarly, we can show that any other arbitrary lottery
P� = fp�1 ; :::; p�ng � ��en + (1� ��)e1, where �� �nPi=1
p�i Ui .
We know from the dominance axiom that P� � P i¤ �� > �,implying
nPi=1p�i Ui >
nPi=1piUi .
So we can de�ne the function
V (p1; :::; pn) =nXi=1
piUi (5)
which implies that P� � P i¤ V (p�1 ; :::; p�n) > V (p1; :::; pn).
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Deriving Expected Utility: The End
The function in (5) is known as von Neumann-Morgensternexpected utility. It is linear in the probabilities and is uniqueup to a linear monotonic transformation.
The intuition for why expected utility is unique up to a lineartransformation comes from equation (2). Here we expresselementary lottery i in terms of the least and most preferredelementary lotteries. However, other bases for ranking a givenlottery are possible.
For Ui = U(xi ), an individual�s choice over lotteries is thesame under the transformation aU(xi ) + b, but not anonlinear transformation that changes the �shape�of U(xi ).
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St. Petersburg Paradox Revisited
Suppose Ui = U(xi ) =pxi . Then the expected utility of the
St. Petersburg payo¤ is
V =nXi=1
piUi =1Xi=1
12ip2i�1 =
1Xi=1
2�12 (i+1) =
1Xi=2
2�i2
= 2�22 + 2�
32 + :::
=1Xi=0
�1p2
�i� 1� 1p
2=
11� 1p
2
� 1� 1p2
=1
2�p2�= 1:707
A certain payment of 1:7072 �= 2:914 ducats has the sameexpected utility as playing the St. Petersburg game.
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Super St. Petersburg
The St. Petersburg game has in�nite expected payo¤ becausethe probability of winning declines at rate 2i , while thewinning payo¤ increases at rate 2i .In a �super�St. Petersburg paradox, we can make thewinning payo¤ increase at a rate xi = U�1(2i�1) to causeexpected utility to increase at 2i . For square-root utility,xi = (2i2)2 = 22i�2; that is, x1 = 1, x2 = 4, x3 = 16, and soon. The expected utility of �super�St. Petersburg is
V =nXi=1
piUi =1Xi=1
12ip22i�2 =
1Xi=1
12i2i�1 =1 (6)
Should we be concerned that if prizes grow quickly enough,we can get in�nite expected utility (and valuations) for anychosen form of expected utility function?
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Von Neumann-Morgenstern Utility
The von Neumann-Morgenstern expected utility can begeneralized to a continuum of outcomes and lotteries withcontinuous probability distributions. Analogous to equation(5) is
V (F ) = E [U (ex)] = Z U (x) dF (x) =ZU (x) f (x) dx (7)
where F (x) is the lottery�s cumulative distribution functionover the payo¤s, x . V can be written in terms of theprobability density, f (x), when F (x) is absolutely continuous.
This is analogous to our previous lottery represented by thediscrete probabilities P = fp1; :::; png.
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Risk Aversion
Diminishing marginal utility results in risk aversion: beingunwilling to accept a �fair� lottery. Why?Let there be a lottery that has a random payo¤, e", where
e" = � "1with probability p"2 with probability 1� p
(8)
The requirement that it be a �fair� lottery restricts itsexpected value to equal zero:
E [e"] = p"1 + (1� p)"2 = 0 (9)
which implies "1="2 = � (1� p) =p, or solving for p,p = �"2= ("1 � "2). Since 0 < p < 1, "1 and "2 are ofopposite signs.
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Risk Aversion and Concave Utility
Suppose a vN-M maximizer with current wealth W is o¤ereda fair lottery. Would he accept it?With the lottery, expected utility is E [U (W + e")]. Withoutit, expected utility is E [U (W )] = U (W ). Rejecting it implies
U (W ) > E [U (W + e")] = pU (W + "1)+ (1� p)U (W + "2)(10)
U (W ) can be written as
U(W ) = U (W + p"1 + (1� p)"2) (11)
Substituting into (10), we have
U (W + p"1 + (1� p)"2) > pU (W + "1)+(1�p)U (W + "2)(12)
which is the de�nition of U being a concave function.
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Risk Aversion , Concavity
A function is concave if a line joining any two points liesentirely below the function. When U(W ) is a continuous,second di¤erentiable function, concavity implies U 00(W ) < 0.
