Risky Assets in the Presence of Background Risk
Gunter Franke 1
, Richard C. Stapleton 2
,
and MartiG. Subrahmanyam.
3
November 2000
1
Fakultat fur Wirtschaftswissenschaften und Statistik, University of Konstanz Email:
guenter.franke@uni-konstanz.de
2
UniversityofStrathclydeEmail: rcs@staplet.demon.co.uk
3
SternSchoolof Business,NewYorkUniversityEmail: m.subrahm@stern.nyu.edu
Standard Risk Aversion and the Demand for Risky Assets in the
Presence of Background Risk
We consider the demandfor state contingent claims in thepresence of a zero-mean, non-
hedgeablebackgroundrisk. Anagentisdenedtobegeneralizedriskaverseifhe/shereacts
to anincreaseinbackgroundriskbychoosingademandfunctionforcontingentclaimswith
a smallerslope. Weshowthattheconditionsforstandardriskaversion: positive,declining
absoluteriskaversionandprudencearenecessaryandsuÆcientforgeneralizedriskaversion.
We also derive a necessary and suÆcient conditionforthe agent's derived risk aversion to
increasewitha simpleincreaseinbackgroundrisk.
"Journal of EconomicLiteratureClassicationNumbers:
D52,D81, G11."
1 Introduction
Recent advances in the theory of risk bearing have concentrated on the eect of a non-
tradeable background risk on the risk aversion of an agent to a second independent risk.
Forexample, GollierandPratt (1996) denearathergeneralclassofutilityfunctionssuch
that risk-averse individualsbecome even more risk averse towards a risk, when a second,
independent,unfairbackgroundriskis added. Theycompare therisk aversion of an agent
with no background risk to that of an agent who faces the background risk. They term
the set of functions under which the agent becomes more risk averse, the class of "risk-
vulnerable" utility functions. The set of risk-vulnerable functions is larger than the set
of proper risk averse functions introduced earlier by Pratt and Zeckhauser (1987), who
considerutility functionssuch thattheexpected utilityof an undesirableriskis decreased
by the presence of an independent, undesirable risk. Kimball (1993) has considered the
eect ofthe[evenlarger]setofexpectedmarginalutilityincreasing backgroundrisks. This
ledhimtodenethemorerestrictiveclassof standardrisk averse utilityfunctions. Standa
rdriskaversioncharacterisesthose functionswheretheindividualrespondsto an expected
marginal utility increasing background risk by reducing the demand for a marketed risk.
Kimballshows thatstandardriskaverse functionsarecharacterizedbypositive,decreasing
absolute risk aversion and absolute prudence. The set of standard risk averse functions
is a subset of the set of proper risk averse functions, which, in turn, are a subset of the
risk vulnerablefunctions,asdiscussedbyGollierand Pratt (1996,pp 1118-9). Inarelated
paper, Eeckhoudt, Gollier and Schlesinger (1996) extend this analysis by considering a
rather general set of changes in background risk, which take the form of rst or second
order stochastic dominance changes. They establish a set of very restrictive conditions
on the utility function such that agents become more risk averse when background risk
increases inthissense.
The purpose of this paper is twofold. First, we consider a smaller set of increases in
background riskthan Eeckhoudt, Gollierand Schlesinger(1996) and derive lessrestrictive
conditionsforanincreaseinbackgroundrisktoincreasethederivedriskaversionofagents.
We restrict the set of increases in the risk of background income y, with E(y) = 0, to
simpleincreases (see alsoEeckhoudt, Gollierand Schlesinger (1995)). A simple increasein
background risk is a change to y such that [=][]0 for y < [=] > y
0
for some y
0
and E() = 0. We derive a necessary and suÆcient condition on the utility function for
a simple increase in background risk to make the agent more risk averse. We show that
standard risk aversion is suÆcient, but not necessary for a simple increase in background
risk to increasederived risk aversion.
functions which guarantee a more risk averse behaviour in the presence of an increased,
independentbackgroundrisk,whentheagentfacesachoicebetweenstate-contingentclaims.
In this setting, changes in risk-averse behaviour are reected in the slope of the demand
function forcontingent claims.
1
Gollier(2000) considersa modelwhere theagent can buy
state-contingentclaimsonconsumption,givennobackgroundrisk. Letbetheprobability
deatedpriceofobtainingone unitofconsumptionifastate occursandnothingotherwise.
Then, in this model, the higher is for a given state, the lower is the agent's demand
for claims on that state, w. In other words, the demandfunction, w(), that relates the
consumption in a state to the price, is downward sloping. Gollier [Proposition51] shows
that, if two agentswith utilityfunctionsu
1 and u
2
have thesame endowment,and ifu
1 is
more risk averse thanu
2
,thenthedemandfunctionof agent 1,w
1
(), `single-crossesfrom
below' thedemandfunctionof agent 2,w
2
(). This single-crossoverpropertyisillustrated
in Figure 1. Gollier goes on to conclude that \risk-vulnerable investors willselect a safer
consumption plan", when they face background risk. Our results, showing the eect of a
simpleincreaseinbackgroundriskonriskaversion,thereforeimplythatanagent facingan
increaseinbackgroundriskwillrespond bychoosinga demandfunctionsimilartoinvestor
1 ratherthanthat chosen byinvestor2, inFigure1.
