Perspectives of risk sharing

KnutK. Aase

Norwegian School of Economics and

Business Administration

5035 Sandviken - Bergen, Norway

May, 2000

Abstract

In this paper we present an overview of the standard risk sharing

modelof insurance. We discussand characterize a competitiveequilib-

rium, Pareto optimality, and representative agent pricing, including its

implicationsfor insurancepremiums. Weonlytouchupontheexistence

problemofacompetitiveequilibrium,primarilybypresentingseveralex-

amples. Risktoleranceandaggregationisthesubjectofonesection. Risk

adjustmentoftheprobabilitymeasureisonetopic,aswellastheinsurance

versionofthecapitalassetpricing model.

Thecompetitiveparadigmmaybealittledemandinginpractice,sowe

alternativelypresentagametheoreticviewofrisksharing,wheresolutions

endupinthecore. Properlyinterpreted,thismaygiverisetoarangeof

pricesofeachrisk,oftenvisualizedinpracticebyanaskpriceand abid

price. Theniceaspectofthisisthatthesepricerangescanbeexplained

by\rstprinciples",notrelyingontransactioncostsorotherfrictions.

We end the paper by indicating the implications of our results for

a pure stock market. In particular we nd it advantageous to discuss

the concepts of incomplete markets in this general setting, where it is

possible to use results for closed, convex subspaces of an L 2

-space to

discussoptimalriskallocationproblemsinincompletenancialmarkets.

KEYWORDS: Reinsurance Model, Equilibrium, Pareto Optimality,

CoreSolution,StockMarket,Complete Model

1 Introduction

Thispaperisprimarilyareviewpaper,wherewepresentthestandardriskshar-

ingmodelofreinsurancemarkets. Themodelconsideredstartswith aset ofI

agents,interpreted as(re)insurers, eachendowed witharandompayo X

i for

agenti, i=1;2;:::;I. Supposingtheagentscannegotiateanyaordablecon-

tractsamongthemselves,resultinginanalportfolioY

i

,oneessentialobjective

is to characterize these random variable Y

i

most preferred by agent i. Other

applicationsaremanifold,sincethismodelisindeedverygeneral. Forinstance,

X

i

mightrepresentrandomlyvarying waterendowments,couldstandforana-

tion'squotainproducingdiversepollutants,couldbetheinitialendowmentof

InvitedlectureatAFIR2000inTroms,Norway,wasbasedlargelyonthepresentpaper.

thelastsectionofthepaper.

We discuss and characterizea competitive equilibrium, Pareto optimality,

andrepresentativeagentpricing,includingitsimplicationsforinsurancepremi-

ums. Weonlytouchupontheexistenceproblem ofacompetitiveequilibrium,

primarily by presenting several examples. Risk tolerance and aggregation is

the subjectof onesection. Risk adjustmentof theprobability measure isone

topic,aswellas theinsuranceversionof thecapitalassetpricingmodelbased

onmultinormality.

The competitive paradigm may be a little demanding in practice, so we

alternativelypresentagametheoreticviewofrisksharing,wheresolutionsend

upin thecore. Properlyinterpreted, thismaygiveriseto arangeof pricesof

eachrisk,oftenvisualizedinpracticebyanaskpriceandabidprice. Thenice

aspect ofthis is that thesepriceranges canbeexplained by\rst principles",

notrelyingontransactioncostsorotherfrictions.

We end the paperby indicating the implications of our results for a pure

stockmarket. Inparticular wend itadvantageousto discussthe conceptsof

incomplete markets in this general setting, where it is possibleto use results

for closed, convexsubspacesof an L 2

-space to discuss optimalrisk allocation

problemsin incompletenancialmarkets.

The paper is organized asfollows: In section 1 we present the basic risk-

exchangemodel, insection2wecharacterizeacompetitiveequilibrium,in sec-

tion 3wecharacterizeaPareto optimum,in section 4weintroducethe repre-

sentativeagent, and is section 5 we discuss existence problems. Section 6 is

devotedtorisktoleranceandaggregation,section7toinsurancepremiumsand

section 8 to risk adjustments of the given probability measure. In section 9

wepresentthe capital assetpricingmodelin insuranceterms. Section 10 isa

game theoreticapproachto theriskallocationproblem,and weend thepaper

insection11,wheretheimplicationsforastockmarketofeÆcientallocationof

risksisdiscussed.

2 The Basic Risk-Exchange Model

Inthisarticlewestudythefollowingmodel: LetI =f1;2;:::;Igbeagroupof

I reinsurers,simplytermedagentsforthetimebeing,havingpreferences

i over

asuitable set of randomvariables, orgambles with realizations(outcomes) in

someAR . ThesepreferencesarerepresentedbyvonNeumann-Morgenstern

expected utility, meaning that there is a set of continuous utility indices u

i :

R ! R , such that X

i

Y if and only if Eu

i

(X) Eu

i

(Y). We assume

monotonicpreferences, andriskaversion,so that, grantedenoughsmoothness,

wehave u 0

i

(w)> 0;u 00

i

(w) 0for allw in therelevant domains.

1

Sometimes

weshallalsorequirestrict riskaversion,meaningstrict concavityforsomeu

i .

Each agent is endowed with a random payo X

i

called his initial portfolio.

Moreprecisely,thereexistsaprobabilityspace(;F;P)suchthatiisentitled

to payo X

i

(!)when ! 2occurs. Thismeans that uncertainty isobjective

andexternal. Andthereisnoinformationalasymmetry. Allpartiesagreeupon

(;F;P) as the probabilistic description of the stochastic environment, the

1

Note thatthe conceptsofmonotonicityand riskaversionmakeperfectlysense without

assumingtheexistenceofthesederivatives.

expectedvaluesandvariancesexistforalltheseinitialportfolios,whichmeans

that allX

i 2L

2

(;F;P),orjust X

i 2L

2

forshort.

Wesupposetheagentscannegotiateanyaordablecontractsamong them-

selves, resulting in a new set of random variables Y

i

;i 2 I, representing the

possiblenalpayoutto thedierentmembersof thegroup,ornalportfolios.

Thetransactionsarecarriedoutrightawayat\marketprices",where(Y)rep-

resentsthe market priceforanyY 2L 2

, i.e., itsigniesthe group'svaluation

of the random variable Y relativeto the other random variables in L 2

. The

essentialobjectiveisthento determine:

(a)Themarketprice(Y)ofany\risk"Y 2L 2

fromthesetofpreferencesof

theagentsandthejointprobabilitydistributionF(x

1

;x

2

;:::;x

I

)oftherandom

vectorX =(X

1

;X

2

;:::;X

I ).

(b) For each i, the nal portfolio Y

i

most preferred by him among those

satisfyinghisbudgetconstraint(Y

i

)(X

i ).

Someobservationsarein order. First,observethatthepossibleeventsF=

F X

:= (X

1

;X

2

;:::;X

I

) is the sigma-eld generated by the initial random

variables X, so that any random variable can be written in the form Y =

f(X

1

;X

2

;:::;X

I

) for f a suitable Borel-measurable function.

2

This means

that theoptimalnal portfoliosY

i

=f

i (X

1

;X

2

;:::;X

I

)forsomeappropriate

functionsf

i

. Inordertoavoidtrivialities,weassumethatF X

iscomplete,i.e.,

augmentedwithalltheP-nullsets.

Second,unlessthefunctionalonL 2

islinear,arbitragewouldbepossible.

Toseethis, considerthecasewheree.g.,(Z+Y)>(Z)+(Y)foranytwo

random variables Z and Y in L 2

. Since we assumeinnite divisibilityof any

portfolio,areinsurercouldinsurethebundle(Z+Y),andthenreinsureZ and

Y separately. Thecash owsfromthesetradeswouldbe

(Z+Y) ((Z)+(Y))>0

attime0,and (Z+Y)(!)+Z(!)+Y(!)=0attime1forany!2. Thus

the reinsurer has made a risk-freeprot whatever the state of nature, which

shouldnotbepossibleinanyconsistentmodelofthismarket. Thusitmustbe

thecasethat islinear,i.e.,itsatises

(aZ+bY)=a(Z)+b(Y)

foranyconstantsa;b2RandrandomvariablesZ ;Y 2L 2

.

