Multivariate comonotonicity,stochastic orders and risk

measures

Alfred Galichon(Ecole polytechnique)

Brussels, May 25, 2012

Based on collaborations with:�A. Charpentier (Rennes) �G. Carlier (Dauphine)�R.-A. Dana (Dauphine) � I. Ekeland (Dauphine)

�M. Henry (Montréal)

This talk will draw on four papers:

[CDG]. �Pareto e¢ ciency for the concave order and mul-tivariate comonotonicity�. Guillaume Carlier, Alfred Gali-chon and Rose-Anne Dana. Journal of Economic Theory,2012.

[CGH] �"Local Utility and Multivariate Risk Aversion�.Arthur Charpentier, Alfred Galichon and Marc Henry.Mimeo.

[GH] �Dual Theory of Choice under Multivariate Risks�.Alfred Galichon and Marc Henry. Journal of EconomicTheory, forthcoming.

[EGH] �Comonotonic measures of multivariate risks�. IvarEkeland, Alfred Galichon and Marc Henry. MathematicalFinance, 2011.

Introduction

Comonotonicity is a central tool in decision theory, insur-ance and �nance.

Two random variables are « comonotone » when they aremaximally correlated, i.e. when there is a nondecreasingmap from one to another. Applications include risk mea-sures, e¢ cient risk-sharing, optimal insurance contracts,etc.

Unfortunately, no straightforward extension to the multi-variate case (i.e. when there are several numeraires).

The goal of this presentation is to investigate what hap-pens in the multivariate case, when there are several di-mension of risk. Applications will be given to:�Risk measures, and their aggregation�E¢ cient risk-sharing�Stochastic ordering.

1 Comonotonicity and its general-

ization

1.1 One-dimensional case

Two random variables X and Y are comonotone if thereexists a r.v. Z and nondecreasing maps TX and TY suchthat

X = TX (Z) and Y = TY (Z) :

For example, if X and Y are sampled from empiricaldistributions, X (!i) = xi and Y (!i) = yi, i = 1; :::; nwhere

x1 � ::: � xn and y1 � ::: � yn

then X and Y are comonotonic.

By the rearrangement inequality (Hardy-Littlewood),

max� permutation

nXi=1

xiy�(i) =nXi=1

xiyi:

More generally, X and Y are comonotonic if and only if

max~Y=dY

EhX ~Y

i= E [XY ] :

Example. Consider

! !1 !2P (!) 1=2 1=2

X (!) +1 �1Y (!) +2 �2~Y (!) �2 +2

X and Y are comonotone.

~Y has the same distribution as Y but is not comonotonewith X.

One has

E [XY ] = 2 > �2 = EhX ~Y

i:

Hardy-Littlewood inequality. The probability space isnow [0; 1]. Assume U (t) = � (t), where � is nonde-creasing.

Let P a probability distribution, and let

X (t) = F�1P (t):

For ~X : [0; 1]! R a r.v. such that ~X � P , one has

E [XU ] =Z 10�(t)F�1P (t)dt � E

h~XU

i:

Thus, letting

%(X) =Z 10�(t)F�1X (t)dt = max

nE[ ~XU ]; ~X =d X

o= max

nE[X ~U ]; ~U =d U

o:

A geometric characterization. Let � be an absolutelycontinuous distribution; two random variables X and Yare comonotone if for some random variable U � �, wehave

U 2 argmax ~U

nE[X ~U ]; ~U � �

o, and

U 2 argmax ~U

nE[Y ~U ]; ~U � �

o:

Geometrically, this means that X and Y have the sameprojection of the equidistribution class of �=set of r.v.with distribution �.

1.2 Multivariate generalization

Problem: what can be done for risks which are multidi-mensional, and which are not perfect substitutes?

Why? risk usually has several dimension (price/liquidity;multicurrency portfolio; environmental/�nancial risk, etc).

Concepts used in the univariate case do not directly ex-tend to the multivariate case.

The variational characterization given above will be thebasis for the generalized notion of comonotonicity givenin [EGH].