- 6
- 6
w
w
2
1
1
2
Figure 1:Demandcurve 1is less Figure2:Demand curve 1 isless
steep thandemandcurve 2every- steepthan demandcurve 2 insome
where rangeand steeperinother ranges
1
In a state-contingent claims model, risk-averse behaviour can be characterized by the slope of the
demandcurveforcontingentclaims. Inthecase ofextremeriskaversion,theagentbuysanequalamount
of claims oneach state, despite the higherprices of the claims onsome of the states. A less risk-averse
agentbuysascheduleofclaimsmoreweightedtowardsclaimsthatarerelativelycheap. InthePratt(1964)
characterization ofriskaversion, themoreriskaverseagentbuyslessofasinglerisky assetandmoreofa
risk-free asset. This alsohasthe eectofproducingademandcurvewith alowerslope. Theequatingof
`less risk-averse behavior'witha smallerslopeof thedemandfunctionforcontingentclaimsis thereforea
However, Gollier'sanalysishighlightsaproblem. Even thoughagent 1is more riskaverse,
he could have a demand function that has a smaller slope at the crossover point, but
has a greater slope over some range of , as Figure 2 illustrates. This means that the
more risk-averse investoractuallyexhibits less risk-averse behaviourover some range . As
Golliernotes,thesingle-crossoverpropertyonlythrowslightonlocalrisk-takingbehaviour
intherange aroundthecrossoverpoint. In thispaper, we wishto lookat localrisk-taking
behaviouroverallranges,henceweemployastricterdenitionofmorerisk-aversebehaviour
inthecontingentclaimsmodel. Wedeneanagent1tobehaveinamoreriskaversemanner
thanan agent 2,ifhisdemandfunctionhasasmaller(absolute) slopethanthatofagent 2
everywhere. Wethenconsiderhowtheslopeofthedemandfunctionchangesasbackground
riskincreases. Iftheagentrespondstoanincreaseinbackgroundriskbychoosingademand
functionwith asmaller slopeeverywhere,we saythat theagent isgeneralized risk averse.
Thisconceptofgeneralizedriskaversionrelates closelytothepreviouslydiscussedconcepts
of `risk vulnerability' and `standard risk aversion'. In the case of `risk vulnerability', an
agent respondsto the introductionof backgroundrisk byreducinghis demandfora single
riskyasset. Inthecaseofstandardriskaversion,anagent respondssimilarlytoamarginal
utility-increasing background risk. In the case of generalized risk aversion the idea of the
responseof risk-takingbehaviour toan increaseinbackgroundriskis extendedto thecase
of state-contingent claims.
2
Weconsidertheeectofanindependentbackgroundriskonthedemandforstate-contingent
claims, using an extension of the analysis of Back and Dybvig (1993), who establish con-
ditionsfor theoptimalityof an agent's demand. We investigate theset of [restrictions on]
utilityfunctions such that the agent responds to monotonic increases in zero-mean back-
groundriskbychoosingademandfunctionthathasasmallerslopeatallpricelevels. Inthe
context ofthischoice problem,weneedto furtherrestrictthesetof changes inbackground
risk that areconsidered to theset of monotonicincreases. A monotonic increase inback-
ground risk y is dened as a change in y, where @=@y 0;8y, and where E() = 0.
Hence, a monotonic increase in background risk is a change, , that itself increases with
y. The simplest example of a monotonic increase is a proportionate increase where is
proportionate to y. Ass uming monotonic increases in background risk, we nd that the
set of generalized risk-averse utility functions is the standard risk-averse class of Kimball
(1993). Hence,risk vulnerabilityis notsuÆcient forbackgroundrisk toreduce theslopeof
the demandfunctionforstate-contingent claims.
2
Variouspapershaveanalysedtheimpactofcertaintypesofincreasesinbackgroundriskonthedemand
for insurance,wheretheamountofinsuranceismeasuredbythecoinsurancerateandthe deductible,see,
for example Eeckhoudtand Kimball(1992) and Meyerand Meyer(1998). Whilethesepapers show that
standardriskaversionissuÆcienttoguaranteeahigherdemandforinsurance,wederiveherenecessaryand
The conditions for standard risk aversion - positive, declining absolute risk aversion, and
positive,decliningabsoluteprudence-aresuÆcientforamonotonicincreaseinbackground
risk to increasederived risk aversion. Theyare also suÆcient for theslope of the demand
function for contingent claims to become smaller everywhere. What is more surprisingis
that these conditionsarealso necessary forgeneralized riskaversion. Necessityarises from
the fact that the slope of the demand function for contingent claims must become less
steep at all possiblevalues of . As Kimball argues, declining absolute risk aversion and
decliningabsolute prudencearenaturalattributes oftheutilityfunction. Theyareshared,
also, by the HARA class of functions with an exponent less than one. The larger set of
risk-vulnerableutilityfunctions,usedbyGollierandPratt, isnotrestrictiveenough, when
we considertheeect of backgroundrisk on theslope of the demandfunction. Our result
adds to the case forthe standardrisk-averse functionsto be thenatural classof functions
to use whenanalysing theimpactof backgroundrisk.