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Risk Aversion , Concavity
To show that concave utility implies rejecting a fair lottery, wecan use Jensen�s inequality which says that for concave U(�)
E [U(~x)] < U(E [~x ]) (13)
Therefore, substituting ~x =W + e" with E [e"] = 0, we haveE [U(W + e")] < U (E [W + e"]) = U(W ) (14)
which is the desired result.
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Risk Aversion and Risk Premium
How might aversion to risk be quanti�ed? One way is tode�ne a risk premium as the amount that an individual iswilling to pay to avoid a risk.Let � denote the individual�s risk premium for a lottery, e". �is the maximum insurance payment an individual would pay toavoid the lottery risk:
U(W � �) = E [U(W + e")] (15)
W � � is de�ned as the certainty equivalent level of wealthassociated with the lottery, e".For concave utility, Jensen�s inequality implies � > 0 when e" isfair: the individual would accept wealth lower than herexpected wealth following the lottery, E [W + e"], to avoid thelottery.
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Risk Premium
For small e" we can take a Taylor approximation of equation(15) around e" = 0 and � = 0.Expanding the left-hand side about � = 0 gives
U(W � �) �= U(W )� �U 0(W ) (16)
and expanding the right-hand side about e" givesE [U(W + e")] �= E �U(W ) + e"U 0(W ) + 1
2e"2U 00(W )� (17)
= U(W ) + 0+ 12�
2U 00(W )
where �2 � E�e"2� is the lottery�s variance.
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Risk Premium cont�d
Equating the results in (16) and (17) gives
� = �12�
2U00(W )U 0(W )
� 12�
2R(W ) (18)
where R(W ) � �U 00(W )=U 0(W ) is the Pratt (1964)-Arrow(1971) measure of absolute risk aversion.
Since �2 > 0, U 0(W ) > 0, and U 00(W ) < 0, concavity of theutility function ensures that � must be positive
An individual may be very risk averse (�U 00(W ) is large), butmay be unwilling to pay a large risk premium if he is poorsince his marginal utility U 0(W ) is high.
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�U 00(W ) and U 0(W )
Consider the following negative exponential utility function:
U(W ) = �e�bW ; b > 0 (19)
Note that U 0(W ) = be�bW > 0 andU 00(W ) = �b2e�bW < 0.Consider the behavior of a very wealthy individual whosewealth approaches in�nity
limW!1
U 0(W ) = limW!1
U 00(W ) = 0 (20)
There�s no concavity, so is there no risk aversion?
R(W ) =b2e�bW
be�bW= b (21)
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Absolute Risk Aversion: Dollar Payment for Risk
We see that negative exponential utility, U(W ) = �e�bW ,has constant absolute risk aversion.
If, instead, we want absolute risk aversion to decline in wealth,a necessary condition is that the utility function must have apositive third derivative:
@R(W )@W
=@ � U 00(W )
U 0(W )
@W= �U
000(W )U 0(W )� [U 00(W )]2[U 0(W )]2
(22)
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R(W )) U(W )
The coe¢ cient of risk aversion contains all relevantinformation about the individual�s risk preferences. Note that
R(W ) = �U00(W )U 0(W )
= �@ (ln [U0(W )])
@W(23)
Integrating both sides of (23), we have
�ZR(W )dW = ln[U 0(W )] + c1 (24)
where c1 is an arbitrary constant. Taking the exponentialfunction of (24) gives
e�RR(W )dW = U 0(W )ec1 (25)
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R(W )) U(W ) cont�d
Integrating once again, we obtainZe�
RR(W )dW dW = ec1U(W ) + c2 (26)
where c2 is another arbitrary constant.
Because vN-M expected utility functions are unique up to alinear transformation, ec1U(W ) + c2 re�ects the same riskpreferences as U(W ).
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Relative Risk Aversion
Relative risk aversion is another frequently used measurede�ned as
Rr (W ) =WR(W ) (27)
Consider risk aversion for some utility functions often used inmodels of portfolio choice and asset pricing. Power utility canbe written as
U(W ) = 1 W
; < 1 (28)
implying that R(W ) = � ( �1)W �2
W �1 = (1� )W and, therefore,
Rr (W ) = 1� .Hence, it displays constant relative risk aversion.
