Reducing Risk in Convex Order

Transcription

1 Reducing Risk in Convex Order Junnan He a, Qihe Tang b and Huan Zhang b a Deartment of Economics Washington University in St. Louis Camus Box 208, St. Louis MO b Deartment of Statistics and Actuarial Science University of Iowa 24 Schaeffer Hall, Iowa City, IA August 8, 204 Abstract Given a risk osition X, we look for conditions under which a random variable Z is a risk reducer for X, that is, the inequality X + Z cx X + E[Z] holds, where cx refers to the convex order. For various cases we show necessary and sufficient conditions for Z to be a risk reducer for X. We are motivated by a recent work of Cheung et al. (204). Keywords: comonotonicity; convex hull; convex order; second order stochastic dominance with the same mean; risk reducer. Introduction For two integrable random variables X and Y defined on the robability sace (Ω, B Ω, P ), say Y is less than X in convex order, denoted by Y cx X, if E [Y ] = E [X] and that E [ ] [ ] (Y d) + E (X d)+ for all d R. Equivalently, it is known that Y cx X if and only if E [v(y )] E [v(x)] for every convex function v such that the two exectations exist. Pioneering works on similar concets include Lehmann (955), Rothschild and Stiglitz (970) and Day (972). Equivalent terminologies such as majorization, mean reserving sread, Rothschild-Stiglitz increase in risk and second order stochastic dominance with the same mean are also oular in mathematics, economics and decision theory. Standard references on these concets include Ross (983), Stoyan (983), Arnold (987), Mas-Colell et al. (995), Denuit et al. (2005) and Marshall and Olkin (20). Recently, Cheung et al. (204) introduced the concet of risk reducer as follows: Definition. For a given random variable X, a random variable Z is said to be its risk reducer, denoted by Z R(X), if X + Z cx X + E [Z]. (.)

2 In addition, Z is said to be a counter-monotonic risk reducer for X if (X, Z) is also countermonotonic. In Theorem of Cheung et al. (204), a sufficient condition for Z to be a countermonotonic risk reducer for X was ut forward. Obviously, in order for (.) to hold, it is not necessary to require Z to be counter-monotonic with X. Take the normal distributions as an examle, it is easy to see that (.) holds if (X, Z) is a mean-zero bivariate normal with correlation coefficient ρ not greater than Var[Z], yielding a much broader range 2 Var[X] than with ρ =. Furthermore, a risk reducer Z can even be ositively deendent on X over a small region. The objective of this aer is to rovide a comlete descrition of the set of risk reducers R(X) in a general setting and also show simle necessary and sufficient conditions for Z R(X) in various secific cases. The rest of the aer is organized as follows: in Section 2 we use convex hull to give a comlete descrition of R(X) for the case of an atomless robability sace and the case of a discrete robability sace, in Section 3 we roose a simle necessary and sufficient condition for Z R(X) when Z is fully deendent on X, in Section 4 we discuss universal risk reducers when the distribution of X is unknown for the single dimensional case, and finally in Section 5 we extend this discussion to the higher dimensional case. 2 Using convex hull to characterize risk reducers 2. Convex hull The main urose of this section is to give a comlete descrition of R(X). For X L (Ω), define the set of less risky ositions than X C(X) = {Y : Y cx X}. Rewriting (.) as X + Z E [Z] cx X, we see that R(X) = C(X) X + R, (2.) and thus we need focus on the descrition of C(X) only. In (2.) we have used the following notation: for two sets A = {a}, B = {b} and an element x, we write A + B = {a + b : a A, b B} and write A x = {a x : a A}. Throughout the aer, we shall use such notation and it should be clear from the context. Let us first recall the concet of convex hull. The convex hull of a set A, denoted as Conv(A), is the set of all finite convex combinations of oints in A. Suose that all random variables under consideration are defined on the robability sace (Ω, B Ω, P ) and are integrable. Denote by D(X) the set of random variables that are identically distributed 2

