Dual moments, volatility, dependence and the copula

Dual moments, volatility, dependence and the copula Dieter Denneberg and Nikola Leufer Universität Bremen September 19, 24 Abstract Walking though the zoo of stochastic volatility and dependence parameters we got some ordering ideas and so detect some new species. The elementary volatility parameters of first and second order in spirit of dual theory of choice under risk or rank dependent expected utility are average absolute deviation and Gini index. Analogous to classical covariance dual dependence parameters will be introduced and investigated in connection with the copula of a multivariate distribution. It is argued that the dual volatility and dependence parameters are better suited than the classical parameters for applications in finance and insurance. From the technical point of view it is fascinating for a Choquet integrator to look at copulas, since for both theories ordering and comonotonicity play important roles. 1 Introduction Predominantly thanks to David Schmeidler s work [12], the dual or rank dependent expected utility theory is now well established and applied in decision under risk and many other fields. Here dual means that not, like in the classical models, the outcomes of the random variables are transformed, but their distributions. The present paper applies the dual view to elementary volatility and dependence parameters. We survey the known first and second order dual volatility parameters, average absolute deviation and Gini index, and propose new rank based dependence parameters. The most common volatility and dependence parameters, standard deviation and correlation coefficient, are of the L 2 -type, i.e. related to the second moments. The product of the random variables entering the respective formulas can hardly be interpreted directly in applications. We propagate parameters of L 1 -type where the product is replaced by the min or a more sophisticated order or lattice relation, which had been investigated by Grabisch (see [5]). Our results apply to the copula of two random variables and generate new concordance parameters. 1

The copula of a random vector is the essence of the common distribution of its components, obtained by normalizing the margins to become uniformly distributed on the unit interval. It is invariant under monotone transformations on the margins, so on the margins only the rank matters. The theory of the copula is well elaborated for continuous random variables. We propose a generalization or slight modification, where it is technically convenient to perceive distribution functions as interval valued functions if they are not continuous, or to attribute the midpoint of the interval. In insurance, independence of the different claims had been, for a long time, a general assumption in the mathematical models. Since in our complex social world the interrelation of different risks increased, there is a new interest in models coping with dependence of random variables. Similarly in finance those models gain importance. During the last decade research on the copula and on concordance and discordance parameters has been intensified. We hope to convince the reader that the dual moments and rank based dependence parameters are better suited for certain applications than the classical parameters. The results of this paper are in the framework of σ-additive probability theory, but methods of non-additive measure and integration come in quite naturally. First of all the dual view to distort probabilities leads to the Choquet integral. Next, comonotonicity of random variables plays a central role in both theories, for the copula and for the Choquet integral. Finally, to use the techniques of the other area will be fruitful for both. In probability theory and for the copula increasing distribution functions dominate, whereas for the Choquet integral decreasing distribution functions are the natural tool. So we employ both and, for the sake of concise formulations, perceive them as interval valued functions if they are not continuous. The authors thank B.S. Rüffer for helpful discussions. 2 Volatility parameters In this section some basic notations are established and some facts about the less known volatility parameters average absolute deviation and Gini index are collected. Throughout the paper we work on a probability space (Ω, A, P ) and denote with L 1 = L 1 (Ω, A, P ) the Lebesgue space of random variables X : Ω R with existing finite expectation (first moment) and with L 2 = L 2 (Ω, A, P ) the space of random variables with finite variance (second central moment). Further F X denotes the increasing distribution correspondence 1 of the ran- 1 A correspondence is a set valued function, here all correspondences happen to be interval valued. A selection h of a correspondence H is a function with h(x) H(x) for all x. If the values of H are closed intervals, the upper selection of H is H(x) := {u u H(x)}, similarly the lower selection H is defined. We use the lattice operators 2