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Logarithmic Utility: Constant Relative Risk Aversion
Logarithmic utility is a limiting case of power utility. Sinceutility functions are unique up to a linear transformation, writethe power utility function as
1 W
� 1 =W � 1
Next take its limit as ! 0. Do so by rewriting thenumerator and applying L�Hôpital�s rule:
lim !0
W � 1
= lim !0
e ln(W ) � 1
= lim !0
ln(W )W
1= ln(W )
(29)Thus, logarithmic utility is power utility with coe¢ cient ofrelative risk aversion (1� ) = 1 since R(W ) = �W �2
W �1 =1W
and Rr (W ) = 1.
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HARA: Power, Log, Quadratic
Hyperbolic absolute-risk-aversion (HARA) utility generalizesall of the previous utility functions:
U(W ) =1�
��W1� + �
� (30)
s:t: 6= 1, � > 0, �W1� + � > 0, and � = 1 if = �1.
Thus, R(W ) =�W1� +
��
��1. Since R(W ) must be > 0, it
implies � > 0 when > 1. Rr (W ) =W�W1� +
��
��1.
HARA utility nests constant absolute risk aversion ( = �1,� = 1), constant relative risk aversion ( < 1, � = 0), andquadratic ( = 2) utility functions.
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Another Look at the Risk Premium
A premium to avoid risk is �ne for insurance, but we may alsobe interested in a premium to bear risk.
This alternative concept of a risk premium was used by Arrow(1971), identical to the earlier one by Pratt (1964).
Suppose that a fair lottery e", has the following payo¤s andprobabilities:
e" = � +� with probability 12
�� with probability 12
(31)
How much do we need to deviate from �fairness� to make arisk-averse individual indi¤erent to this lottery?
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Risk Premium v2
Let�s de�ne a risk premium, �, in terms of probability ofwinning p:
� = Prob(win)� Prob(lose) = p � (1� p) = 2p � 1 (32)
Therefore, from (32) we have
Prob(win) � p = 12 (1+ �)
Prob(lose) = 1� p = 12 (1� �)
We want � that equalizes the utilities of taking and not takingthe lottery:
U(W ) =12(1+ �)U(W + �) +
12(1� �)U(W � �) (33)
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Risk Aversion (again)
Let�s again take a Taylor approximation of the right side,around � = 0
U(W ) =12(1+ �)
�U(W ) + �U 0(W ) + 1
2 �2U 00(W )
�(34)
+12(1� �)
�U(W )� �U 0(W ) + 1
2 �2U 00(W )
�= U(W ) + ��U 0(W ) + 1
2 �2U 00(W )
Rearranging (34) implies
� = 12 �R(W ) (35)
which, as before, is a function of the coe¢ cient of absoluterisk aversion.
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Risk Aversion (again)
Note that the Arrow premium, �, is in terms of a probability,while the Pratt measure, �, is in units of a monetary payment.
If we multiply � by the monetary payment received, �, thenequation (35) becomes
�� = 12 �2R(W ) (36)
Since �2 is the variance of the random payo¤, e", equation (36)shows that the Pratt and Arrow risk premia are equivalent.Both were obtained as a linearization of the true functionaround e" = 0.
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A Simple Portfolio Choice Problem
Let�s consider the relation between risk aversion and anindividual�s portfolio choice in a single period context.Assume there is a riskless security that pays a rate of returnequal to rf and just one risky security that pays a stochasticrate of return equal to er .Also, let W0 be the individual�s initial wealth, and let A be thedollar amount that the individual invests in the risky asset atthe beginning of the period. Thus, W0 � A is the initialinvestment in the riskless security.Denote the individual�s end-of-period wealth as ~W :
~W = (W0 � A)(1+ rf ) + A(1+ ~r) (37)
= W0(1+ rf ) + A(~r � rf )
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Single Period Utility Maximization
A vN-M expected utility maximizer chooses her portfolio bymaximizing the expected utility of end-of-period wealth:
maxAE [U( ~W )] = max
AE [U (W0(1+ rf ) + A(~r � rf ))] (38)
Maximization satis�es the �rst-order condition wrt. A:
EhU 0�~W�(~r � rf )
i= 0 (39)
Note that the second order condition
EhU 00�~W�(~r � rf )2
i� 0 (40)
is satis�ed because U 00�~W�� 0 from concavity.
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Obtaining A� from FOC
If E [~r � rf ] = 0, i.e., E [~r ] = rf , then we can show A=0 is thesolution.