Third, the pricing functional should be positive, meaning simply that

(Z) 0 for any Z 0 P-a.s. In other words, a random variable that is

non-negativewithprobability1,should haveanon-negativemarketprice.

Fromfunctionalanalysisitisknownthatapositive,linearfunctionalonan

L p

-space is bounded (1 p <1), and hence also continuous, in which case

wecan use theRiesz representation theorem and concludethat there exists a

uniquerandomvariable2L 2

suchthat

(Z)=E(Z) forallZ2L 2

:

This randomvariable,the Rieszrepresentation, weshallsometimes refertoas

thestate-price deator. Atthemomentwecanonlyconcludethatthere exists

2

Thisisaresultthatisknownfrommeasuretheory,e.g.,Tucker(1967),Theorem1.1.

1 2 I

Riesz representation. Our aim is now to characterizethis particular f, and

also the f

i

-functions corresponding to the optimal Y

i

;i 2 I. The following

notationalconventionwillbeused: IfX andY aretworandomvariables,then

byXY wemeanthat (Y X)0P-a.s.,i.e.,therandomvariable(Y X)

isnon-negativealmostsurely.

Denition1 An allocation Z=(Z

1

;Z

2

;:::;Z

I

)iscalledfeasibleif

I

X

i=1 Z

i

I

X

i=1 X

i :=X

M :

Theproblemeachagentissupposed tosolveisthefollowing:

sup

Zi2L 2

Eu

i (Z

i

) subjectto (Z

i

)(X

i

): (1)

An importantissueis,ofcourse,existence(anduniqueness)ofsolutionsto(1).

Weshallnotelaborate onthis here,suÆceitistonotethefollowing: If

fZ

i 2L

2

:Eu

i (Z

i

)<1; (Z

i

)(X

i )g

isbounded(inL 2

-norm),thenexistenceisguaranteed.

3

Also,astrictlyconcave

u

i

suÆcesforuniqueness.

Denition2 Acompetitive equilibrium isa collection (;Y

1

;Y

2

;:::;Y

I )con-

sistingofapricefunctionalandafeasibleallocationY =(Y

1

;Y

2

;:::;Y

I )such

thatforeachi,Y

i

solvestheproblem(1)andmarketsclear;

P

I

i=1 Y

i

= P

I

i=1 X

i .

4

Weclosethesystembyassumingrationalexpectations. Thismeansthatthe

marketclearingprices impliedby agentbehaviorisassumed tobethesame

asthepricefunctionalonwhichagentdecisionsarebased. Themainanalytic

issueisthenthedeterminationofequilibrium pricebehavior.

Inthemicroeconomicliterature thereare colorfuldescriptionsofhowsuch

an equilibrium might result, involving e.g., the Walrasian auctioneer, in the

caseofnouncertainty. Inthereinsurancemarketitisperhapsmorerealisticto

think ofbilateraltradesbetweenreinsurers.

Wenoticethat theconceptofWalrasianequilibriumis widelyemployedin

consumertheory,althoughtheanalysiscanbehardandtheconclusionsrequire

consumers who are extraordinarily sophisticated. There is, however, a lot of

experimental evidence, where anumber of researchershave attempted to see

whether markets perform under controlled conditions in the way economists

assume theydo. Theresults obtainedare usually strikingin theirsupport of

Walrasianequilibrium.

When an insurer is invited to cover a large risk, he may decide that he

cannot,or doesnotwanttodoso entirely. Hemayrathercovermerelypartof

the risk, saya fraction,against thecorrespondingpart ofthe premium. This

3

Byi.a.,theBanach-AlaogherTheorem

4

Marketclearingisusuallydenedby P

I

i=1 Y

i

P

I

i=1 X

i

. Sincewehavestrictlymono-

tonicpreferences,equalitywillresultinequilibrium.

the rest of the risk. From the 1680's he knewthat he could nd these other

insurersatthecoeehouseofEdwardLloydinLondon.

Lloyd's of London still operatesin this way. To buy insuranceat Lloyd's

onehas to contact abrokerwhois accredited at Lloyd's. The brokertakesa

\slip", which containsall relevantinformation about the risk, to oneormore

underwriterswhospecializesin risksof this type. The underwriterwhooers

thebestterms,willsetarateandaccepttocoveracertainpartoftherisk. The

brokerwillnextcontactotherunderwritersuntiltheslipislled. Usuallythese

underwriterswillfollowtheratesetbythe\leadingunderwriter",butthatmay

notbethecase.

Theproceduredescribedabovemayseemcumbersome,anditcanbecostly.

Itserves,however,toillustratehowthecompetitiveequilibrium(CE)ofDeni-

tion2mayresult,orbewellapproximated,inpracticeforareinsurancemarket.

OneisleadtobelievethatthenotionofaCEmaybeespeciallyfruitfulforthis

typeofmarkets,andgivesreasonablepredictionsofwhatprices\ought"tobe.

Finally let us comment on the assumption of homogeneous beliefs. This

assumptionseemsreasonableforareinsurancemarket,wheretradeistradition-

ally supposed to takeplace under the conditionsof umberrimae dei, and no

informationissupposed tobehidden.

Premiumsofrisksinreinsurancemarketsarelikelytoinuencepremiumsin

thedirectmarketforinsurance,wherethisassumptionseemslessrealistic. The

causeforthismaybethatthedierentagentshavedierentinformationabout

therisks. Itseemslikelythat thebuyersofinsurancepossessmoreinformation

about the risk that they try to get rid of, than the insurers. This potential

asymmetricinformationgivesrisetotheselectionproblem oradverseselection.

In addition, the buyers may often directly, or indirectly be able to inuence

events so that the probability distributions of the insured risks are altered.

Thismayhappenbecausetheinsurerisusuallyunabletoperfectly monitorall

theactionsoftheinsured,aphenomenon givingrisetomoral hazard.

Whereastheproblemofmoralhazarddoesnotseemofparticularimportance

in a reinsurance market, the problem of adverse selection may occur since a

ceding company usually hasmore detailed information about the risks it has

underwritten, and subsequently tries to get rid of in the reinsurance market,

than the reinsurers. It may of course be tempting for a direct insurer to get

rid of some \bad risks". For this reason the reinsurance industry makes use

of a detailed rating system for insurance companies, through e.g., Insurance

SolvencyInternational,whichmaypenalizesuchactions. Ifaninsurergetsabad

reputation,hemaygetalowclassicationbysuchratingagencies,implyingthat

hewillfacetougherconditionsinthereinsurancemarket,likehigherpremiums.

Theveryexistenceofsuchrating companiesisanindication oftheseverityof

theselectionproblem. In anycase,weshallabstract fromboththese problem

areas.

Theabovemodelisformulatedintermsofareinsurancesyndicate,butother

applicationsaremanifold,sincethemodelisindeedverygeneral. Forinstance,

X

i

mightbetherandomlyvaryingwaterendowmentsofagriculturalregion

(orhydro-electricpowerstation)i;

X

i

couldstandfornationi'sstate-dependentquotasin producingdiverse

pollutants(orincatchingvariousshspecies);

i

portationrmimustbringfromvarious originstospecieddestinations;

X

i

couldbetheinitialendowmentsofsharesinastockmarket,inunitsof

aconsumptiongood.

Thislatterapplication wewill returnto insomedetaillater. Forinstance,

thepresentformulationallowsusto emphasizeandstudy theconceptofcom-

plete nancialmarkets,andtheeconomicvalue,orrathertherationalebehind

contingent claims,suchase.g.,optionsandfutures contracts.

3 The characterization of a competitive equilib-

rium

In thissection we characterizeaCE assumingthat itexists. In theliterature

citedattheend thereaderwillndseveral referencestotheexistenceissue.

5

Wetakeitthat theinitialportfoliosarenotidentically equaltozero,andthat

auniqueequilibriumexists. Wealsoassumequitenaturallythat (X

i

)>0for

eachi. Infact, itseemsreasonablethat eachagentisrequiredto bringto the

market an initial \endowment" of positive value.

