This document discusses the generalization of comonotonicity to multivariate risks. [1] Comonotonicity in one dimension means two risks are maximally correlated through a common underlying risk factor. The document explores generalizing this concept to multiple dimensions when risks have several components. [2] -Comonotonicity is introduced as a generalization where two multivariate risks are -comonotonic if they can be expressed as functions of a common underlying risk vector through convex functions. [3] -Comonotonicity reduces to classical comonotonicity in one dimension but depends on the baseline distribution - in higher dimensions. Applications to risk measures and efficient risk sharing are discussed.

1. 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)

2. 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 Economic
Theory, forthcoming.
[EGH] “Comonotonic measures of multivariate risks”. Ivar
Ekeland, Alfred Galichon and Marc Henry. Mathematical
Finance, 2011.

3. Introduction
Comonotonicity is a central tool in decision theory, insur-
ance and …nance.
Two random variables are « comonotone » when they are
maximally correlated, i.e. when there is a nondecreasing
map 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.

4. 1 Comonotonicity and its general-
ization
1.1 One-dimensional case
Two random variables X and Y are comonotone if there
exists a r.v. Z and nondecreasing maps TX and TY such
that
X = TX (Z ) and Y = TY (Z ) :
For example, if X and Y are sampled from empirical
distributions, X (! i) = xi and Y (! i) = yi, i = 1; :::; n
where
x1 ::: xn and y1 ::: yn
then X and Y are comonotonic.

5. By the rearrangement inequality (Hardy-Littlewood),
n
X n
X
max xiy (i) = xi y i :
permutation
i=1 i=1
More generally, X and Y are comonotonic if and only if
h i
~
max E X Y = E [XY ] :
~
Y =d Y

6. Example. Consider
! !1 !2
P (! ) 1=2 1=2
X (! ) +1 1
Y (! ) +2 2
~
Y (! ) 2 +2
X and Y are comonotone.
~
Y has the same distribution as Y but is not comonotone
with X .
One has
h i
E [XY ] = 2 > ~
2 = E XY :

7. Hardy-Littlewood inequality. The probability space is
now [0; 1]. Assume U (t) = (t), where is nonde-
creasing.
Let P a probability distribution, and let
X (t) = FP 1(t):
~ ~
For X : [0; 1] ! R a r.v. such that X P , one has
Z 1 h i
1 ~
E [XU ] = (t)FP (t)dt E XU :
0
Thus, letting
Z 1 n o
1 ~ ~
%( X ) = (t)FX (t)dt = max E[XU ]; X =d X
0 n o
~ ~
= max E[X U ]; U =d U :

8. A geometric characterization. Let be an absolutely
continuous distribution; two random variables X and Y
are comonotone if for some random variable U , we
have
n o
~
~ ~
U 2 argmaxU E[X U ]; U , and
n o
~
~ ~
U 2 argmaxU E[Y U ]; U :
Geometrically, this means that X and Y have the same
projection of the equidistribution class of =set of r.v.
with distribution .

9. 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.

10. The variational characterization given above will be the
basis for the generalized notion of comonotonicity given
in [EGH].
De…nition ( -comonotonicity). Let be an atomless
probability measure on Rd. Two random vectors X and
Y in L2 are called -comonotonic if for some random
d
vector U , we have
n o
~
~
U 2 argmaxU E[X U ]; ~
U , and
n o
~
U 2 argmax ~ E[Y U ]; ~
U
U
equivalentely:
X and Y are -comonotonic if there exists two convex
functions V1 and V2 and a random variable U such
that
X = rV1 (U )
Y = rV2 (U ) :
Note that in dimension 1, this de…nition is consistent with
the previous one.

11. Monge-Kantorovich problem and Brenier theorem
Let and P be two probability measures on Rd with
second moments, such that is absolutely continuous.
Then
sup E [hU; Xi]
U ;X P
where the supremum is over all the couplings of and P if
attained for a coupling such that one has X = rV (U )
almost surely, where V is a convex function Rd ! R
which happens to be the solution of the dual Kantorovich
problem
Z Z
inf V (u) d (u) + W (x) dP (x) :
V (u)+W (x) hx;ui
Call QP (u) = rV (u) the -quantile of distribution P .

12. Comonotonicity and transitivity.
Puccetti and Scarsini (2010) propose the following de…n-
ition of comonotonicity, called c-comonotonicity: X and
Y are c-comonotone if and only if
n o
~
~ ~
Y 2 argmaxY E[X Y ]; Y Y
or, equivalently, i¤ there exists a convex function u such
that
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 are
c-comonotone and Y and Z are c-comonotone, and if the
distributions 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 be
seen as desirable.

