Claims Reserving: New Developments of Mario V. Wüthrich

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Markov Chain Monte Carlo Methods for Claims Reserving

Markov chain Monte Carlo (MCMC) methods are very powerful simulation tools that allow to generate samples from general densities. In claims reserving MCMC methods are used to determine posterior distributions of outstanding loss liabilities given an upper claims development triangle of observations. Due to its generality MCMC methods can be applied to almost any such Bayesian claims reserving problem. The following papers give some insight how MCMC methods are used.

• Model uncertainty in claims reserving within Tweedie's compound Poisson models.
  (with G.W. Peters and P.V. Shevchenko) Astin Bulletin 39 (2009), no. 1, 1--33.


• Bayesian overdispersed Poisson model and the Bornhuetter-Ferguson claims reserving method.
  (with P. England and R. Verrall) pdf-Preprint, 2011. To appear in Annals of Actuarial Science.


• Challenges with non-informative gamma priors in the Bayesian over-dispersed Poisson reserving model.
  pdf-Preprint, 2012.

MCMC methods allow for modeling dependence between accident years through an accounting year and inflation parameter in the chain ladder model.

• Accounting year effects modelling in the stochastic chain ladder reserving method.
  North American Actuarial J.14 (2010), no. 2, 235--255.


• Bayesian prediction of disability insurance frequencies using economic factors.
  (with C. Donnelly) pdf-Preprint, 2012. To appear in Annals of Actuarial Science.

The reversible jump Markov chain Monte Carlo (RJMCMC) method gives a mathematical tool for parameter reduction and tail factor modeling using parametric curves.

• Reversible jump Markov chain Monte Carlo method for parameter reduction in claims reserving.
  (with R. Verrall) pdf-Preprint, 2012. To appear in North American Actuarial J.

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Credibility Chain Ladder Method and the Claims Development Result (CDR)

We set the classical distribution-free chain ladder (CL) method into a Bayesian framework. This approach allows for a natural study of prediction uncertainty and predictive modeling. Moreover, we provide the one-year solvency view and compare it to the traditional long term uncertainty view.

• Recursive credibility formula for chain ladder factors and the claims development result.
  (with H. Bühlmann, M. De Felice, A. Gisler, F. Moriconi) Astin Bulletin 39 (2009), no. 1, 275--306.


• Credibility for the chain ladder reserving method.
  (with A. Gisler) Astin Bulletin 38 (2008), no. 2, 565--600.


• Modelling the claims development result for solvency purposes.
  (with M. Merz) CAS E-Forum, Fall 2008, 542--568.


• Higher moments of the claims development result in general insurance.
  (with R. Salzmann and M. Merz) pdf-Preprint, 2010. To appear in Astin Bulletin.

• Spreadsheet: Example from Merz-Wüthrich, CAS E-Forum, Fall 2008

The Bayes chain ladder method assumes that accident years are independent, conditionally given the model parameters. This conditional independence does not allow for the modelling of inflation and accounting year effects which act on all accident years simultaneously. In the paper below we extend the Bayes chain ladder model so that it also allows for dependence modelling within accounting years. This model is then studied with the Markov chain Monte Carlo (MCMC) simulation methodology.

• Accounting year effects modelling in the stochastic chain ladder reserving method.
  North American Actuarial J.14 (2010), no. 2, 235--255.


• Prediction of disability frequencies in life insurance.
  (with B. König and F. Weber) pdf manuscript, 2011. Zavarovalniski horizonti 7 (2011), no. 3, 5--23.


• Bayesian prediction of disability insurance frequencies using economic factors.
  (with C. Donnelly) pdf-Preprint, 2012. To appear in Annals of Actuarial Science.

We extend the classical chain ladder algorithm so that it also allows to integrate additional information such as number of reported claims and number of payments. This additional information allows for model validation using graphical tools.

• Chain ladder method and individual claims development analysis.
  (with M.D. Martinez-Miranda and J.P. Nielsen) pdf-Preprint, 2012.

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Paid-Incurred Chain Claims Reserving Method

We develop a new model for the outstanding loss liability prediction which is based both on cumulative payments information and incurred losses information. This leads to a so-called Paid-Incurred Chain (PIC) claims reserving method. The PIC model combines these two different sources of information using appropriate credibility weights, and the PIC model allows for claims prediction and the quantification of its prediction uncertainty.