In section 2, we look again at the eect of an increase in background risk on the risk
aversionof thederivedutilityfunction. Herewe areconcerned, aswereEeckhoudt, Gollier
and Schlesinger (1996) with changes in background risk. However, in order to avoid the
restrictive conditions on utilitythey found, we restrict the analysis to simple increases in
background risk. In section 3,we thenintroducetheproblemof analysingtheslope of the
demandfunction forcontingent claims. We then present our mainresult: agentschoosing
state-contingent claims become more risk averse in their choice, if and only if they are
standard risk averse, i.e. positiveand decliningabsolute risk aversion and prudenceis the
necessary and suÆcientconditionforgeneralized riskaversion.
2 The Eect of an Increase in Background Risk on Derived
Risk Aversion
We consider an individual agent who can buy a set of contingent claims on future con-
sumption and faces background risk. The agent's total income at the end of the period,
W,istherefore composedof anincome fromtradeable claims,w, plusthebackgroundrisk
income y, i.e. W = w+y. We assume that background risk, y, has a zero mean, and
is bounded from below, y a. Moreover we assume that y is distributed independently
of w. A state of the world determinesboth theagent's income from tradeable claims and
the background risk income. Let (;F;P) be the probabilityspace on which the random
variablesare dened.
strictly increasing,strictly concave, and four times dierentiable on W"(W;1), where W
is thelowerboundof W. We assume that there exist integrable functions on!", u
0 and
u
1
suchthat
u
0
(!)u(W)u
1 (!)
Wealso assumethat similarconditionshold forthederivativesu 0
(W),u 00
(W)and u 000
(W).
The agent's expected utility,conditional on w, is given by thederived utilityfunction, as
dened byKihlstrom etal. (1981) and Nachman (1982):
(w)=E
y
[u(W)]E[u(w+y)jw] (1)
whereE
y
indicatesanexpectation takenoverdierentoutcomesofy. Thus,theagentwith
backgroundriskandavonNeumann-Morgensternconcaveutilityfunctionu(W)actslikean
individualwithoutbackgroundriskand aconcave utilityfunction(w).
3
ThecoeÆcientof
absolute risk aversion isdened asr(W)= u 00
(W)=u 0
(W) and the coeÆcient of absolute
prudence as p(W) = u 000
(W)=u 00
(W). From Kimball (1993), the agent is standard risk
averse if and only if r(W) and p(W) are both positive and declining. The absolute risk
aversion oftheagent's derived utilityfunctionisdened asthenegative of theratioof the
second derivative to the rst derivative of the derived utility function with respect to w,
i.e.,
^ r(w)=
00
(w)
0
(w)
= E
y [u
00
(W)]
E
y [u
0
(W)]
(2)
We rst investigate the question of how an agent's derived risk aversion is aected by a
\simple increase" inbackground risk. A simple increase in background risk, which Eeck-
houdt,GollierandSchlesinger(1995)term'asimplespreadacrossy
0
',isdenedasachange
iny,, such thatfora giveny
0 ,
[=][]0; ify<[=][>]y
0
andE()=0:
Notsurprisingly,theconditionforanagent'sderivedriskaversiontoincrease,whenthereis
a marginalincreaseinzero-meanbackgroundrisk,isstronger thantheconditionofGollier
and Pratt(1996). This is because the \risk vulnerability" condition of Gollier and Pratt
only considers changes inbackground risk from zero to a nitelevel, whereaswe consider
anychanges inbackgroundrisk.
3
See,forexample,Eeckhoudt,GollierandSchlesinger(1996),p. 684.
It is worth noting that, in the absence of background risk , r(w)^ is equal to r(W), the
coeÆcient of absolute riskaversion of theoriginalutilityfunction. In thepropositionthat
follows,wecharacterizethebehaviorofr(w)^ inrelationtor(W),andexploretheproperties
of derived risk aversion in the presence of increasing zero-mean background risk. We will
proceed by proving a proposition about the condition under which any marginal increase
in background riskraises derived riskaversion. Since theconditionholdsfor anymarginal
increase in background risk, the same condition must hold for a nite increase to raise
derived risk aversion. It is convenient to dene an index of background risk,sR +
,where
s= 0 if no background risk exists. A marginalincrease in background risk is represented
byamarginalincreaseins. We assumethatthebackgroundriskincome yis dierentiable
ins.
4
Proposition 1 (Derived Risk Aversion and SimpleIncreases in BackgroundRisk)
If u 0
(W)>0 and u 00
(W)<0, then
@^r(w)
@s
>[=][<]0;8(w;s)()
u 000
(W
2 ) u
000
(W
1
)<[=][>] r(W)[u 00
(W
2 ) u
00
(W
1 )];
8(W;W
1
;W
2 );W
1
W W
2
Proof: See Appendix1.