dom variable X, F X (x) := [ ] P (X < x), P (X x) (1) and G X := 1 F (X) the decreasing one. The usual right continuous increasing distribution function is the upper selection of the distribution correspondence, F X (x) = P (X x). Similarly the left continuous increasing distribution function is the lower selection F X (x) = P (X < x). Let Ȟ denote the inverse correspondence of a correspondence H, i.e. the graph of Ȟ is the graph of H after interchanging the coordinate axes (for the explicit formulas see e.g. [5] Section 3). Again the pseudo inverse function of a distribution function is a selection of the inverse of the corresponding distribution correspondence. Let H be a strongly increasing correspondence from an interval I R to R, i.e. x 1 < x 2 implies y 1 y 2 for all y i H(x i ). Distribution correspondences and their inverses have this property. The integral of H is defined as H(x) dx := h(x) dx, h a selection of H. (2) I I This definition is unambiguous since H is single-valued except an at most countable set in I. The latter derives from the fact that on the real axis any family of disjoint open intervals is finite or countable. As usual EX := XdP = 1 ˇF (p) dp = 1 Ǧ(p) dp (see (2)) denotes the expectation of X and MX := ˇF (.5) the median. The median is an interval in general and the subsequent equation (3) is valid for any point in this interval. Average absolute deviation of X from its median, τ(x) := X MX dp = X MX 1, (3) is the first absolute central moment of X, hence related to the Lebesgue space L 1. Similarly, standard deviation σ(x) = X EX 2 is related to the L 2 -norm and variance var(x) = σ(x) 2 is the second central moment of X. Recall that MX minimizes X c 1 for all constants c R like σ(x) minimizes X c 2. It is well known ([17] (9) implicitly, [16], [2], [4] Example 8.2) that average absolute deviation τ(x) and Gini index gini(x) are volatility parameters that can be expressed by means of the Choquet integral w.r.t. a piecewise linear or a quadratic distortion γ i P of P, γ i : [, 1] [, 1], respectively, τ(x) = EX X d(γ 1 P ), with γ 1 (p) := (2p 1), (4) gini(x) := EX X d(γ 2 P ), with γ 2 (p) := p 2, (5) and to denote max and min or sup and inf, respectively. 3

Obviously, τ(x) exists and is finite iff X L 1. The same holds for gini(x) since the distortion γ 2 has bounded derivative (see (14) in the proof of Proposition 5.2). Notice that in this paper gini(x) is not normalized like the usual Gini index gini(x)/ex of welfare theory, measuring inequality of the income distribution X in a population Ω. One might regard (4) as first and (5) as second order dual moments. Here dual refers to the functional XdP of the two variables X and P, where the piecewise linear or quadratic transforms are applied to the variable X or, dually, to P. In fact the first absolute central moment (3) coincides with the first dual moment (4) and there is an L 2 -formula for the second dual moment (5) as we will see in Proposition 5.2. There we need the fact that quantiles behave a.s. multiplicative for nonnegative comonotonic random variables (see [3] or Section 4 for definitions of comonotonicity), which is of interest for its own. Proposition 2.1 Suppose X, Y are nonnegative and comonotonic, then almost everywhere ǦXY = ǦXǦY and ˇF XY = ˇF X ˇFY. Proof We will apply ([3] Proposition 4.1) Ǧ u X = u(ǧx) a.e. for continuous increasing u : R R. X, Y being comonotone, there exist (see [3] Proposition 4.5) a random variable Z and continuous increasing functions u, v on R such that X = u Z and Y = v Z. We may suppose that u, v, otherwise replace them by their positive parts. Then the function w(z) := u(z)v(z) is increasing, too, and we get ǦXY = Ǧw Z = w(ǧz) = u(ǧz)v(ǧz) = ǦXǦY a.e.. (x, p) belongs to the graph of F X iff (x, 1 p) belongs to the graph of G X, hence ǦX(p) = ˇF X (1 p) and the formulas for increasing respectively decreasing distribution correpondences are equivalent. Proposition 2.1 sheds some more light on the analogy between dual moments and the classical ones. In the dual case comonotonicity plays the role, independence plays for the classical case. The location parameter median M X, which is related to τ(x) (see (3)), behaves essentially multiplicative for comonotonic random variables, whereas expected value EX, related to variance var(x), behaves multiplicative for independent random variables. The quantile function behaves also comonotonic additive ([3] Corollary 4.6). This fact implies comonotonic additivity of the Choquet integral, hence by (4), (5) also comonotonic additivity of the dual parameters τ(x) and gini(x). These facts are again analogous to independence additivity of the L 2 -parameter var(x). Finally we remark that τ(x) and gini(x) are subadditive since γ 1 and γ 2 are convex. 4

3 Dependence parameters We introduce two dependence parameters, related to the volatility parameters τ and gini in the sam way as covariance is related to variance. The most popular dependence parameter of a random 2-vector (X 1, X 2 ) is its covariance cov(x 1, X 2 ) := (X 1 EX 1 )(X 2 EX 2 )dp. It is the inner product of the centralized random variables in the Hilbert space L 2 and var(x) = X EX 2 2 = cov(x, X) for X : Ω R. (6) In (3) we have seen that average absolute deviation τ(x) is the L 1 - norm of the centralized random variable. For defining the corresponding dependence parameter we have to find a substitute for the inner product, which does not exist in the Banach space L 1. For this purpose we have to replace the product in the definition of covariance with a new operation having the property x = x x. (7) The required operation had been introduced by Grabisch and is called bipolar meet in [5], { x y if sign x = sign y x y := ( x y ) else., x, y R. We emphasize that for positive x, y the bipolar meet is just the min (or meet in lattice terminology). The bipolar meet is commutative and associative. Now, in analogy with covariance which could be called 2-covariance in our context, we define 1-covariance cov 1 (X 1, X 2 ) := (X 1 ṀX 1) (X 2 ṀX 2)dP. Here and in the sequel a dot on an interval denotes its barycenter, I := a + b 2 if I = [a, b]. Like in (6) we get, using (7), τ(x) = X ṀX 1 = cov 1 (X, X). 5

