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On the Importance of Dispersion Modeling for Claims Reserving: Application of the Double GLM Theory Danal Davidov under the supervision of Jean-Philippe Boucher Dpartement de mathmatiques Universit du Qubec Montral August 2009

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On the Importance of Dispersion Modeling for Claims Reserving: Application of the Double GLM Theory

Danaïl Davidov under the supervision of Jean-Philippe Boucher

Département de mathématiques Université du Québec à Montréal

August 2009

  • D. Davidov (UQAM)

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Table of Contents

1

Introduction

2

Tweedie GLM Model Definition Log-likelihood Overdispersion

3

Variance Modeling Variance in regressions Dispersion Models Double GLMs

4

Swiss Motor Industry Example Analysis of results

5

Conclusion

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Introduction

Definition

A loss reserve is a provision for an insurer’s liability for claims

Notes

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Introduction

Definition

A loss reserve is a provision for an insurer’s liability for claims A stochastic model uses random variables in a regression framework

Notes

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Introduction

Definition

A loss reserve is a provision for an insurer’s liability for claims A stochastic model uses random variables in a regression framework

Notes

Claims reserves models presented here use GLM theory as introduced in England, Verrall 2002

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Model Definition

Notations

Ci,j Incremental payments

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Model Definition

Notations

Ci,j Incremental payments wi,j Exposure

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Model Definition

Notations

Ci,j Incremental payments wi,j Exposure ri,j Incremental number of payments

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Model Definition

Notations

Ci,j Incremental payments wi,j Exposure ri,j Incremental number of payments Yi,j Normalized incremental payments

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Model Definition

Notations

Ci,j Incremental payments wi,j Exposure ri,j Incremental number of payments Yi,j Normalized incremental payments Yi,j = Ci,j

wi,j

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Model Definition

Hypotheses

Ci,j is a compound Poisson-Gamma distribution

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Model Definition

Hypotheses

Ci,j is a compound Poisson-Gamma distribution Frequency ∼ Poisson with mean ϑi,jwi,j and variance ϑi,jwi,j

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Model Definition

Hypotheses

Ci,j is a compound Poisson-Gamma distribution Frequency ∼ Poisson with mean ϑi,jwi,j and variance ϑi,jwi,j Severity ∼ Gamma with mean τi,j and variance ντ 2

i,j

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Model Definition

Hypotheses

Ci,j is a compound Poisson-Gamma distribution Frequency ∼ Poisson with mean ϑi,jwi,j and variance ϑi,jwi,j Severity ∼ Gamma with mean τi,j and variance ντ 2

i,j

Using the following parametrisation p = ν + 2 ν + 1 , p ∈ (1, 2) µ = ϑτ φ = ϑ1−pτ 2−p (2 − p)

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Model Definition

Tweedie Model

Yi,j ∼ Tweedie(µi,j, p, φ, wi,j)

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Model Definition

Tweedie Model

Yi,j ∼ Tweedie(µi,j, p, φ, wi,j) µi,j = eXβ

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Model Definition

Tweedie Model

Yi,j ∼ Tweedie(µi,j, p, φ, wi,j) µi,j = eXβ E[Yi,j] = µi,j , Var[Yi,j] =

φ wi,j µp i,j

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Log-likelihood function

Tweedie Model

l =

  • i,j

ri,j log

  • (wi,j/φ)ν+1yν

i,j

(p − 1)ν(2 − p)

  • − log
  • ri,j!Γ
  • ri,jν
  • yi,j
  • + wi,j

φ

  • yi,j

µ1−p

i,j

1 − p − µ2−p

i,j

2 − p

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Parameters

FIGURE: Parameter main influence

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Parameter p

Tweedie Model

Can be estimated only when the number of payments is known

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Parameter p

Tweedie Model

Can be estimated only when the number of payments is known Otherwise, it’s supposed fixed and known

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Parameter p

Tweedie Model

Can be estimated only when the number of payments is known Otherwise, it’s supposed fixed and known p and φ need both to be estimated at the same time

