Download Tails of Copulas

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
2
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
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
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
5
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
6
Example C(u,v) Functions

Frank: -a-1ln[1 + gugv/g1], with gz = e-az – 1
a
 t(a) = 1 – 4/a + 4/a2 0 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
7
Copulas Differ in Tail Effects
Light Tailed Copulas Joint Lognormal
Frank Joint Unit Lognormal Density Tau = .35
Normal Joint Unit Lognormal Density Tau = .35
10
8.9
7.8
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
0.153-0.17
6.7
0.136-0.153
0.119-0.136
0.102-0.119
5.6
0.085-0.102
0.068-0.085
0.051-0.068
4.5
0.034-0.051
0.017-0.034
0-0.017
3.4
2.3
1.2
Guy Carpenter
0.1
1.2
2.3
3.4
8
0.1
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
Copulas Differ in Tail Effects
Heavy Tailed Copulas Joint Lognormal
HRT Joint Unit Lognormal Density Tau = .35
Gumbel Joint Unit Lognormal Density Tau = .35
10
8.9
7.8
0.187-0.204
0.17-0.187
6.7
0.153-0.17
0.136-0.153
0.119-0.136
5.6
0.102-0.119
0.085-0.102
0.068-0.085
4.5
0.051-0.068
0.034-0.051
0.017-0.034
3.4
0-0.017
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
2.3
1.2
9
Guy Carpenter
0.1
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
10
LR Functions for Tau = .35
0.8
LR Function
(L below ½, R above)
0.7
0.6
0.5
Gum
HRT
Frank
0.4
Max
Power
Clay
Norm
0.3
0.2
0.1
11
Guy Carpenter
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Auto and Fire Claims in French Windstorms
Guy Carpenter
12
MLE Estimates of Copulas
Gumbel Normale HRT
Paramètre
Log Vraisemblance
t de Kendall
Guy Carpenter
Frank
Clayton
1,323
0,378
1,445
2,318
3,378
77,223
55,428
84,070
50,330
16,447
0,244
0,247
0,257
0,245
0,129
13
Modified Tail Concentration Functions

Modified function is R(z)/(1 – z)

Both MLE and R function show that HRT fits best
1000
La fonction R(z)
100
Gumbel
Frank
Clayton
HRT
10
Empirique
1
0,5
0,6
0,7
0,8
0,9
1
0,1
Guy Carpenter
14
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
15
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 chisquared 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
16
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
17
Cumulative tau for data and fit (US cat data)
Guy Carpenter
18
Conditional expected value data vs. fits
Guy Carpenter
19
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
Correlation of Losses and ALAE
10,000,000
1,000,000
100,000
10,000
1,000
100
10
10Guy
Carpenter 100
1,000
10,000
Variable1
100,000
1,000,000
10,000,000
20
Probability Correlation
Scatter Plot
1.00
0.90
0.80
0.70
Variable 2
0.60
0.50
0.40
0.30
0.20
0.10
-
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
Variable1
Guy Carpenter
21
Fit Some Copulas
Copula
Log-Likelihood
function
Gumbel
HRT
181.52
Normal
159.18
Frank
179.51
Clayton
165.02
Flipped Gumbel
128.71
96.01
Tail Concentration Functions
L for z<1/2, R for z>1/2
Gumbel
HRT
Empirical
Frank
Clayton
Flipped Gumbel
0.83
L/R
0.62
0.42
0.21
-
Guy Carpenter
-
0.10
0.20
0.30
0.40
0.50
z
0.60
0.70
0.80
0.90
1.00
22
Best Fitting
Tail Concentration Functions
L for z<1/2, R for z>1/2
0.90
0.75
0.60
Gumbel
L/R
Empirical
Frank
0.45
Normal
t
0.30
0.15
-
0.10
Guy Carpenter
0.20
0.30
0.40
0.50
z
0.60
0.70
0.80
0.90
1.00
23
Fit Severity
1
Survival Funtions Loss Expense
0.1
0.01
Empirical
Lognormal
Lognormal - Inverse Exponential
Inverse Exponential
0.001
0.0001
100
Guy Carpenter
1,000
10,000
100,000
1,000,000
10,000,000
24
Loss
1
Survival Funtions Loss
0.1
Empirical
Lognormal
Lognormal - Inverse Exponential
0.01
0.001
100
Guy Carpenter
Inverse Weibull
1,000
10,000
100,000
1,000,000
10,000,000
100,000,000
25
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) , v = FY(y) = e-(c/y) ,

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
q
p
26
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)
r
by e-c(1+d/x) /y if you wanted to


p
p
r
Then with FX(x) = e-(b/x) , F(x,y) = e-(b/x) e-c(1+d/x) /y
p
Inverse Weibull u = FX(x) = e-(b/x) , v = FY(y) = e-c/y, FX-1(u) = b(-ln(u))-1/p
FY-1(v) = -c/ln(v) so
1/p])r

C(u,v) = uv[1+(d/b)(-ln(u))

Can fit that copula to data. Tried by fitting to the LR function
Guy Carpenter
27
Adding Our Own Copula to the Fit
Tail Concentration Functions
L for z<1/2, R for z>1/2
0.90
0.75
0.60
Gumbel
L/R
Empirical
Frank
0.45
Normal
t
P-fit
0.30
0.15
-
0.10
Guy Carpenter
0.20
0.30
0.40
0.50
z
0.60
0.70
0.80
0.90
1.00
28
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
29
finis
Guy Carpenter
30