correlation & dependency structurescorrelation as the measure of dependence. • the mean does...
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Correlation & DependencyStructures
GIRO - October 1999
Andrzej Czernuszewicz
Dimitris Papachristou
Why are we interested incorrelation/dependency?
• Risk management
• Portfolio management
• Reinsurance purchase
• Pricing
How does dependency arise in theinsurance world?
Related Events
• insurance cycle
• economic factors
• physical events
• social trends
• reinsurance failure
Items to be covered:
1. Simulations of correlated variables
2. Problems with the traditional approach
3. Copulas
Statement of problem
Suppose we have n classes of business witheach class of business having its ownmarginal distribution.
How do we model the portfolio?
Traditionally this problem is tackled usingcorrelation as the measure of dependence.
• The mean does not tell everything abouta distribution.
• The coefficient of correlation does nottell everything about the dependencystructure.
Standard Simulation Technique
Step 1
Simulate an n-dimensional multivariate normal distributionwith correlation matrix ρ
Simulate Multivariate Normal Variables
Simulated Independent Normal
-5
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
N1
N2
Simulated Multivariate Normal
-5
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
N 1
Step 2generate n series of the appropriate marginal distributions andput into a matrix
Step 3
Apply Normal multivariate dependency structure from step 1
Standard Simulation Technique
Simulated Independent Gamma Variables
0
0.5
1
1.5
2
2.5
3
3.5
4
0 0.5 1 1.5 2 2.5 3 3.5 4
X
Y
Simulated Multivariate Normal
-5
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
N 1
0
0.5
1
1.5
2
2.5
3
3.5
4
0 0.5 1 1.5 2 2.5 3 3.5 4X
Y
Simulated Independent Normal
-5
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
N1
N2
Standard Simulation Technique
Observations
• The Spearman rank correlations are matchedrather then the Pearson correlations
• The solution is not unique
• In particular use of normal distributioninfluences the tail of the modelled portfolio
The same Correlation, butDifferent Dependency Structures
0
0.5
1
1.5
2
2.5
3
3.5
4
0 0.5 1 1.5 2 2.5 3 3.5 4
X
Y
number of points in uper quartile = 9
0
0.5
1
1.5
2
2.5
3
3.5
4
0 0.5 1 1.5 2 2.5 3 3.5 4
X
Y
number of points in uper quartile = 23
Observation
In the majority of applications a symmetricapproach is used in determining the dependency.
However in practice this need not be the case.
Example: An earthquake may cause a stockmarketcrash but a stockmarket crash will not cause anearthquake.
Skewed distributions may be used to get aroundthis problem.
Asymmetric Dependency Structure
0
0.5
1
1.5
2
2.5
3
3.5
4
0 0.5 1 1.5 2 2.5 3 3.5 4
X
Ynumber of points in uper quartile =37
The same Correlation, butDifferent Dependency Structures
0
0.5
1
1.5
2
2.5
3
3.5
4
0 0.5 1 1.5 2 2.5 3 3.5 4X
Y
Two FallaciesFallacy 1
Marginal distributions and correlationsdetermine the joint distribution
Fallacy 2 (not for rank correlation)
F1, F2 marginal distribution for X1, X2
∀ ∈ [-1,1] ∃ F such that F is the jointdistribution and X1, X2 have Pearsoncorrelation
ρ
ρ
Other problems with Pearson correlation
• A correlation of zero does not indicateindependence of risk.
Problem 2 (not for rank)
• Correlation is not invariant undertransformations of the risk.
Problem 3
• Correlation is not an appropriate dependencemeasure for very heavily-tailed distributions.
Problem 1
Why dependency structures?
We need to amend our concept ofdependency to allow for desirable features
• In particular we have to introduce non-lineardependency. For example:
• Need to reflect special features of taildependence.
• In general, single numeric measures ofdependency are insufficient.
