Multivariate comonotonicity, stochastic orders and risk measures

Transcription

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 multivariate comonotonicity. Guillaume Carlier, Alfred Galichon and Rose-Anne Dana. Journal of Economic Theory, [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, insurance 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 measures, e cient risk-sharing, optimal insurance contracts, etc. Unfortunately, no straightforward extension to the multivariate case (i.e. when there are several numeraires). The goal of this presentation is to investigate what happens in the multivariate case, when there are several dimension of risk. Applications will be given to: Risk measures, and their aggregation E cient risk-sharing Stochastic ordering.

4 1 Comonotonicity and its generalization 1.1 One-dimensional case Two random variables X and Y are comonotone if there exists a r.v. Z and nondecreasing maps T X and T Y such that X = T X (Z) and Y = T Y (Z) : For example, if X and Y are sampled from empirical distributions, X (! i ) = x i and Y (! i ) = y i, i = 1; :::; n where x 1 ::: x n and y 1 ::: y n then X and Y are comonotonic.

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

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 E [XY ] = 2 > 2 = E h X ~Y i :

7 Hardy-Littlewood inequality. The probability space is now [0; 1]. Assume U (t) = (t), where is nondecreasing. 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 E [XU] = Z 1 0 (t)f 1 P (t)dt E h ~XU i : Thus, letting Z 1 %(X) = (t)f 1 0 X (t)dt = max n E[ ~XU]; = max n E[X ~U]; ~U = d U o : ~X = d X o

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 U 2 argmax ~ U U 2 argmax ~ U n E[X ~U]; ~U o, and n E[Y ~U]; ~U o : 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 multidimensional, 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 extend 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 R d. Two random vectors X and Y in L 2 d are called -comonotonic if for some random vector U, we have U 2 argmax ~ U U 2 argmax ~ U n E[X ~U]; ~U o, and n E[Y ~U]; ~U o equivalentely: X and Y are -comonotonic if there exists two convex functions V 1 and V 2 and a random variable U such that X = rv 1 (U) Y = rv 2 (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 R d 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 R d! 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 Q P (u) = rv (u) the -quantile of distribution P.

12 Comonotonicity and transitivity. Puccetti and Scarsini (2010) propose the following de nition of comonotonicity, called c-comonotonicity: X and Y are c-comonotone if and only if n Y 2 argmax Y ~ E[X ~Y ]; ~Y Y o or, equivalently, i there exists a convex function u such that Y (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 continuous, then X and Z are -comonotome. Indeed, express -comonotonicity of X and Y : for some U, X = rv 1 (U) Y = rv 2 (U) and by -comonotonicity of Y and Z, for some ~U, Y = rv 2 ~ U Z = rv 3 ~ 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 comonotonicity 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 R d. 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 + u 0 where > 0 and u 0 2 R d. In other words, comonotonicity is an invariant of the locationscale family classes.

15 2 Applications to risk measures 2.1 Coherent, regular risk measures (univariate case) Following Artzner, Delbaen, Eber, and Heath, recall the classical risk measures axioms: Recall axioms: De nition. A functional % : L 2 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.

16 De nition. % : L 2! 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)f 1 0 X (t)dt; where F X denotes the cumulative distribution functions of the random variable X (thus Q X (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 Conditional VaR or TailVaR): (t) = ftg : Then %(X) := 1 1 Z 1 F 1 X (t)dt:

18 Kusuoka s result, intuition. Law invariance ) %(X) = FX 1 Comonotone additivity+positive homogeneity ) is linear w.r.t. FX 1 : FX 1 R = 10 (t)fx 1 (t)dt. Monotonicity ) is nonnegative Subadditivity ) is increasing Unfortunately, this setting does not extend readily to multivariate 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 X 1 ; :::; X N. Eg. investments portfolio of a fund of funds. True economic risk of the fund X 1 + ::: + X N. Business units: portfolio of (contingent) losses X i report a summary of the risk %(X i ) to management. Manager has limited information: 1) does not know what is the correlation of risks - and more broadly, the dependence structure, or copula between X 1 ; :::; X N. 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 %(X 1 ) + ::: + %(X N ) to shareholders.

20 Reported risk: %(X 1 )+:::+%(X N ); true risk: %(X 1 + ::: + X N ). Requirement: management does not understate risk to shareholders. Summarized by %(X 1 ) + ::: + %(X N ) %( ~X 1 + ::: + ~X N ) (*) whatever the joint dependence (X 1 ; :::; X N ) 2 (L 1 d )2. But no need to be overconservative: %(X 1 )+:::+%(X N ) = where denotes equality in distribution. sup %(X 1 +:::+X N ) ~X 1 X 1 ;:::; ~X N X N De nition. A functional % : L 2 d! R is called a strongly coherent risk measure if it is convex continuous and for all (X i ) in 2 N, L 2 d %(X 1 )+:::+%(X N ) = sup n %( ~X 1 + ::: + ~X N ) : ~X i X i o :

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

22 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 m X : ~Y i, thus %(X)+%(Y ) = sup n %(X) + E h m X : ~Y i + o () : ~Y Y o thus %(Y ) = sup n E h m X : ~Y i : ~Y Y o 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 contingent state: X = X 1 + X 2 X 1 and X 2 are said to form an allocation of X. Agents are risk averse in the sense of stochastic dominance: 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 (Y 1 ; :::; Y p ) such that X i Y i = X: Given two allocations of X, Allocation (Y i ) dominates allocation (X i ) whenever E 2 4 X i 3 u i (Y i ) 5 E 2 4 X i u i (X i ) for every continuous concave functions u 1 ; :::; u p. The domination is strict if the previous inequality is strict whenever the u i s are strictly concave. 3 5 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 theory, 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. Moreover, this dominance can be made strict unless X is already 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 (X i ) solves X i X i = ru i (U) X i = X for convex functions u i : R d! R, with U. Hence X = ru (U) with u = P i u i. That is U = ru (X) ; hence e cient allocations are such that X i = ru i ru (X) : This result opens the way to the investigation of testable implication of e ciency in risk-sharing in an risky endowment 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 ranking of risk aversion in the rank dependent utility framework. MMPIR is the mean preserving version of Bickel- Lehmann dispersion, which we now de ne. De nition. Let Q X and Q Y 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 Q Y (u) Q X (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 - MMP IR. MMPIR is a stronger ordering than concave ordering in the sense that X % MMP 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 : R d! R such that the -quantiles Q X and Q Y of X and Y satisfy Q Y (u) Q X (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 Q Y (u) Q X (u) =

29 rv (u) and Q Z (u) Q Y (u) = rw (u). Therefore, Q Z (u) Q X (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 comonotonicity to higher dimension: -comonotonicity. This concept 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 axioms 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 interpretation of mu? computational issues? empirical testability? case of heterogenous beliefs?