When A=0, ~W =W0 (1+ rf ) and, therefore,
U 0�~W�= U 0 (W0 (1+ rf )) is nonstochastic. Hence,
EhU 0�~W�(~r � rf )
i= U 0 (W0 (1+ rf ))E [~r � rf ] = 0.
Next, suppose E [~r � rf ] > 0.A = 0 is not a solution becauseEhU 0�~W�(~r � rf )
i= U 0 (W0 (1+ rf ))E [~r � rf ] > 0 when
A = 0.
Thus, when E [~r ]� rf > 0, let�s show that A > 0.
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Why must A > 0?
Let rh denote a realization of ~r > rf , and let W h be thecorresponding level of ~W
Also, let r l denote a realization of ~r < rf , and let W l be thecorresponding level of ~W .
Then U 0(W h)(rh � rf ) > 0 and U 0(W l )(r l � rf ) < 0.
For U 0�~W�(~r � rf ) to average to zero for all realizations of
~r , it must be that W h >W l so that U 0�W h�< U 0
�W l�due
to the concavity of the utility function.
Why? Since E [~r ]� rf > 0, the average rh is farther above rfthan the average r l is below rf . To preserve (39), themultipliers must satisfy U 0
�W h�< U 0
�W l�to compensate,
which occurs when W h >W l and which requires that A > 0.
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How does A change wrt W0?
We�ll use implicit di¤erentiation to obtain dA(W0)dW0
:
De�ne f (A;W0) � EhU�fW�i and let
v (W0) = maxAf (A;W0) be the maximized value of expected
utility when A, is optimally chosen.
Also de�ne A (W0) as the value of A that maximizes f for agiven value of the initial wealth parameter W0.
Now take the total derivative of v (W0) with respect to W0 byapplying the chain rule:dv (W0)dW0
= @f (A;W0)@A
dA(W0)dW0
+ @f (A(W0);W0)@W0
.
However, @f (A;W0)@A = 0 since it is the �rst-order condition for a
maximum.
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How does A change wrt W0 cont�d
The total derivative simpli�es to dv (W0)dW0
= @f (A(W0);W0)@W0
:
Thus, the derivative of the maximized value of the objectivefunction with respect to a parameter is just the partialderivative with respect to that parameter.
Second, consider how the optimal value of the controlvariable, A (W0), changes when the parameter W0 changes.
Derive this relationship by taking the total derivative of theF.O.C. (39), @f (A (W0) ;W0) =@A = 0, with respect to W0:
@(@f (A(W0);W0)=@A)@W0
= 0 =@2f (A(W0);W0)@A2
dA(W0)dW0
+@2f (A(W0);W0)@A@W0
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How does A change wrt W0 cont�d
Rearranging the above gives us
dA (W0)
dW0= �@
2f (A (W0) ;W0)
@A@W0=@2f (A (W0) ;W0)
@A2(41)
We can then evaluate it to obtain
dAdW0
=(1+ rf )E
hU 00( ~W )(~r � rf )
i�E
hU 00( ~W )(~r � rf )2
i (42)
The denominator of (42) is positive because of concavity.Therefore, the sign of dA
dW0depends on the numerator.
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Implications for dAdW0
with DARA
Consider an individual with absolute risk aversion that isdecreasing in wealth. Assuming E [~r ] > rf so that A > 0:
R�W h�< R (W0(1+ rf )) (43)
where, as before, R(W ) = �U 00(W )=U 0(W ).Multiplying both terms of (43) by �U 0(W h)(rh � rf ), which isa negative quantity, the inequality sign changes:
U 00(W h)(rh � rf ) > �U 0(W h)(rh � rf )R (W0(1+ rf )) (44)
Then for A > 0, we have W l <W0(1+ rf ). If absolute riskaversion is decreasing in wealth, this implies
R(W l ) > R (W0(1+ rf )) (45)
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Implications for dAdW0
with DARA
Multiplying (45) by �U 0(W l )(r l � rf ), which is positive, sothat the sign of (45) remains the same, we obtain
U 00(W l )(r l � rf ) > �U 0(W l )(r l � rf )R (W0(1+ rf )) (46)
Inequalities (44) and (46) are the same whether therealization is ~r = rh or ~r = r l .