Proposition 3.1 Suppose, X 1 is symmetrically distributed on R, i.e. P (X 1 ṀX 1 x 1 ) = P (X 1 ṀX 1 x 1 ) for all x 1 R. If X 1, X 2 L 1 are independent, then cov 1 (X 1, X 2 ) =. For classical covariance this result holds for X 1, X 2 L 2 without the symmetry assumption. The restricted validity of Proposition 3.1 does not harm us so much since our main concern is to apply it to copulas, where the assumption holds in the essential cases. Proof We may suppose Ω = R 2, X i (x 1, x 2 ) = x i and ṀX i =. By the independence assumption P = P (X 1,X 2 ) is the product of P X 1 and P X 2 so that we can apply Fubini s Theorem. We decompose R 2 in three parts: A := {x R 2 x 2 x 1 }, B := {x R 2 x 2 x 1 } and C := R 2 \ (A B). We are done if we show that the integral of X 1 X 2 on each of these parts vanishes. First on A we get x 1 x 2 = x 1 so that X 1 X 2 dp = x 1 1 A (x 1, x 2 ) dp X 2 (x 2 ) dp X 1 (x 1 ) A = = =. x 1 1 A (x 1, x 2 ) dp X 2 (x 2 ) dp X 1 (x 1 ) x 1 P (X 2 x 1 ) dp X 1 (x 1 ) For the last equation we needed that P X 1 is symmetrically distributed around and that the function x 1 P (X 2 x 1 ) is an odd function of the variable x 1. On B we get x 1 x 2 = x 1 and the proof runs like for A. Finally, on C we get x 1 x 2 = x 2 sign (x 1 ) and we proceed like for A but change the order of integration. Then the inner integral sign (x 1)1 C (x 1, x 2 )P X 1 (x 1 ) vanishes by the symmetry assumption. It is well known (e.g. [17]) that the Gini index has also a L 1 -representation, namely gini(x) = 1 2 X Y 1 for iid X, Y : Ω R. So in the spirit of cov 1 we define Gini covariance 2 cogini(x 1, X 2 ) := 1 2 (X 1 Y 1 ) (X 2 Y 2 ) dp, 2 We attributed the name Gini to this parameter only by analogy. Gini himself introduced a different dependence parameter indice di cograduazione simplice nowadays called Gini s γ (see [1]). 6

where (Y 1, Y 2 ) is an iid copy of (X 1, X 2 ) : Ω R 2. With X 1 = X 2 we get, using (7), gini(x 1 ) = cogini(x 1, X 1 ). Concerning independence, we get a better result for cogini than for cov 1. Proposition 3.2 cogini(x 1, X 2 ) = if X 1, X 2 L 1 are independent. Proof Since the pair X 1, Y 1 is independent, their difference X 1 Y 1 is symmetrically distributed around and we can proceed like for Proposition 3.1. Like gini and cogini another pair of volatility and dependence parameters could be constructed with the squared L 2 -norm and the inner product of L 2. We wonder if these parameters have been applied somewhere. The parameters τ, cov 1, gini and cogini exist and have finite values for random variables in L 1, whereas var and cov in general only for random variables in L 2 L 1. By normalization L 2 -correlation, L 1 -correlation and Gini correlation are defined, ρ(x 1, X 2 ) := cov(x 1, X 2 ) σ(x 1 ) σ(x 2 ), ρ 1(X 1, X 2 ) := cov 1(X 1, X 2 ) τ(x 1 ) τ(x 2 ), giniρ(x 1, X 2 ) := cogini(x 1, X 2 ) gini(x 1 ) gini(x 2 ) if the denominator does not vanish, i.e. X 1 and X 2 are not constant a.e.. Like ρ all these correlations have their values in the interval [ 1, 1]. Proposition 3.3 If the correlations exist, ρ 1 (X 1, X 2 ), giniρ(x 1, X 2 ) [ 1, 1]. Proof Using x y = x y one gets cov 1 (X 1, X 2 ) (X 1 ṀX 1) (X 2 ṀX 2) dp = X 1 ṀX 1 X 2 ṀX 2 dp X 1 ṀX 1 dp X2 ṀX 2 dp = τ(x 1 ) τ(x 2 ). Similarly cogini(x 1, X 2 ) gini(x 1 ) gini(x 2 ). Also, like for ρ we get ρ 1 (X 1, X 2 ) = ρ 1 (X 2, X 1 ), ρ 1 (X, X) = 1, ρ 1 (X, X) = 1, ρ 1 (X 1, X 2 ) = ρ 1 (X 1, X 2 ). and the corresponding equations for giniρ. 7

