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?