UNSW Actuarial Symposium Nov 2004
Download
Report
Transcript UNSW Actuarial Symposium Nov 2004
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 X1 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 EX 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 CF1 x1 , F2 x2 ,...,Fn xn
Copulas
Copulas of extreme dependence
Independence
copula
Cu1 ,...,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)
Cu1,...,un Tv tv1 u1 ,...,tv1 un
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
0.95 1.001.05 1.101.15 1.20
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
Fire..ISR
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
1.45
1.20
0.95
0.70
0.45
0.20
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
1.02
0.5
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