Gary G Venter

Tails of Copulas

Guy Carpenter 2

Correlation Issues

Correlation is stronger for large events

Can model by copula methods

Quantifying correlation– Degree of correlation– Part of spectrum correlated

Guy Carpenter 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

Guy Carpenter 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]

Guy Carpenter 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

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

(a) = 1 – 4/a + 4/a2 0a t/(et-1) dt

Gumbel: exp{- [(- ln u)a + (- ln v)a]1/a}, a 1 (a) = 1 – 1/a

HRT: u + v – 1+[(1 – u)-1/a + (1 – v)-1/a – 1]-a (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 (a) = 2arcsin(a)/

Guy Carpenter 8

Copulas Differ in Tail EffectsLight Tailed Copulas Joint Lognormal

0.1 1.2 2.3 3.4 4.5 5.6 6.7 7.8 8.9 100.1

1.2

2.3

3.4

4.5

5.6

6.7

7.8

8.9

10Normal Joint Unit Lognormal Density Tau = .35

0.153-0.17

0.136-0.153

0.119-0.136

0.102-0.119

0.085-0.102

0.068-0.085

0.051-0.068

0.034-0.051

0.017-0.034

0-0.017

0.1 1.2 2.3 3.4 4.5 5.6 6.7 7.8 8.9 100.1

1.2

2.3

3.4

4.5

5.6

6.7

7.8

8.9

10Frank Joint Unit Lognormal Density Tau = .35

0.187-0.204

0.17-0.187

0.153-0.17

0.136-0.153

0.119-0.136

0.102-0.119

0.085-0.102

0.068-0.085

0.051-0.068

0.034-0.051

0.017-0.034

0-0.017

Guy Carpenter 9

Copulas Differ in Tail EffectsHeavy Tailed Copulas Joint Lognormal

0.1 1.2 2.3 3.4 4.5 5.6 6.7 7.8 8.9 10

0.1

1.2

2.3

3.4

4.5

5.6

6.7

7.8

8.9

10HRT Joint Unit Lognormal Density Tau = .35

0.187-0.204

0.17-0.187

0.153-0.17

0.136-0.153

0.119-0.136

0.102-0.119

0.085-0.102

0.068-0.085

0.051-0.068

0.034-0.051

0.017-0.034

0-0.017

0.1 1.2 2.3 3.4 4.5 5.6 6.7 7.8 8.9 100.1

1.2

2.3

3.4

4.5

5.6

6.7

7.8

8.9

10Gumbel Joint Unit Lognormal Density Tau = .35

0.187-0.204

0.17-0.187

0.153-0.17

0.136-0.153

0.119-0.136

0.102-0.119

0.085-0.102

0.068-0.085

0.051-0.068

0.034-0.051

0.017-0.034

0-0.017

Guy Carpenter 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

Guy Carpenter 11

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

Pareto(1,4) with h=(uv)̂ .3

00.5

11.5

22.5

33.5

44.5

5

0 1 2 3 4 5

Pareto(1,4) with h=(uv)̂ .3

0.00001

0.0001

0.001

0.01

0.1

1

10

0.00001 0.0001 0.001 0.01 0.1 1 10

Guy Carpenter 12

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

PP Max Data Pairs t = .5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

Guy Carpenter 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)

Guy Carpenter 14

Tau’s

h(u)=ua, (a)= (a+1)-4/3 +8/[(a+1)(a+2)2(a+3)]

h(u)=(u>k), (k) = (1 – k)4

h(u)=h0.5, (h) = (h2+2h)/3

h(u)= h0.5ua(u>k), (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)

Guy Carpenter 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

Guy Carpenter 16

LR Functions for Tau = .35

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Gum

HRT

Frank

Max

Power

Clay

Norm

LR Function(L below ½, R above)

Guy Carpenter 17

R as a Function of Tau

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

Tau

R

Gumbel

HRT

Power

Max

R usually above tauR usually above tau

Guy Carpenter 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

Loss Median Loss Tail Expense Median Expense Tail

Frees & Valdez 12,000 1.12 5500 2.12

Klugman & Parsa 12,275 1.05 5875 1.58

All moments less than tail parameter convergeAll moments less than tail parameter converge

Guy Carpenter 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

Guy Carpenter 20

Example: 2 States’ Hurricanes

MD & DE Joint Empirical Probabilities

DE vs. MD copula

-

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

0.900

1.000

- 0.200 0.400 0.600 0.800 1.000

Guy Carpenter 21

L and R Functions, Tau = .45

R looks about .25, which is >0, <tau, so none of our copulas match

DE and MD L(z) & R(z)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Guy Carpenter 22

Fits

LR Function for DE/MD and Fits

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Data

Frank

Normal

PP Power

HRT Gumbel Frank Normal Flipped Gumbel

Parameter 0.968 1.67 4.92 0.624 1.68

Ln Likelihood 124 157 183 176 161

Tau 0.34 0.40 0.45 0.43 0.40

Guy Carpenter 23

Auto and Fire Claims in French Windstorms

Guy Carpenter 24

MLE Estimates of Copulas

Gumbel Normale HRT Frank Clayton

Paramètre 1,323 0,378 1,445 2,318 3,378

Log Vraisemblance 77,223 55,428 84,070 50,330 16,447

de Kendall 0,244 0,247 0,257 0,245 0,129

Guy Carpenter 25

Modified Tail Concentration Functions Both MLE and R function show that HRT fits

best

Guy Carpenter 26

Conclusions

Copulas allow correlation of different parts of distributions

Tail functions help describe and fit

Guy Carpenter 27

finis