3 as X and by Conv(D(X)) the convex hall of D(X); that is, D(X) = {X : X = d X}, { m Conv(D(X)) = a i X i : m N, a i 0, i= } m a i = and X i D(X). i= Furthermore, denote by Conv(D(X)) the closure of Conv(D(X)) in L sace. 2.2 A general discussion For two random variables X and Y uniformly distributed on two n-dimensional real vectors x = (x,..., x n ) and y = (y,..., y n ), resectively, according to.a.3 of Marshall et al. (20), Y cx X if and only if y is in the convex hull of n! ermutations of x. A similar result about random variables in the [0, ] sace can be found in Theorem 5 of Ryff (965). We first resent a closely related result in which no restriction on the underlying robability sace is assumed: Lemma 2. The set C(X) is convex and closed. Lemma 2. leads to the following result: Corollary 2. Conv(D(X) {E[X]}) C(X). We construct a simle examle to illustrate that E[X] is not a trivial addition above and that the reverse inclusion does not hold in general. Consider a robability sace consisting of only two atoms and define two random variables X and Y as follows: Ω ω ω 2 P /3 2/3 X 2 Y /2 In this setting, D(X) is only the singleton {X}, and so is Conv(D(X)). Thus, E[X] = 0 / Conv(D(X)). Moreover, Y cx X but Y / Conv(D(X) {E[X]}). In the following of this section, we consider the case of an atomless robability sace and the case of a discrete robability sace, resectively, and we ursue a comlete descrition of C(X) for each case. 3

4 2.3 The case of an atomless robability sace First consider an atomless robability sace. For this case, we will directly aly Theorem 5 of Ryff (965) to show that under mild additional conditions, Conv(D(X)) is identical to C(X). To this end, we introduce several concets in analysis. See e.g. age??? and??? of Kechris (994) for details. A maing f from one toological sace to another toological sace is called a Borel isomorhism if it is a bijection and both f and f are Borel measurable. A toological sace is called Polish if it is searable and comletely metrizable. A measurable sace (Ω, B Ω ) is called a standard Borel sace if Ω is endowed with a Polish toology and B Ω is the Borel σ-algebra generated. Any robability measure P is called atomless if for any measurable set A Ω with P (A) > 0 there exists a measurable subset B of A such that 0 < P (B) < P (A). The following result is an immediate extension of Theorem 5 of Ryff (965). Lemma 2.2 Let (Ω, B Ω, P ) be an atomless standard Borel sace. Then C(X) = Conv(D(X)). Lemma 2.2 and relation (2.) lead to the first main result of this section: Theorem 2. Let (Ω, B Ω, P ) be an atomless and standard Borel sace. Then R(X) = Conv(D(X)) X + R. 2.4 The case of a discrete robability sace Let (Ω, B Ω, P ) and (Ω 2, B Ω2, P 2 ) be two robability saces. A measurable ma f : Ω Ω 2 is called measure reserving if P (f (B)) = P 2 (B) for all B B Ω2. See, e.g. age 292 of Stein and Shakarchi (2005) for this concet. The urose of this subsection is to resent a comlete descrition of C(X) for the case of a discrete robability sace. The idea of the following result is to switch discussions from a discrete robability sace to a atomless robability sace through a measure reserving maing. The equivalence of (a) and (e) gives a comlete descrition of C(X). Lemma 2.3 Let ( Ω, 2 Ω, P ) be a discrete robability sace and let X, Y be two random variables defined on (Ω, P ). Then there is a measure reserving maing f : [0, ] Ω with Lebesgue measure λ on [0, ] such that the following statements are equivalent: (a) Y C(X) on (Ω, P ); (b) Y f C (X f) on ([0, ], λ); (c) Y f Conv (D(X f)); (d) Y = d E λ [Y f] for some Y Conv (D(X f)); (e) There exists Y Conv (D(X f)) such that Y (ω) = E λ [Y f = ω] for all ω Ω. Lemma 2.3 and relation (2.) lead to the second main result of this section: Theorem 2.2 Let (Ω, P ) be a discrete robability sace. Then R(X) = { Y : Y ( ) = E λ [Y f = ] for Y Conv(D(X f) } X + R. 4

