Economic Capital and the Aggregation of Risks Using Copulas
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Transcript of Economic Capital and the Aggregation of Risks Using Copulas
Economic Capital and the Aggregation of Risks Using Copulas
Dr. Emiliano A. Valdez and Andrew Tang
Motivation and aims
Technical background - copulas
Numerical simulation
Results of simulation
Key findings and conclusions
Overview
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
Where Xi represents the loss variable from line i.
Xis are dependent
nXXXZ ...21
Multi-Line Insurers
Dependencies between Xis ignored
E.g., APRA Prescribed Method
Dependencies modelled using linear correlations
Inadequate
Non-linear dependence
Tail dependence
Multi-Line Insurers
Capital risk measures
Capital requirements
Value-at-Risk (VaR) – quantile risk measure
Tail conditional expectation (TCE)
RZ :
qxFxXVaR Xq inf
XVaRXXEXTCE qq
RZK
Multi-Line Insurers
Diversification benefit
q = 97.5% and 99.5%
n
iiXZ
1
01
n
iiXZDB
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)
nnn xFxFxFCxxxF ,...,,,...,, 221121
Copulas
Copulas of extreme dependence
Independence copula
Archimedean copulas
Gumbel-Hougaard copula
Frank copula
Cook-Johnson copula
nn uuuuC ...,..., 11
Copulas
Elliptical copulas / variants of the student-t copula
Gaussian “Normal” copula (infinite df)
Student-t copula (3 & 10 df)
Cauchy copula (1 df)
Where Tv(.) and tv(.) denote the multivariate and univariate Student-t distribution with v degrees of freedom respectively.
nvvvn ututTuuC 11
11 ,...,,...,
Copulas
Tail dependence (Student-t copulas)
where t* denotes the survivorship function of the Student-t distribution with n degrees of freedom.
1/112 * nt n
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
Industry data between 1992 and 2002
Semi-annual
SAS/IML (Interactive Matrix Language)
ti
titi EP
ICLR
,
,,
Numerical Simulation
Five lines of business
Motor: domestic & commercial
Household: buildings & contents
Fire & ISR
Liability: public, product, WC & PI
CTP
Numerical Simulation
Correlation matrix input
Line of Business
Motor Household Fire & ISR Liability CTP
Motor 100%
Household 20% 100%
Fire & ISR 20% 50% 100%
Liability 10% 0% 20% 100%
CTP 20% 0% 0% 25% 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
Motor
0.8
0.9
1.0
1.1
1.2
0.95
1.00
1.05
1.10
1.15
1.20
0.9550.9600.9650.9700.975
0.8 0.9 1.0 1.1 1.2
CTP
Household
0.5650.5700.5750.5800.5850.590
0.951.001.051.101.151.20
Liability
0.955
0.960
0.965
0.970
0.975
0.565
0.570
0.575
0.580
0.585
0.590
Fire..ISR
0.4
0.5
0.6
0.7
0.8
0.9
0.4 0.5 0.6 0.7 0.8 0.9
Results of Simulation
Student-t (3 df) copula
Motor
0.5
0.7
0.9
1.1
1.3
1.5
0.6
0.8
1.0
1.2
1.4
1.6
0.91 0.93 0.95 0.97 0.99
0.5 0.7 0.9 1.1 1.3 1.5
CTP
Household
0.540.560.580.600.620.640.66
0.6 0.8 1.0 1.2 1.4 1.6
Liability
0.91
0.93
0.95
0.97
0.99
0.540.560.580.600.620.640.66
Fire..ISR
0.20
0.45
0.70
0.95
1.20
1.45
0.200.450.700.951.201.45
Results of Simulation
Student-t (10 df) copula
Motor
0.8
0.9
1.0
1.1
1.2
0.9
1.0
1.1
1.2
0.952 0.962 0.972 0.982
0.8 0.9 1.0 1.1 1.2
CTP
Household
0.56 0.57 0.58 0.59 0.60
0.9 1.0 1.1 1.2
Liability
0.952
0.962
0.972
0.982
0.56
0.57
0.58
0.59
0.60
Fire..ISR
0.4
0.5
0.6
0.7
0.8
0.9
0.4 0.5 0.6 0.7 0.8 0.9
Results of Simulation
Cauchy copula
Motor
0.1
0.6
1.1
1.6
2.1
0.50.70.91.11.31.5
0.800.850.900.951.001.051.10
0.1 0.6 1.1 1.6 2.1
CTP
Household
0.4 0.5 0.6 0.7 0.8
0.5 0.7 0.9 1.1 1.3 1.5
Liability
0.800.850.900.951.001.051.10
0.4
0.5
0.6
0.7
0.8
Fire..ISR
0.0
0.5
1.0
1.5
0.0 0.5 1.0 1.5
Results of Simulation
Independence copula
Motor
0.85
0.95
1.05
1.15
0.9
1.0
1.1
1.2
0.9550.9600.9650.9700.975
0.85 0.95 1.05 1.15
CTP
Household
0.562 0.572 0.582 0.592
0.9 1.0 1.1 1.2
Liability
0.955
0.960
0.965
0.970
0.975
0.562
0.572
0.582
0.592
Fire..ISR
0.4
0.5
0.6
0.7
0.8
0.9
0.4 0.5 0.6 0.7 0.8 0.9
Results of Simulation
Aggregated loss, Z, under each copula0
.84
0.8
5
0.8
6
0.8
7
0.8
8
0.8
8
0.8
9
0.9
0
0.9
1
0.9
2
0.9
2
0.9
3
0.9
4
Normal Copula
0.00
0.02
0.04
0.06
0.08
0.10
0.8
0
0.8
3
0.8
6
0.8
8
0.9
1
0.9
4
0.9
7
1.0
0
1.0
3
1.0
6
1.0
8
1.1
1
1.1
4
Student 3 Copula
0.0
0.1
0.2
0.3
0.8
4
0.8
5
0.8
6
0.8
7
0.8
8
0.8
9
0.9
0
0.9
1
0.9
2
0.9
3
0.9
4
0.9
5
0.9
6
Student 10 Copula
0.00
0.04
0.08
0.12
0.7
2
0.7
5
0.7
8
0.8
1
0.8
4
0.8
7
0.9
0
0.9
3
0.9
6
0.9
9
1.0
2
1.0
5
1.0
8
Cauchy Copula
0.0
0.1
0.2
0.3
0.4
0.5
0.8
5
0.8
5
0.8
6
0.8
7
0.8
7
0.8
8
0.8
9
0.8
9
0.9
0
0.9
0
0.9
1
0.9
2
0.9
2
Independence Copula
0.00
0.02
0.04
0.06
0.08
0.10
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
0.90
0.92
0.94
0.96
0.98
1.00
1.02
1.04
1.06
1.08
0 1 2 3 4 5
Risk Measure
CR
Normal
t (3 df)
t (10 df)
Cauchy
Independence
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
0%
2%
4%
6%
8%
10%
12%
14%
0 1 2 3 4 5
Risk Measure
DB
Normal
t (3 df)
t (10 df)
Cauchy
Independence
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
% Savings 7.84% 2.79% 6.51% -2.35% 7.73%
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
% Savings 7.85% 4.31% 6.89% 1.54% 19.45%
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
% Savings 7.00% -3.46% 6.31% -12.75% -17.21%
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 between copulas
Not 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