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
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
(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)/
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
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 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 12
Auto and Fire Claims in French Windstorms
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
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
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+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.
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 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
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
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
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
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
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
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 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
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
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