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Parameter φ

Optimizing φ using the likelihood principle

  • φp

= −

i,j wi,j

  • yi,j

µ1−p

i,j

1−p − µ2−p

i,j

2−p

  • (1 + ν)

i,j ri,j

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Parameter φ

FIGURE: Optimizing p using the likelihood principle for φ

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Parameter φ

Optimizing φ using the deviance principle

  • φp

=

  • i,j

2 N − Q

  • yi,j

y1−p

i,j

− µ1−p

i,j

1 − p − y2−p

i,j

− µ2−p

i,j

2 − p

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Parameter φ

FIGURE: Optimizing p using the deviance principle for φ

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Parameter wi,j

Note

The exposure has been incorporated in the begining, within the initial hypothesis

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Parameter wi,j

Note

The exposure has been incorporated in the begining, within the initial hypothesis Different ways of incorporating the exposure within the initial hypothesis would lead to different models

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Frequency vs Severity

Compound Poisson Model

Y =

N

  • k=1

Xi

Case 1 Case 2 Case 3 E[N] 10 20 10 Var[N] 10 20 10 E[X] 10 10 20 Var[X] 100 100 400 E[Y] 100 200 200 Var[Y] 2000 4000 8000 TABLE: Mean and variance of total costs for various situations

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Frequency vs Severity

Typical situation in a long-tail business

Decreasing average frequency

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Frequency vs Severity

Typical situation in a long-tail business

Decreasing average frequency Increasing average severity

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Frequency vs Severity

Typical situation in a long-tail business

Decreasing average frequency Increasing average severity Increasing variance in the severity

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Impact of the Distribution

FIGURE: Fitted curve for Normal, Poisson and Gamma models

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Dispersion Models

Model Definition

Yi,j ∼ Tweedie(µi,j, p, φi,j, wi,j)

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Dispersion Models

Model Definition

Yi,j ∼ Tweedie(µi,j, p, φi,j, wi,j) µi,j = eXβ

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Dispersion Models

Model Definition

Yi,j ∼ Tweedie(µi,j, p, φi,j, wi,j) µi,j = eXβ E[Yi,j] = µi,j , Var[Yi,j] = φi,jµp

i,j

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Dispersion Models

Model Definition

Yi,j ∼ Tweedie(µi,j, p, φi,j, wi,j) µi,j = eXβ E[Yi,j] = µi,j , Var[Yi,j] = φi,jµp

i,j

φi,j = eVγ

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Dispersion Models

Log-likelihood

l =

  • i,j

ri,j log

  • (wi,j/φi,j)ν+1yν

i,j

(p − 1)ν(2 − p)

  • − log
  • ri,j!Γ
  • ri,jν
  • yi,j
  • + wi,j

φi,j

  • yi,j

µ1−p

i,j

1 − p − µ2−p

i,j

2 − p

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Dispersion Models

Notes

The p parameter is optimized at the same time as all other parameters using implicitly the likelihood principle

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Dispersion Models

Notes

The p parameter is optimized at the same time as all other parameters using implicitly the likelihood principle Accident years do not have a significant impact on the dispersion parameter

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Dispersion Models

Notes

The p parameter is optimized at the same time as all other parameters using implicitly the likelihood principle Accident years do not have a significant impact on the dispersion parameter Due to lack of data in the last column, the model was build so that the last two columns have the same dispersion parameter

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Dispersion Models

Notes

The p parameter is optimized at the same time as all other parameters using implicitly the likelihood principle Accident years do not have a significant impact on the dispersion parameter Due to lack of data in the last column, the model was build so that the last two columns have the same dispersion parameter Possibility to incorporate trends in the dispersion parameter by using the Hoerl’s curve parametrisation

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Double GLMs

Algorithm

1

Start with the initial exposure and find the normalized incremental payments

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Double GLMs

Algorithm

1

Start with the initial exposure and find the normalized incremental payments

2

Find the deviance between fitted and observed values

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Double GLMs

Algorithm

1

Start with the initial exposure and find the normalized incremental payments

2

Find the deviance between fitted and observed values

3

Model the deviance

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Double GLMs

Algorithm

1

Start with the initial exposure and find the normalized incremental payments

2

Find the deviance between fitted and observed values

3

Model the deviance

4

Establish the new exposure and start over

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Double GLMs

FIGURE: Inter-relationship between the two sub-models

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Iterative Weighted Least Squares

