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Beyond correlation: Recent developments in the modelling of claims dependenciesBenjamin Avanzi, School of Risk and Actuarial Studies, UNSW and Département de Mathématiques et de Statistique, Université de MontréalGreg Taylor, Bernard Wong,School of Risk and Actuarial Studies, UNSWInsights, 5 April 2016

Beyond correlation: Recent developments in the modelling of claims dependencies

Context

Actuarial: Dependence modelling

Our team: Linkage project

On correlations

Introduction

Practice

Correlation pitfalls

So what?

Dependence modelling approaches

Implicit vs explicit modelling of dependence

Micro vs Macro modelling

Discussion

Recent developments and case studies

Multivariate Tweedie reserving model

Lévy Copulas - dependence modelling for Lévy processes Cox process approach to the micro-modelling of insurance claims

Dependence modelling using Cox processes

Summary and Conclusions

List of (some) recent references

Details of all references

1/61 Beyond correlation: Recent developments in the modelling of claims dependencies

Context

Context

Actuarial: Dependence modelling

Our team: Linkage project

On correlations

Introduction

Practice

Correlation pitfalls

So what?

Dependence modelling approaches

Implicit vs explicit modelling of dependence

Micro vs Macro modelling

Discussion

Recent developments and case studies

Multivariate Tweedie reserving model

Lévy Copulas - dependence modelling for Lévy processes Cox process approach to the micro-modelling of insurance claims

Dependence modelling using Cox processes

Summary and Conclusions

List of (some) recent references

Details of all references

2/61 Beyond correlation: Recent developments in the modelling of claims dependencies

Context

Actuarial: Dependence modelling

Context

Actuarial: Dependence modelling

Our team: Linkage project

On correlations

Introduction

Practice

Correlation pitfalls

So what?

Dependence modelling approaches

Implicit vs explicit modelling of dependence

Micro vs Macro modelling

Discussion

Recent developments and case studies

Multivariate Tweedie reserving model

Lévy Copulas - dependence modelling for Lévy processes Cox process approach to the micro-modelling of insurance claims

Dependence modelling using Cox processes

Summary and Conclusions

List of (some) recent references

Details of all references

2/61 Beyond correlation: Recent developments in the modelling of claims dependencies

Context

Actuarial: Dependence modelling

Significance of stochastic dependence

Practically speaking, dependence structures can have a major impact in several areas of the business of an insurance company (see also IAA, 2004, 2009), including I determining actuarial reserves (IBNR): quantile, or central estimate (the mean) plus margin; I determining a risk based capital for solvency assessment In all cases, structures with less than perfect dependence will lead to diversification b enefits , whose accurate estimation is crucial for I capital efficiency (not underestimate); I solvency of the insurance company (not overestimate). 2/61 Beyond correlation: Recent developments in the modelling of claims dependencies

Context

Our team: Linkage project

Context

Actuarial: Dependence modelling

Our team: Linkage project

On correlations

Introduction

Practice

Correlation pitfalls

So what?

Dependence modelling approaches

Implicit vs explicit modelling of dependence

Micro vs Macro modelling

Discussion

Recent developments and case studies

Multivariate Tweedie reserving model

Lévy Copulas - dependence modelling for Lévy processes Cox process approach to the micro-modelling of insurance claims

Dependence modelling using Cox processes

Summary and Conclusions

List of (some) recent references

Details of all references

3/61 Beyond correlation: Recent developments in the modelling of claims dependencies

Context

Our team: Linkage project

Modelling claim dependencies for the general insurance industry with economic capital in view: An innovative approach with stochastic processes "The project will develop progressive methods to b etterrep resent the fine, complex structures driving the significant dependencies relevant to the Enterprise Risk Management of general insurers Collaborative between, and jointly funded by Government, industry (Allianz, IAG, Suncorp) and academia

See also

a rticlein the A ctuariesMagazine, August 2014 (A vanzi,

Taylor, and Wong, 2014)back

3/61 Beyond correlation: Recent developments in the modelling of claims dependencies

On correlations

Context

Actuarial: Dependence modelling

Our team: Linkage project

On correlations

Introduction

Practice

Correlation pitfalls

So what?