6

In this case wehave the

following:

Theorem1 Suppose the preferences of the agentsare monotonic, i.e., u 0

i

>0

for all i2I. The equilibrium isthen characterized by the existenceof positive

constants

i

,i2I,suchthatfor the equilibriumallocation (Y

1

;Y

2

;:::;Y

I )

u 0

i (Y

i )=

i

; a:s: for all i2I; (2)

where isthe Rieszrepresentation ofthe pricing functional.

ProofRecallthatmax

Zi Eu

i (Z

i

)s.t. h(Z

i

)0,whereh(Z

i

):=(Z

i ) (X

i ),

is a nice optimization problem: The objective is concave and the constraint

function h (the feasible set) is convex. For such problems the Kuhn-Tucker

Theorem saysthat,grantedasuitableconstraintqualication,anyoptimalso-

lution Y

i

will be supported by aLagrange multiplier

i

: That is, there exists

i

0suchthattheLagrangian

L

i (Z

i

;

i )=Eu

i (Z

i

)

i h(Z

i )

ismaximalinZ

i atZ

i

=Y

i

. Moreover,complementaryslacknessholds:

i h(Y

i )=

0. Thesaidqualicationcouldbeh(Z 0

i

)<0forsomeZ 0

i

. (Thisisthesocalled

Slatercondition.) HereletZ 0

i

= 1

2 X

i .

NextweexplorewhatmaximalityofL

i (;

i )atY

i

means. Forthatpurpose

deneavariation

~

Y

i :=Y

i

+tZwhereY

i

istheoptimalsolutionof(1),t2Risa

scalardummyvariableandZ2L 2

isanarbitraryrandomvariable. According

to our conditions the function f(t;Z) := L

i (

~

Y

i

;

i

) attains its maximum for

t=0forallZ2L 2

,andconsequentlymust

f 0

(0;Z)=EfZ(u 0

i (Y

i

)

i

)g=0 forall Z2L 2

; (3)

5

ExistenceofArrow-Debreuequilibriaininnite-dimensionalsettingsseemstohavebeen

rsttreatedinBewley(1972).

6

ThisisofcourseaweakerrequirementthanthepositivityassumptionX

i

0P-a.s. for

allifoundinconsumertheory.

whichimpliesthatu

i (Y

i

)

i

=0a.s.

Finally, since u 0

i

> 0 for all i, the shadow price > 0 a.s., otherwise the

problem (1)cannothaveasolution,contrarytoourassumptionthat anequi-

librium exists. Fromtherstordercondition(2)itthen followsthat

i

>0of

alli.

Notice that in an equilibrium of the above type only relative prices are

determined. Weget

u 0

i (Y

i (!))

u 0

i (Y

i (!

0

))

= (!)

(!

0

)

foralmostall !;! 0

2:

Thus the rate of substitution betweenstates of nature is constantacross the

agents.

Consideranequilibriumwhere =(

1

;

2

;::: ;

I

)aretheassociatedpos-

itiveconstants. Thenthesame equilibriumis obtainedforthe ray^ =c for

c >0apositivescalar. Inthe lattercaseall thepricesare obtainedfrom the

formeraftermultiplicationbytheconstant1=c. Thustheequilibriumallocation

(Y

1

;Y

2

;::: ;Y

I

)remainsinvarianttomultiplicationoftheraybyanormaliz-

ing constant c. Ingeneralpricesare determinedby auniqueequilibrium only

moduloanormalization.

Oneshould perhapsnotloosetouchwiththesituationofthemorefamiliar

Euclidean space. If the set of states of theworld is nite, we are basically

back in nitedimensional Euclidean space, ifwe takepropercareof thestate

probabilities. The result of this theorem is then analogous to the geometri-

cal interpretation that the state-price vectoris asuitable normalizedpositive

vectororthogonalto the budgetsets ofthe agents. Moreprecisely, for almost

everystatetherealized marginalutility is(ex-post) perpendicular tothe real-

ized budget set. There is an important point here. Theutility maximization

within budget yields an expected marginal utility Eu 0

i (Y

i

)which is normalto

the expected budget set, i.e., Eu 0

i (Y

i ) =

i

(1) for some

i

0. The more

interestingfact isthatthis conditiondisintegratesto haveu 0

i (Y

i )=

i

almost

surely.

Oneshould observethat several main features of thegeometrical interpre-

tations from Euclidean spacescarry overto thepresent Hilbert space L 2

. For

example,theargumentleadingtoequation(3)isverysimpleandintuitive,and

canofcoursebemoreformallyexplainedintermsofdirectionalderivatives: We

dene

5L

i (Y

i

;Z)= lim

t#0 +

L

i (Y

i

+tZ;

i ) L

i (Y

i

;

i )

t

;

where5L

i (Y

i

;Z)iscalledthedirectionalderivativeofL

i (Y

i

;

i

)inthedirection

Z. L

i

is dierentiableat Y

i

means that 5L

i (Y

i

;Z)exists for allZ 2L 2

, and

thefunctionalZ !5L

i (Y

i

;Z)is linear. This functional,thegradientofL

i at

Y

i

,isdenoted by5L

i (Y

i

). Itcanherebeshownto begivenby

(5L

i (Y

i

))(Z)=Ef(u 0

i (Y

i

)

i

)Zg: (4)

A necessaryconditionfor amaximumof L

i at Y

i

isthat thelinear functional

in equation(4)iszeroin alldirectionsZ,whichleadsdirectlytothecondition

(2).

Example1. Consider the case with negative exponential utility functions,

with marginal utilities u 0

i

(z) = e z=ai

;i 2 I, where a 1

i

is the absolute risk

aversion ofagenti,ora

i

isthecorrespondingrisktolerance. Usingthecharac-

terization (2),weget

i e

Yi=ai

=; a:s:; where

i

= 1

i

; i2I:

After taking logarithms in this relation, and summing overi, market clearing

implies

=e (K X

M )=A

; a:s: where K:=

I

X

i=1 a

i ln

i

; A:=

I

X

i=1 a

i :

Furthermore,fromthesamerstorderconditionswealsogetthattheoptimal

portfolioscanbewritten

Y

i

= a

i

A X

M +b

i

; where b

i

=a

i ln

i a

i K

A

; i2I:

Thus the reinsurance contracts involveoptimal sharing rules which are aÆne

in X

M

. Contractsofthis typebelongto theclassof proportional reinsurance.

Theconstantsof proportionality a

i

=A aresimply equaltoto eachagent'srisk

tolerance, measured relative to the market. In order to compensate for the

fact that the least risk-averse reinsurerwill hold the larger proportion of the

market,zero-sumsidepaymentsoccurbetweenthereinsurers,hererepresented

bythe termsb

i

. Without thesesidepaymentsanagent,witha\small" initial

endowment but with alargerisk tolerance, would end upwith a\large"nal

endowment,butthiscouldnotpossiblybeconsistentwithhisbudgetconstraint.

Thiskindoftreatyseemscommonin reinsurancepractice.

Inordertodeterminetheray=(

1

;:::;

I

),weemploythebudgetcon-

straints:

E(Y

i e

(K XM)=A

)=E(X

i e

(K XM)=A

); i2I;

whichgivethat

b

i

= EfX

i e

X

M

=A a

i

A X

M e

X

M

=A

g

Efe XM=A

g

; i2I:

Hence the optimalsharing rulesY

i

are completely determined in termsof the

givenprimitivesofthemodel. Nowtheraycanalsobedeterminedmoduloa

normalization. LettingK= P

I

i=1 a

i ln

i

denotethis normalization,then

i

=e b

i

=a

i

e K=A

; i2I:

IfweimposethenormalizationEfg=1ofthestatepricedeator,weobtain

e K=A

=Efe XM=A

g,in whichcasetheconstantsaregivenby

i

= e

b

i

=a

i

Efe XM=A

g

; i2I:

arenowgivenby

(Z)=

EfZe XM=A

g

Efe XM=A

g

; forany Z 2L 2

: (5)

Pricesgivenbyanexpressionlike(5)aresometimesreferredtoasthe\Esscher

principle" in actuarialmathematics, but then with the important distinction

thattheaggregatemarketindexX

M

in(5)issubstitutedbytheriskZitself. For

thislatterprinciplethepriceruleisofcoursenolongeralinearfunctional,which

then can, unfortunately, lead to arbitrage possibilities and other anomalies.