4 Concordance and comonotonicity For applications the events of assuming simultaneously positive (or very large) values or simultaneously negative values are of special interest. In this context concordance and comonotonicity are important issues. A similar structure between Kendall s tau and Gini covariance become apparent. Formulas analogous to var(x + Y ) = var(x) + var(y ) + 2 cov(x, Y ) are derived. For this purpose cov 1 and cogini have to be defined on the discordance event. We denote with Q 13 R 2 the union of the first and third open quadrants, and with Q 24 = R \ Q 13 the union of the second and fourth open quadrants. A pair x, y R 2 of 2-vectors is called concordant if x y Q 13 and discordant if x y Q 24. Now let X, Y be a pair of random 2-vectors. The difference of the probabilities of concordance and discordance is commonly denoted Q(X, Y ) in the context of copulas, Q(X, Y ) := P (X Y Q 13 ) P (X Y Q 24 ) = sign [(X 1 Y 1 ) (X 2 Y 2 )] dp = sign [(X 1 Y 1 ) (X 2 Y 2 )] dp. The coordinates X 1, X 2 of X are called comonotonic iff X Y Q 13 for an iid copy Y of X. More general, the random variables X 1, X 2 are called P -comonotonic iff P (X Y Q 13 ) = 1 for an iid copy Y of X. Other characterisations of comonotonicity can be found in [3] Proposition 4.5. Inserting in Q(X, Y ) for Y a certain random vector related to the random 2-vector X defines further dependence parameters. First taking the constant vector Y = ṀX = (ṀX 1, ṀX 2 ) results in Blomquist s beta Next Kendall s tau is defined as β(x 1, X 2 ) := Q(X, ṀX). Ktau(X 1, X 2 ) := Q(X, Y ) with X, Y iid. For continuously distributed random variables Ktau(X 1, X 2 ) = 1 is equivalent with P -comonotonicity. Observe the similarity of the integrals in the definitions of Kendall s tau and Gini covariance, only sign has to be added in the case of Kendall s tau. So we might regard Gini covariance as a dependence parameter one level above Kendall s tau. 3 Blomquist s β and 1-covariance cov 1 are related 3 One might ask for the one variable parameter corresponding to Kendall s tau. Clearly, Ktau(X 1, X 1) = P (X 1 Y 1) with X 1, Y 1 being iid. Thus Ktau(X 1, X 1) = 1 iff X 1 is continuously distributed and = iff X 1 is essentially constant. In some sense Ktau(X 1, X 1) measures the distance of X 1 from being a constant. The correlation coefficient corresponding to Kendall s tau is, for continuously distributed random variables, Kendall s tau itself. 8

in a similar manner. We define 1-covariance on event A of a random vector (X, Y ) R 2, cov 1 (X, Y ; A) := (X ṀX) (Y ṀY )dp, A and similarly cogini(x, Y ; A), Gini covariance on event A. One easily verifies for x, y R { x + y + 2(x y) if (x, y) Q24 x + y = x + y else and this implies τ(x + Y ) = τ(x) + τ(y ) + 2 cov 1 (X, Y ; Q 24 ) (8) if Ṁ(X + Y ) = ṀX + ṀY, gini(x + Y ) = gini(x) + gini(y ) + 2 cogini(x, Y ; Q 24 ). (9) These equations again establish that τ and gini are comonotonic additive. But cov 1 (X, Y ; Q 24 ) = and cogini(x, Y ; Q 24 ) = are weaker conditions than comonotonicity of X, Y. Notice that x y = x y < for (x, y) Q 24. Hence cov 1 (X, Y ; Q 24 ) and cogini(x, Y ; Q 24 ), which again proves subadditivity of τ and gini. 5 The uniformisation of a random variable The combination of a random variable with her distribution function is a basic tool for the copula. Some technicalities are given here and they will be used to prove a general L 1 -representation of the Gini index. In the context of the copula it is most convenient to use the barycenter F X = 1 2 (F X(x) + F X (x)). (1) of the interval valued distribution correspondence F X (x) = [F X (x), F X (x)] of a random variable X. We refer to it as the (midpoint) distribution function of X. So we avoid the use of interval valued random objects which would blow up the technicalities. Crucial for the copula is the (up-)uniformisation (or probability integral transform in some literature) of X, U X := F X X. If we use the decreasing distribution correspondence G X = 1 F X of X, we get the down-uniformisation V X := ĠX X = 1 U X of X. Our name uniformisation anticipates the following well known result (i). 9