13. In the case of -comonotonicity, transitivity holds: if X
and Y are -comonotone and Y and Z are -comonotone,
and if the distributions of X , Y and Z are absolutely con-
tinuous, then X and Z are -comonotome.
Indeed, express -comonotonicity of X and Y : for some
U ,
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.

14. Importance of . In dimension one, one recovers the
classical 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 measures
on Rd. Then:
- In dimension d = 1, -comonotonicity always implies
-comonotonicity.
- In dimension d 2, -comonotonicity implies -comonotonicity
if and only if = T # for some location-scale transform
T (u) = u + u0 where > 0 and u0 2 Rd. In other
words, comonotonicity is an invariant of the location-
scale family classes.

15. 2 Applications to risk measures
2.1 Coherent, regular risk measures (uni-
variate case)
Following Artzner, Delbaen, Eber, and Heath, recall the
classical risk measures axioms:
Recall axioms:
De…nition. A functional % : L2 ! R is called a coherent
d
risk 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.

16. De…nition. % : L2 ! R is called a regular risk measure
if 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 increasing
transformation 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 % is
regular if and only if for some increasing and nonnegative
function on [0; 1], we have
Z 1
%(X ) := (t)FX 1(t)dt;
0
where FX denotes the cumulative distribution functions
of the random variable X (thus QX (t) = FX 1(t) is the
associated quantile).
% is called a Spectral risk measure. For reasons explained
later, also called Maximal correlation risk measure.

17. Leading example: Expected shortfall (also called Con-
ditional VaR or TailVaR): (t) = 1 1 1ft g: Then
Z 1
1
%(X ) := FX 1(t)dt:
1

18. Kusuoka’ result, intuition.
s
Law invariance ) %(X ) = FX 1
Comonotone additivity+positive homogeneity )
is linear w.r.t. FX 1:
R1
FX 1 = 0 (t)FX 1(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 in
a way that will lend itself to easier multivariate extension.

19. 2.2 Alternative set of axioms
Manager supervising several N business units with risk
X1; :::; XN .
Eg. investments portfolio of a fund of funds. True
economic risk of the fund X1 + ::: + XN .
Business units: portfolio of (contingent) losses Xi report
a summary of the risk %(Xi) to management.
Manager has limited information:
1) does not know what is the correlation of risks - and
more broadly, the dependence structure, or copula be-
tween X1; :::; XN . Maybe all the hedge funds in the
portfolio have the same risky exposure; maybe they have
independent risks; or maybe something inbetween.
2) aggregates risk by summation: reports %(X1) + ::: +
%(XN ) to shareholders.

20. Reported risk: %(X1)+ ::: + %(XN ); true risk: %(X1 +
::: + XN ).
Requirement: management does not understate risk to
shareholders. Summarized by
%(X1) + ::: + %(XN ) ~ ~
%(X1 + ::: + XN ) (*)
whatever the joint dependence (X1; :::; XN ) 2 (L1)2.
d
But no need to be overconservative:
%(X1)+:::+%(XN ) = sup %(X1+:::+XN )
~ ~
X1 X1 ;:::;XN XN
where denotes equality in distribution.
De…nition. A functional % : L2 ! R is called a strongly
d
coherent risk measure if it is convex continuous and for
N
all (Xi)i N 2 L2 ,
d
n o
~ ~ ~
%(X1)+:::+%(XN ) = sup %(X1 + ::: + XN ) : Xi Xi :

21. A representation result.
The following result is given in [EGH].
Theorem. The following propositions about the func-
tional % on L2 are equivalent:
d
(i) % is a strongly coherent risk measure;
(ii) % is a max correlation risk measure, namely there
exists U 2 L2 , such that for all X 2 L2 ,
d d
n o
~ ~
%(X ) = sup E[U X ] : X X ;
(iii) There exists a convex function V : Rd ! R such
that
%(X ) = E [U rV (U )]

22. n
~
Idea of the proof . One has %(X )+ %(Y ) = sup %(X + Y ) :
~ ~
But %(X + Y ) = %(X ) + D%X (Y ) + o ( )
h i
~ ~
By the Riesz theorem (vector case) D%X (Y ) = E mX :Y ,
thus
n h i o
~ ~
%(X )+ %(Y ) = sup %(X ) + E mX :Y + o ( ) : Y Y
thus
n h i o
~ ~
%(Y ) = sup E mX :Y : Y Y
therefore % is a maximum correlation measure.