In theabove examplewe were able to completely specify the equilibrium,

giventhattherelevantexpectationsarewelldened. Wemaysafelyconjecture

that if the side payments b

i

's can be computed, an equilibrium exists and is

unique.

We notice that both the optimal, nal portfolios Y

i

and the state-price

deator depend upon the initial portfolios X

i

only through the aggregate

X

M

= P

I

i=1 X

i

. In other words, = f(X

1

;X

2

;:::;X

I

) = g(X

M ), and

Y

i

=f

i (X

1

;X

2

;:::;X

I )=g

i (X

M

)forsomefunctionsg andg

i

. Onemaywon-

derhowgeneralthis is. Inthis particularexamplegturned outtobesmooth,

and the g

i

-functions are even linear. We know that non-proportional reinsur-

ance is another of themain classes of contractsprevailing in real reinsurance

markets,sothelinearityofthecontractsmaynotbeallthat general.

Beforeweinvestigatethesemattersanyfurther,weintroducetheconceptof

(strong)Pareto optimality ofanallocation.

Denition3 AfeasibleallocationY =(Y

1

;Y

2

;:::;Y

I

)iscalledParetooptimal

if there is no feasible allocation Z =(Z

1

;Z

2

;:::;Z

I

)with Eu

i (Z

i )Eu

i (Y

i )

for alli andwithEu

j (Z

j )>Eu

j (Y

j

)for somej.

A famous neoclassicalresult is that any competitiveequilibrium is Pareto

optimal, sometimes also termed eÆcient. Not surprisingly, the same result

obtainshere:

Theorem2 Suppose (Y

1

;Y

2

;:::;Y

I

) is a competitive equilibrium allocation.

Then itisPareto optimal.

Proof. Let (;Y

1

;Y

2

;:::;Y

I

) denote the equilibrium, and suppose that Z is a

Paretodominatingallocation. SinceEu

j (Z

j )>Eu

j (Y

j

)forsomej,itmustbe

thecasethat(Z

j )>(Y

j

)forthesej. Considertheotheriwhereweonlyhave

equality in expected utilities. It must be thecase that (Z

i

)(Y

i

)also for

thesei. Supposetheopposite. Thenbylocalinsatiability(andafortioribystrict

monotonicity)anysuchagentiwouldbeabletoachievealargerexpectedutility

thatEu

i (Y

i

)byusingalltheavailablebudget(Y

i

),implyingthattheresulting

expected utility would bestrictly largerthan Eu

i (Y

i

), and the corresponding

allocationmeatsthebudgetconstraint,acontradictiontotheoptimalityofY

i .

Accordinglywehavethat(Z

i )(Y

i

)foralli,andwithstrictinequalityfor

somej. Butthen

( I

X

i=1 X

i )(

I

X

i=1 Z

i )=

I

X

i=1 (Z

i )>

I

X

i=1 (Y

i )=(

I

X

i=1 Y

i )=(

I

X

i=1 X

i );

feasibleandispositiveandlinear,thestrictinequalityfollowsfromwhathas

justbeendemonstrated,andthelastequalityfollowssinceY clearsthemarket.

Hence Y mustbeParetooptimal.

In consumption theory the preceding theorem is known as First Welfare

Theorem.

4 The characterization of a Pareto optimum

Aconsequenceof thelasttheoremisthat Paretooptimaarealsocharacterized

bytheequations(2),atleastthoseallocationsthatarealsoequilibria. Itturns

out that this include most Pareto optima. Before we show this, we turn to

another useful characterization of Pareto optimum. Here we shall employ a

versionof one of the most useful mathematical tools in microeconomics, The

Separating HyperplaneTheorem: SupposeX andY areconvex,disjointsubsets

of R I

. Then there exists a non-trivial linear functional on R I

such that

(x) = P

I

i=1

i x

i

P

I

i=1

i y

i

=(y)for allx 2X and y 2Y. Moreover,if

x 2 int(X) or y 2 int(Y)then (x) <(y). In thefollowingweassumethat

alltheportfoliosZc,where cissomeconstant. Inaone-periodmodel,ifwe

interprettheportfolioofanagentas\wealth",itmaysometimesbediÆcultto

giveanymeaningto negativewealth,whichthennecessitatesanassumptionof

thiskindwherec=0. Wenowshowthefollowing.

Theorem3 Suppose u

i

are concave and increasing for all i. Then Y is a

Pareto optimal allocation if and only if there exists a nonzerovector of agent

weights 2R I

+

suchthat Y =(Y

1

;Y

2

;:::;Y

I

)solves the problem

sup

(Z1;:::;ZI) I

X

i=1

i Eu

i (Z

i

) subject to I

X

i=1 Z

i

I

X

i=1 Y

i

=X

M

: (6)

Proof. First,assume(Y

1

;Y

2

;:::;Y

I

)is Pareto optimal,and dene twosets A

andB in R I

asfollows:

A:=fa2R I

:a

i

Eu

i (Z

i ) Eu

i (Y

i

);i2I;Z 2Zg

where Z denotes the set of feasible allocations Z = (Z

1

;:::;Z

I

) such that

P

Z

i

X

M

,and B :=fb2R I

+

:b6=0g. Thenthe set Ais convex,sinceall

theu

i

areassumed concave,andA\B =;, sinceY is Paretooptimal. Thus

we know that there exists a separatinghyperplane, i.e., there exists a vector

2R I

, 6=0such that ab 8a2A and b2B. Giventhe nature of

B,cannothaveanegativecoordinate,hence0. Since02cl(B)wehave

that a08a2A,thus

I

X

i=1

i Eu

i (Y

i )

I

X

i=1

i Eu

i (Z

i

); 8Z2Z;

whichistheconclusion.

Theotherdirection iseasytoshow.

The fact that some of the weights

i

may be zero in thecharacterization

of Theorem3maybeillustratedasfollows: ImaginesharingacakebetweenI

cake is in fact Pareto optimal, including the \sharing"giving the whole cake

to onesingleperson. This correspondsto onlytheweightof thispersonbeing

positive, all the other weights being zero. In this case the concept of Pareto

optimalityisvoid,but thatis notthetypicalcasewithmultiple goodsand/or

manystatesoftheworld.

5 Representative agent pricing

In this section we introduce the representative agent, and demonstrate what

implicationshehasforthepricingofinsurancecontracts,aswellasforhowthe

optimalcontractsareobtained.

Wehavealreadybrieymetthis agentin equation(6)ofTheorem 3: Con-

siderforeachnonzerovector2R I

+

ofagentweightsthefunctionu

():R!R

dened by

u

(v):= sup

(z1;:::;zI) I

X

i=1

i u

i (z

i

) subjectto I

X

i=1 z

i

v: (7)

Asthenotationindicates,thisfunctiondependsonlyonthevariablev,meaning

that ifthesupremumis attainedat thepoint(y

1

;:::;y

I

), allthese y

i

=y

i (v)

and u

(v) =

P

I

i=1

i u

i (y

i

(v)). It is a consequence of the Implicit Function

Theorem that under ourassumptions,the functionu

()istwotimesdieren-

tiable in v. In particular it follows that u 0

(v) =

P

I

i=1

i u

0

i (y

i (v))y

0

i

(v), and

hencethat allthefunctionsy

i

(v)arealsodierentiableinv. Moreimportantly

in thepresent situation,wewant to show that for appropriate thefunction

u 0

(v)=g(v)=(v),i.e.,thereisadirectconnectiontothestate-pricedeator.

Accordinglyweareinterestedin theproblem

Eu

(V):= sup

(Z

1

;:::;Z

I )

I

X

i=1

i Eu

i (Z

i

) subjectto I

X

i=1 Z

i

V: (8)

whereZ

i 2L

2

foralli.

Theorem4 Assumeu 0

i

>0;u 00

i

0for alli,andsuppose(;Y

1

;Y

2

;:::;Y

I )is

acompetitiveequilibrium. Then

(i)Thereexists anonzerovector of agentweights =(

1

;:::;

I ),

i 0

foralli,suchthattheequilibriumallocation(Y

1

;Y

2

;:::;Y

I

)solvestheallocation

problem(8)atV =X

M

= P

I

i=1 X

i

inwhichcaseEu

(X

M )=

P

I

i=1

i Eu

i (Y

i ).

