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