23. 3 Application to e¢ cient risk-sharing
Consider a risky payo¤ X (for now, univariate) to be
shared 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 expected
utility 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. These
are incomplete preferences: it can be impossible to rank
X and Y.

24. One wonders what is the set of e¢ cient allocations, i.e.
allocations that are not dominated w.r.t. the concave
order for every agent.
Dominated allocations. Consider a random variable X
(aggregate risk). An allocation of X among p agents is
a set of random variables (Y1; :::; Yp) such that
X
Yi = X:
i
Given two allocations of X , Allocation (Yi) dominates
allocation (Xi) whenever
2 3 2 3
X X
E 4 ui (Yi)5 E 4 ui (Xi)5
i i
for every continuous concave functions u1; :::; up. The
domination is strict if the previous inequality is strict
whenever the ui’ are strictly concave.
s
Comonotone allocations. In the single-good case, it is
intuitive that e¢ cient sharing rules should be such that in

25. “better”states of the world, every agent should be better
of than in “worse” state of the world – otherwise there
would be some mutually agreeable transfer.
This leads to the concept of comonotone allocations. The
precise connection with stochastic dominance is due to
Landsberger and Meilijson (1994). Comonotonicity has
received a lot of attention in recent years in decision the-
ory, insurance, risk management, contract theory, etc.
(Landsberger and Meilijson, Ruschendorf, Dana, Jouini
and Napp...).
Theorem (Landsberger and Meilijson). Any allocation
of 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 allocations
of X coincides with the set of comonotone allocations.
This result generalizes well to the multivariate case. Up
to technicalities (see [CDG] for precise statement), ef-
…cient allocations of a random vector X is the set of

26. -comonotone allocations of X , hence (Xi) solves
Xi = rui (U )
X
Xi = X
i
for convex functions ui : Rd ! R, with U . Hence
X = ru (U )
P
with u = i 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 testable
implication of e¢ ciency in risk-sharing in an risky endow-
ment economy.

27. 4 Application to stochastic orders
Quiggin (1992) shows that the notion of monotone mean
preserving increases in risk (hereafter MMPIR) is the
weakest 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 functions
of the random variables X and Y . X is said to be
Bickel-Lehmann less dispersed, denoted X %BL Y , if
QY (u) QX (u) is a nondecreasing function of u on
(0; 1). The mean preserving version is called monotone
mean preserving increase in risk (MMPIR) and denoted
-M M P IR.
MMPIR is a stronger ordering than concave ordering in
the sense that X %M M P IR Y implies X %cv Y .

28. The following result is from Landsberger and Meilijson
(1994):
Proposition (Landsberger and Meilijson). A random
variable X has Bickel-Lehmann less dispersed distribution
than a random variable Y if and only i¤ there exists Z
comonotonic with X such that Y =d X + Z .
The concept of -comonotonicity allows to generalize this
notion to the multivariate case as done in [CGH].
De…nition. A random vector X is called -Bickel-Lehmann
less dispersed than a random vector Y , denoted X % BL
Y , if there exists a convex function V : Rd ! R such
that the -quantiles QX and QY of X and Y satisfy
QY (u) QX (u) = rV (u) for -almost all u 2 [0; 1]d.
As de…ned above, -Bickel-Lehmann dispersion de…nes a
transitive binary relation, and therefore an order. Indeed,
if X % BL Y and Y % BL Z , then QY (u) QX (u) =

29. rV (u) and QZ (u) QY (u) = rW (u). Therefore,
QZ (u) QX (u) = r(V (u) + W (u)) so that X % BL
Z . When d = 1, this de…nition simpli…es to the classical
de…nition.
[CGH] propose the following generalization of the Landsberger-
Meilijson characterization .
Theorem. A random vector X is -Bickel-Lehmann less
dispersed than a random vector Y if and only if there
exists a random vector Z such that:
(i) X and Z are -comonotonic, and
(ii) Y =d X + Z .

30. 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 down
to 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 between
e¢ ciency of risk-sharing and -comonotonicity.
We have extended existing reults on functions increasing
with respect to the Bickel-Lehman order.
Interesting questions for future research: behavioural in-
terpretation of mu? computational issues? empirical
testability? case of heterogenous beliefs?