(ii) There existsa nonzerovector of agentweights =(

1

;:::;

I

), where

i

0foralli,suchthat(;X

M

)isanequilibriuminthesingle-agenteconomy

(u

;X

M

). Thelinear pricingfunctional isthengiven by

(Z)=E(u 0

(X

M

)Z) 8Z2L 2

;

that isu 0

(X

M

)= a.s.

Remarks: 1)Theequilibriumin thesingleagenteconomymustbeunderstood

as a consistency requirement, since \the representative agent" has no oneto

tradewith.

struction enablesus to nd the pricesin the original economy, since is the

sameinthesetwoeconomies. Theconvenienceofaccommodatingarepresenta-

tiveagentisrelatedtothefactthat anequilibriumproblemthusreducestoan

optimizationproblem.

3) We now see that the Riesz representation, the state price deator, or

the shadow price = u 0

(X

M

), so = f(X

1

;X

2

;:::;X

I

) = g(X

M

) is true

in general, for X

M

= P

I

i=1 X

i

, and the function g(x) = u 0

(x); x 2 R, is

determinedfrom (7).

4) Given the probability distribution function F, the optimal equilibrium

allocations Y

i

depend on the initial portfolios (X

1

;X

2

;:::;X

I

) only through

theaggregateX

M

= P

I

i=1 X

i

aswell,orY

i

=f

i (X

1

;X

2

;:::;X

I )=g

i (X

M )is

trueingeneral,sinceY

i

=(u 0

i )

1

(

i

)followsdirectlyfromthecharacterization

in Theorem1,and dependsonlyontheaggregateriskX

M

asjustnoticed.

7

ProofofTheorem 4 It follows from Theorem 1 that there exist Lagrange

multipliers

i

>0suchthat Y

i

solvestheproblem

sup

Z2L 2

Efu

i

(Z)

i

(Z X

i

)g; (9)

and the budget conditionsthus hold with equality, i.e., E(Y

i

) =E(X

i );8i.

Nowchoose

i

= 1

i

;8i. Foranyfeasible(Z

1

;:::;Z

I

)wethenhave

I

X

i=1

i Eu

i (Y

i )=

I

X

i=1

i (Efu

i (Y

i

)

i (Y

i X

i )g)

I

X

i=1

i ( Efu

i (Z

i

)

i (Z

i X

i )g)=

I

X

i=1

i Eu

i (Z

i )

I

X

i=1 Ef(Z

i X

i )g

I

X

i=1

i Eu

i (Z

i ):

Therstinequalityfollowsfrom(9),andthesecondfollowsfromthefeasibility

of(Z

1

;:::;Z

I

)andthepositivityof a.s. Thuswehavefoundaset ofstrictly

positiveagentweights

i

suchthat(Y

1

;:::;Y

I

)solvesallocationproblem(8)at

V =X

M

= P

I

i=1 X

i .

Next, in order to prove(ii) we must showthat no trade is optimalin the

singleagenteconomy,wheretheagenthasutilityindexu

()andinitialportfolio

X

M

. Ifthis werenotthecase,there would9Z

M 6=X

M

such that

Eu

(Z

M )>Eu

(X

M

) and E(Z

M

)E(X

M ):

From the denition of u

, this would imply the existence of an allocation

(Z

1

;Z

2

;:::;Z

I )with

P

Z

i Z

M

such that

I

X

i=1

i Eu

i (Z

i )>

I

X

i=1

i Eu

i (Y

i )

7

Thefunction (u 0

i )

1

(x)denotes the inverse function ofu 0

i

(x),which existsforall i ac-

cordingtoourassumptions.

I

X

i=1

i

i E(Z

i )=E(

X

Z

i

)E(Z

M

)E(X

M )=

I

X

i=1

i

i E(X

i ):

Puttingthesetwoinequalitiestogetherweget

I

X

i=1

i [Eu

i (Z

i

)

i Ef(Z

i X

i )g]>

I

X

i=1

i [Eu

i (Y

i

)

i Ef(Y

i X

i )g];

whichcontradictsthefactthat(Y

i

)solvestheproblem(9).

It remains to show that =u 0

(X

M

). From (ii) we know that X

M is the

solutionoftheproblem

sup

Z2L 2

Eu

(Z) subjectto (Z)(X

M );

wheretheLagrangianisgivenby

L(Z;)=Eu

(Z) (E(Z) E(X

M )):

By the Kuhn-TuckerTheorem anecessary (and suÆcient)condition for opti-

malityofX

M

isgivenbytherstordercondition

u 0

(X

M

)=; a:s:;

whichnowfollowspreciselyasintheproofofTheorem1. Noticethatu 0

(X

M )>

0a.s. followsfrom strictmonotonicpreferencesofallthereinsurers,and>0

a.s. must hold since the present optimization problem is known to have a

solution. Hence >0, = 1

u

0

(X

M

) a.s., and bya renormalizationwe now

havethatu 0

(X

M

)=

AconsequenceofRemark4)aboveisthatthereinsurerscanhandinalltheir

initial portfoliosX

i

to apool,and after ! 2 isrealized, letthe pool's clerk

distribute partsofthetotalX

M

(!)backto thesyndicatesmembersaccording

totheoptimalsharingrulesY

i (!)=g

i (X

M

(!)). Inthisrespectthecompetitive

solutioncontains,perhapssurprisingly, an element ofcooperation, i.e., that of

pooling.

6 The existence of optimal allocations

InthissectionweprovideconditionsfortheexistenceofaParetooptimum,and

wealsobrieystudy theexistenceofacompetitiveequilibrium.

Theextentto which aParetooptimal allocationcanalso be consideredas

acompetitiveequilibriumisthecontentsofourrsttheorem. Inthetheoryof

consumptionitisknownasTheSecondWelfare Theorem:

Theorem5 Under the assumptions of Theorem 3, let (Y

1

;Y

2

;:::;Y

I ) be a

Pareto optimal allocation. Then there exists a re-allocation (

~

X

1

;

~

X

2

;::: ;

~

X

I ),

satisfying P

~

X

i

= P

Y

i

=X

M

,suchthatY

i solves

sup

Z

i 2L

2 Eu

i (Z

i

) subjectto E(Z

i u

0

(X

M

))E(

~

X

i u

0

(X

M

)) (10)

for all i,where the function u 0

is dened through (8) with V =X

M

a.s., and

the nonzeroweightsfollow fromthe characterization inTheorem 3.

R I

+

ofagentweightssuchthat

Eu

(X

M )=

I

X

i=1

i Eu

i (Y

i

): (11)

(These weightsare used to dene the function u 0

in (10).) We haveto show

that (Y

1

;Y

2

;:::;Y

I

) satises(10). Suppose theopposite, i.e., foreachfeasible

(

~

X

1

;

~

X

2

;:::;

~

X

I ),

P

~

X

i

=X

M

,thereexistsafeasibleallocation(Z

1

;Z

2

;:::;Z

I ),

satisfyingthebudgetconstraintsin(10),andwhichisnotequalto(Y

1

;Y

2

;:::;Y

I )

a.s.,such thatforalli

Eu

i (Z

i

)

i Efu

(X

M )(Z

i

~

X

i

)gEu

i (Y

i

)

i Efu

(X

M )(Y

i

~

X

i )g;

(12)

forall

i

,where theinequalityisstrict foratleastsomej. Inparticularthese

inequalitiesholdfor

i

=1=

i

(wemayusetheusualconventionthat10=0.

8

Nowwehavethat

I

X

i=1

i

i E(u

0

(X

M )Z

i

)=E(u 0

(X)

I

X

i=1 Z

i

) (13)

E(u 0

(X

M )

I

X

i=1

~

X

i )=

I

X

i=1

i

i E(u

0

(X

M )

~

X

i )

Further,forthesame-vectorwehave

I

X

i=1

i Eu

i (Z

i )=

I

X

i=1

i h

E(u

i (Z

i

)

i Efu

0

(X

M )(Z

i

~

X

i )g

i

>

I

X

i=1

i h

E(u

i (Y

i

)

i Efu

0

(X

M )(Y

i

~

X

i )g

i

= I

X

i=1

i Eu

i (Y

i )

for allmarket clearing

~

X-allocations. Therstequalityfollowssinceboththe

Z-and

~

X-allocationsarefeasiblewithequality,theinequalityfollowsfromthe

twoinequalities(12)and(13)put together, andthelast equality followssince

the Y-allocation is feasible with equality, i.e., P

Y

i

= P

~

X

i

= P

Z

i

= X

M ,

and

i

i

=1forall i. Butthis iscontrary to thefact that (Y

1

;Y

2

;:::;Y

I ) is

Paretooptimal.

