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Paid-incurred chain claims reserving method. (English) Zbl 1231.91217
Summary: We present a novel stochastic model for claims reserving that allows us to combine claims payments and incurred losses information. The main idea is to combine two claims reserving models (J. Hertig’s model [ASTIN Bull. 15, No. 2, 171–183 (1985)] and D. Gogol’s model [Insur. Math. Econ. 12, No. 3, 297–299 (1993; Zbl 0800.62678)]) leading to a log-normal paid-incurred chain (PIC) model. Using a Bayesian point of view for the parameter modelling we derive in this Bayesian PIC model the full predictive distribution of the outstanding loss liabilities. On the one hand, this allows for an analytical calculation of the claims reserves and the corresponding conditional mean square error of prediction. On the other hand, simulation algorithms provide any other statistics and risk measure on these claims reserves.

##### MSC:
 91B30 Risk theory, insurance (MSC2010) 60K10 Applications of renewal theory (reliability, demand theory, etc.) 62F15 Bayesian inference 62P05 Applications of statistics to actuarial sciences and financial mathematics
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##### References:
 [1] Asmussen, S.; Glynn, P.W., Stochastic simulation, (2007), Springer · Zbl 1126.65001 [2] Bühlmann, H.; De Felice, M.; Gisler, A.; Moriconi, F.; Wüthrich, M.V., Recursive credibility formula for chain ladder factors and the claims development result, Astin bulletin, 39, 1, 275-306, (2009) · Zbl 1205.91078 [3] Bühlmann, H.; Gisler, A., () [4] Dahms, R., A loss reserving method for incomplete claim data, Bulletin swiss association of actuaries, 127-148, (2008) · Zbl 1333.62231 [5] Dahms, R.; Merz, M.; Wüthrich, M.V., Claims development result for combined claims incurred and claims paid data, Bulletin francais d’actuariat, 9, 18, 5-39, (2009) [6] Gilks, W.R.; Richardson, S.; Spiegelhalter, D.J., Markov chain Monte Carlo in practice, (1996), Chapman & Hall · Zbl 0832.00018 [7] Gisler, A.; Wüthrich, M.V., Credibility for the chain ladder reserving method, Astin bulletin, 38, 2, 565-600, (2008) · Zbl 1274.91486 [8] Gogol, D., Using expected loss ratios in reserving, Insurance: mathematics and economics, 12, 3, 297-299, (1993) · Zbl 0800.62678 [9] Halliwell, L.J., 1997. Cojoint prediction of paid and incurred losses. CAS Forum Summer. pp. 241-379. [10] Halliwell, L.J., 2009. Modeling paid and incurred losses together. CAS E-Forum Spring. [11] Hertig, J., A statistical approach to the IBNR-reserves in marine insurance, Astin bulletin, 15, 2, 171-183, (1985) [12] Liu, H., Verrall, R.J., 2008. Bootstrap estimation of the predictive distributions of reserves using paid and incurred claims. In: Conference Paper, 38th Astin Colloquium 2008. Manchester, UK. [13] Mack, T., Distribution-free calculation of the standard error of chain ladder reserve estimates, Astin bulletin, 23, 2, 213-225, (1993) [14] Merz, M.; Wüthrich, M.V., A credibility approach to the Munich chain-ladder method, Blätter DGVFM, Band XXVII, 619-628, (2006) [15] Merz, M., Wüthrich, M.V., 2010. Estimation of tail factors in the paid – incurred chain reserving method, Preprint (submitted for publication). [16] Posthuma, B., Cator, E.A., Veerkamp, W., Zwet, van E.W., 2008. Combined analysis of paid and incurred losses. CAS E-Forum Fall. pp. 272-293. [17] Quarg, G.; Mack, T., Munich chain ladder, Blätter DGVFM, Band XXVI, 597-630, (2004) [18] Roberts, G.O.; Gelman, A.; Gilks, W.R., Weak convergence and optimal scaling of random walks metropolis algorithm, Annals of applied probability, 7, 110-120, (1997) · Zbl 0876.60015 [19] Scollnik, D.P.M., Actuarial modeling with MCMC and BUGS, North American actuarial journal, 5, 2, 96-125, (2001) · Zbl 1083.62543 [20] Venter, G.G., 2008. Distribution and value of reserves using paid and incurred triangles. CAS E-Forum Fall. pp. 348-375. [21] van Dyk, D.A.; Meng, X.-L., The art of data augmentation, Journal of computational and graphical statistics, 10, 1, 1-50, (2001) [22] Verdier, B., Klinger, A., 2005. JAB chain: a model-based calculation of paid and incurred loss development factors. In: Conference Paper, 36th Astin Colloquium 2005. Zürich, Switzerland. [23] Williams, D., () [24] Wüthrich, M.V.; Merz, M., Stochastic claims reserving methods in insurance, (2008), Wiley · Zbl 1273.91011
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