In order to interpret the necessary and suÆcient condition under which an increase in a
zero-mean, background risk willraisethe riskaversion of thederived utilityfunction, rst
consider the special case in which background risk changes from zero to a small positive
level. This is the case analysed previously by Gollier and Pratt (1996). In this case, we
have
Corollary 1 In the caseof small risks, Proposition 1 becomes
^
r(w)>[=][<]r(W) i
@
@W
<[=][>]0;8W
4
This assumption in no way restricts the type of background risk increases assumed in the analysis.
Consider, forexample, jumpsinbackgroundrisk. These canbeanalysed assumsof smallincreases. Our
proofderivesconditionsforthederivedriskaversiontochangeinacertainmanner,givenasmallincrement
inbackgroundrisk. Thesameconditions assurethat derivedrisk aversion changesinasimilarmannerin
where (W)u 000
(W)=u 0
(W).
Proof: Let W
2 W
1
! dW. In this case, u 000
(W
2
) u
000
(W
1 ) ! u
0000
(W)dW. Similarly
u 00
(W
2 ) u
00
(W
1 )!u
000
(W)dW.
Hence,the conditionin Proposition1yields, inthiscase, u 0000
(W)<[=][>] r(W)u 000
(W).
This isequivalent to @=@W <[=][>]0;8W.2
InCorollary1,wedeneanadditionalcharacteristicoftheutilityfunction(W)=u 000
(W)=u 0
(W)
as a combined prudence/risk aversion measure. This measure is dened by theproductof
thecoeÆcientofabsolute prudenceand thecoeÆcient ofabsoluteriskaversion. Thecorol-
lary says that for a small background risk derived risk aversion exceeds [is equal to] [is
smaller than] risk aversion if and only if (W) decreases [stays constant] [increases] with
W. Hence, it is signicant that neither decreasing prudence nor decreasing absolute risk
aversionis necessaryforderivedrisk aversion to exceedriskaversion. However, thecombi-
nation of these conditions is suÆcient for theresult to hold, sincethe requirement is that
the product of the two must be decreasing. The condition is thus weaker than standard
risk aversion,whichrequiresthat both absoluterisk aversion andabsolute prudenceshould
be positive and decreasing. Note that the conditionin this case is the same as the'local
riskvulnerability'conditionderived byGollierandPratt (1996). Localriskvulnerabilityis
r 00
>2rr 0
,whichisequivalentto 0
<0. WenowapplyProposition1toshowthatstandard
risk aversion is a suÆcient, but not a necessary condition, for an increase in background
risk to cause an increase in the derived risk aversion [see also Kimball(1993)]. We state
thisas
Corollary 2 Standard riskaversion isa suÆcient,butnotnecessary,conditionforderived
risk aversion toincreasewith a simple increase in backgroundrisk.
Proof: Standardriskaversion requiresbothpositive,decreasingabsolute riskaversion and
positive decreasing absolute prudence. Further, r 0
(W) < 0 ) p(W) > r(W). Also, stan-
dard risk aversion requires u 000
(W) > 0. It follows that the condition in Proposition1 for
an increase inthederived risk aversion can bewrittenas 5
u 000
(W
2 ) u
000
(W
1 )
u 00
(W
2 ) u
00
(W
1 )
< r(W
1 )
5
Notethatwheneverr 0
(W)hasthesamesignforallW,thethree-stateconditioninProposition1(i.e.
theconditiononW,W ,andW )canbereplacedbyatwo-statecondition(aconditiononW andW ).
or, alternatively,
p(W
1 )
1 u
000
(W
2 )
[u 000
(W
1 )
=
1 u
00
(W
2 )
[u 00
(W
1 )
>r(W
1 )
Since p(W
1
) >r(W
1
), a suÆcientconditionis that theterm inthesquare bracketexceeds
1. This,in turn, follows from decreasing absolute prudence, p 0
(W) <0. Hence, standard
risk aversion is asuÆcient condition.
Toestablishthatstandardriskaversionisnotnecessary,consideracasethatisnotstandard
risk averse. Suppose, in particular, that u 000
(W) < 0;u 0000
(W) < 0, that is, the utility
function exhibits increasing risk aversion and negative prudence.
6
In this case, it follows
from Proposition1 that@^r=@s>0;8(w;s). 2
Inorderto obtainmoreinsightintothemeaningoftheconditioninProposition1,consider
the case where the increase inbackground risk raises derived risk aversion. Dening y 0
=
@y=@s,
^
r(w) =E
y
"
u 0
(W)
E
y [u
0
(W)]
r(W)
#
;
@^r(w)
@s
=E
y
"
u 0
(W)
E
y [u
0
(W)]
r 0
(W)y 0
#
+E
y
"
r(W)
@
@y
"
u 0
(W)
E
y [u
0
(W)]
#
y 0
#
(3)
Asshowninappendix1,itsuÆcesto considerathree-pointdistributionofbackgroundrisk
with y
1
<0;y
2
>0;y
1
<y
0
<y
2 and y
0
0
=0;y 0
1
<0;y 0
2
>0. The rst terminequation (3)
is positive whenever r is declining and convex. This follows since E(y 0
) = 0 and y 0
2
> y 0
1
impliesthat E[r 0
(W)y 0
]0. Since u 0
(W) is declining,it follows thatthe rst term in(3)
is positive. Now consider the second term: @[u 0
(W)=E
y [u
0
(W)]]=@y is positive for y
1 and
negative for y
2
and has zero expectation. Therefore a declining r impliesthat the second
termispositive. HenceasuÆcientconditionfor@r(w)=@s^ 0isadecliningandconvexr.