Remark.Letus noteherethatKarl Borch(1960, 62)used aslightlydier-

entdenition of Pareto optimality than ourDenition 3. Inhis denition no

exchangeistobecarriedoutunlessallreinsurersgainfromit:

Denition4 A feasibleallocation Y =(Y

1

;Y

2

;:::;Y

I

)is (weakly)Pareto op-

timal if there is no feasible allocation Z = (Z

1

;Z

2

;::: ;Z

I

) with Eu

i (Z

i ) >

Eu

i (Y

i

)for all i.

8

Notethatifweavoid\cornerallocations",i.e.,situationswheresomeYi=0a.s.,wemay

safelyassumethat

i

>0foralli.

Borch then showedthat, under our conditions u

i

>0;u

i

< 0for all i, an

allocation(Y

1

;Y

2

;:::;Y

I

)is(weakly)ParetooptimalinthesenseofDenition

4ifandonlyif

u 0

i (Y

i )=k

i u

0

1 (Y

1

); a:s: forall i2I; (14)

where k

1

= 1and k

i

> 0 for all i, 9

or equivalently, if and only if (2) holds

withconstants

i

>0foralli. MattiRuohonen(1979)hasfurthershownthat,

under ourconditionsontheu

i

-functions,this theoremisalsotruefor(strong)

Pareto optimalityof our original Denition 3. Thus these twodenitions are

equivalent under our conditions. This equivalence fails when not all the u

i

arestrictlymonotonic,whilethetheoremremainsvalidforthe(strong)Pareto

optimalityofDenition 3.

Usingtheorems 3and5we maynowsaysomethingaboutthe existenceof

a competitive equilibrium. We note that the allocation problem (6) is also a

niceoptimizationproblem. AccordingtotheSaddle PointTheorem, granteda

suitableconstraintqualication,anyoptimalsolutionY willbesupportedbya

(stochastic) Lagrangemultiplier (X

M )2 L

2

. That is, there exists arandom

variable(X

M

)withnitevariance,(X

M

)0a.s.,suchthat theLagrangian

L(Z

1

;Z

2

;:::;Z

I

;(X

M

))=Ef I

X

i=1

i u

i (Z

i

) (X

M )

I

X

i=1 (Z

i X

i )g

ismaximalinZatZ =Y. Moreover,complementaryslacknessholds. Therst

orderconditionsforthisoptimizationproblemare:

i u

0

i (Y

i

)=(X

M

) a:s: 8i; (15)

which are seen to be identical to the rst order conditions (2) of Theorem 1

with some reinterpretations. Here the Lagrange multiplier (X

M

) associated

with the problem (6) can be seento be the sameas the Riesz representation

(X

M

)inthepricingrepresentationforacompetitiveequilibrium,or,whatwe

havealsocalled thestatepricedeator,and, asusual

i

=1=

i

. Thisexplains

KarlBorch'scharacterizationofaParetooptimalsolution: Giventheexistence

ofasolutiontotheallocationproblem(6),anecessaryandsuÆcientcondition

foraParetooptimumisgiven,underourassumptions,bytheconditionsin(15).

Weargueintermsofdirectionalderivatives: Dene

5L((Y

1

;:::;Y

I );(Z

1

;:::;Z

I ))=

lim

t#0 +

L(Y

1 +tZ

1

;:::;Y

I +tZ

I

;(X

M

)) L(Y

1

;:::;Y

I

;(X

M ))

t

;

where 5L(Y;Z) is the directional derivative of L(Y;(X

M

)) in the direction

Z = (Z

1

;:::;Z

I

). L is dierentiable at Y = (Y

1

;:::;Y

I

) now means that

5L(Y;Z) exists for all Z

i 2 L

2

;i = 1;2;:::;I, and the functional Z !

5L(Y;Z) is linear. This functional, the gradient of L at Y, we denote by

5L(Y). Itisgivenby

(5L(Y))(Z)=Ef I

X

i=1 (

i u

0

i (Y

i

) (X

M ))Z

i

g: (16)

9

Adetailedtechnical proofofthistheoremisprovidedbyDuMouchel(1968). Notethat

theseauthorshavedisregardedcornersolutions.

equation (16)is zero in alldirections Z, which leadsdirectly to thecondition

(15).

Onemaynowwonderifthereexist Paretooptimalsolutionsto theriskex-

changeproblemintherstplace. ThisproblemhasbeenstudiedbyDuMouchel

(1968), who has shown that if all u 0

i

(x) are continuous and the ranges of the

functions

i u

0

i

(x) havea common, non-empty intersection, then this problem

hasasolution. TheseconditionsfortheexistenceofaParetooptimalsolution

are very weak indeed. In particular, in the case treated here - where all the

utilityfunctionsarestrictly monotonic-wecanalwayschoosethe

i

>0,pro-

vided westay awayfromcorner solutions,suchthat there isaPareto optimal

solution. Thusthere will alsoexist acompetitive equilibrium, possibly aftera

re-allocationoftheinitialportfoliosX

i .

6.1 The existence of an equilibrium

Given an initial allocation X = (X

1

;:::;X

I

), one would presume that each

reinsurerwouldrequireatleastindividual rationality, i.e.,

Eu

i (Y

i )Eu

i (X

i

); 8i; (17)

for the nal allocations Y

i

, i = 1;2;:::;I. This requirement will naturally

excludemanyoftheParetooptimalpoints,whichdonotreallytakeintoaccount

improvementsfromtheinitialportfoliosX

i

,onlytakingasitspointofreference

theaggregateX

M .

Acompetitiveequilibrium satisesindividual rationality, andwenowturn

to theexistenceof anequilibrium forthegiven initialportfolios. This subject

happenstobearatherdelicatematter,usuallyrequiringx-pointtheoremsor

other rather technical, mathematical machinery. Matters are further compli-

catedbytheinnitedimensionalityofthespaceL 2

. SincetheinteriorofL 2

+ is

empty, wewill usuallyhaveproblems to ndanon-zero pricingfunctional us-

ingseparationarguments,sincee.g.,theseparatinghyperplanecannotbeused

directly in this situation. Note, however, that we have not insisted that our

portfolio space is L 2

+

. We will not elaborate on this issue here, but shall be

contentwithreferringtoonetheoremin thisregard.

Mas-Colell(1986)hascomeupwithaconceptcalled properness whichcan

beusedin thepresentmodel. ReturningtoourconditionsbehindTheorem1,

thefollowinghasbeenshown(Aase(1993a)),whichwepresentwithoutproof:

Theorem6 Supposeu 0

i

>0;u 00

i

<0,and(X

i

)>0for all i. If X

M

>0a.s.,

and there exists an allocation Z, Z

i

0 a.s., with P

I

i=1 Z

i

= X

M

a.s. and

E(u 0

i (Z

i ))

2

<1for alli,thenthereexistsacompetitive equilibrium.

It seems natural to check the initial portfolio X if it satises the above

requirements. NotethatitfollowsfromtheabovetheoremandfromTheorem1

thatifX

i

0a.s. andE(u 0

i (X

i ))

2

<1,foralli,thenanequilibriumallocation

Y exists suchthat E(u 0

i (Y

i ))

2

<1foralli,sinceweknowthat 2L 2

. Letus

consider someexamples.

Example2.Wereturnto thesituationin Example1,andassumethateach

X

i

isexponentiallydistributedwithparameter

i

,i2I. SinceX

M

= P

X

i

>0

u 0

i (X

i

)=exp( X

i

=a

i )and

E( u 0

i (X

i ))

2

=E

e 2

a

i X

i

=

i

i +2=a

i

<1 foralli

fortherisktoleranceparametersa

i

>0.

Nowconsiderthenormaldistribution,andassumethateachX

i isN(

i

;

i )-

distributed,andfurthermorethatX isjointlynormal. Inthiscase

E(u 0

i (X

i ))

2

=E

e 2

a

i Xi

=exp 2

i

a

i

2

2

i

a

i

!