Proposition 5.1 (i) If F X is a continuous function, then U X is uniformly distributed on [, 1]. (ii) Let ϕ : X(Ω) R be an injective and increasing function, i.e. x 1 < x 2 implies ϕ(x 1 ) < ϕ(x 2 ). Then U ϕ(x) = U X. Proof (i) Applying [3] Proposition 4.1 with G X as continuous transformation function we get ǦV X (v) = ǦG X X(v) = G X ǦX(v) = v for almost all v. Hence V X is uniformly distributed and so is U X. (ii) Go back to the definition of the uniformisation and use {ϕ(x) ϕ(x(ω))} = {X X(ω)} and {ϕ(x) < ϕ(x(ω))} = {X < X(ω)}. The random variable U X is not uniformly distributed in general. But in any case we get E(U X ) = 1/2, var(u X ) = 1/12 if F X is continuous, (11) Ṁ(U X ) = 1/2 if X is symmetrically distributed, τ(u X ) = 1/4 if Ṁ(U X ) = 1/2, (12) gini(u X ) = 1/6 if F X is continuous. (13) Since the uniformisation of U X is U X itself, (13) computes easily with the subsequent L 2 -representation of the Gini index. It is known in case of the uniform distribution on a finite set [1] or a continuous random variable [9] (citations from [15]). The last paragraph of the proof will show us that defining the uniformisation with the midpoint distribution function (1) instead of another selection, say F X, of the distribution correspondence F X is essential for the validity of Proposition 5.2 in the general case. So there is no arbitrariness in defining the uniformisation U X of X as one might believe at first sight. Proposition 5.2 A random variable X and its uniformisation U X are comonotonic and gini(x) = 2 cov(x, U X ) for X L 1, X. Proof Comonotonicity of X and U X is plain since U X is an increasing transform of X (see e.g. [3] Proposition 4.5). For evaluating the Choquet integral in formula (5) for gini(x) we need the decreasing distribution correspondence G γ2 P,X = γ 2 G P,X of X w.r.t. the distorted probability γ 2 P. Since γ 2 (t) = t 2 is invertible on [, 1] we get 1

Ǧ γ2 P,X = ǦP,X γ2 1. This formula together with the substitution t = p2 imply (notice that we write also like above G X for G P,X ) gini(x) = EX = EX = EX 2 = EX 2 1 = 2 1 1 1 Ǧ γ2 P,X(t) dt = EX Ǧ X (p) dp 2 = EX 2 ( EX 1 1 ˇF X (1 p) p dp = EX 2 1 ˇF X (q) q dq EX. On the other hand we get ) ˇF X (q) q dq 2 cov(x, U X ) = 2E(XU X ) 2EXEU X. Since EU X = 1/2, it is sufficient to show 1 ˇF X (q)q dq = 1 Ǧ X γ2 1 (t) dt Ǧ X (p) p dp (14) 1 ˇF X (q) (1 q) dq ˇF XUX (q) dq = E(XU X ). (15) If F X is a continuous function we know from Proposition 5.1 ˇF UX (q) = q. Then, by Proposition 2.1 (here we need X ), both integrands in (15) coincide a.e. and the proof is complete. Notice that we got cov(x, U X ) R only supposing X L 1. But also in the general case, in the left hand side integral, we can replace the factor q with the quantile correspondence ˇF UX (q) of U X without changing the value of the integral. If q is an inner point of the range of U X in [, 1], clearly q = ˇF UX (q). If not, regard the interval [a, b] := F X (x ) [, 1] with q [a, b], i.e. ˇFX (q) = x. It is sufficient to show b a ˇF X (t)t dt = b a ˇF X (t) ˇF UX (t) dt. This is an easy computation since for all t [a, b] we get ˇF X (t) = x and ˇF UX (t) = (a + b)/2 so that b ˇF b a X (t)t dt = x a t dt = x (b 2 a 2 )/2 = x (b + a)(b a)/2 = b a x (b + a)/2 dt = b ˇF a X (t) ˇF UX (t) dt. 6 The copula of a random vector A general definition and some elementary properties of the copula are given here, including the case of non-continuous distributions. 11