Dependence modelling approaches

Implicit vs explicit modelling of dependence

Micro vs Macro modelling

Discussion

Recent developments and case studies

Multivariate Tweedie reserving model

Lévy Copulas - dependence modelling for Lévy processes Cox process approach to the micro-modelling of insurance claims

Dependence modelling using Cox processes

Summary and Conclusions

List of (some) recent references

Details of all references

4/61 Beyond correlation: Recent developments in the modelling of claims dependencies

On correlations

Introduction

Context

Actuarial: Dependence modelling

Our team: Linkage project

On correlations

Introduction

Practice

Correlation pitfalls

So what?

Dependence modelling approaches

Implicit vs explicit modelling of dependence

Micro vs Macro modelling

Discussion

Recent developments and case studies

Multivariate Tweedie reserving model

Lévy Copulas - dependence modelling for Lévy processes Cox process approach to the micro-modelling of insurance claims

Dependence modelling using Cox processes

Summary and Conclusions

List of (some) recent references

Details of all references

4/61 Beyond correlation: Recent developments in the modelling of claims dependencies

On correlations

Introduction

At our last

'Insights" Session (Octob er2015) , we concluded: I

Correlation depends on your model

I

Correlation happens as a result of real phenomena

I The more of those phenomenons you can explain in your model, the less dependent your residuals will look like I We wiped away all correlation from the AUSI dataset I But what you can use to explain past data may not necessarily be available to explain (predict) the future I Correlation is all but one way of specifying dependence

See also

I article in the Actuaries Magazine, September 2015 (Avanzi, Taylor, and Wong, 2015a) I article in BusinessThink, December 2015 (BusinessThink, 2015) I academic article in ASTIN Bulletin, in press (Avanzi, Taylor, and Wong, 2015b) 4/61 Beyond correlation: Recent developments in the modelling of claims dependencies

On correlations

Introduction

Correlation

Definition:

I

We refer to 'Pearson correlation"

I

This is a measure of

linea r dep endence,which is symmetric around the mean I

Hence, it is tied to elliptical distributions

(Normal, Student) I In fact, it completely specifies the dependence structure of

Normal and Student distributions

Why then, could there be a need to move beyond correlation? 5/61 Beyond correlation: Recent developments in the modelling of claims dependencies

On correlations

Practice

Context

Actuarial: Dependence modelling

Our team: Linkage project

On correlations

Introduction

Practice

Correlation pitfalls

So what?

Dependence modelling approaches

Implicit vs explicit modelling of dependence

Micro vs Macro modelling

Discussion

Recent developments and case studies

Multivariate Tweedie reserving model

Lévy Copulas - dependence modelling for Lévy processes Cox process approach to the micro-modelling of insurance claims

Dependence modelling using Cox processes

Summary and Conclusions

List of (some) recent references

Details of all references

6/61 Beyond correlation: Recent developments in the modelling of claims dependencies

On correlations

Practice

Risk margins

When calculating risk margins, usual practice would be to follow

these steps:1Estimate the mean and variance of liability for each LoB;2Estimate the associated correlation matrix;3Hence estimate the mean and variance of the total liability

across all LoBs;4Assume some convenient distribution for this total liability, usually log normal;5Calculate the 75-percentile from this distribution. Frequent assumption for step [2] would be consistent with Bateup and Reed (2001) and/or Collings and White (2001). 6/61 Beyond correlation: Recent developments in the modelling of claims dependencies

On correlations

Practice

Capital margins

When calculating (high percentile) capital margins, usual practice

would be to follow these steps:1Estimate the distribution of liability for each LoB;2Assume a copula across the LoBs, most commonlyt-copula;3Perform a multivariate simulation of liabilities for all LoBs;4Form the replicates of total liabilities across LoBs, and read off

required percentile. 7/61 Beyond correlation: Recent developments in the modelling of claims dependencies

On correlations

Correlation pitfalls

Context

Actuarial: Dependence modelling

Our team: Linkage project

On correlations

Introduction

Practice

Correlation pitfalls

So what?

Dependence modelling approaches

Implicit vs explicit modelling of dependence

Micro vs Macro modelling

Discussion

Recent developments and case studies

Multivariate Tweedie reserving model

Lévy Copulas - dependence modelling for Lévy processes Cox process approach to the micro-modelling of insurance claims

Dependence modelling using Cox processes

Summary and Conclusions

List of (some) recent references

Details of all references

8/61 Beyond correlation: Recent developments in the modelling of claims dependencies

On correlations

Correlation pitfalls

Assume high correlation when there should be none2000 2002 2004 2006 2008

29 30 31 32 33 34

Year

Pounds

300 400 500 600 700 800 900

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