1. Using Copulas

2. 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

3. 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

4. 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

5. 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

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)]
Guy Carpenter

7. 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

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

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

10. 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

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

12. Auto and Fire Claims in French Windstorms
Guy Carpenter

13. MLE Estimates of Copulas
Guy Carpenter

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

15. 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

16. 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

17. 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

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

19. Conditional expected value data vs. fits
Guy Carpenter

20. 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

21. Probability Correlation
Guy Carpenter

22. 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

23. Best Fitting
Guy Carpenter

24. Fit Severity
Guy Carpenter

25. Loss
Guy Carpenter

26. 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

27. 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

28. Adding Our Own Copula to the Fit
Guy Carpenter

29. 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

30. finis
Guy Carpenter