Algorithm additional specifications

Analogous to Fisher’s weighted scoring method for optimization

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Iterative Weighted Least Squares

Algorithm additional specifications

Analogous to Fisher’s weighted scoring method for optimization IWLS implicitly uses the deviance principle for estimating φ

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Restricted Maximum Likelihood

Notes

Maximum likelihood estimators are biased downwards when the number of estimators is large compared to the number of

  • bservations
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Restricted Maximum Likelihood

Notes

Maximum likelihood estimators are biased downwards when the number of estimators is large compared to the number of

  • bservations

REML produces estimators which are approximately and sometimes exactly unbiased

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Restricted Maximum Likelihood

Notes

Maximum likelihood estimators are biased downwards when the number of estimators is large compared to the number of

  • bservations

REML produces estimators which are approximately and sometimes exactly unbiased Approximately maximizes the penalized log-likelihood l ∗p (y; γ; p) = l(y; βγ; γ; p) + 1 2 log

  • X TWX
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Optimizing p

Algorithm

1

Suppose p fixed and known

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Optimizing p

Algorithm

1

Suppose p fixed and known

2

Evaluate all the other parameters using the DGLM

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Optimizing p

Algorithm

1

Suppose p fixed and known

2

Evaluate all the other parameters using the DGLM

3

Evaluate the penalized log-likelihood

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Optimizing p

Algorithm

1

Suppose p fixed and known

2

Evaluate all the other parameters using the DGLM

3

Evaluate the penalized log-likelihood

4

Start over with different values for p and compare the penalized log-likelihood

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Reserve Variability

Mean Square Error of Prediction

Dispersion Models : overdispersion included implicitly in the parameter covariance matrix

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Reserve Variability

Mean Square Error of Prediction

Dispersion Models : overdispersion included implicitly in the parameter covariance matrix GLMs : overdispersion is included manually in the parameter covariance matrix

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Data Analyzed

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Data Analyzed

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Data Analyzed

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Optimizing p

FIGURE: Restricted log-likelihood for various p in a DGLM

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Analysis of Results

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Analysis of Results

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Analysis of Results

FIGURE: Adjusting the most deviant observations has a bigger influence on the deviance principle

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Analysis of Results

FIGURE: Contribution of each cell to the dispersion. p = 1.1741 , wi,j ≡ 1

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Conclusion

Further Discussion

Lack of observations and abundance of parameters is a hostile environment for DGLMs

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Conclusion

Further Discussion

Lack of observations and abundance of parameters is a hostile environment for DGLMs The deviance cannot be estimated in the last column for a DGLM and is hence ignored when estimating φ

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Conclusion

Further Discussion

Lack of observations and abundance of parameters is a hostile environment for DGLMs The deviance cannot be estimated in the last column for a DGLM and is hence ignored when estimating φ The algorithm does not converge for p fixed when there are more than 7 parameters

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Conclusion

Further Discussion

Lack of observations and abundance of parameters is a hostile environment for DGLMs The deviance cannot be estimated in the last column for a DGLM and is hence ignored when estimating φ The algorithm does not converge for p fixed when there are more than 7 parameters The algorithm does converge for p fixed for one parameter, but p cannot be optimized

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Conclusion

Further Discussion

Lack of observations and abundance of parameters is a hostile environment for DGLMs The deviance cannot be estimated in the last column for a DGLM and is hence ignored when estimating φ The algorithm does not converge for p fixed when there are more than 7 parameters The algorithm does converge for p fixed for one parameter, but p cannot be optimized The "bounds" of convergence need to be explored further

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References

England, PD and Verrall, RJ (2002). Stochastic Claims Reserving in General Insurance. Institute of Actuaries and Faculty of Actuaries Smyth, G. and Jorgensen, B. (2002). Fitting Tweedie’s Compound Poisson Model to Insurance Claims Data : Dispersion Modelling. ASTIN Bulletin, 1, 143–157 Wüthrich, MV (2003). Claims Reserving Using Tweedie’s Compound Poisson Model. ASTIN Bulletin, 331–346

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The end

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