<1 8i:

However, the positivity requirements are not met. Still all the computations

of theequilibrium arewell dened,thestate-price deator(X

M

)is astrictly

positiveelementofL 2

+

,andpricescanreadilybecomputed. Weconcludethat

an equilibrium exists even if the positivity requirementsare not satised. It

mayadmittedlybeunclearwhatnegativewealth should meanin aoneperiod

model,butasidefromthistherearenoformaldiÆcultieswiththiscaseaslong

asutilityiswelldenedforallpossiblevaluesofwealth.

SupposethateachX

i

isParetodistributedwithprobabilitydensityfunction

(seee.g.,Johnsonet. al. (1994))

f

Xi (x)=

i c

i

i

x 1+i

; c

i

x<1;

i

;c

i

2(0;1):

This is known as the Pareto distribution of the rst kind, also borrowing its

namefromtheItalian-bornSwissprofessorofeconomics,VilfredoPareto(1848-

1923). In thiscaseEX

i

existsonly if

i

>1,and varX

i

exists only if

i

>2,

etc. The moment generating functions '

i

() = Ee X

i

of these distributions

exist for0,sotheabovecriteriaaremetforZ =X. Accordingly,forthese

distributions acompetitiveequilibriumexists.

Wenowturntothecasethecasewheretherelativeriskaversionsofallthe

reinsurersareconstants:

Example3.Considerthecaseofpowerutility,whereu

i

(x)=(x 1 ai

1)=(1

a

i

)forx>0;a

i

6=1andu

i

(x)=ln (x)forx>0anda

i

=1,where thenatural

logarithmresultsasalimitwhena

i

!1. Thisexampleonlymakessenseinthe

no-bankruptcycasewhereX

i

>0a.s. foralli. Theparametersa

i

>0arethen

therelative risk aversions oftheagents,whicharegivenbypositiveconstants

forthisclassofpreferences.

Considerrstthecasewhere a

1

=a

2

=:::=a

I

=a. Hereallthemarginal

utilitiesaregivenbyu 0

i (x)=x

a

,andusingTheorem1weget

u 0

i (Y

i (X

M )=

i (X

M

); a:s: foralli;

which implies that Y

i (X

M ) =

1=a

i (X

M )

1=a

, a.s., and using the market

clearingX

M

= P

i2I Y

i (X

M

),a.s.,weget

u 0

(X

M

)=(X

M )=(

X

i2I

1=a

i )

a

X a

M

a:s:;

i i

isof thesametypeasthat oftheindividualagents. Theoptimalsharingrules

arelinear,andgivenby

Y

i (X)=

1=a

i

P

j2I

1=a

j X

M

a:s: foralli:

Theweights

i

aredeterminedbythebudgetconstraints,implyingthat

i

=k

E(X

i X

a

M )

E(X 1 a

M )

a

; i2I;

or,

i

isdeterminedmodulotheproportionalityconstantk=( P

j2I

1=a

j )

a

for

eachi.

IfwenormalizesuchthatE(u 0

(X

M

))=1wendthatk=1=E(X a

M )and

the\pricingprinciple"

(Z)=

E ZX a

M

E(X a

M )

; forany Z2L 2

(18)

results.

Whenitcome to existence, letus check ourcriterionin the casewhere all

theX

i

areexponentiallydistributed. Inthiscasewehavetochecktheintegrals

E(X 2ai

i )=

Z

1

0 x

2a

i

i e

i x

dx<1;

which converge (near zero) when a

i

< 1=2. An equilibrium may still exist

outsidethisregiondependinguponthestochasticinterdependencebetweenthe

initialportfolios. Empiricalstudiessuggestthattheinterestingvaluesofa

i may

beintherangebetweenoneandthree,say.

Letus consider a situation where there exists afeasible allocationZ asin

Theorem 6,where theZ

i

componentsare i.i.d. exponentiallydistributed with

parameter . Let X = AZ where A is an I I-matrix with elements a

i;j

satisfying P

i a

i;j

=1forallj,sothatX

M

= P

I

i=1 Z

i :=Z

M

. This yieldsan

initialallocationX ofdependentportfolios,whichwemustrequireinarealistic

modelofareinsurancemarket,anditmeansthattheX

i

portfoliosaremixtures

ofexponentialdistributionswith afairly arbitrarydependence structure. Now

it turns out that we can still compute the

i

-weightsin the region a <I. In

this case X

M

has a Gamma distribution with parameters I and , and the

expectations E(X 1 a

M

) and E(Z

i X

a

M

) both exist for a < I 1. In order to

verify this, we note that the joint distribution of Z

i and X

M

is given by the

probabilitydensity

f(z

i

;x)= 2

e x

((x z

i ))

I 2

(I 2)!

; z

i

x<1;0z

i

<1:

Sowehaveto checktheintegral

E(Z

i X

a

M )=

Z

1

0 Z

1

zi z

i x

a

2

e x

((x z

i ))

I 2

(I 2)!

dz

i dx:

testyieldsthatwhen(1 a+I 2)> 1,thisintegralisnite. Fromthisitis

obviousthat theexpectationsE(X

i X

a

M

)alsoconvergeinthe sameregion, by

thelinearityofexpectation,sincetheX

i

= P

j a

i;j Z

j .

Similarlywehavetocheckthefollowingexpectation:

E(X 1 a

M )=

Z

1

0 x

1 a

e x

(x) I 1

(I 1)!

dx:

Nearzerothepossibleproblemagainoccurs,andthestandardcomparisontest

givesconvergence when(1 a+I 1)> 1. SowhenI >maxfa;a 1g=a,

both expectations exist, suggestingthat an equilibrium will also exist in the

interestingregionfortheparameterawhenthenumberofreinsurersI 4.

Letus considerthecaseofParetodistributions aswell. Nowtheintegrals

E(X 2ai

i )=

c 2ai

i (1+

2a

i

i )

1

<1:

Sincemin

i2I

i

>0therearenoproblemswithconvergence,andanequilibrium

existsinthiscaseregardlessofthevaluesoftherelativeriskaversionparameters.

In this latter caseall the portfolios are bounded away from zerowhich helps

on the existence problem for power utility, while the exponentialdistribution

has moreprobability massnear zero, potentially causing some problems with

existenceofequilibrium.

WhensharingrulesareaÆne,itispossibletotoreachaParetooptimumby

anexchangeoffractions oftheinitialportfolios,sometimesalsowith zero-sum

sidepayments. AÆnesharingrulesareoptimalwhen theindividualutilityin-

dicesaremembersof theHyperbolicAbsolute Risk Aversion (HARA) class. In

areinsurancemarketthismeansthatthereshouldbenoneedformorethanthe

standardproportionalreinsurancecontractwhenthisistrue. Appliedtoastock

market the assumptionmeans that there should be no need fortrading other

securitiesthan ordinaryshares(commonstock). Non-proportionalreinsurance

and securities such ascontingentclaims(e.g., options) both exist and areim-

portant, sowemust conclude that the preferences of the decision makers are

atleastsodiversethattheycannotberepresentedbyHARA-utilityfunctions

only. For somereasonmany economistsused to referto a market in which it

isimpossibletoreachaParetooptimumthroughanexchangeofproportionsof

theinitialportfolioasan\incompletemarket".

Ournextexample illustratesa situation where thePareto optimalsharing

rulesarenotaÆne:

Example4.Considerpowerutility whenthe exponentsarenot equal,e.g.,

u

i (x)=x

ai

;a

i

2(0;1);i2I. Therstorder conditionsgive

Y

i (X)=

u 0

(X

M )

i a

i

1

(a

i 1)

a:s: i2I;

where the state-price deatoris implicitly determined bythe market clearing

condition, and the budget constraints determine the agent weights modulo a

normalizingconstant.

1 2

utilityoftherepresentativeagentequals

u 0

(X

M )=

p

h+ p

h+4X

M

2X

M

!