7
The rst termishigher, themore convexis r. Therefore, @^r(w)=@s0 is alsopossiblefor
an increasing r,ifconvexity is suÆcientlyhigh. Therefore, there are utilityfunctionswith
6
Asanexample,considertheutilityfunction
u(W)=
1
A+ W
1
;where2(1;2);W <A( 1)
This utility function exhibits increasing risk aversion and negative prudence. Still, (W) decreases with
wealtheveninthiscaseandthederivedriskaversionincreaseswithbackgroundrisk.
7
increasing risk aversion which still imply that simple increases in zero-mean background
risk raisethederived riskaversion.
3 The Eect of Changes in Background Risk on the Optimal
Demand Function for Contingent Claims
Intheprevioussectionwederivedtheconditionunderwhichanincreaseinbackgroundrisk
increasestheagent's derivedriskaversion. Aswillbeshown,thisconditionisnotsuÆcient
to guarantee thatthe increaseinbackgroundrisk reduces theslope ofthe agent's demand
curve for state-contingent claims, everywhere, i.e., it is not suÆcient for generalized risk
aversion. In this section we derive the necessary and suÆcient condition for the utility
functionto exhibitgeneralized risk aversion.
Weassumethatthecapitalmarketisperfect. Astate ofnaturedeterminesboththeagent's
tradeableincomewandhisbackgroundincomey. Wepartitionthestatespaceintosubsets
ofstatesthatdieronlyinthebackgroundincome,y. Wecallthesesubsets\tradedstates"
sincetheyrepresentstatesonwhichstate-contingentclaimscanbetraded. Weassumethere
is a continuum of such states and, for convenience, we label these states by a continuous
variable xR +
. We assume the market, in the traded states, iscomplete. We also assume
that there exists a pricing kernel, = (x) with the property > 0, where (x) is a
continuousfunction.
8
Let w = g(x) be the agent's income from the purchase of state-contingent claims. The
agent chooses w = g(x), subject to the constraint that the cost of acquiring this set of
claims is equal to his/her initial endowment. The agent's consumption at the end of the
singleperiod,W,isequaltothechosenmarketedclaim,w,plusanindependent,zero-mean
background risk y, i.e. W = w+y. The background risk y aects his/her choice of the
function w =g(x). We assume that the agent has suÆcient endowment to ensure that w
can be chosen to obtainW W inall traded states. We also assumecertain propertiesof
8
ThemarketiscompleteinthesenseofNachman(1988). Theagentcanbuyadigitaloptionwhichpays
oneunitofconsumption,ifxk,and0otherwise, 8kR +
. Thepriceofsuchanoption is
Z
1
k
(x)f(x)d(x);
where (x) is the pricing kernel and the probability density function is f(x). A contingent claim is a
contract(aportfolioofdigitaloptions)payingoneunitofconsumptionifx[k;k+)andnothingotherwise,
for positive,innitelysmall.
the utilityfunction. First, themarginalutilityhasthelimits:
u 0
(W)!1ifW !W;
u 0
(W)!0ifW !1:
Second,therisk aversiongoesto zero at highlevels of income,i.e.
r(W)!0ifW !1:
Thesereasonablerestrictionsaresatised,forexample,bytheHARAclasswithanexponent
less than1.
The agent solves thefollowingmaximizationproblem:
max
w=g(x) E
x
[(w)]=E
x
[(g(x))] (4)
s.t. E
x h
(g(x) g 0
(x))(x) i
=0
In thebudgetconstraint, w 0
=g 0
(x) istheagent's endowmentof claims. (x), thepricing
kernel, is given exogenously. The maximisation problem(4)is a standard state-preference
maximisation problem. The expectation, E
x
(:),is taken onlyover thetraded states. Note
that the background income, y, has only an indirect impact on problem (4) through its
eect on thederivedutilityfunction. This isdenedbyequation(1)asthe expectedvalue
of utilityoverdierent outcomes ofy,given thetraded incomew.
The rst orderconditionfora maximumis
0
(g(x))=(x);
orsimply
0
(w) =; (5)
where is a positive Lagrange multiplier which reects the tightness of the budget con-
straint. Equation (5) holdsas an equality since, by assumption, u 0
(W) !1 forW ! W
and u 0
(W) ! 0 for W ! 1. The demand for claims in equation (5) can be shown to
be optimal and unique under some further niteness restrictions.
9
This follows from the
resultsof Back andDybvig (1993).
9
E[w]<1forany>0andeachwsatisfying(5)isassumed.
