TAILS OF COPULAS

The various copulas in the actuarial and statistical literature differ not so much in the degree of association they allow, but rather in which part of the distributions the association is strongest. In property and casualty applications there is interest in copulas that emphasize correlation among large losses, i.e., in the right tails of the distributions. Several copulas that have this characteristic are discussed. In addition, univariate functions of copulas are introduced that describe various aspects of the copulas, including tail concentration. Univariate descriptive functions can be thought of as an intermediate step between the several zero-dimensional measures of association (Kendall, Spearman, Gini, etc.) and the multi-dimensional copula function itself. Guy Carpenter Instrat Tails of Copulas Gary G. Venter Guy Carpenter Instrat Copulas are just joint distributions of unit uniform variates. They become especially useful when the unit uniform variates are viewed as probabilities from some other distributions. Then the percentiles of the other distributions can be recovered by inverting these probabilities. Copulas thus provide a ready method for simulating correlated variables, e.g., in a DFA model. Quite a few copulas are available for this, and they have differing characteristics that lead to different relationships among the variables generated. This paper reviews several popular copulas, introduces some others, and discusses methods for selecting those that may be most appropriate for a given application. In particular, the behavior of the copulas in the right and left tails is used as a way to distinguish among copulas that produce the same overall correlation. The organization of the paper is first to review copula methods in general, then examine several specific copulas, and finally look at measures that can be used to identify key characteristics of copulas. These are applied to some correlated loss data as an example. 1 . G e n e ra l C o n s i d e r a t i o n s Copulas Defined A joint distribution function F(x,y) can be expressed as a function of FX(x) and FY(y), the individual (or marginal) distribution functions for X and Y, i.e., as F(x,y) = C(FX(x),FY(y)). There has to be a function C defined on the unit square that makes this work, because FX(x) and FY(y) are order preserving maps of the real line or some segment of it to the unit interval. This can be defined by C(u,v) = F(FX-1(u),FY-1(v)). Then C(FX(x),FY(y)) = F(FX1(FX(x)),FY-1(FY(y))) = F(x,y). The function C(u,v) is called a copula. For many bivariate distributions, the copula form is the easiest way to express and generate the joint probabilities. Copulas work in the multi-variate context as well, but this paper mainly focuses on joint distributions of two variates. A copula is a joint distribution of two uniform random variates U and V with C(u,v) = Pr( U≤ u, V≤v). Also, c(u,v) will be used to denote the corresponding probability density, which is the mixed second partial derivative of C(u,v). The simplest copula is the uniform density for independent draws, i.e., c(u,v) = 1, C(u,v) = uv. Conditioning with Copulas The conditional distribution can be defined using copulas. Let C1(u,v) denote the first partial derivative of C(u,v). When the joint distribution of X and Y is given by F(x,y) = C(FX(x),FY(y)), then the conditional distribution of Y|X=x is given by: FY|X(y) = C1(FX(x),FY(y)) For example, in the independent case C(u,v) = uv, the conditional distribution of V given that U=u is C1(u,v) = v = Pr(V<v|U=u). This is of course independent of u. If C1 is simple enough to invert algebraically, then the simulation of joint probabilities can be done using the derived conditional distribution. That is, first simulate a value of U, then simulate a value of V from C1, the conditional distribution of V|U. Correlation The usual correlation coefficient based on the covariance of two variates is not preserved by copulas. That is, two pairs of correlated variates with the same copula can have different correlations. However, the Kendall correlation, usually denoted by τ, is a constant of the copula. That is, any correlated variates with the same copula will have the τ of that copula. There are different ways of defining τ, but the simplest may be τ = 4E[C(u,v)] – 1. For independent variates with C(u,v) = uv, E[C(u,v)] = 1⁄4, so τ = 0. Also, for perfectly correlated variates U = V, E[C(u,v)] = 1⁄2, so τ will be 1. Thus the scaling makes τ look like a correlation coefficient. The key measure though is E[C(u,v)], which is a basic constant of a copula and generalizes to the case of several variates. 