0,],1,1[u~ 2 ==− ρXYX
The Copula approach
• Operates by separating marginal distributionsfrom dependency structures
• Combination of copula and marginal will yieldoriginal distribution exactly
• No longer have the problems associated withcorrelation
Definition of Copula For m-variate distribution F with j th univariant margin Fj
the copula associated with F is a distribution function
(Note: if F is a continuous m-variate distribution the copula associated with F is unique)
))(),..,(()(
satisfiesthat]1,0[]1,0[:
11 mm
m
XFXFCXF
C
=→
This can also be represented via a density function
1,0,),(
),( <<∂∂
∂= vuvu
vuCvuc
Construction of a Copula
• Construction of a copula– parametric
– non-parametric
• parametric form needed for higher dimensionproblem
Example of Parametric Copula (1)
Independent Copula
1,0,1),(
1,0,),(
≤≤=≤≤=
vuvuc
vuuvvuC
Example of Parametric Copula (2)
Normal copula
ncorrelatiotheisand
)1,0(is)(),(
where
]}[2
1exp{]}.2[)1(
2
1exp{)1(),,(
11
2222122
12
ρ
ρρρρ
Nvyux
yxxyyxvuc
ΦΦ=Φ=
+−+−−−=
−−
−−
Example of Parametric Copula (3)
Gumbel Copula
For ∞<≤δ1
{ } 1,0,))log()log((exp);,(1
≤≤−+−−= vuvuvuC δδδδ
Note: this is an extreme value copula
Independent (Product) Copula Density
0.0
1
0.1
0.1
9
0.2
8
0.3
7
0.4
6
0.5
5
0.6
4
0.7
3
0.8
2
0.9
1 1
0.01
0.2
0.39
0.58
0.77
0.96
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Density
X
Y
Independent (Product) Copula
0.9-1
0.8-0.9
0.7-0.8
0.6-0.7
0.5-0.6
0.4-0.5
0.3-0.4
0.2-0.3
0.1-0.2
0-0.1
0.0
05
0.0
75
0.1
45
0.2
15
0.2
85
0.3
55
0.4
25
0.4
95
0.5
65
0.6
35
0.7
05
0.7
75
0.8
45
0.9
15
0.9
85
0.005
0.185
0.365
0.545
0.725
0.905
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
X
Y
Bivariate Normal Copula (Density)
0.18-0.2
0.16-0.18
0.14-0.16
0.12-0.14
0.1-0.12
0.08-0.1
0.06-0.08
0.04-0.06
0.02-0.04
0-0.02
Bivariate Normal Copula (Density)
0.0
1
0.0
8
0.1
5
0.2
2
0.2
9
0.3
6
0.4
3
0.5
0.5
7
0.6
4
0.7
1
0.7
8
0.8
5
0.9
2
0.9
9
0.01
0.19
0.37
0.55
0.73
0.91
-1
1
3
5
7
9
11
13
15
17
19
21
23
25
27
29
31
33
35
Copula Density Function
X
Y
Gumbel Copula (Density)
33-35
31-33
29-31
27-29
25-27
23-25
21-23
19-21
17-19
15-17
13-15
11-13
9-11
7-9
5-7
3-5
1-3
-1-1
Gumbel Copula (Density)
Simulation of a copula
• simulate a value u1 from U (0,1)
• simulate a value u2 from C2 (u2 u1)
• simulate a value un from Cn (un u1…. un-1)
where Ci = C(u1,….,ui,1,….,1) for i=2,….,n
• Stop Loss on two classes of business
• Assumptions
• Limit: 15%
• Deductible: 107.5%
• Expected L/R: 91.5%
Example: Reinsurance Pricing
Class A Class Bmean L/R 91.14% 91.85%st. dev. of L/R 10.98% 9.21%Premium 100 100
gumbel delta 1.2rank correlation 0.25
Distribution of Loss Ratios for Classes A and B
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
50% 70% 90% 110% 130% 150% 170%
Loss Ratio
Cu
mu
lati
ve D
istr
ibu
tio
n
Class A
Class B
Loss Ratios are assumed to be lognormally distributed
The same rank correlation ( =0.25), but differentdependency structures especially at the tail
ρ
Distribution of Losses to the Stop Loss Layer
95%
96%
97%
98%
99%
100%
0 5 10 15 20 25
Amount of Loss
Cu
mu
lati
ve D
istr
ibu
tio
n
gumbel dependence
independent
MN dependence
Distribution of Losses to the Stop Loss Layer
Results
Expected amount of loss to the layer isunderestimated approximately by 50%, if amultivariate Normal dependency structure isused.Losses to the Stop Loss
Gumbel Independent Mult.Normalmean 0.325 0.116 0.220standard deviation 2.297 1.169 1.735Rate on Line 1.1% 0.4% 0.7%
Conclusions
• Correlation is not a sufficient measure ofdependence.
• Opportunities may be missed by remaining inthe correlation framework.
• True dependency reflected in the copulaapproach by separating marginal distributionfrom dependency structures.
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