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Using Copulas

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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 - PowerPoint PPT Presentation

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Page 1: Using Copulas

Using Copulas

Page 2: Using Copulas

Guy Carpenter 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

Page 3: Using Copulas

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

Page 4: Using Copulas

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

Page 5: Using Copulas

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

Page 6: Using Copulas

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)]

Page 7: Using Copulas

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

– FV|U(v) = g1e-au/(g1+gugv)

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)

– 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 (a) = 2arcsin(a)/

Page 8: Using Copulas

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

Page 9: Using Copulas

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

Page 10: Using Copulas

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

Page 11: Using Copulas

Guy Carpenter 11

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)

Page 12: Using Copulas

Guy Carpenter 12

Auto and Fire Claims in French Windstorms

Page 13: Using Copulas

Guy Carpenter 13

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

Page 14: Using 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

La fonction R(z)

0,1

1

10

100

1000

0,5 0,6 0,7 0,8 0,9 1

Gumbel

Frank

Clayton

HRT

Empirique

Page 15: Using Copulas

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

Page 16: Using Copulas

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

Page 17: Using Copulas

Guy Carpenter 17

Other descriptive functions

tau is defined as –1+4010

1 C(u,v)c(u,v)dvdu.

cumulative tau: J(z) = –1+40z0

z C(u,v)c(u,v)dvdu/C(z,z)2.

expected value of V given U<z. M(z) = E(V|U<z) = 0z0

1 vc(u,v)dvdu/z.

Page 18: Using Copulas

Guy Carpenter 18

Cumulative tau for data and fit (US cat data)

Page 19: Using Copulas

Guy Carpenter 19

Conditional expected value data vs. fits

Page 20: Using Copulas

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 positiveCorrelation of Losses and ALAE

10

100

1,000

10,000

100,000

1,000,000

10,000,000

10 100 1,000 10,000 100,000 1,000,000 10,000,000

Variable1

Vari

ab

le 2

Losses

Page 21: Using Copulas

Guy Carpenter 21

Probability Correlation

Scatter Plot

-

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

- 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

Variable1

Var

iab

le 2

Page 22: Using Copulas

Guy Carpenter 22

Fit Some Copulas

Copula Gumbel HRT Normal Frank Clayton Flipped GumbelLog-Likelihood

function 181.52

159.18

179.51

165.02

128.71

96.01

Tail Concentration Functions

L for z<1/2, R for z>1/2

-

0.21

0.42

0.62

0.83

- 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 z

L/R

Gumbel

HRT

Empirical

Frank

Clayton

Flipped Gumbel

Page 23: Using Copulas

Guy Carpenter 23

Best Fitting

Tail Concentration FunctionsL for z<1/2, R for z>1/2

-

0.15

0.30

0.45

0.60

0.75

0.90

- 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00z

L/R

Gumbel

Empirical

Frank

Normal

t

Page 24: Using Copulas

Guy Carpenter 24

Fit Severity

Survival Funtions Loss Expense

0.0001

0.001

0.01

0.1

1

100 1,000 10,000 100,000 1,000,000 10,000,000

Empirical

Lognormal

Lognormal - Inverse Exponential

Inverse Exponential

Page 25: Using Copulas

Guy Carpenter 25

LossSurvival Funtions Loss

0.001

0.01

0.1

1

100 1,000 10,000 100,000 1,000,000 10,000,000 100,000,000

Empirical

Lognormal

Lognormal - Inverse Exponential

Inverse Weibull

Page 26: Using Copulas

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

.

Page 27: Using Copulas

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

Page 28: Using Copulas

Guy Carpenter 28

Adding Our Own Copula to the FitTail Concentration Functions

L for z<1/2, R for z>1/2

-

0.15

0.30

0.45

0.60

0.75

0.90

- 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00z

L/R

Gumbel

Empirical

Frank

Normal

t

P-fit

Page 29: Using Copulas

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

Page 30: Using Copulas

Guy Carpenter 30

finis