Negative (lu and Ul) Tail Dependence Using Copulae

This paper studies the four types: lower, upper, lower-upper and upper-lower focusing on developping theory to measure the negative tail dependence Introduction As a consequence of globalization and relaxed regulations on financial and insurance markets the dependencies between financial time series have increased during recent times. Particularly in extreme events such as economic crisis, these dependencies affect the profit of companies as well as the financial stability of the financial sector. Risk measures such as VaR (Value at Risk) also depend on the dependence structure of extreme values. In order to measure these dependencies, tail dependence measures are used, these measures rely on copulae, tail copulae and the tail dependence coefficient (TDC) in order to explain the dependence structure of extreme values (see Embrechts et al. (1997)). In the study of financial time series it has become increasingly important to distinguish between different types of dependence. The structure dependence of time series have been studied for a long time, traditionally through the use of correlation. Due to drawbacks of this measure new methodologies have been developped, in particular the use of copulae has proved to be the way forward (see McNeil et al. (2005), Chapter 5), however this analysis mainly focuses on positive correlation by the use of usual copulae and survival copulae. The use of copulae has been particularly succesful to measure dependence on the extreme values in what is known as tail dependence through the use of tail copulae and the tail dependence coefficients (TDCs), but yet again the analysis has been mostly directed to the left 1 2 YURI SALAZAR FLORES (lower tail dependence) and the right tail (upper tail dependence). Many times, when analysing financial time series it happens that two series present the same type of extreme behaviour, either upper or lower tail dependence. However it can also happen that a value in the left tail in one series may occur at the same time when a value in the right tail of the other series appears. This can be observed in prices of stocks that affect a certain portfolio as well as in stock indices returns and other financial time series. We will refer to this as lower-upper (LU) tail dependence or upper-lower (UL) tail dependence. Although UL tail dependence may seem to be covered by LU tail dependence, in most of the cases, it is worth to be studied separately. The use of estimators for the UL and LU TDCs has already observed in finance (see Zhang (2007)), In this paper we define new copulae associated to the sample that enable us to capture the whole dependence structure of the series. This work is divided in two sections:In the first section the theory of tail dependence is studied and theory on LU and UL tail dependence is developed from probability functions, copulae, tail copulae and the TDC, mathematical proofs are provided on main results. In order to study LU and UL tail dependence it is necessary to work with certain probability functions, which we call LU and UL probability functions.Using this probability functions we introduce new types of copulae, LU and UL copulae. The first results are related to the equalities connecting these copulae and then to connect it to usual and survival copulae. Given that the usual copula is the copula of distribution functions, to differentiate it from survival and other copulae, we will refer to it as distributional copula. The boundaries of copulae are used to restrict LU and UL probability functions and results on exchangeable copulae are presented. In the second section upper, lower, LU and UL tail dependence are modelled, we revise some of the most important copulae models discussed in literature. The first examples of copulae we study are the fundamental copulae which encompasses NEGATIVE (LU AND UL) TAIL DEPENDENCE USING COPULAE 1 three cases: the independent case with the independent copula, the perfect positive dependence case with the comonotonicity copula and the perfect negative dependence case with the countercomonotonicity copula. We then study two examples of implicit copulae, for which there is no closed form, the Gaussian and the Student’s t copula. After that we study the archimidean copulae, such as the Gumbel, Clayton, Frank and the Generalized Clayton copula. Finally we also study a non-archimidean copula which is the Marshal-Olkin copula. For all examples we present their corresponding tail copulae and TDCs to see if they account for tail dependence. 1. Copulae, Tail Copulae and the Tail Dependence Coefficients 1.1. Copula. In order to define and study the dependence structure between two random variables we use the concept of copula. The following study is based on copulae which describe the dependence structure of multidimensional random variables. Here we restrict to the continuous two dimensional case. We first study the theory of copula and introduce the LU and UL copulae along with results regarding these copulae and then we focus on their relationship with distributional and survival copulae. After this we study Tail Copulae and the TDC for the four types of copula: distributional, survival, LU and UL. The concept of copula was first introduced by Sklar (1959), and is now a cornerstone topic in finance (see Nelsen (1999) or McNeil et al. (2005), Chapter 5), a two dimensional copula is defined in the following way: Definition 1. A two dimensional copula C(u, v) is a distribution function on [0, 1] with standard uniform marginal distributions. In the two dimensional case copulae functions C : [0, 1] → [0, 1] are used to link bivariate distribution functions with their corresponding marginal distributions. On the other hand, survival copulae: C : [0, 1] → [0, 1] link bivariate survival functions with their corresponding marginal survival functions. 