Economic Capital and the Aggregation of Risks Using Copulas Dr.
Emiliano A. Valdez and Andrew Tang
Slide 2
Motivation and aims Technical background - copulas Numerical
simulation Results of simulation Key findings and conclusions
Overview
Slide 3
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
Slide 4
Multi-Line Insurers Increasingly prominent Diverse range
insurance products Aggregate loss, Z Where X i represents the loss
variable from line i. X i s are dependent
Slide 5
Multi-Line Insurers Dependencies between X i s ignored E.g.,
APRA Prescribed Method Dependencies modelled using linear
correlations Inadequate Non-linear dependence Tail dependence
Slide 6
Multi-Line Insurers Capital risk measures Capital requirements
Value-at-Risk (VaR) quantile risk measure Tail conditional
expectation (TCE)
Slide 7
Multi-Line Insurers Diversification benefit q = 97.5% and
99.5%
Slide 8
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
Slide 9
Copulas Individual line losses - X 1, X 2, , X n Joint
distribution is F(x 1,x 2,,x n ) Marginal distributions are F 1 (x
1 ), F 2 (x 2 ), , F n (x n ) A copula, C, is a function that
links, or couples the marginals to the joint distribution Sklar
(1959)
Slide 10
Copulas Copulas of extreme dependence Independence copula
Archimedean copulas Gumbel-Hougaard copula Frank copula
Cook-Johnson copula
Slide 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 T v (.) and t v (.) denote the
multivariate and univariate Student-t distribution with v degrees
of freedom respectively.
Slide 12
Copulas Tail dependence (Student-t copulas) where t* denotes
the survivorship function of the Student-t distribution with n
degrees of freedom. n\ 00.50.91 10.290.50.781 30.120.310.671
100.010.080.461 infinity0001
Slide 13
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)
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Numerical Simulation Five lines of business Motor: domestic
& commercial Household: buildings & contents Fire & ISR
Liability: public, product, WC & PI CTP
Slide 15
Numerical Simulation Correlation matrix input Line of
BusinessMotorHouseholdFire & ISRLiabilityCTP Motor100%
Household20%100% Fire & ISR20%50%100% Liability10%0%20%100%
CTP20%0% 25%100%
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Numerical Simulation Marginal distribution input Line of
businessMarginal distribution MotorGamma HouseholdGamma Fire &
ISRLog-normal LiabilityLog-normal CTPLog-normal
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Results of Simulation Normal copula
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Results of Simulation Student-t (3 df) copula
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Results of Simulation Student-t (10 df) copula
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Results of Simulation Cauchy copula
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Results of Simulation Independence copula
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Results of Simulation Aggregated loss, Z, under each
copula
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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.
Slide 24
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.
Slide 25
Results of Simulation Comparison with Prescribed Method (PM)
industry portfolio Normalt (3 df)t (10 df)CauchyIndependence PM
CR1.0102911.0102331.0088571.0025360.999034 VaR 99.5%
CR0.9310900.9820050.9431311.0261400.921855 Excess
Capital0.0792010.0282280.065726-0.0236040.077179 %
Savings7.84%2.79%6.51%-2.35%7.73%
Slide 26
Results of Simulation Comparison with Prescribed Method (PM)
short tail portfolio Normalt (3 df)t (10 df)CauchyIndependence PM
CR0.9516090.9520250.9511910.9486281.093202 VaR 99.5% CR
0.8768920.9110360.8857010.9340660.880529 Excess Capital
0.0747170.0409890.0654900.0145620.212673 %
Savings7.85%4.31%6.89%1.54%19.45%
Slide 27
Results of Simulation Comparison with Prescribed Method (PM)
long tail portfolio Normalt (3 df)t (10 df)CauchyIndependence PM
CR1.0983141.0975431.0953571.0833990.857781 VaR 99.5% CR
1.0213801.1355601.0262401.2215001.005440 Excess Capital
0.076934-0.0380170.069117-0.138101-0.147659 %
Savings7.00%-3.46%6.31%-12.75%-17.21%
Slide 28
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 insurers 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
Slide 29
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
Slide 30
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
