Economic Capital and the Aggregation
of Risks Using Copulas
Dr. Emiliano A. Valdez and Andrew Tang
Overview
Motivation and aims
Technical background - copulas
Numerical simulation
Results of simulation
Key findings and conclusions
Capital
 Buffer
A rainy day fund, so when bad things happen, there is
money to cover it
Quoted from the IAA Solvency Working Party (2004) – “A Global Framework for Solvency Assessment”
 Solvency and financial strength indicator
 Economic capital - worst tolerable value of the risk
portfolio
Multi-Line Insurers
 Increasingly prominent
 Diverse range insurance products
 Aggregate loss, Z
Z  X 1  X 2  ...  X n
Where Xi represents the loss variable from line i.
 Xis are dependent
Multi-Line Insurers
 Dependencies between Xis ignored
 E.g., APRA
Prescribed Method
 Dependencies modelled using linear correlations
 Inadequate
 Non-linear
 Tail
dependence
dependence
Multi-Line Insurers
 Capital risk measures
 :Z  R
 Capital requirements
K   Z  R
 Value-at-Risk (VaR) – quantile risk measure
VaRq  X   inf x FX x  q 
 Tail conditional expectation (TCE)
TCEq  X   EX X  VaRq  X 
Multi-Line Insurers
 Diversification benefit
n
 Z      X i 
i 1
n
DB   Z      X i   0
i 1
 q = 97.5% and 99.5%
Aims
 Study the capital requirements (CRs) under different
copula aggregation models
 Study the diversification benefits (DBs) under
different copula aggregation models
 Compare the CRs from copula models to the
Prescribed Method (PM) used by APRA
Copulas
 Individual line losses - X1, X2, …, Xn
 Joint distribution is F(x1,x2,…,xn)
 Marginal distributions are F1(x1), F2(x2), …, Fn(xn)
 A copula, C, is a function that links, or couples the
marginals to the joint distribution
 Sklar
(1959)
F x1 , x2 ,..., xn   C F1 x1 , F2 x2 ,..., Fn xn 
Copulas
 Copulas of extreme dependence
 Independence
copula
C u1 ,..., un   u1...un
 Archimedean copulas
 Gumbel-Hougaard
 Frank
copula
copula
 Cook-Johnson
copula
Copulas
 Elliptical copulas / variants of the student-t copula
 Gaussian
“Normal” copula (infinite df)
 Student-t
copula (3 & 10 df)
 Cauchy
copula (1 df)
 u ,..., t u 
C u1 ,..., un   Tv t
1
v
1
1
v
n
Where Tv(.) and tv(.) denote the multivariate and
univariate Student-t distribution with v degrees of
freedom respectively.
Copulas
 Tail dependence (Student-t copulas)

  2t *n n  1 1   / 1  

where t* denotes the survivorship function of the
Student-t distribution with n degrees of freedom.
n\
0
0.5
0.9
1
1
0.29
0.5 0.78
1
3
0.12 0.31 0.67
1
10
0.01 0.08 0.46
1
infinity
0
0
0
1
Numerical Simulation
 1 year prospective gross loss ratios for each line of
business
LRi ,t 
ICi ,t
EPi ,t
 Industry data between 1992 and 2002
 Semi-annual
 SAS/IML (Interactive Matrix Language)
Numerical Simulation
 Five lines of business
 Motor:
domestic & commercial
 Household:
 Fire
& ISR
 Liability:
 CTP
buildings & contents
public, product, WC & PI
Numerical Simulation
 Correlation matrix input
Line of Business
Motor
Household
Fire & ISR
Liability
Motor
100%
Household
20%
100%
Fire & ISR
20%
50%
100%
Liability
10%
0%
20%
100%
CTP
20%
0%
0%
25%
CTP
100%
Numerical Simulation
 Marginal distribution input
Line of business
Marginal distribution
Motor
Gamma
Household
Gamma
Fire & ISR
Log-normal
Liability
Log-normal
CTP
Log-normal
Results of Simulation
 Normal copula
0.8
0.9
1.0
1.1
1.2
0.951.001.051.101.151.20
0.975
0.970
Motor
0.965
0.960
0.955
1.2
1.1
1.0
CTP
0.9
0.8
0.590
0.585
0.580
0.575
0.570
0.565
Household
1.20
1.15
1.10
Liability
1.05
1.00
0.95
0.9
0.8
Fire..ISR
0.7
0.6
0.5
0.4
0.9550.9600.9650.9700.975
0.565
0.570
0.575
0.580
0.585
0.590
0.4 0.5 0.6 0.7 0.8 0.9
Results of Simulation
 Student-t (3 df) copula
0.5 0.7 0.9 1.1 1.3 1.5
0.6 0.8 1.0 1.2 1.4 1.6
0.99
0.97
Motor
0.95
0.93
0.91
1.5
1.3
1.1
CTP
0.9
0.7
0.5
0.66
0.64
0.62
0.60
0.58
0.56
0.54
Household
1.6
1.4
1.2
Liability
1.0
0.8
0.6
1.45
1.20
Fire..ISR
0.95
0.70
0.45
0.20
0.91 0.93 0.95 0.97 0.99
0.540.560.580.600.620.640.66
0.200.450.700.951.201.45
