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Multivariate comonotonicity, stochastic orders and risk measures - - PDF document
Multivariate comonotonicity, stochastic orders and risk measures - - PDF document
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)
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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.
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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.
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By the rearrangement inequality (Hardy-Littlewood), max
permutation n
X
i=1
xiy(i) =
n
X
i=1
xiyi: More generally, X and Y are comonotonic if and only if max
~ Y =dY
E
h
X ~ Y
i
= E [XY ] :
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- 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
E [XY ] = 2 > 2 = E
h
X ~ Y
i
:
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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) = F 1
P (t):
For ~ X : [0; 1] ! R a r.v. such that ~ X P, one has
E [XU] =
Z 1
0 (t)F 1 P (t)dt E
h ~
XU
i
: Thus, letting %(X) =
Z 1
0 (t)F 1 X (t)dt = max
n
E[ ~
XU]; ~ X =d X
- =
max
n
E[X ~
U]; ~ U =d U
- :
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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 U 2 argmax ~
U
n
E[X ~
U]; ~ U
- , and
U 2 argmax ~
U
n
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 .
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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.
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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
d are called -comonotonic if for some random
vector U , we have U 2 argmax ~
U
n
E[X ~
U]; ~ U
- , and
U 2 argmax ~
U
n
E[Y ~
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.
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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
U;XP
E [hU; Xi] 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 inf
V (u)+W(x)hx;ui
Z
V (u) d (u) +
Z
W (x) dP (x) : Call QP (u) = rV (u) the -quantile of distribution P.
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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 Y 2 argmax ~
Y
n
E[X ~
Y ]; ~ Y Y
- r, 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.
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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.
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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
- n 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.
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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
d ! R is called a coherent
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.
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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 %(X) :=
Z 1
0 (t)F 1 X (t)dt;
where FX denotes the cumulative distribution functions
- f the random variable X (thus QX (t) = F 1
X (t) is the
associated quantile). % is called a Spectral risk measure. For reasons explained later, also called Maximal correlation risk measure.
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Leading example: Expected shortfall (also called Con- ditional VaR or TailVaR): (t) =
1 11ftg: Then
%(X) := 1 1
Z 1
F 1 X (t)dt:
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Kusuoka’s result, intuition. Law invariance ) %(X) =
- F 1
X
- Comonotone additivity+positive homogeneity )
is linear w.r.t. F 1
X :
- F 1
X
- =
R 1
0 (t)F 1 X (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.
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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.
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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
d )2.
But no need to be overconservative: %(X1)+:::+%(XN) = sup
~ X1X1;:::; ~ XNXN
%(X1+:::+XN) where denotes equality in distribution. De…nition. A functional % : L2
d ! R is called a strongly
coherent risk measure if it is convex continuous and for all (Xi)iN 2
- L2
d
N,
%(X1)+:::+%(XN) = sup
n
%( ~ X1 + ::: + ~ XN) : ~ Xi Xi
- :
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A representation result. The following result is given in [EGH].
- Theorem. The following propositions about the func-
tional % on L2
d are equivalent:
(i) % is a strongly coherent risk measure; (ii) % is a max correlation risk measure, namely there exists U 2 L2
d, such that for all X 2 L2 d,
%(X) = sup
n
E[U ~
X] : ~ X X
- ;
(iii) There exists a convex function V : Rd ! R such that %(X) = E[U rV (U)]
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Idea of the proof . One has %(X)+%(Y ) = sup
n
%(X + ~ Y ) : But %(X + ~ Y ) = %(X) + D%X( ~ Y ) + o () By the Riesz theorem (vector case) D%X( ~ Y ) = E
h
mX: ~ Y
i
, thus %(X)+%(Y ) = sup
n
%(X) + E
h
mX: ~ Y
i
+ o () : ~ Y Y
- thus
%(Y ) = sup
n
E
h
mX: ~ Y
i
: ~ Y Y
- therefore % is a maximum correlation measure.
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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.
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One wonders what is the set of e¢cient allocations, i.e. allocations that are not dominated w.r.t. the concave
- rder 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
i
Yi = X: Given two allocations of X, Allocation (Yi) dominates allocation (Xi) whenever
E
2 4X
i
ui (Yi)
3 5 E 2 4X
i
ui (Xi)
3 5
for every continuous concave functions u1; :::; up. The domination is strict if the previous inequality is strict whenever the ui’s are strictly concave. Comonotone allocations. In the single-good case, it is intuitive that e¢cient sharing rules should be such that in
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“better” states of the world, every agent should be better
- f 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-
- ry, insurance, risk management, contract theory, etc.
(Landsberger and Meilijson, Ruschendorf, Dana, Jouini and Napp...). Theorem (Landsberger and Meilijson). Any allocation
- f X is dominated by a comonotone allocation. More-
- ver, this dominance can be made strict unless X is al-
ready comonotone. Hence the set of e¢cient allocations
- f 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
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- comonotone allocations of X, hence (Xi) solves
Xi = rui (U)
X
i
Xi = X for convex functions ui : Rd ! R, with U . Hence X = ru (U) with u = P
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.
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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
- f 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
- MMPIR.
MMPIR is a stronger ordering than concave ordering in the sense that X %MMPIR Y implies X %cv Y .
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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) =
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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.
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