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

,

,,


Five lines of business

Motor: domestic & commercial

Household: buildings & contents

Fire & ISR

Liability: public, product, WC & PI

CTP


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%


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


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


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


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


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


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


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


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


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%


Comparison with Prescribed Method (PM) – short tail portfolio


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%


Comparison with Prescribed Method (PM) – long tail portfolio


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

economic capital and the aggregation of risks using copulas

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