Using Copulas

Multi-variate Distributions Usually the distribution of a sum of random variables is needed When the distributions are correlated, getting the distribution of the sum requires calculation of the entire joint probability distribution F(x, y, z) = Probability (X < x and Y < y and Z < z) Copulas provide a convenient way to do this calculation From that you can get the distribution of the sum Guy Carpenter

Correlation Issues In many cases, correlation is stronger for large events Can model this by copula methods Quantifying correlation Degree of correlation Part of distribution correlated Can also do by conditional distributions Say X and Y are Pareto Could specify Y|X like F(y|x) = 1 – [y/(1+x/40)]-2 Conditional specification gives a copula and copula gives a conditional, so they are equivalent But the conditional distributions that relate to common copulas would be hard to dream up Guy Carpenter

Modeling via Copulas Correlate on probabilities Inverse map probabilities to correlate losses Can specify where correlation takes place in the probability range Conditional distribution easily expressed Simulation readily available Guy Carpenter

Formal Rules – Bivariate Case F(x,y) = C(FX(x),FY(y)) Joint distribution is copula evaluated at the marginal distributions Expresses joint distribution as inter-dependency applied to the individual distributions C(u,v) = F(FX-1(u),FY-1(v)) u and v are unit uniforms, F maps R2 to [0,1] Shows that any bivariate distribution can be expressed via a copula FY|X(y|x) = C1(FX(x),FY(y)) Derivative of the copula gives the conditional distribution E.g., C(u,v) = uv, C1(u,v) = v = Pr(V<v|U=u) So independence copula Guy Carpenter

Correlation Kendall tau and rank correlation depend only on copula, not marginals Not true for linear correlation rho Tau may be defined as: –1+4E[C(u,v)] Guy Carpenter

Example C(u,v) Functions Frank: -a-1ln[1 + gugv/g1], with gz = e-az – 1 t(a) = 1 – 4/a + 4/a2 0a t/(et-1) dt FV|U(v) = g1e-au/(g1+gugv) Gumbel: exp{- [(- ln u)a + (- ln v)a]1/a}, a  1 t(a) = 1 – 1/a HRT: u + v – 1+[(1 – u)-1/a + (1 – v)-1/a – 1]-a t(a) = 1/(2a + 1) FV|U(v) = 1 – [(1 – u)-1/a + (1 – v)-1/a – 1]-a-1 (1 – u)-1-1/a Normal: C(u,v) = B(p(u),p(v);a) i.e., bivariate normal applied to normal percentiles of u and v, correlation a t(a) = 2arcsin(a)/p Guy Carpenter

Copulas Differ in Tail Effects Light Tailed Copulas Joint Lognormal Guy Carpenter

Copulas Differ in Tail Effects Heavy Tailed Copulas Joint Lognormal Guy Carpenter

Quantifying Tail Concentration L(z) = Pr(U<z|V<z) R(z) = Pr(U>z|V>z) L(z) = C(z,z)/z R(z) = [1 – 2z +C(z,z)]/(1 – z) L(1) = 1 = R(0) Action is in R(z) near 1 and L(z) near 0 lim R(z), z->1 is R, and lim L(z), z->0 is L Guy Carpenter

LR Function (L below ½, R above) Guy Carpenter

Auto and Fire Claims in French Windstorms Guy Carpenter

MLE Estimates of Copulas Guy Carpenter

Modified Tail Concentration Functions Modified function is R(z)/(1 – z) Both MLE and R function show that HRT fits best Guy Carpenter

The t- copula adds tail correlation to the normal copula but maintains the same overall correlation, essentially by adding some negative correlation in the middle of the distribution Strong in the tails Some negative correlation Guy Carpenter

Easy to simulate t-copula Generate a multi-variate normal vector with the same correlation matrix – using Cholesky, etc. Divide vector by (y/n)0.5 where y is a number simulated from a chi-squared distribution with n degrees of freedom This gives a t-distributed vector The t-distribution Fn with n degrees of freedom can then be applied to each element to get the probability vector Those probabilities are simulations of the copula Apply, for example, inverse lognormal distributions to these probabilities to get a vector of lognormal samples correlated via this copula Common shock copula – dividing by the same chi-squared is a common shock Guy Carpenter

Other descriptive functions tau is defined as –1+40101 C(u,v)c(u,v)dvdu. cumulative tau: J(z) = –1+40z0z C(u,v)c(u,v)dvdu/C(z,z)2. expected value of V given U<z. M(z) = E(V|U<z) = 0z01 vc(u,v)dvdu/z. Guy Carpenter

Cumulative tau for data and fit (US cat data) Guy Carpenter

Conditional expected value data vs. fits Guy Carpenter

Loss and LAE Simulated Data from ISO Cases with LAE = 0 had smaller losses Cases with LAE > 0 but loss = 0 had smaller LAE So just looked at case where both positive Guy Carpenter

Probability Correlation Guy Carpenter

Log-Likelihood function Fit Some Copulas Copula Gumbel HRT Normal Frank Clayton Flipped Gumbel Log-Likelihood function 181.52 159.18 179.51 165.02 128.71 96.01 Guy Carpenter

Best Fitting Guy Carpenter

Fit Severity Guy Carpenter

Loss Guy Carpenter

Joint Distribution F(x,y) = C(FX(x),FY(y)) Gumbel: exp{-[(- ln u)a + (- ln v)a]1/a}, a  Inverse Weibull u = FX(x) = e-(b/x)p, v = FY(y) = e-(c/y)q, F(x,y) = exp{-[(b/x)ap + (c/y)aq]1/a} FY|X(y|x) = exp{-[(b/x)ap + (c/y)aq]1/a}[(b/x)ap + (c/y)aq]1/a-1(b/x)ap-pe(b/x)p . Guy Carpenter

Make Your Own Copula If you know FX(x) and FY|<X(y|X<x) then the joint distribution is their product. If you also know FY(y) then you can define the copula C(u,v) = F(FX-1(u),FY-1(v)) Say for inverse exponential FY(y) = e-c/y, you could define FY|<X(y|X<x) by e-c(1+d/x)r/y if you wanted to Then with FX(x) = e-(b/x)p, F(x,y) = e-(b/x)pe-c(1+d/x)r/y Inverse Weibull u = FX(x) = e-(b/x)p, v = FY(y) = e-c/y, FX-1(u) = b(-ln(u))-1/p FY-1(v) = -c/ln(v) so C(u,v) = uv[1+(d/b)(-ln(u))1/p])r Can fit that copula to data. Tried by fitting to the LR function Guy Carpenter

Adding Our Own Copula to the Fit Guy Carpenter

Conclusions Copulas allow correlation of different parts of distributions Tail functions help describe and fit For multi-variate distributions, t-copula is convenient Guy Carpenter

finis Guy Carpenter