>From the rst order condition (5), it follows that we can dene a function w = w() =
0( 1)
(). Hence,giventhederivedutilityfunctionandtheinitialendowment,thedemand
forclaimscontingentonatradedstatexdependsonlyon(x). Thusw()isadeterministic
function relating the demand for state-contingent claims to the pricingkernel. It follows
from ourassumptionsthat w() isa twice dierentiablefunctionof . 10
OuraimistondthenecessaryandsuÆcientconditionsontheutilityfunction,whichguar-
anteethattheagent's demandfunctionbecomeslesssteepwhenbackgroundriskincreases.
Firstwedene
Denition 1 Anagentisgeneralizedrisk-averseiftheabsolutevalueoftheslopeofhis/her
demand function for state-contingent claims w() becomes smaller for all , given an in-
crease in backgroundrisk.
Dierentiating equation (5) with respect to , for a given level of background risk, and
dividingby, yieldstheslopeof thedemandfunction
@w
@
= 1=
^ r(w)
;8 (6)
Suppose that background risk increases the derived risk aversion of the agent, r(w).^ It
followsfromequation(6)thatthebackgroundriskaectstheslopeofthedemandfunction.
We now considerthe eect of changes in the level of background risk, assuming that the
pricingfunctionis given. Fromequation (6)itappearsat rst sightthattheslopeof the
demandfunctionbecomeslesssteepwhenevertheincreaseinbackgroundriskincreasesthe
agent's derived riskaversion. Infact, it followsfrom Gollier(2000, Proposition51) that:
Proposition 2 Suppose that an increasein backgroundrisk raises the agent'sderived risk
10
Considerthefunction
F(w;;s)= 0
(w) =0:
The partial derivative F
w
exists and is continuous, since the utility function u(w+y) and its rst three
derivativesare assumedto exist and to beintegrable. Also Fw 6=0 for w < 1. Hence, by the implicit
functiontheorem,thefunctionw=w()isdierentiablewith
@w=@= F
F
w :
Also, sincey=y(s)isdierentiable,and sinceF
andF
w
aredierentiableiny,thenF
andF
w
arealso
2
aversion, everywhere. Then the new demand curve for contingent claims intersects the
original one oncefrom below.
Proof: At anintersectionofthenew demandcurve,w
1
(),and theoriginaldemandcurve,
w
0 (),w
1
=w
0
sothat,byequation(6),@w
1
=@>@w
0
=@followsfrom^r
1
>r^
0
. Asecond
intersection would require @w
1
=@ < @w
0
=@, which contradicts (6). Also, at least one
intersectionmust exist,inorder forthebudget constraint to be satised.2
However, asnoted byGollier(2000),theone-intersectionpropertydoesnotimplythat the
newdemandcurveislesssteep thantheoriginalone, everywhere. Thisisbecause achange
in backgroundrisk, aects ^r(w) bothdirectly and through theinducedchange inw. This
is statedinthefollowingproposition.
Proposition 3 For the slope of the demand function for contingent claims to become
smaller with an increase in background risk (generalized risk aversion), it is necessary,
but not suÆcient for the absolute risk aversion of the derived utility function to increase
with background risk. Thatis
d
ds
@w
@
0)
@r(w)^
@s
0; (7)
but
@^r(w)
@s 0
does not imply
d
ds
@w
@
0:
Proof: Totallydierentiatingequation (6)with respectto syields, since1=is given,
d
ds
@w
@
= 1=
[^r(w)]
2 dr(w)^
ds
: (8)
Since
1=
[^r(w)]
2
>0;
d
@w
0, d^r(w)
=
@^r(w)
+
@^r(w)@w
0: (9)
Given thebudget constraint, @w=@shas to bepositiveinsome traded statesand negative
inothers. It followsimmediately that@^r(w)=@s0is notsuÆcient to ensurethat
d
ds
@w
@
0:
Now to establishnecessity,supposethat
d
ds
@w
@
0
forall , thensincethesign of
@r(w)^
@w
@w
@s
depends onthesign of @w=@s, which can bepositive ornegative,
d
ds
@w
@s
0)
@r(w)^
@s 0:
2
Havingshownthatincreasedderivedriskaversionisanecessary,butnotsuÆcientcondition
forgeneralized riskaversion,wecan nowestablish ourmainresult. Inorder to analyse the
impactofbackgroundriskontheslopeoftheagent'sdemandfunctionforcontingentclaims
we need to make stronger assumptions. Regarding the background risk we now assume
monotonic changes in background risk. This is a somewhat stronger than the previous
assumption of simple increasesin backgroundrisk. Firstwe dene monotonicincreases in
background risk.