Results of Simulation
 Student-t (10 df) copula
0.9
0.8 0.9 1.0 1.1 1.2
1.0
1.1
1.2
0.982
0.972
Motor
0.962
0.952
1.2
1.1
CTP
1.0
0.9
0.8
0.60
0.59
Household
0.58
0.57
0.56
1.2
1.1
Liability
1.0
0.9
Fire..ISR
0.952 0.962 0.972
0.982
0.56 0.57 0.58 0.59 0.60
0.4 0.5 0.6 0.7 0.8 0.9
0.9
0.8
0.7
0.6
0.5
0.4
Results of Simulation
 Cauchy copula
0.1
0.6
1.1
1.6
2.1
0.5 0.7 0.9 1.1 1.3 1.5
1.10
1.05
1.00
0.95
0.90
0.85
0.80
Motor
2.1
1.6
CTP
1.1
0.6
0.1
0.8
0.7
Household
0.6
0.5
0.4
1.5
1.3
1.1
0.9
0.7
0.5
Liability
1.5
1.0
Fire..ISR
0.5
0.0
0.800.850.900.951.001.051.10
0.4
0.5
0.6
0.7
0.8
0.0
0.5
1.0
1.5
Results of Simulation
 Independence copula
0.85
0.95
1.05
1.15
0.9
1.0
1.1
1.2
0.975
0.970
Motor
0.965
0.960
0.955
1.15
1.05
CTP
0.95
0.85
0.592
0.582
Household
0.572
0.562
1.2
1.1
Liability
1.0
0.9
0.9
0.8
Fire..ISR
0.7
0.6
0.5
0.4
0.9550.9600.9650.9700.975
0.562 0.572 0.582 0.592
0.4 0.5 0.6 0.7 0.8 0.9
0.92
0.92
0.08
0.06
0.04
0.02
0.00
0.10
Independence Copula
1.08
1.05
0.5
1.02
1.14
1.11
1.08
1.06
1.03
1.00
0.97
0.94
0.91
0.88
0.86
0.83
0.80
Normal Copula
0.99
0.96
0.93
0.90
0.87
0.84
0.81
0.78
0.00
0.75
0.72
0.08
0.91
Student 10 Copula
0.90
0.94
0.93
0.92
0.92
0.00
0.90
0.96
0.95
0.94
0.91
0.06
0.89
0.89
0.88
0.87
0.87
0.86
0.85
0.12
0.93
0.90
0.89
0.88
0.88
0.87
0.86
0.85
0.84
0.10
0.85
0.92
0.91
0.90
0.89
0.88
0.87
0.86
0.85
0.84
Results of Simulation
 Aggregated loss, Z, under each copula
0.3
Student 3 Copula
0.08
0.2
0.04
0.02
0.1
0.0
0.4
Cauchy Copula
0.3
0.04
0.2
0.1
0.0
Results of Simulation
 Capital requirements (CRs)
Note: risk measures 1 – 4 are VaR(97.5%), VaR(99.5%),TCE(97.5%) and TCE(99.5%) respectively.
Effect of Copula Assumption on CR
1.08
1.06
1.04
1.02
CR
Normal
1.00
t (3 df)
t (10 df)
Cauchy
0.98
Independence
0.96
0.94
0.92
0.90
0
1
2
3
Risk Measure
4
5
Results of Simulation
 Diversification benefits (DBs)
Note: risk measures 1 – 4 are VaR(97.5%), VaR(99.5%),TCE(97.5%) and TCE(99.5%) respectively.
Effect of Copula Assumption on DB
14%
12%
10%
Normal
8%
DB
t (3 df)
t (10 df)
Cauchy
6%
Independence
4%
2%
0%
0
1
2
3
Risk Measure
4
5
Results of Simulation
 Comparison with Prescribed Method (PM) – industry
portfolio
Normal
t (3 df)
t (10 df)
Cauchy
Independence
PM CR
1.010291
1.010233
1.008857
1.002536
0.999034
VaR 99.5% CR
0.931090
0.982005
0.943131
1.026140
0.921855
Excess Capital
0.079201
0.028228
0.065726
-0.023604
0.077179
7.84%
2.79%
6.51%
-2.35%
7.73%
% Savings
Results of Simulation
 Comparison with Prescribed Method (PM) – short tail
portfolio
Normal
t (3 df)
t (10 df)
Cauchy
Independence
PM CR
0.951609
0.952025
0.951191
0.948628
1.093202
VaR 99.5%
CR
0.876892
0.911036
0.885701
0.934066
0.880529
Excess
Capital
0.074717
0.040989
0.065490
0.014562
0.212673
7.85%
4.31%
6.89%
1.54%
19.45%
% Savings
Results of Simulation
 Comparison with Prescribed Method (PM) – long tail
portfolio
Normal
t (3 df)
t (10 df)
Cauchy
Independence
PM CR
1.098314
1.097543
1.095357
1.083399
0.857781
VaR 99.5%
CR
1.021380
1.135560
1.026240
1.221500
1.005440
Excess
Capital
0.076934
-0.038017
0.069117
-0.138101
-0.147659
7.00%
-3.46%
6.31%
-12.75%
-17.21%
% Savings
Key Findings
 Choice of copula matters dramatically for both CRs
and DBs
 More
tail dependent  higher CR
 More
tail dependent  higher DB
 Need
to select the correct copula for the insurer’s
specific dependence structure
 CR and DB shares a positive relationship
 PM is not a “one size fits all” solution
 Mis-estimations
of the true capital requirement
Limitations
 Simplifying assumptions
 Underwriting
risk only
 Ignores
impact of reinsurance
 Ignores
impact of investment
 Results do not quantify the amount of capital required
 Comparison
 Not
between copulas
comparable with results of other studies
Further Research
 Other copulas
 Isaacs
(2003) used the Gumbel
 Other types of risk dependencies
 E.g.,
between investment and operational risks
 Relax some assumptions
 Include
reinsurance
 Factor
in expenses
 Factor
in investments