1=2

a:s:

wherewehavearbitrarilyset

2

=3=4,whichwecandosinceonlytheratioof

thetwoweightsmatters. Here

h=

a

1

a

2

1

2

4

:

Inthiscasetheoptimalsharingrulesare

Y

1 (X

M )=

1

2

p

h 2

+4hX

M h

;Y

2

(X)=X

M +

1

2

h p

h 2

+4hX

M

;

a.s. Finally, one of the budget constraints is now enough to determine the

remainingunknownconstanth,inwhichcaseeverythingisdeterminedinterms

oftheprimitivesofthemodel.

It should be clear that this Pareto optimum can not be achieved by an

exchangeofproportionalreinsurancecontracts.

7 Risk tolerance and aggregation

The risk tolerance function of an agent (x) : R ! R

+

, is dened by the

reciprocal of the absolute risk aversion function R (x) = u

00

(x)

u 0

(x)

, or (x) =

1=R (x). Thereisaneatresultconnectingtherisktolerancesofalltheagentsin

themarkettotherisktoleranceoftherepresentativeagentinaParetooptimal

allocation. Itgoesasfollows: InaParetooptimumweknowthat

u 0

i (Y

i

(x))=

i u

0

(x); x2R :

Because of oursmoothness assumptions, both sides ofthe aboveequation are

real, dierentiablefunctions a.e. (the right-hand-side because of the implicit

function theorem),sotakingderivativesofbothsidesgives

u 00

i (Y

i (x))Y

0

i

(x)=

i u

00

(x); x2R :

Dividing the second equation by the rst, we obtain the following non-linear

dierentialequationforthePareto optimalallocationfunctionY

i (x):

Y 0

i (x)=

R

(x)

R

i (Y

i (x))

; x2R ; (19)

whereR

(x)=

u 00

(x)

u 0

(x)

istheabsoluteriskaversionfunctionoftherepresentative

agent,andR

i (Y

i (x))=

u 0 0

i (Y

i (x))

u 0

i (Y

i (x))

istheabsoluteriskaversionofagentiatthe

ParetooptimalallocationfunctionY

i

(x),i2I. Since P

i2I Y

0

i

(x)=1,wenow

getbysummationin (19)

(X)=

X

i2I

i (Y

i (X

M

)) a:s:; (20)

oftheindividualagentsinaParetooptimum. Theaboveresulthasbeenfound

by Borch (1985); see also Bhlmann(1980) for the special caseof exponential

utilityfunctions.

Example5.ReturningtoExample 1where u 0

i (x)=e

x=ai

foralli2I, we

getthat

i (x)=a

i

forallx2R ,i.e.,therisktolerancefunction of eachagent

is aconstant. Usingthe result(20), we getthat

(x)=

P

i2I a

i

=A for all

x, also a constant. That

(x) = A can easily be veried by going back to

Example1,whereweshowedthatu 0

(x)= =exp((K x)=A):

Imaginethat agentj is risk neutral, meaning that

j (Y

j

) =1, while the

othersareriskaverse. Fromtheresult(20)itfollowsthat

=1aswell,i.e.,

therepresentativeagentisthenalsoriskneutral. Fromtherelation(19)itmay

be seenthat this implies that Y 0

j

(x) = 1for all x, meaning that agentj will

thencarryalltheriskin themarket. Inother words,wehaveshownthat ina

Pareto optimum allrisk shouldbecarriedby therisk neutralparticipant.

Example6. In order to illustrate this last point, consider a case where

u

1

(x) = x and u

2

(x) = 2 p

x, and I = 2. Here agent 1 is risk neutral. The

rstorderconditionsgive

1=

1

;

1

Y

2 (X

M )

=

2

; a:s:

implying that = 1

1

, aconstant, and p

Y

2 (x)=

1

2

= 2

1

, anotherconstant.

Theoptimalsharingrulesarethus

Y

1 (X

M )=X

M

2

1

2

;Y

2 (X

M )=

2

1

2

;a:s:

andtheutilityfunctionoftherepresentativeagentisgivenby

u

(x)=

1 Y

1 (x)+

2 2

p

Y

2 (x)=

1 x+

2

2

1 :

ThusfromtworiskyprojectsbroughttothemarkethavingpayosX

i

,i=1;2,

theriskneutralagenttakesalltherisk,leavingaxedamount,oradeterministic

salary, to the risk averse agent. The representative agent is seen to be risk

neutral in accordance withthe abovetheory, andthe state-price deator =

u 0

(X

M )=

1

,aconstant. Thebudgetconstraintsdeterminetheratiosbetween

theagentweightsasfollows:

2

1

= p

E(X

2 ) :

Ifwenormalize such thatEu 0

(X

M

)=1,thensince =u 0

(X

M )=

1 ,

1

=1

and

2

= p

E(X

2

) .

Onemay wonderwhat happens whenmore thanoneagentis riskneutral.

Intheaboveexample,ifbothagentsareriskneutraltheycannotbothassume

alltherisk. Inthiscasetheriskneutralagentsasagrouppresumably endup

withalltherisk,wheretheyareindierenttoanysplit ofthetotalriskamong

themthat doesnotchangeeachindividual'sexpectedpayo.

The foregoing has been formulated in terms of portfolios and market values

of net reserves. To obtain market premiums of insurance contracts, we note

thenet reservesofinsurer iconsists ofassetsa

i

lessofliabilities Z

i

under the

insurancecontractsheld by theinsurer. Assume forsimplicity that theassets

a

i

areriskless. Thenwemayapplytheforegoingtheoryto

X

i

=a

i Z

i

; i2I:

Wenotethatthemarketvaluesof theinitialportfolioscanbewritten

(X

i )=a

i (Z

i )=a

i E( u

0

(a Z

M )Z

i );

wherea= P

a

i andZ

M

= P

Z

i

. Wemaydenethemarketdisutility ofclaim

paymentsbythefunctionv

(z),wherev 0

(z)=u

0

(a z). Fromourassumption

itfollowsthat v

(x)isadecreasingfunction in zandv 0 0

(z)= u 00

(a z)>0.

Theaboveformulasimplysaysthatthemarketvalueoftheinsurer'sportfolio

is equal to his riskless assets less the market premium for insurance of the

liabilities. Thisformulamakesiteasytotranslateresultsexpressedintermsof

valuesofnetreservesinto insurancepremiums. Noticein particularthat iffor

someportfolioX

i

themarketvalue(X

i

)<E(X

i

),thenwegetfromtheabove

formulathat the correspondinginsurancepremium (Z

i

)>E(Z

i

)so that the

economicriskpremium ((Z

i

) E(Z

i

))ofthisinsurancecontractispositive.

Usingthenormalization Ev 0

(a Z

M

)=1,(meaningthat therisk-freein-

terestrateequalszero),wendthattheriskpremiumcaningeneralbewritten

asfollows:

(Z

i

) E(Z

i

)=cov( Z

i

;v 0

(Z

M

)): (21)

Sincethemarginaldisutilityoftherepresentativeagentisanincreasingfunction

ofz,from (21)onemaybeledto believethat forclaimsZ

i

thatarepositively

correlated with the aggregate claims Z

M

in the market, the risk premium is

positive, and for claims that are negatively correlated with Z

M

the risk pre-

mium is negative. This is, however,onlytrue in generalwhen (Z

1

;:::;Z

I ) is

multinormally distributed. Thereexist jointdistributions fortheclaims where

thismaynotbetrue. Hereonehastorememberthatcovarianceisameasureof

linearstatisticaldependence,andcanaccordinglyonlybeconsideredasagood

measure of\stochasticassociation"undermultinormality.

Onecan ofcoursearguethatin insuranceanassumptionofjointnormality

is not veryrealistic, sincefor once claims canonly be non-negative. Wemay

therefore be reluctant to use the nice theoretical results obtainable from this

assumption in insurance. Here we must remember, however, that the normal

distributionis commonlyused withgreat success tomodelanumberof quan-

tities,liketheheights,orweightsof recruits,and manyother quantities which

are clearly non-negative. The point is that the resulting parameterestimates

will usuallyyieldacompletelynegligibleprobabilityoffallingin theforbidden

regions. This is one of the reasons why we still nd it fruitful to return to

the situation with a multinormal distribution forthe net reservesin the next

section.

Althoughthepresentreformulationisstraight-forward,onehastobecareful

whenmodelingclaimsizedistributions. Inpracticeinsuranceclaimsarealways