Let X = (X 1,, X n ) R n be a random vector. Like in (1) the increasing distribution correspondence of a random vector is defined as F X (x) := [P (X 1 < x 1,, X n < x n ), P (X 1 x 1,, X n x n )], x R n. It is well known that the lower and upper selections F X and F X coincide on a dense subset of R n and are continuous there. The distribution P X of X is uniquely determined by F X. Denote with U X := (U 1,, U n ) [, 1] n, or U for short, the vector of the uniformisations U i := U Xi of the coordinates X i. If the X i are continuously distributed the U i are uniformly distributed on [, 1] (Proposition 5.1). This is often supposed in the literature on the copula. The copula C X of the random vector X is the increasing distribution correspondence of the random vector U X, C X := F UX. C X is just another way to prescribe the distribution P U of U = U X, which might be called the copula distribution of X. The usual definition of the copula for continuously distributed X i (see e.g. [1]) is the upper selection C X = F U of our one. Since our definition differs slightly from the usual one, we reprove some basic facts about the copula. The copula distribution P U can also be characterized as the image measure of P X under the measurable application P U = (P X ) Γ (16) Γ : R n [, 1] n, Γ(x) := ( F 1 (x 1 ),, F n (x n )). As always in this paper F i := F Xi denotes the midpoint distribution function of X i. Observe that U = Γ X. The image A := {Γ(x) x R n } of Γ is the cartesian product n i=1 A i of sets A i which consist of at most countable many intervals on the u i -axis. Hence A is Borel measurable. P U lives on A, i.e. P U (A) = 1. In the other direction we introduce the application Ξ : [, 1] n R n, Ξ(u) := ( ˇF1 (u 1 ),, ˇFn (u n )). Recall that ˇF i is the inverse correspondence of F i and ˇFi (u i ) is the midpoint of the interval ˇFi (u i ). 4 Again Ξ is measurable and its image B := {Ξ(u) 4 If X i is essentially bounded below, ˇF i() is an unbounded interval ], b]. Then we set ˇFi() := b 1 in order to remain real valued. Similarly proceed with ˇFi(1). 12

u [, 1] n } is a Borel measurable set. The functions F i and ˇFi are not surjective, but if we restrict each of both to the image of the other, we get bijections, ˇF i F i (x i ) = x for x i im ˇFi, F i ˇFi (u i ) = u i for u i imf i. Hence Ξ Γ(x) = x, x B R n, Γ Ξ(u) = u, u A [, 1] n and the applications Ξ A : A B and Γ B : B A are inverse to each other. Then the measures (P U ) Ξ and P X coincide on measurable subsets of B, especially P X (B) = (P U ) Ξ (B) = P U (Ξ 1 (B)) = P U (A) = 1, so they are identical on the Borel σ-algebra of R n, P X = (P U ) Ξ. (17) Hence in addition to (16), we can also recover P X from the copula distribution P U by means of the applications Ξ or Γ, which are determined through the marginal distributions of X alone. Less abstract this result reads as follows. The distribution of the random vector X, i.e. the common distribution of X 1,, X n, can be reconstructed from its copula by means of the distribution functions F i of its components. The copula is invariant under increasing transformations on the margins. This fact follows immediately from Proposition 5.1 (ii). Proposition 6.1 Suppose ϕ i : X i (Ω) R are injective and increasing functions and set ϕ(x) := (ϕ 1 (x 1 ),, ϕ n (x n )) R n for x X 1 (Ω) X n (Ω) R n. Then C ϕ(x) = C X. The copula of a continuously distributed random vector X is well known if its components are independent or comonotonic. In the first case P U is the uniform distribution on [, 1] n. If the components of X are P -comonotonic, P U is the uniform distribution on the main diagonal of [, 1] n, the corresponding copula being called Fréchet-Höffding upper copula. Here is another basic special case which is not of the continuous type. Example 6.1 Suppose, X 2 is constant and X 1 arbitrary with continuous distribution function. Then U 2 1/2 and U 1 is uniformly distributed. Hence P U is the uniform distribution on the horizontal line {(u 1,.5) u 1 [, 1]} [, 1] 2. Notice that X 1 and X 2 are independent and comonotonic simultaneously. 13