De�nition (�-comonotonicity). Let � be an atomlessprobability measure on Rd. Two random vectors X andY in L2d are called �-comonotonic if for some randomvector U � �, we have

U 2 argmax ~U

nE[X � ~U ]; ~U � �

o, and

U 2 argmax ~U

nE[Y � ~U ]; ~U � �

oequivalentely:

X and Y are �-comonotonic if there exists two convexfunctions V1 and V2 and a random variable U � � suchthat

X = rV1 (U)Y = rV2 (U) :

Note that in dimension 1, this de�nition is consistent withthe previous one.

Monge-Kantorovich problem and Brenier theorem

Let � and P be two probability measures on Rd withsecond moments, such that � is absolutely continuous.Then

supU��;X�P

E [hU;Xi]

where the supremum is over all the couplings of � and P ifattained for a coupling such that one has X = rV (U)almost surely, where V is a convex function Rd ! Rwhich happens to be the solution of the dual Kantorovichproblem

infV (u)+W (x)�hx;ui

ZV (u) d� (u) +

ZW (x) dP (x) :

Call QP (u) = rV (u) the �-quantile of distribution P .

Comonotonicity and transitivity.

Puccetti and Scarsini (2010) propose the following de�n-ition of comonotonicity, called c-comonotonicity: X andY are c-comonotone if and only if

Y 2 argmax ~YnE[X � ~Y ]; ~Y � Y

oor, equivalently, i¤ there exists a convex function u suchthat

Y 2 @u (X)

that is, whenever u is di¤erentiabe at X,

Y = ru (X) :

However, this de�nition is not transitive: if X and Y arec-comonotone and Y and Z are c-comonotone, and if thedistributions of X, Y and Z are absolutely continuous,then X and Z are not necessarily c-comonotome.

This transivity (true in dimension one) may however beseen as desirable.

In the case of �-comonotonicity, transitivity holds: if Xand Y are �-comonotone and Y and Z are �-comonotone,and if the distributions ofX, Y and Z are absolutely con-tinuous, then X and Z are �-comonotome.

Indeed, express �-comonotonicity of X and Y : for someU � �,

X = rV1 (U)Y = rV2 (U)

and by �-comonotonicity of Y and Z, for some ~U � �,

Y = rV2�~U�

Z = rV3�~U�

this implies ~U = U , and therefore X and Z are �-comonotone.

Importance of �. In dimension one, one recovers theclassical notion of comotonicity regardless of the choice of�. However, in dimension greater than one, the comonotonic-ity relation crucially depends on the baseline distribution�, unlike in dimension one. The following lemma from[EGH] makes this precise:

Lemma. Let � and � be atomless probability measureson Rd. Then:- In dimension d = 1, �-comonotonicity always implies�-comonotonicity.- In dimension d � 2, �-comonotonicity implies �-comonotonicityif and only if � = T#� for some location-scale transformT (u) = �u + u0 where � > 0 and u0 2 Rd. In otherwords, comonotonicity is an invariant of the location-scale family classes.

2 Applications to risk measures

2.1 Coherent, regular risk measures (uni-

variate case)

Following Artzner, Delbaen, Eber, and Heath, recall theclassical risk measures axioms:

Recall axioms:De�nition. A functional % : L2d ! R is called a coherentrisk measure if it satis�es the following properties:- Monotonicity (MON): X � Y ) %(X) � %(Y )- Translation invariance (TI): %(X+m) = %(X)+m%(1)- Convexity (CO): %(�X + (1� �)Y ) � �%(X) + (1��)%(Y ) for all � 2 (0; 1).- Positive homogeneity (PH): %(�X) = �%(X) for all� � 0.

De�nition. % : L2 ! R is called a regular risk measureif it satis�es:- Law invariance (LI): %(X) = %( ~X) when X � ~X.- Comonotonic additivity (CA): %(X + Y ) = %(X) +

%(Y ) when X;Y are comonotonic, i.e. weakly increasingtransformation of a third randon variable: X = �1 (U)

and Y = �2 (U) a.s. for �1 and �2 nondecreasing.

Result (Kusuoka, 2001). A coherent risk measure % isregular if and only if for some increasing and nonnegativefunction � on [0; 1], we have

%(X) :=Z 10�(t)F�1X (t)dt;

where FX denotes the cumulative distribution functionsof the random variable X (thus QX (t) = F

�1X (t) is the

associated quantile).

