Economic Capital and the Aggregation of Risks Using Copulas
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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

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Economic Capital and the Aggregation of Risks Using Copulas

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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


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)


Multi-Line Insurers

  • Diversification benefit

  • 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)


Copulas

  • Copulas of extreme dependence

    • Independence copula

  • Archimedean copulas

    • Gumbel-Hougaard copula

    • Frank 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)

      Where Tv(.) and tv(.) denote the multivariate and univariate Student-t distribution with v degrees of freedom respectively.


Copulas

  • Tail dependence (Student-t copulas)

    where t* denotes the survivorship function of the Student-t distribution with n degrees of freedom.


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)


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


Numerical Simulation

  • Marginal distribution input


Results of Simulation

  • Normal copula


Results of Simulation

  • Student-t (3 df) copula


Results of Simulation

  • Student-t (10 df) copula


Results of Simulation

  • Cauchy copula


Results of Simulation

  • Independence copula


Results of Simulation

  • Aggregated loss, Z, under each copula


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.


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.


Results of Simulation

  • Comparison with Prescribed Method (PM) – industry portfolio


Results of Simulation

  • Comparison with Prescribed Method (PM) – short tail portfolio


Results of Simulation

  • Comparison with Prescribed Method (PM) – long tail portfolio


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


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