5 3 Fully deendent risk reducers Deterministic transformation of risks and its alications in comarative statics of risk has been studied by Meyer and Ormiston (989), Quiggin (99) and Levy and Wiener (998), among others. Motivated by these works, we now restrict the discussion on risk reducers that are fully deendent on the given random osition. As mentioned before, Cheung et al. (204) have recently studied counter-monotonic risk reducers. However, a fully deendent risk reducer need not be a conter-monotonic one. For examle, suose X is uniformly distributed on [0, ] and let Z := α sin(2πx). It can be seen that the correlation coefficient of X and Z is zero, but for α > 0 small enough, Z is a risk reducer of X. This examle is in essense the same as the ictorial examle rovided in Mas-Colell et al. (995 age 99). In this section we consider a comaratively more general situation in which Z = h(x) for some measurable but not necessarily monotone function h. The main result in this section will also generalize Theorem of Meyer and Ormiston (989). First, introduce some related notation. For a non-decreasing function g : R R, define its two generalized inverses as, for y R, g (y) = inf {x R : g(x) y}, g (y) = su {x R : g(x) y}, where we have followed the usual conventions inf = and su =. In the literature, these inverses are often written as g and g +, resectively. In articular, for a distribution function F, its inverse F (q) for 0 < q < is the well-known quantile function. Next, recall the well-known concet of comonotonicity. A random vector (X,..., X n ) is said to be (almost surely) comonotonic if there is a null set N such that, for any i, j {,..., n}, (X i (ω) X i (ω )) (X j (ω) X j (ω )) 0, ω, ω Ω\N. When we quote almost sure comonotonicity, we shall dro the words almost sure for simlicity. A air of random variables (X, Y ) is said to be counter-monotonic if (X, Y ) is comonotonic. See Dhaene et al. (2002a, 2002b, 2006) for the theory of comonotonicity and its alications to various areas of economics. To develo the main result of this section, we need to add a condition that (X, X + Z) is comonotonic. In the insurance context, if X is interreted as a claim amount and Z as the amount aid by the insurer, then X + Z is the amount retained to the insured. Thus, the comonotonicity of (X, X + Z) is naturally required to revent moral hazard. Recall that Cheung et al. (204) conducted their study under the condition that (Z, X + Z) is countermonotonic, which, as discussed there, imlies the comonotonicity of (X, X + Z) but not vice versa. It is noteworthy that Theorem of Meyer and Ormiston (989) roves one direction of the following theorem for a secial case with the random osition X distributed on [0, ] and with the transformation h(x) nondecreasing, continuous, and iecewise differentiable. 5

6 Theorem 3. Let X and Z be two integrable random variables such that Z = h(x) for some measurable (but not necessarily monotone) function h and (X, X + Z) is comonotonic. Then the following are equivalent: (a) Z R(X); (b) E[Z X > d] E[Z] holds for every d R. The following lemma lays a crucial role in roving Theorem 3.; a similar result is Theorem of Dhaene et al. (2002): Lemma 3. Let X be a real-valued random variable and let Y = g(x) for some measurable function g. (a) If g is non-decreasing, then FY () = g F X () almost everywhere for (0, ); (b) If g is non-increasing, then FY ( ) = g F X () almost everywhere for (0, ). 4 Universal risk reducers under all robability measures Denote the set of states of the world by Ω, and by X the (say, monetary) ayoff function that secifies losses or gains in each state. In many situations, an individual decision maker knows about the ayoff function X, but not the underlying robability measure. The following result describes comletely what tye of risk hedging behaviors, interreted as an insurance olicy Z, can be alied so that the individual s risk will be reduced regardless of the underlying robability measure. Such a random variable Z is called a universal risk reducer for X. In this and the next sections, we need to enhance the concet of comonotonicity. A random vector (X,..., X n ) is said to be strictly comonotonic if, for any i, j {,..., n}, (X i (ω) X i (ω )) (X j (ω) X j (ω )) 0, ω, ω Ω. Similarly, a air of random variables (X, Y ) is said to be strictly counter-monotonic if (X, Y ) is strictly comonotonic. A real function f : A R is said to be -Lischitz over a subset A of R if f(x) f(y) x y for all x, y A. For any real-valued random variable X, define H(X) = {h(x) h : X(Ω) R is non-increasing and -Lischitz}. It turns out that under some mild conditions H(X) is recisely the set of universal risk reducers for X. Here is the main result of this section: Theorem 4. Let (Ω, B Ω ) be a measurable sace in which every singleton {ω} Ω is measurable and let X be a given random variable. Then Z R(X) under every robability measure P such that E P [ X ], E P [ Z ] < if and only if Z H(X). 6

