1 / 27

Tails of Copulas

Tails of Copulas. Gary G Venter. Correlation Issues. Correlation is stronger for large events Can model by copula methods Quantifying correlation Degree of correlation Part of spectrum correlated. Modeling via Copulas. Correlate on probabilities

lilian
Download Presentation

Tails of Copulas

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Tails of Copulas Gary G Venter

  2. Correlation Issues • Correlation is stronger for large events • Can model by copula methods • Quantifying correlation • Degree of correlation • Part of spectrum correlated

  3. 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

  4. What is a copula? • A way of specifying joint distributions • A way to specify what parts of the marginal distributions are correlated • Works by correlating the probabilities, then applying inverse distributions to get the correlated marginal distributions • Formally they are joint distributions of unit uniform variates, as probabilities are uniform on [0,1]

  5. Formal Rules • 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] • FY|X(y) = C1(FX(x),FY(y)) • Derivative of the copula is the conditional distribution • E.g., C(u,v) = uv, C1(u,v) = v = Pr(V<v|U=u) • So independence copula

  6. 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)]

  7. Example C(u,v) Functions • Frank: -a-1ln[1 + gugv/g1], with gz = e-az – 1 • t(a) = 1 – 4/a + 4/a20at/(et-1) dt • 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) • 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

  8. Copulas Differ in Tail EffectsLight Tailed Copulas Joint Lognormal

  9. Copulas Differ in Tail EffectsHeavy Tailed Copulas Joint Lognormal

  10. Partial Perfect Correlation Copulas of Kreps • Each simulated probability pair is either identical or independent depending on symmetric function h(u,v), often =h(u)h(v) • h(u,v) –> [0,1], e.g., h(u,v) = (uv)3/5 • Draw u,v,w from [0,1] • If h(u,v)>w, drop v and set v=u • Simulate from u and v, which might be u

  11. Simulated Pareto (1,4) h(u)=u0.3(Partial Power Copula)

  12. Partial Cutoff Copula h(u)=(u>k)

  13. Partial Perfect Copula Formulas • For case h(u,v)=h(u)h(v) • H’(u)=h(u) • C(u,v) = uv – H(u)H(v) + H(1)H(min(u,v)) • C1(u,v) = v – h(u)H(v) + H(1)h(u)(v>u)

  14. Tau’s • h(u)=ua, t(a)= (a+1)-4/3 +8/[(a+1)(a+2)2(a+3)] • h(u)=(u>k), t(k) = (1 – k)4 • h(u)=h0.5, t(h) = (h2+2h)/3 • h(u)= h0.5ua(u>k), t(h,a,k) = h2(1-ka+1)4(a+1)-4/3 +8h[(a+2)2(1-ka+3)(1-ka+1)–(a+1)(a+3)(1-ka+2)2]/d where d = (a+1)(a+2)2(a+3)

  15. 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

  16. LR Function(L below ½, R above)

  17. R usually above tau

  18. Example: ISO Loss and LAE • Freez and Valdez find Gumbel fits best, but only assume Paretos • Klugman and Parsa assume Frank, but find better fitting distributions than Pareto All moments less than tail parameter converge

  19. Can Try Joint Burr, from HRT • F(x,y) = 1–(1+(x/b)p)-a –(1+(y/d)q)-a +[1+(x/b)p +(y/d)q]-a • E.g. F(x,y)=1–[1+x/14150]-1.11–[1+(y/6450)1.5]-1.11 +[1+x/14150 +(y/6450)1.5]-1.11 • Given loss x, conditional distribution is Burr: • FY|X(y|x) = 1–[1+(y/dx)1.5]–2.11 • with dx = 6450 +11x 2/3

  20. Example: 2 States’ Hurricanes

  21. L and R Functions, Tau = .45 • R looks about .25, which is >0, <tau, so none of our copulas match

  22. Fits

  23. Auto and Fire Claims in French Windstorms

  24. MLE Estimates of Copulas

  25. Modified Tail Concentration Functions • Both MLE and R function show that HRT fits best

  26. Conclusions • Copulas allow correlation of different parts of distributions • Tail functions help describe and fit

  27. finis

More Related