% is called a Spectral risk measure. For reasons explainedlater, also called Maximal correlation risk measure.

Leading example: Expected shortfall (also called Con-ditional VaR or TailVaR): �(t) = 1

1��1ft��g: Then

%(X) :=1

1� �

Z 1�F�1X (t)dt:

Kusuoka�s result, intuition.

� Law invariance ) %(X) = ��F�1X

�

� Comonotone additivity+positive homogeneity ) �

is linear w.r.t. F�1X :��F�1X

�=R 10 �(t)F

�1X (t)dt.

� Monotonicity ) � is nonnegative

� Subadditivity ) � is increasing

Unfortunately, this setting does not extend readily to mul-tivariate risks. We shall need to reformulate our axioms ina way that will lend itself to easier multivariate extension.

2.2 Alternative set of axioms

Manager supervising several N business units with riskX1; :::; XN .Eg. investments portfolio of a fund of funds. Trueeconomic risk of the fund X1 + :::+XN .

Business units: portfolio of (contingent) losses Xi reporta summary of the risk %(Xi) to management.

Manager has limited information:1) does not know what is the correlation of risks - andmore broadly, the dependence structure, or copula be-tween X1; :::; XN . Maybe all the hedge funds in theportfolio have the same risky exposure; maybe they haveindependent risks; or maybe something inbetween.

2) aggregates risk by summation: reports %(X1) + :::+%(XN) to shareholders.

Reported risk: %(X1)+:::+%(XN); true risk: %(X1+:::+XN).

Requirement: management does not understate risk toshareholders. Summarized by

%(X1) + :::+ %(XN) � %( ~X1 + :::+ ~XN) (*)

whatever the joint dependence (X1; :::; XN) 2 (L1d )2.

But no need to be overconservative:

%(X1)+:::+%(XN) = sup~X1�X1;:::; ~XN�XN

%(X1+:::+XN)

where � denotes equality in distribution.

De�nition. A functional % : L2d ! R is called a stronglycoherent risk measure if it is convex continuous and forall (Xi)i�N 2

�L2d�N,

%(X1)+:::+%(XN) = supn%( ~X1 + :::+ ~XN) : ~Xi � Xi

o:

A representation result.

The following result is given in [EGH].

Theorem. The following propositions about the func-tional % on L2d are equivalent:(i) % is a strongly coherent risk measure;(ii) % is a max correlation risk measure, namely thereexists U 2 L2d, such that for all X 2 L2d,

%(X) = supnE[U � ~X] : ~X � X

o;

(iii) There exists a convex function V : Rd ! R suchthat%(X) = E[U � rV (U)]

Idea of the proof . One has %(X)+�%(Y ) = supn%(X + � ~Y ) : ~Y � Y

o.

But %(X + � ~Y ) = %(X) + �D%X( ~Y ) + o (�)

By the Riesz theorem (vector case)D%X( ~Y ) = EhmX : ~Y

i,

thus

%(X)+�%(Y ) = supn%(X) + �E

hmX : ~Y

i+ o (�) : ~Y � Y

othus

%(Y ) = supnEhmX : ~Y

i: ~Y � Y

otherefore % is a maximum correlation measure.

3 Application to e¢ cient risk-sharing

Consider a risky payo¤ X (for now, univariate) to beshared between 2 agents 1 and 2, so that in each contin-gent state:

X = X1 +X2

X1 and X2 are said to form an allocation of X.

Agents are risk averse in the sense of stochastic domi-nance: Y is preferred to X if every risk-averse expectedutility decision maker prefers Y to X:

X �cv Y i¤ E[u(X)] � E[u(Y )] for all concave u

Agents are said to have concave order preferences. Theseare incomplete preferences: it can be impossible to rankX and Y.

One wonders what is the set of e¢ cient allocations, i.e.allocations that are not dominated w.r.t. the concaveorder for every agent.

Dominated allocations. Consider a random variable X(aggregate risk). An allocation of X among p agents isa set of random variables (Y1; :::; Yp) such thatX

i

Yi = X:

Given two allocations of X, Allocation (Yi) dominatesallocation (Xi) whenever

E

24Xi

ui (Yi)

35 � E24Xi

ui (Xi)

35for every continuous concave functions u1; :::; up. Thedomination is strict if the previous inequality is strictwhenever the ui�s are strictly concave.