7 Theorem 2 of Cheung et al. (204) has a similar flavor, which shows that in order for X + Z cx Y + Z to hold for all Y satisfying X cx Y, it is necessary and sufficient that Z = a.s. Z and Z H(X). The following result, which will be used in roving Theorem 4., is in essence the same as Proosition 3 of Cheung et al. (204): Lemma 4. For a given random variable X, a random variable Z belongs to H(X) if and only if (Z, X + Z) is strictly counter-monotonic. 5 Extension to the higher dimensional case In this section we are interested in a situation where the given random osition is decided by multile random factors, but risk reducers, due to the limitation of knowledge or a certain regulatory requirement, can only be constructed based on one of the random factors. We ask under what conditions such a risk reducer reduces the overall risk of the given random osition. We use catastrohe (CAT) bonds as an illustrative examle. Let f(θ) be the ayoff of a CAT bond at exiration that is linked to a risk factor θ. The risk factor θ can be chosen to be the magnitude of a major catastrohe during a secific eriod in a secific region. Let θ be defined on a hysical robability sace (Ω, P, {F t }), while let the interest rate rocess {r t, t 0} be defined on another risk-neutral robability sace (Ω 2, P 2, {F 2 t }) corresonding to the financial market. Assuming indeendence between occurrences of the catastrohe and evolvements of the financial market, the rice at time t [0, ] of the CAT bond is a stochastic rocess defined on the roduct sace (Ω Ω 2, P P 2 ) adated to F t F 2 t. See Cummins and Geman (995), Vaugirard (2003), Zimbidis et al. (2007) and Nowak and Romaniuk (203) for related discussions. Let Z be defined on (Ω 2, P 2 ) be a derivative used to hedge the interest rate risk, such as interest forward. It is interesting to study under what conditions Z reduces the overall risk of the CAT bond in the sense of convex order. It is without loss of generality to restrict our discussion to 2 dimensions. We will show that if the risks factors are indeendent, the revious Theorem 4. can be generalized directly. In order to fix our terminology, let X = X(ω, ω 2 ) and Z = Z(ω 2 ) be two random variables defined on (Ω Ω 2, P P 2 ) and (Ω 2, P 2 ), resectively. With ω Ω fixed, define X ω : Ω 2 R to be X ω (ω 2 ) = X(ω, ω 2 ). In the sum X + Z aearing below, Z is understood as Z(ω 2 ) = Z(ω 2 ) Ω (ω ). Theorem 5. For i =, 2, let (Ω i, B Ωi ) be a measurable sace in which every singleton {ω i } Ω i is measurable. Let X : Ω Ω 2 R and Z : Ω 2 R be two random variables. Then Z is a risk reducer for X under every roduct robability measure P = P P 2 if and only if Z ω Ω H(X ω ). 7

