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

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

Overview


Capital

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

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 insurers1

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 insurers2

Multi-Line Insurers

  • Capital risk measures

  • Capital requirements

  • Value-at-Risk (VaR) – quantile risk measure

  • Tail conditional expectation (TCE)


Multi line insurers3

Multi-Line Insurers

  • Diversification benefit

  • q = 97.5% and 99.5%


Economic capital and the aggregation of risks using copulas

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

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)


Copulas1

Copulas

  • Copulas of extreme dependence

    • Independence copula

  • Archimedean copulas

    • Gumbel-Hougaard copula

    • Frank copula

    • Cook-Johnson copula


Copulas2

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.


Copulas3

Copulas

  • Tail dependence (Student-t copulas)

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


Numerical simulation

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 simulation1

Numerical Simulation

  • Five lines of business

    • Motor: domestic & commercial

    • Household: buildings & contents

    • Fire & ISR

    • Liability: public, product, WC & PI

    • CTP


Numerical simulation2

Numerical Simulation

  • Correlation matrix input


Numerical simulation3

Numerical Simulation

  • Marginal distribution input


Results of simulation

Results of Simulation

  • Normal copula


Results of simulation1

Results of Simulation

  • Student-t (3 df) copula


Results of simulation2

Results of Simulation

  • Student-t (10 df) copula


Results of simulation3

Results of Simulation

  • Cauchy copula


Results of simulation4

Results of Simulation

  • Independence copula


Results of simulation5

Results of Simulation

  • Aggregated loss, Z, under each copula


Results of simulation6

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 simulation7

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 simulation8

Results of Simulation

  • Comparison with Prescribed Method (PM) – industry portfolio


Results of simulation9

Results of Simulation

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


Results of simulation10

Results of Simulation

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


Key findings

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

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

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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