Therefore, if we take expectations over all realizations of ~r , weobtain
EhU 00( ~W )(~r � rf )
i> �E
hU 0( ~W )(~r � rf )
iR (W0(1+ rf ))
(47)
The �rst term on the right-hand side is just the FOC.
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Implications for risk-taking with ARA/RRA
Inequality (47) reduces to
EhU 00( ~W )(~r � rf )
i> 0 (48)
Thus, DARA ) dA=dW0 > 0: amount invested A increases ininitial wealth.What about the proportion of initial wealth? To analyze this,de�ne
� �dAdW0AW0
=dAdW0
W0
A(49)
which is the elasticity measuring the proportional increase inthe risky asset for an increase in initial wealth.
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Implications for risk-taking with RRA
Adding 1� AA to the right-hand side of (49) gives
� = 1+(dA=dW0)W0 � A
A(50)
Substituting dA=dW0 from equation (42), we have
� = 1+W0(1+ rf )E
hU 00( ~W )(~r � rf )
i+ AE
hU 00( ~W )(~r � rf )2
i�AE
hU 00( ~W )(~r � rf )2
i(51)
Collecting terms in U 00( ~W )(~r � rf ), this can be rewritten as
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Implications for risk-taking with RRA
� = 1+EhU 00( ~W )(~r � rf )fW0(1+ rf ) + A(~r � rf )g
i�AE
hU 00( ~W )(~r � rf )2
i (52)
= 1+EhU 00( ~W )(~r � rf ) ~W
i�AE
hU 00( ~W )(~r � rf )2
i (53)
The denominator in (53) is positive for A > 0 by concavity.Therefore, � > 1, so that the individual invests proportionallymore in the risky asset with an increase in wealth, ifEhU 00( ~W )(~r � rf ) ~W
i> 0.
Can we relate this to the individual�s risk aversion?George Pennacchi University of Illinois
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Implications for risk-taking with DRRA
Consider an individual whose relative risk aversion isdecreasing in wealth.
Then for A > 0, we again have W h >W0(1+ rf ). WhenRr (W ) �WR(W ) is decreasing in wealth, this implies
W hR(W h) <W0(1+ rf )R (W0(1+ rf )) (54)
Multiplying both terms of (54) by �U 0(W h)(rh � rf ), which isa negative quantity, the inequality sign changes:
W hU 00(W h)(rh�rf ) > �U 0(W h)(rh�rf )W0(1+rf )R (W0(1+ rf ))(55)
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Implications for risk-taking with DRRA
For A > 0, we have W l <W0(1+ rf ). If relative risk aversionis decreasing in wealth, this implies
W lR(W l ) >W0(1+ rf )R (W0(1+ rf )) (56)
Multiplying (56) by �U 0(W l )(r l � rf ), which is positive, sothat the sign of (56) remains the same, we obtain
W lU 00(W l )(r l�rf ) > �U 0(W l )(r l�rf )W0(1+rf )R (W0(1+ rf ))(57)
Inequalities (55) and (57) are the same whether therealization is ~r = rh or ~r = r l .
Therefore, taking expectations over all realizations of ~r yields
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Implications for risk-taking with DRRA
Eh~WU 00( ~W )(~r � rf )
i> �E
hU 0( ~W )(~r � rf )
iW0(1+rf )R(W0(1+rf ))
(58)
The �rst term on the right-hand side is just the FOC, soinequality (58) reduces to
Eh~WU 00( ~W )(~r � rf )
i> 0 (59)
Hence, decreasing relative risk aversion implies � > 1 so anindividual invests proportionally more in the risky asset aswealth increases.The opposite is true for increasing relative risk aversion: � < 1so that this individual invests proportionally less in the riskyasset as wealth increases.
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Risk-taking with ARA/RRA
The main results of this section can be summarized as:
Risk Aversion Investment BehaviorDecreasing Absolute @A
@W0> 0
Constant Absolute @A@W0
= 0Increasing Absolute @A
@W0< 0
Decreasing Relative @A@W0
> AW0
Constant Relative @A@W0
= AW0
Increasing Relative @A@W0
< AW0
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Conclusions
We have shown:
� Why expected utility, rather than expected value, is a bettercriterion for choosing and valuing assets.
� What conditions preferences can satisfy to be represented byan expected utility function.
� The relationship between a utility function, U(W ), and riskaversion.
� How ARA/RRA a¤ects the choice between risky and risk-freeassets.
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