2 YURI SALAZAR FLORES Let (X;Y )0 be a random vector with joint distribution function F (x, y) = P (X ≤ x, Y ≤ y), marginals G(x) = P (X ≤ x), H(y) = P (Y ≤ y), survival function F (x, y) = P (X > x, Y > y) and marginal survival functions , G(x) = P (X > x) and H(y) = P (Y > y). Two versions of Sklar’s theorem guarantee the existence and uniqueness of copulae C and C (see Schweizer and Sklar (1983)) F (x, y) = C(G(x),H(y)), which is equivalent to C(u, v) = F (G−1(u),H−1(v)) (1.1) and F (x, y) = C(G(x),H(y)), which is equivalent to C(u, v) = F (G −1 (u),H −1 (v)). (1.2) Given that G −1 (u) = G−1(1 − u), equation (1.2) is also equivalent to C(u, v) = F (G−1(1− u),H−1(1− v)). 1.1.1. Sklar’s Theorem for LU and UL Probability Functions. In this case we refer to C as distributional copula and to C as survival copula. In the same way that a distributional copula and a survival copula explain the dependence structure of two random variables between their left and right tails respectively, we now introduce a new type of copula to explain the dependence structure of two random variables between the left tail of the first one and the right tail of the other one, and vice versa. Definition 2. Let (X,Y )0 be a random vector, its lower-upper (LU) and upperlower (UL) probability funtions are FLU(x, y) = P (X ≤ x, Y > y) and FUL(x, y) = P (X > x, Y ≤ y). If G, H, G and H are the disbtributions and survival functions of X and Y we refer to G and H as the marginals of FLU and to G and H as the marginals of FUL NEGATIVE (LU AND UL) TAIL DEPENDENCE USING COPULAE 3 The copulas we consider are CLU and CUL that link the functions in definition (2) with their corresponding marginals. We refer to them as copulae of the LU and UL probability functions or simply LU and UL copulae. The following version of Sklar’s theorem guarantees the existence and uniqueness of CLU and CUL in the continuous case. A more general version can be stated following the same reasoning of the proof of Skar’s theorem in the non-continuous case, but here we restrict to this case (see Schweizer and Sklar (1983) or Nelsen (1999), p. 18). Theorem 1. Sklar’s theorem for lower-upper and upper-lower probability funtions. Let (X,Y )0 be a random vector, FLU and FUL its LU and UL probability functions as in definition (2) and the distribution functions of X and Y , G and H be continuous, then there exist unique copulae CLU and CUL : [0, 1] → [0, 1], such that, for all x and y in [−∞,∞], FLU(x, y) = CLU(G(x), H(y)), (1.3) FUL(x, y) = CUL(G(x), H(y)). (1.4) Conversely, if we have any copulae CLU and CUL satisfying definition (1) and G, H, G and H univariate disbtributions and its survival functions then, considering the previous equations, FLU defines a LU probability function with marginals G and H and FUL defines an UL probability function with marginals G and H. Proof. The proof of this theorem is analogous to the proof of Sklar’s theorem in the continuous case, (see McNeil et al (2005), p.186). From Defintion (2) and considering that P (X ≤ x) = P (F (X) ≤ F (x)) for any distribution function F , we have that for any x and y in [−∞,∞] FLU(x, y) = P (G(X) ≤ G(x),H(Y ) ≤ H(y)). Using the continuity of G and H (see McNeil et al. (2005), proposition (5.2 (2)), p.185), G(X) and H(Y ) are uniformly distributed, which implies 1−H(Y ) is uniformly distributed, so we have that both G(X) and H(Y ) have standard uniform 4 YURI SALAZAR FLORES distributions. Definition (1) implies that the distribution function of (G(x), H(y)) is a copula. We denote this copula by CLU , yielding equation (1.3). Evaluating this equation in the generalized inverses G←(u) and H←(1−v) for u, v ∈ [0, 1] and using the fact that one of the properties of the generalized inverse is that when T is continuous T ◦ T←(x) = x, we get: CLU(u, v) = FLU(G ←(u),H←(1− v)) , which explicitly represents CLU in terms of FLU and its marginals implying its uniqueness. For the converse statement of the theorem let CLU be any copula satisfying definition (1), W = (U, V ) a random vector with distribution function CLU and G and H univariate distribution and survival functions. We now define Z = (X,Y ) := (G←(U), H←(1 − V )). Considering that another property of the generalized inverse is that if T is right continuous, like distribution functions, T (x) ≥ y ⇐⇒ T←(y) ≤ x, the LU probabilty function of Z is P (X ≤ x, Y > y) = CLU(G(x),H(y)). This implies that FLU defined in (1.3) is the LU probability function of Z with marginals P (X ≤ x) = P (G←(U) ≤ x) = P (U ≤ G(x)) = G(x) and P (Y > y) = P (H←(1− V ) > y) = P (V ≤ H(y)) = H(y). Note that for this theorem we refered to generalized inverses rather than inverse functions, as the first are more general. However throughout this work, whenever we are not proving a general property, we assume the distribution functions have inverse functions. For the properties of the generalized inverse function used in this proof (see McNeil et al (2005), proposition (A.3)) ¤ Note that this theorem implies that in the continuous case CLU and CUL are the LU and UL probability functions of (G(X), H(Y )) and (G(x), H(y)) characterized as: NEGATIVE (LU AND UL) TAIL DEPENDENCE USING COPULAE 5 FLU(x, y) = CLU(G(x),H(y)), CLU(u, v) = FLU(G(u),H(1− v)) (1.5)