Denition 2 (Monotonic Increases in Background Risk)
Let y
i
(s) denote a realisation i = 1;:::;j of background risk income, given the index of
background risk, s. Suppose that
y
1
(s)y
2
(s):::y
i
(s):::y
j (s)
with y
i
(0)=0;8i. Then, increases in background risk are monotonic, iffor any s>s0,
y
1 (s) y
1
(s)y
2 (s) y
2
(s):::y
i (s) y
i
(s):::y
j (s) y
j (s)
Theeect ofassumingmonotonicincreasesinbackgroundriskisthattherankorderof the
outcomes y
1
;y
2
;::: is preserved under a monotonic increasein backgroundrisk. The main
Proposition 4 (Generalized Risk Aversion)
Assumeanymonotonicincreaseinanindependent,zero-meanbackgroundrisk. Letu 0
(W)>
0 and u 00
(W) < 0, where W"(W;1). Suppose that u 0
(W) ! 1 for W ! W and that
u 0
(W)!0 and r(W)!0, for W !1. Then
d
ds
@w
@
0; 8; 8probabilitydistributionsof
()
utilityisstandardriskaverse:
We rstestablish three lemmaswhich arerequiredintheproof. We have
Lemma 1 Suppose that u 0
(W) ! 1 for W ! W, then r(W) ! 1 and p(W) ! 1 for
W !W.
Proof: u 0
(W) ! 1, for W ! W, implies@lnu 0
(W)=@W ! 1, and hence r(W) ! 1.
Also, sinceforW !W,r 0
<0;p>r,and hencep(W)!1. 2
The second lemma establishes the equivalenceof declining risk aversion and declining de-
rivedrisk aversion. We have:
Lemma 2 ^r 0
(w)0 for any background risk ,r 0
(W)0
Proof: Kihlstromet. al. (1981)andNachman(1982)haveshownthatdecliningriskaversion
impliesdeclining derived risk aversion. Conversely, decliningderived risk aversion implies
decliningrisk aversionof u(W). This follows fromthe caseof smallbackground risks.2
The thirdlemma establishes a condition for declining prudence, in the case of monotonic
changes inbackgroundrisk:
Lemma 3 For monotonic increases in background risk,
d
d
@ 0
(w)=@s
@ 0
(w)=@w
0,p 0
(W)0
Proof: See Appendix2.
We now presentthe proof ofProposition(4).
Proofof Proposition(4) : Totallydierentiatingequation (5) withrespectto syields
@ 0
(w)
@s +
@ 0
(w)
@w
@w
@s
= d
ds
: (10)
Substitutingfrom equation(5) thenyields
@ 0
(w)
@s +
@ 0
(w)
@w
@w
@s
= dln
ds
0
(w):
Hence,the eect ofthe backgroundriskon thedemandforclaimsis givenby
@w
@s
=
dln
ds 1
^ r(w)
@ 0
(w)=@s
@ 0
(w)=@w
: (11)
The Propositionis concernedwiththeconditions underwhich
d
ds
@w
@
= d
d
@w
@s
0:
We investigate these conditions bylooking at the behaviour of the two terms in equation
(11).
SuÆciencyof Standard Risk Aversion: First,weshowthatthersttermin(11)isnegative,
while the second term is positive. In order to satisfy the budget constraint, @w=@s has
to be positive in some traded states and negative in others. Given positive prudence,
@ 0
(w)=@s > 0, so that the second term in (11) is positive. It follows that the rst term
must benegative. We can nowinvestigate
d
d
@w
@s
;
bytaking the two terms in(11) one-by-one. First,the (negative)rst term increaseswith
, since
@r^
=
@r^@w
ispositive. Thisfollowsfrom @w=@<0(seeequation (6))and@r=@w^ 0(whichinturn
follows from @r=@w0 and Lemma2). Second,the(positive)second termincreases in,
given decliningprudence(see Lemma3). Hence
d
d
@w
@s
is positivegiven standardrisk aversion.
Necessity of Standard Risk Aversion: We establish necessity of standard risk aversion by
taking the specialcase of a small background risk. Also, we assume converges in prob-
ability to a degenerate distribution,
0
. By assuming w(
0
) is, in turn, large [small], we
showthattherst[second]termin(11)dominates. Fortherst termin(11)toincreasein
, decliningriskaversionisrequired. Forthesecondtermin(11)to increasein, declining
prudence is required. Hence,to cover bothof these possibilities,standard risk aversion is
required. First,weconsidertheterm dln=ds.
We have from equation (5),
E[
0
(w)]=E[u 0
(w+y)]=E()=
and
d
ds
= d
ds E[u
0
(w+y)]= d
ds E[u
0
(w )];
where = (w) is theprecautionarypremiumasdened byKimball(1990). Hence,
d
ds
=E
u 00
(w )
@w
@s
@
@s
@
@w
@w
@s
:
Assume that we start from a position of no background risk. In this case, s =0, = 0,
and @ =@w = 0. Since, for small background risks with variance 2
, the precautionary
premiumis 11
= 1
2 p(w)
2
;
it followsthat
d
ds
=E
u 00
(w)
@w
@s
@
@s
=E
u 00
(w)
@w
@s 1
2 p(w)
2
:
11
ThisfollowsbyanalogywiththePratt-Arrowargumentfortheriskpremium,sinceinitially,s=0.