8 Proof. Assume that Z is a risk reducer for X for every P = P P 2. For every ω Ω, choose P such that P ({ω }) =. Then by Theorem 4., we have Z H(X ω ). Conversely, assume Z H(X ω ) for every ω Ω, let v : R R be a convex function, and let P = P P 2 be an arbitrary roduct measure under which the exectations aearing below are finite. By Fubini s theorem and Theorem 4., we have E[v(X + Z)] = v(x(ω, ω 2 ) + Z(ω 2 ))P {dω dω 2 } Ω Ω 2 ( ) = v(x ω (ω 2 ) + Z(ω 2 ))P 2 {dω 2 } P {dω } Ω Ω ( 2 ) v(x ω (ω 2 ) + E[Z])P 2 {dω 2 } P {dω } Ω 2 Ω = E[v(X + E[Z])]. Thus, Z is a risk reducer for X under P = P P 2. Some observations on Corollary 5. follow below: First, the intersection ω Ω H(X ω ) may reduce to a trivial set consisting of deterministic constants only, meaning that there is simly no random risk reducer under every roduct robability measure P = P P 2. For examle, let X and X 2 be defined on the robability saces (Ω, P ) and (Ω 2, P 2 ), resectively, and let X = X X 2, (5.) which is the roduct of two indeendent random variables under P P 2. For this case, the intersection ω Ω H(X ω ) reduces to a trivial set if X takes both ositive and negative values and X 2 is not degenerate. Actually, let ω + and ω in Ω are such that X (ω + ) > 0 and X (ω ) 0, then + ) H(X ω ). ω Ω H(X ω ) H(X ω However, the intersection H(X ω+ ) H(X ω ) consists of deterministic constants only because each of its elements Z must be both strictly counter-monotonic and strictly comonotonic with X 2. Nevertheless, still confined to (5.), following the same analysis as above we see that the intersection ω Ω H(X ω ) will be nontrivial as long as X (Ω ) is contained in an interval that is away from 0. Second, the membershi Z ω Ω H(X ω ) does not imly that Z is a risk reducer for X under every robability measure P on Ω Ω 2. For examle, let X and X 2 be defined on the robability saces (Ω, P ) and (Ω 2, P 2 ), resectively, and let X = X + X 2. Hence, ω Ω H(X ω ) = H(X 2 ). 8

9 Choose a robability measure P concentrated on the set {(ω, ω 2 ) : X (ω ) = 2X 2 (ω 2 )} so that under P we have X = X + X 2 = a.s X 2. For this case, Z ω Ω H(X ω ) = H(X 2 ) actually increases the risk of X since Z and X 2 must be strictly comonotonic. 6 Aendix 6. Proof of Lemma 2. Proof. To establish he convexity of C(X), suose Y, Y 2 cx X and a [0, ]. Since for all d R, E[(aY + ( a)y 2 ) d) + ] ae[(y d) + ] + ( a)e[(y 2 d) + ] E[(X d) + ], it follows that ay + ( a)y 2 cx X. To rove that C(X) is closed, suose for a random variable Y, there exists a sequence of random variables {Y n, n N} from C(X) such that E [ Y n Y ] 0 as n. Then, for every d R, [ ] [ ] E (Yn d) + E (Y d)+ E [ ] (Yn d) + (Y d) + = E [ ] (Y d) {Yn d} {Y >d} + (Y n d) {Yn>d} {Y d} + Y n Y {Yn>d} {Y >d} E [ Y n Y ] 0, as n. Hence, E [ (Y d) + ] E [ (X d)+ ]. 6.2 Proof of Lemma 2.2 Proof. In view of Corollary 2., it suffices to show the inclusion C(X) Conv(D(X)). By Theorem 7.4 of Kechris (994), there is a Borel isomorhism f : Ω [0, ] such that P (B) = λ (f (B)) for any B B Ω, where λ is the Lebesgue measure. Let Y be an integrable random variable defined on (Ω, B Ω, P ). As illustrated in the diagram below, through this maing f we construct an integrable random variable Y f defined on ( [0, ], B [0,], λ ). Y Ω R f f Y f [0, ] For any B B R, P (Y B) = P ( Y (B) ) = λ ( f ( Y (B) )) = λ ( Y f B ). (6.) This imlies that Y f = d Y. 9