Comonotone allocations. In the single-good case, it isintuitive that e¢ cient sharing rules should be such that in

�better�states of the world, every agent should be betterof than in �worse� state of the world � otherwise therewould be some mutually agreeable transfer.

This leads to the concept of comonotone allocations. Theprecise connection with stochastic dominance is due toLandsberger and Meilijson (1994). Comonotonicity hasreceived a lot of attention in recent years in decision the-ory, insurance, risk management, contract theory, etc.(Landsberger and Meilijson, Ruschendorf, Dana, Jouiniand Napp...).

Theorem (Landsberger and Meilijson). Any allocationof X is dominated by a comonotone allocation. More-over, this dominance can be made strict unless X is al-ready comonotone. Hence the set of e¢ cient allocationsof X coincides with the set of comonotone allocations.

This result generalizes well to the multivariate case. Upto technicalities (see [CDG] for precise statement), ef-�cient allocations of a random vector X is the set of

�-comonotone allocations of X, hence (Xi) solves

Xi = rui (U)Xi

Xi = X

for convex functions ui : Rd ! R, with U � �. Hence

X = ru (U)

with u =Pi ui. That is

U = ru� (X) ;

hence e¢ cient allocations are such that

Xi = rui � ru� (X) :

This result opens the way to the investigation of testableimplication of e¢ ciency in risk-sharing in an risky endow-ment economy.

4 Application to stochastic orders

Quiggin (1992) shows that the notion of monotone meanpreserving increases in risk (hereafter MMPIR) is theweakest stochastic ordering that achieves a coherent rank-ing of risk aversion in the rank dependent utility frame-work. MMPIR is the mean preserving version of Bickel-Lehmann dispersion, which we now de�ne.

De�nition. Let QX and QY be the quantile functionsof the random variables X and Y . X is said to beBickel-Lehmann less dispersed, denoted X %BL Y , ifQY (u) � QX(u) is a nondecreasing function of u on(0; 1). The mean preserving version is called monotonemean preserving increase in risk (MMPIR) and denoted-MMPIR.

MMPIR is a stronger ordering than concave ordering inthe sense that X %MMPIR Y implies X %cv Y .

The following result is from Landsberger and Meilijson(1994):

Proposition (Landsberger and Meilijson). A randomvariableX has Bickel-Lehmann less dispersed distributionthan a random variable Y if and only i¤ there exists Zcomonotonic with X such that Y =d X + Z.

The concept of �-comonotonicity allows to generalize thisnotion to the multivariate case as done in [CGH].

De�nition. A random vectorX is called �-Bickel-Lehmannless dispersed than a random vector Y , denotedX %�BLY , if there exists a convex function V : Rd ! R suchthat the �-quantiles QX and QY of X and Y satisfyQY (u)�QX(u) = rV (u) for �-almost all u 2 [0; 1]d.

As de�ned above, �-Bickel-Lehmann dispersion de�nes atransitive binary relation, and therefore an order. Indeed,ifX %�BL Y and Y %�BL Z, thenQY (u)�QX(u) =

rV (u) and QZ(u) � QY (u) = rW (u). Therefore,QZ(u)�QX(u) = r(V (u)+W (u)) so that X %�BLZ. When d = 1, this de�nition simpli�es to the classicalde�nition.

[CGH] propose the following generalization of the Landsberger-Meilijson characterization .

Theorem. A random vector X is �-Bickel-Lehmann lessdispersed than a random vector Y if and only if thereexists a random vector Z such that:

(i) X and Z are �-comonotonic, and

(ii) Y =d X + Z.

Conclusion

We have introduced a new concept to generalize comonotonic-ity to higher dimension: ��-comonotonicity�. This con-cept is based on Optimal Transport theory and boils downto classical comonotonicity in the univariate case.

We have used this concept to generalize the classical ax-ioms of risk measures to the multivariate case.

We have extended existing results on equivalence betweene¢ ciency of risk-sharing and �-comonotonicity.

We have extended existing reults on functions increasingwith respect to the Bickel-Lehman order.

Interesting questions for future research: behavioural in-terpretation of mu? computational issues? empiricaltestability? case of heterogenous beliefs?