0.01 0.1 1 10 Frank's Copula Density on Log Scale τ=.5 2 . S o m e P a r t i c u l a r C o p u l a s Some well-known copulas and a few designed particularly for loss severity distributions are reviewed here. Frank’s Copula Define gz = e-az – 1. Then Frank’s copula with parameter a ≠ 0 can be expressed as: C(u,v) = -a-1ln[1 + gugv/g1], with conditional distribution C1(u,v) = [gugv+gv]/[gugv+g1] c(u,v) = -ag1(1+gu+v)/(gugv+g1)2 and Kendall’s τ of τ(a) = 1 – 4/a + 4/a2 ∫0 a t/(et-1) dt For a<0 this will give negative values of τ. C1 can be inverted, so correlated pairs u,v can be simulated using the conditional distribution. First simulate u and p by random draws on [0,1]. Here p is considered a draw from the conditional distribution of V|u. Since this has distribution function C1, v can then be found as v = C1-1(p|u). The formula for this, which can be found from the formula for C1, is: v = -a-1ln{1+pg1/[1+gu(1–p)]} Once u and v have been simulated, the variables of interest X and Y can be simulated by inverting the marginal distributions, i.e., x = FX-1(u) and y = FY-1(v). The copula density is graphed here. As with many copulas there is a degree of concentration near 0,0 and 1,1. This often weakens considerably in the inversion back to X and Y, which is usually a non-linear transformation. Some of the copulas below have even greater concentration in the tails. 0.001 0 . 0 1 0.1 1 1 0 100 Gumbel Copula Log Scale τ=0.5 Gumbel Copula This copula has more probability concentrated in the tails than does Frank’s. It is also asymmetric, with more weight in the right tail. It is given by: C(u,v) = exp{[(ln u)a + (ln v)a]1/a}, a ≥ 1. C1(u,v) = C(u,v)[(ln u)a + (ln v)a]-1+1/a(-ln u)a-1/u c(u,v) = C(u,v)u-1v-1[(-ln u)a +(-ln v)a]-2+2/a[(ln u)(ln v)]a-1{1+(a-1)[(-ln u)a +(-ln v)a]-1/a} τ(a) = 1 – 1/a Unfortunately, C1 is not invertible, so another method is needed to simulate variates. Embrechts, etal.1 discuss the Gumbel copula and give a procedure to simulate uniform deviates from it. First simulate two independent uniform deviates u and v. Next solve numerically for s>0 with ues = 1 + as. Then the pair [exp(-sva), exp(-s(1v)a)] will have the Gumbel copula distribution. As shown in this graph, the Gumbel copula has a high degree of tail concentration, with more emphasis in the right tail. Heavy Right Tail Copula and Joint Burr For some applications actuaries need a copula with less correlation in the left tail, but high correlation in the right tail, i.e., for the large losses. Here is one: C(u,v) = u + v – 1 + [(1 – u)-1/a + (1 – v)-1/a – 1] a>0 C1(u,v) = 1 – [(1 – u)-1/a + (1 – v)-1/a – 1] (1 – u)-1-1/a c(u,v) = (1+1/a)[(1 – u)-1/a + (1 – v)-1/a – 1] [(1 – u)(1 – 1 Embrechts, McNeil and Strauman Correlation and Dependency in Risk Management in the XXX ASTIN Colloquium Papers 0.1 1 10 100 Heavy Right Tail Copula Log Scale τ = .5 v)]-1-1/a τ(a) = 1/(2a + 1) The conditional distribution given by the derivative C1(u,v) can be solved in closed form for v, so simulation can be done by conditional distributions as in Frank’s copula. Frees and Valdez2 show how this copula can arise in the production of joint Pareto distributions through a common mixture process. Generalizing this slightly, a joint Burr distribution is produced when the a parameter of both Burrs is the same as that of the heavy right tail copula. Given two Burr distributions, F(x) = 1 – (1 + (x/b)p)-a and G(y) = 1 – (1 + (y/d)q)-a, the joint Burr distribution from the heavy right tail copula is: F(x,y) = 1 – (1 + (x/b)p)-a – (1 + (y/d)q)-a + [1 + (x/b)p + (y/d)q]-a The conditional distribution of y|X=x is also Burr: FY|X(y|x) = 1 – [1 + (y/dx)q], where dx = d[1 + (x/b)p/q] By analogy to the joint normal, this can be called the joint Burr because the marginal and conditional distributions are all Burr. In practice, the degree of correlation can be set with the a parameter, leaving the p and q parameters to fit the tails, and b and d to set the scales of the two distributions. Kreps’ Partial Perfect Correlation Copula Generator 2 EW Frees and EA Valdez, “Understanding Relationships Using Copulas,” North American Actuarial Journal, vol.2, #1, pp. 1-25 Pareto(1,4) with h=(uv)^.3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5