10 Hence if Y cx X then Y f cx X f. By Theorem 5 of Ryff (965), Y f Conv (D(X f )). Thus, there is a sequence of random variables Yn Conv (D(X f )) defined on [0, ] such that lim Yn Y f dλ = 0. n [0,] For each Y n, there are some ositive integer m, nonnegative numbers {a n,..., a nm } with m i= a ni =, and random variables {X n,..., X nm} defined on [0, ] identically distributed as X f such that Y n = Then Y n f = m i= a ni (X ni f). Note that m a ni Xni. i= X ni f = d X ni = d X f = d X, where the first equality can be verified similarly as above. Thus, Yn f Conv (D(X)). Moreover, lim Yn f Y dp = lim Yn n Ω n [0,] Y f dλ = 0. Hence, Y Conv(D(X)). 6.3 Proof of Lemma 2.3 Proof. We assume without loss of generality that Ω = {ω i : i I} for some index set I N. Define { ω, x [0, ( (ω )], f (x) = i ω i, x j= (ω j), ] i j= (ω j) for i I {}. It is easy to see that f is a measure reserving maing from [0, ] to Ω. (a) (b): For a Borel set B B [0,], since f is a measure reserving maing, Y = d Y f can be verified by 6.. In the same way, X = d X f. Since convex order is law invariant, we see the equivalence of (a) and (b). (b) (c): This is a direct consequence of Lemma 2.2. (c)= (d): Choose Y = Y f Conv (D(X f)). Since Y f is σ(f) measurable, we have E λ [Y f] = E λ [Y f f] = Y f = d Y, where the last ste has been roved above. This roves (d). (d)= (a): For a convex function v( ) for which the exectations involved below are finite, we have E P [v(y )] = E λ [v(e λ [Y f])] E λ [E λ [v(y ) f]] = E λ [v(y )] E λ [v(x f)] = E P [v(x)], 0

11 where the second ste is due to Jensen s inequality and the fourth ste due to Lemma 2.2. This roves (a). (c)= (e): Choose Y = Y f Conv (D(X f)). Then it holds for every ω Ω that This roves (e). (e)= (d): Trivial. E λ [Y f = ω] = E λ [Y f f = ω] = Y (ω). 6.4 Proof of Lemma 3. Proof. (a) Since g is non-decreasing, it is easy to see that, for every y g(r), x < g (y) = g(x) y = x g (y). (6.2) The first imlication in (6.2) gives that P (g(x) y) P (X < g (y)). Hence, F Y () = inf {y : P (g(x) y) } inf {y : P (X < g (y)) }. Letting z = g (y) and noticing that this imlies y g(z + 0), we have F Y () inf {g(z + 0) : P (X < z) } g (F X () + 0), (6.3) where the last ste is due to the fact that P (X FX ()). Symmetrically, the second imlication in (6.2) gives that F Y () = inf {y : P (g(x) y) } inf {y : P (X g (y)) }. Letting z = g (y) and noticing that this imlies y g(z 0), we have F Y () inf {g(z 0) : P (X z) } g (F X () 0). (6.4) Combining (6.3) (6.4) yields that g (F X () 0) F Y () g (F X () + 0). Since g has at most countably many jums, FY () = g F X () holds almost everywhere. (b) First notice a fact. For a random variable Z and 0 < <, it holds that FZ ( ) = inf {z : P (Z z) } = inf {z : P (Z > z) } = inf {z : P ( Z < z) } = su {z : P ( Z < z) } = F Z(). Thus, FZ ( ) = F Z () holds almost everywhere over 0 < <. By this and (a) we have, almost everywhere of (0, ), F Y ( ) = F g(x)() = ( g) F X () = g F X (). This comletes the roof of Lemma 3..

12 6.5 Proof of Theorem 3. Proof. In this roof, we introduce the following intermediate condition: (c) It holds for every (0, ) that h F X (t)dt E[Z]. (6.5) Then we formulate the roof of Theorem 3. into the following three stes: (a) (c): Introduce g(x) = x+h(x) : R R. Due to the comonotonicity of (X, X+Z), the function g( ) is non-decreasing over the range of X. By Theorem 2.5 of Bäuerle and Müller (2006), (a) holds if and only if F g(x)(t)dt F X+E[Z](t)dt, (0, ). By Lemma 3. and the identity FX+E[Z] (t) = F X (t) + E[Z], this inequality is rewritten as g F X (t)dt F X (t)dt + E[Z], (0, ), which is equivalent to (c). (c)= (b): For arbitrarily fixed d R, let = F X (d). Then FX () d and F X(FX ()) = F X (d) =. Hence, E [Z X > d] = = = h(x)df X (x) F X (d) d h(x)df X (x) FX () h F X (t)dt, (6.6) where the second ste due to P (FX () < X d) = 0 and the last ste is due to the change of variables x = FX (t). This roves the imlication. (b)= (c): If F X (R), which means that = F X (d) for some d R, then the derivation of (6.6) is still valid and, hence, by (b), inequality (6.5) holds. Now let (0, ) F X (R). Define d = FX (), which is a discontinuity oint of F, and define = P (X < d), 2 = F X (d). Clearly, < 2. In order to rove inequality (6.5) for this, first look at it at and 2. Inequality (6.5) already holds for 2 since 2 F X (R). Choose a sequence d n F X (R), n N, aroaching d = FX () from below and write q n = P (X d n ), n N, which aroaches = P (X < d). By (6.6) and (b), h F X (t)dt = lim n q n Clearly, FX () d for < 2, imlying that function in [, 2 ]. Hence, for [, 2 ), { h(fx (t))dt max h(fx (t))dt, This comletes the roof of Theorem 3.. h FX (t)dt = lim E [Z X > d n ] E[Z]. q n n h(f X (t))dt is a monotonic 2 2 } h(fx (t))dt E[Z]. 2