7 Dependence parameters and the copula The correlation parameters presented in Section 3 are now applied to the copula distribution of a random 2-vector. Since the uniformizations of the coordinates are bounded, no requirements on the random variables, except not being constant, are needed. Denoting again with U i the uniformization F Xi X i of the random variable X i, i = 1, 2, we define the copula 1-correlation of the random 2-vector (X 1, X 2 ) COR 1 (X 1, X 2 ) := ρ 1 (U 1, U 2 ). This definition is the L 1 analogue to Spearman s rho COR 2 (X 1, X 2 ) := ρ(u 1, U 2 ). Similarly we define the copula Gini correlation CORG(X 1, X 2 ) := giniρ(u 1, U 2 ). Since U 1, U 2 are bounded random variables, all these parameters exist with finite values provided that X 1, X 2 are not constant a.e., i.e. the denominator in the correlation coefficients on the right hand sides does not vanish. Using (12), (11) and (13) we get COR 1 (X 1, X 2 ) = 4 cov 1 (U 1, U 2 ) if ṀU i = 1, i = 1, 2, (18) 2 COR 2 (X 1, X 2 ) = 12 cov(u 1, U 2 ) if F X is continuous, (19) CORG(X 1, X 2 ) = 6 cogini(u 1, U 2 ) if F X is continuous. (2) From Kendall s tau one gets no new parameter (cf. footnote 3), Ktau(X 1, X 2 ) = Ktau(U 1, U 2 ) if F X is continuous. For a proof see [1] Theorem 5.1.1. COR 2 and Ktau are measures of concordance as defined by Scarsini [11] (cf. also [1] Theorem 5.1.9). Also COR 1 and CORG have the properties of a measure of concordance. Especially, by propositions 3.1 and 3.2, Proposition 7.1 Suppose X 1, X 2 are independent and not constant a.e., then (i) COR 1 (X 1, X 2 ) = if U 1 is symmetrically distributed, which is the case if X 1 is continuously distributed, (ii) CORG(X 1, X 2 ) =. A similarly important result, not being one of Scarsini s axioms, is 14

Proposition 7.2 Suppose X 1, X 2 are P -comonotonic, not constant a.e. and continuously distributed, then COR 1 (X 1, X 2 ) = 1, COR 2 (X 1, X 2 ) = 1, CORG(X 1, X 2 ) = 1 Of course, for Spearman s rho COR 2 this result is known already [6] Theorem 3. Proof Since X 1 and X 2 are P -comonotonic, the common distribution P U, U = (U 1, U 2 ), of U 1 and U 2 is the uniform distribution on the diagonal D [, 1]. Hence by the transformation rule for the second equation below and (7) for the third cov 1 (U 1, U 2 ) = (u 1 1/2) (u 2 1/2) dp (U 1,U 2 ) (u 1, u 2 ) = = D 1 1 = τ(u 1 ). (u 1 1/2) (u 1 1/2) du 1 u 1 1/2 du 1 Now ρ 1 (U 1, U 2 ) = 1 and the first equation is proved. Replacing the bipolar meet with the product in the argument above we get cov 1 (U 1, U 2 ) = var(u 1 ) and ρ(u 1, U 2 ) = 1, the second equation. For proving the third equation let Y be an iid copy of X. Then U and V := (U Y1, U Y2 ) are likewise iid, so that as above and with Fubini s Theorem and the notation u = (u 1, u 2 ), v = (v 1, v 2 ) we get cogini(u 1, U 2 ) = (u 1 v 1 ) (u 2 v 2 ) dp U P V (u, v) = (u 1 v 1 ) (u 2 v 2 ) dp U (u) dp V (v) = = D D 1 1 1 1 = gini(u 1 ). (u 1 v 1 ) (u 1 v 1 ) du 1 dv 1 u 1 v 1 du 1 dv 1 Now giniρ(u 1, U 2 ) = 1 and the proof is complete. Again the question arises, if the assumptions in propositions 7.1 and 7.2 can be weakened. Example 6.1 shows that cov i (X 1, X 2 ) = and cogini(x 1, X 2 ) = can happen for comonotone random variables. In this case COR 1 (X 1, X 2 ), COR 2 (X 1, X 2 ) and CORG(X 1, X 2 ) are not defined. We conjecture that at least the continuity assumption for COR 1, CORG in Proposition 7.2 can be weakened. 15