• Paid-incurred chain claims reserving method.
  (with M. Merz) Insurance: Math. Econom. 46 (2010), no. 3, 568--579.


• Paid-incurred chain reserving method with dependence modeling.
  (with S. Happ) SSRN Preprint, 2011.


• Claims development result in the paid-incurred chain reserving method.
  (with S. Happ and M. Merz) SSRN Preprint, 2011. To appear in Insurance: Math. Econom.

• Spreadsheet: Excel Example

The PIC model also allows for the estimation of tail development factors by considering incurred-paid ratios in a natural and consistent way.

• Estimation of tail development factors in the paid-incurred chain reserving method.
  (with M. Merz) pdf-Preprint, 2010.

• Spreadsheet: Excel Example

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Market-Value Margin (Risk Margin, Cost-of-Capital Margin) in Claims Reserving

We construct a mathematically consistent approach for the calculation of the cost-of-capital margin (similar to a market-value margin) for a non-life insurance runoff and compare it to proxies used in practice.

• Cost-of-capital margin for a general insurance liability runoff.
  (with R. Salzmann) Astin Bulletin 40 (2010), no. 2, 415--451.


• Runoff of the claims reserving uncertainty in non-life insurance: a case study.
  Zavarovalniski horizonti 6 (2010), no. 3, 5--18. Journal of the Slovenian Insurance Assocation.

• Spreadsheet: Excel Example

A second approach, based on economic theory, models the risk aversion of financial agents through probability distortions. This way we obtain in a rather natural way a market-value margin which basically is obtained by putting more probability weight to adverse scenarios (under risk aversion).

• Risk margin for a non-life insurance run-off.
  (with P. Embrechts and A. Tsanakas) Statistics & Risk Modeling 28 (2011), no. 4, 299-317.

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Bornhuetter-Ferguson Method and the Overdispersed Poisson Model

We provide a stochastic framework for the calculation of the prediction uncertainty in the Bornuetter-Ferguson (BF) method. Our method is based on the overdispersed Poisson (ODP) model with maximum likelihood estimators (MLE) which provides a claims development pattern identical to the one obtained from the chain-ladder method. In this setup we derive an estimator for the conditional mean square error of prediction (MSEP) for the BF method.

• Mean square error of prediction in the Bornhuetter-Ferguson claims reserving method.
  (with D. Alai and M. Merz) Annals of Actuarial Science 4 (2009), no. 1, 7--31.


• Prediction uncertainty in the Bornhuetter-Ferguson claims reserving method: revisited.
  (with D. Alai and M. Merz) Annals of Actuarial Science 5 (2010), no. 1, 7--17.

• Spreadsheet: Excel Example

The BF method considered above is inconsistent in the sense that it considers prior information for the estimation of the accident year exposures, however it neglects this information for the estimation of the claims development pattern. In the next paper we present a consistent estimation method, i.e. all available information is considered also for the estimation of the claims development pattern.

• Development pattern and prediction error for the stochastic Bornhuetter-Ferguson claims reserving model.
  (with A. Saluz and A. Gisler) Astin Bulletin 41 (2011), no. 2, 279--317.

We put the BF method into a full Bayesian model. That is, we gather all prior information on the accident year exposures and the developement pattern in appropriate prior distributions for these (unknown) parameters. Using Bayesian inference methods then allow to calculate the full predictive distribution of the outstanding loss liabilities.

• Bayesian overdispersed Poisson model and the Bornhuetter-Ferguson claims reserving method.
  (with P. England and R. Verrall) pdf-Preprint, 2011. To appear in Annals of Actuarial Science.


• Challenges with non-informative gamma priors in the Bayesian over-dispersed Poisson reserving model.
  pdf-Preprint, 2012.

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Complementary Loss Ratio Method

The complementary loss ratio method (CLRM) provides a claims reserving method that allows for (i) the combination of claims incurred and claims payments data, (ii) handling incomplete claims development triangles.

• R. Dahms (2008). A loss reserving method for incomplete data.
  Mitteilungen-Bulletin SAV, Heft 1&2 (2008), 127--148.


• Claims development result for combined claims incurred and claims paid data.
  (with R. Dahms and M. Merz) Bulletin Francais d'Actuariat 9 (2009), no. 18, 5--39.