13 6.6 Proof of Lemma 4. Proof. Suose Z H(X), or, equivalently, Z = h(x) for some non-increasing and - Lischitz function h : X(Ω) R. For any ω, ω 2 Ω, assume without loss of generality that X(ω ) X(ω 2 ). Then Z(ω ) Z(ω 2 ) and (X(ω ) + Z(ω )) (X(ω 2 ) + Z(ω 2 )) = (X(ω ) X(ω 2 )) (h(x(ω 2 )) h(x(ω ))) 0. Hence, (Z, X + Z) is strictly counter-monotonic. Conversely, suose that (Z, X + Z) is strictly counter-monotonic. First, we need to show that Z can be written as a function of X. Assume by contradiction that there exists ω, ω 2 Ω such that X(ω ) = X(ω 2 ) but Z(ω ) Z(ω 2 ). Then (Z(ω ) Z(ω 2 )) ((X(ω ) + Z(ω )) (X(ω 2 ) + Z(ω 2 ))) = (Z(ω ) Z(ω 2 )) 2 > 0, which contradicts to the strict counter-monotonicity of (Z, X + Z). Hence, Z can be written as a function of X, say, Z = h(x). Second, it is easy to see that the function h is nonincreasing. Finally, it follows again from the strict counter-monotonicity of (Z, X + Z) that, for every x, x 2 X(Ω), (h(x 2 ) h(x ))((x 2 + h(x 2 )) (x + h(x ))) 0. Rearrange this inequality as (h(x 2 ) h(x )) 2 (x x 2 ) (h(x 2 ) h(x )), we see that h is -Lischitz. 6.7 Proof of Theorem 4. Proof. Suose that Z is a risk reducer for X under every robability measure P under which X and Z are integrable. By Lemma 4., it suffices to rove that (Z, X + Z) is strictly counter-monotonic. Assume by contradiction that (Z, X +Z) is not strictly countermonotonic. Then there exist ω, ω 2 Ω such that Z(ω ) > Z(ω 2 ) and X(ω ) + Z(ω ) > X(ω 2 ) + Z(ω 2 ). For notational convenience, write z i = Z (ω i ), x i = X(ω i ) and s i = X(ω i )+Z(ω i ) for i =, 2, so z > z 2 and s > s 2. Define a robability measure P satisfying P ({ω }) = P ({ω 2 }) = for some (0, ). If x x 2, we easily deduce that s s 2 > x x 2. Under P, the variance of X which equals (x x 2 ) 2 ( ) is strictly smaller than the variance of X + Z which equals (s s 2 ) 2 ( ), contradicting to X + Z cx X + E P [Z]. If x < x 2, due to the inequality s > s 2 and the identity E P [X + Z] = E P [X + E P [Z]], it is always the case that min{s, x 2 + E P [Z]} > max{s 2, x + E P [Z]} := d. Introduce a convex function v(t) = (t d) +. Noting that E P [Z] = ( )(s x )+(s 2 x 2 ) and that x + E P [Z] s > s 2 as 0, it holds for all small > 0 that E P [v(x + Z)] = (s d)( ) = (s (x + E P [Z])) ( ) = (s x ( )(s x ) (s 2 x 2 )) ( ) = ((x 2 x ) + (s s 2 )) ( ) 3

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