8 Applications to insurance and finance Premium functionals using average absolute deviation τ or, more general, distorted probabilities can be found in the literature, [2], [13] and [4] Section 8. [14] gives an axiomatization based on Schmeidler s work [12]. A survey on the use of the copula in insurance and finance is given in [7]. We regard a simple example to illustrate the usefulness of our new dependence parameters. In insurance a claim X often has high probability of no payments being due, say P (X = ) >.5. Then MX = and τ(x) = E(X). Regard a portfolio of two random variables X 1, X 2 of this type. The 1-covariance is cov 1 (X 1, X 2 ) = X 1 X 2 dp. High values of cov 1 (X 1, X 2 ) indicate that both claims in the portfolio assume large values simultaneously with high probability, unfavorable for the insurance company since this fact excludes diversification. If one relates large and small to τ(x 1 ) τ(x 2 ) = EX 1 EX 2 the same indication is done by 1-correlation ρ 1 (X 1, X 2 ). The classical covariance cov and correlation ρ are not useful here, since the product of the claim amounts of the insurance contracts has no immediate interpretation, but the minimum has. The transformation F Xi transforms a set [x, [ of high claims of X i to the set [ F Xi (x ), 1] of high quantiles. So, like above, values of COR 1 (X 1, X 2 ) close to 1 indicate that both claims in the portfolio have high quantiles simultaneously with high probability. [8] contains an example from life insurance, where X 1 X 2, the integrand in cov 1, has a direct interpretation. 9 Conclusions and outlook We have sketched an approach to dual moments, especially to dependence parameters, and to their application in connection with the copula. Many details and more complex applications still have to be investigated. Our dependence parameters cov 1 (X 1, X 2 ) and cogini(x 1, X 2 ) are hybrid ones as they combine order relations and algebraic operations. The generalization from 2-vectors to n-vectors seems to be promising and feasible since the bipolar meet is commutative and associative. We looked at first and second moments and their dual. What about null moments? The space L (Ω, A, P ) of measurable random variables is a metric space with the Ky Fan metric X Y and an ordinal analogue of an inner product can be defined by means of the Fan-Sugeno functionals in [5]. So -covariance cov and copula -correlation COR might be defined. But also the the popular value at risk given a security level α [, 1], VaR α (X) = ǦX(α) 16

might be perceived as dual null moment. It is the Choquet integral w.r.t. a distorted probability with {, 1}-valued, hence piecewise constant distortion function. 5 But this distortion function is not convex like the distortions in (4), (5). This fact is the source for the shortcomings of VaR α in the applications to finance. References [1] S. Anand: Inequality and Poverty in Malaysia: Measurement and Decomposition, New York: Oxford University Press, 1983. [2] D. Denneberg: Premium calculation: why standard deviation should be replaced by absolute deviation. ASTIN Bulletin, vol. 2, pp. 181-19, 199. [3] D. Denneberg: Non-additive Measure and Integral. Theory and Decision Library Series B, vol. 27, Kluwer Academic, Dordrecht, Boston, 1994 (2nd ed. 1997). [4] D. Denneberg: Non-additive measure and integral, basic concepts and their role for applications. In M. Grabisch, T. Murofushi, M. Sugeno (eds.): Fuzzy Measures and Integrals - Theory and Applications. Studies in Fuzzyness and Soft Computing, vol. 4, Physica-Verlag, Heidelberg, 2. [5] D. Denneberg, M. Grabisch: Measure and integral with purely ordinal scales. Journal of Mathematical Psychology, vol. 48, pp. 15-27, 24. [6] P. Embrechts, A. McNeil, D. Straumann: Correlation and Dependence n Risk Management: Properties and Pitfalls. In Risk Management: Value at Risk and Beyond, ed. by M. Dempster and H.K. Moffatt, Cambridge University Press, 21. [7] P. Embrechts, F. Lindskog, A.J. McNeil: Modelling dependence with copulas and applications to risk management. In S.T. Rachev (ed.): Handbook of heavy tailed distributions in finance. Elsevier/North- Holland, Amserdam, 23. [8] W. Hürlimann: Fair pricing using deflators and decrement copulas: the unit linked endowment approach. Blätter DGVFM 26, 421-437, 24. [9] R.I. Lerman and S. Yitzhaki: A Note on the Calculation and Interpretation of the Gini Index. Economics Letters, vol. 15, pp. 363-368, 1984. 5 W.r.t. this distorted probability, since it is {, 1}-valued, Choquet integral and Sugeno integral coincide for [, 1]-valued random variables. 17

[1] R. Nelsen: An Introduction to Copulas. LN Statistics 139, Springer,, New York, 1999. [11] M. Scarsini: On measures of concordance. Stochastica 8, 21-218, 1984. [12] D. Schmeidler: Subjective probability and expected utility without additivity. Econometrica. vol. 57, pp. 571-587, 1989. [13] S.S. Wang: Premium calculation by transforming the layer premium density. ASTIN Bulletin, vol. 26, pp. 71-92, 1996. [14] S.S. Wang, V.R. Young and H.H. Panjer: Axiomatic characterization of insurance prices. Insurance: Mathematics and Economics, vol. 21, pp. 173-183, 1997. [15] Kuan Xu: How Has the Literature on Gini s Index Evolved in the Past 8 Years? China Economic Quarterly 2, 757-778, 23. [16] M.E. Yaari: A controversial proposal concerning inequality measurement. Journal of Economic Theory, vol. 44, pp. 381-397, 1988. [17] D. Zagier: Inequalities for the Gini Coefficient of composite populations. Journal of Mathematical Economics 12, 13-118, 1983. 18