Now we assume that converges to the degenerate distribution
0
, in probability. Since
wecan write
d
ds
=E[f()];
where f isa continuous,uniformlyintegrable function, thenitfollows that
d
ds
!u 00
(w
0 )
1
2 p(w
0 )
2
;
where w
0
= w(
0
), since @w
0
=@s = 0, for the case of the degenerate distribution,
0 .
Dividingby=u 0
(w
0 ),
dln
ds
! u
00
(w
0 )
u 0
(w
0 )
1
2 p(w
0 )
2
and hence
dln
ds
! r(w
0 )
1
2 p(w
0 )
2
:
Substitutingin(11), wenowhave
@w
@s
!r(w
0 )
1
2 p(w
0 )
2
1
^ r(w)
@ 0
(w)=@s
@ 0
(w)=@w :
Starting withno backgroundrisk,theterm
@ 0
()=@s
@ 0
()=@w
= 1
2 p(w)
2
;
since@ =@w=0. Hence,wecan write
@w
@s
!r(w
0 )
1
2 p(w
0 )
2
1
^ r(w)
+ 1
2 p(w)
2
: (12)
Dierentiating (12), wethen have
d
ds
@w
@
= d
d
@w
@s
!
r(w
0 )
1
2 p(w
0 )
2
^ r 0
(w)
^ r(w)
2 +
1
2 p
0
(w) 2
@w
@
Since @w=@<0,the conditionfora smaller slopebecomes
r(w
0 )
1
p(w
0 )
2
^ r 0
(w)
2 +
1
p 0
(w) 2
0: (13)
Toestablish thenecessityofdecliningabsolute riskaversion,we choose
0
suchthat w
0
!
W. By Lemma 1 hence r(w) ! 1 and p(w
0
) ! 1, for w ! W. Therefore, r^ 0
(w) > 0
implies that the rst term in equation (13) ! 1. Then, since the second term in (13) is
independentofw
0
;r^ 0
0andbyLemma2,r 0
0isrequiredforthecondition(13)tohold.
r 0
0 also establishesthenecessityof positiveprudence, p>0.
To establish necessity of declining absolute prudence, we choose
0
such that w
0
! 1
and hence, by assumption r(w
0
) ! 0. Then r 0
(w
0
) = r(w
0 )[r(w
0
) p(w
0
)] ! 0 implies
r(w
0 )p(w
0
)!0. Hencetherstterminequation(13)!0. Then,sincethesecondtermin
(13)is independentofw
0 ,p
0
0isrequiredforthecondition(13)to hold. Hencestandard
risk aversion is anecessary conditionforasmaller slope.2
Proposition(4)allows usto analyzethe eect ofa marginalincreaseina zero-mean, inde-
pendent backgroundrisk, given that thisincrease hasa negligible impact on the prices of
state-contingent claims. Since a nite increasein background risk is the sum of marginal
increases,theconditioninProposition(4)also holdsforniteincreasesinbackgroundrisk.
Proposition (4) says that an increase in background risk will reduce the steepness of the
slopeofthisagent'sdemandfunction. AscanbeseenfromProposition(4),theagentreacts
to a monotonicincreaseinbackgroundrisk bypurchasingmore claimsintraded states for
whichthepriceishigh,nancingthepurchasebysellingsomeclaimsinthetradedstates
with low prices. Proposition(4) can also be interpretedby comparing, within an equilib-
rium,thedemandofagents, whodieronlyinthesizeoftheirrespectivebackgroundrisks.
Proposition(4)suggests thatagents withhigherbackground riskwilladjust theirdemand
functions by buyingstate-contingent claims on high-price traded states and selling claims
on low-price traded states. This is illustrated in Franke, Stapleton and Subrahmanyam
(1998), for an economy in which all agents have the same type HARA-class utility func-
tion, exhibiting declining absolute risk aversion. These functions are standard risk averse
and hencegeneralized risk averse. Inthis economy, agents withhigh background riskbuy
options from those with relatively low background risk. The latter agents sell portfolio
insuranceto theformer withrelativelyhigh backgroundrisk.
4 Conclusions
Themainconclusionsregardingtheeectsofanincreaseinbackgroundrisk,onriskaversion
and on the demand for contingent claims, are summarised in the four propositions of the
paper. Proposition1 provides a necessary and suÆcient condition forsimple increases in
background risk to increase the derived risk aversion of agents. The condition on utility
vulnerability. By considering onlythe setof simpleincreasesinbackgroundrisk,wenda
larger set of utility functions which satisfythe criterion of increasedderived risk aversion,
thanthoseofEeckhoudt,GollierandSchlesinger. Wethenproceedtoexaminethecondition
for'generalizedriskaversion',wherebyagentsreacttoincreasedbackgroundriskbyreducing
the slope of the demandcurve forstate contingent claims. We ndin Proposition3 that
increasedderivedriskaversionis necessary,butnotsuÆcient,forgeneralized riskaversion.
The stronger requirement forgeneralized risk aversion isshown forthe case of monotonic
increasesinbackgroundriskinProposition4. Standardriskaversion,i.e. positive,declining
absolute risk aversion and absolute prudence, is a necessary and suÆcient condition for
generalized riskaversion.
