×

Full and 1-year runoff risk in the credibility-based additive loss reserving method. (English) Zbl 06292442

Summary: In this paper, we consider the additive loss reserving (ALR) method in a Bayesian and credibility setup. The classical ALR method is a simple claims reserving method that combines prior information (e.g., premiums, number of contracts, market statistics) with claims observations. The Bayesian setup, which we present, in addition, allows for combining the information from a single runoff portfolio (e.g.,company-specific data) with the information from a collective (e.g.,industry-wide data) to analyze the claims reserves and the claims development result. However, in insurance practice, the associated distributions are usually unknown. Therefore, we do not follow the full Bayesian approach but apply credibility theory, which is distribution free and where we only need to know the first and second moments. That is, we derive the credibility predictors that minimize the expected squared loss within the class of affine-linear functions of the observations (i.e.,we derive linear Bayesian predictors). Using non-informative priors, we link our credibility-based ALR method to the classical ALR method and show that the credibility predictors coincide with the predictors in the classical ALR method. Moreover, we quantify the 1-year risk and the full reserve risk by means of the conditional mean square error of prediction.

MSC:

62-XX Statistics
62P20 Applications of statistics to economics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] CAS, Foundations of Casualty Actuarial Science (1990)
[2] Teugels, Encyclopedia of Actuarial Science 1 (2004)
[3] Bühlmann, Recursive credibility formula for chain ladder factors and the claims development result, Astin Bulletin 39 (1) pp 275– (2009) · Zbl 1205.91078
[4] Bühlmann, A Course in Credibility Theory and Its Applications (Universitext) (2005)
[5] Hickman, Credibility theory: the cornerstone of actuarial science, North American Actuarial Journal 3 (2) pp 1– (1999) · Zbl 1082.62533
[6] Mack, Distribution-free calculation of the standard error of chain ladder reserve estimates, Astin Bulletin 23 (2) pp 213– (1993)
[7] Wüthrich, Stochastic Claims Reserving Methods in Insurance (2008) · Zbl 1273.91011
[8] Schmidt, Methods and models of loss reserving based on run-off triangles: a unifying survey, CAS Forum (Fall) pp 269– (2006)
[9] Gisler, Credibility for the chain ladder reserving method, Astin Bulletin 38 (2) pp 565– (2008) · Zbl 1274.91486
[10] AISAM-ACME AISAM-ACME study on non-life long tail liabilities 2007
[11] Merz, Modelling the claims development result for solvency purposes, Casualty Actuarial Society E-Forum (Fall) pp 542– (2008)
[12] England, Predictive distributions of outstanding liabilities in general insurance, Annals of Actuarial Science 1 (2) pp 221– (2007)
[13] Bühlmann, Mathematical Methods in Risk Theory (1970)
[14] Mack, The prediction error of Bornhuetter-Ferguson, Astin Bulletin 38 (1) pp 87– (2008) · Zbl 1169.91426
[15] Gilks, Markov Chain Monte Carlo in Practice (1996)
[16] Vylder De, Estimation of IBNR claims by credibility theory, Insurance: Mathematics and Economics 1 (1) pp 35– (1982) · Zbl 0504.62091
[17] Neuhaus, Another pragmatic loss reserving method or Bornhuetter-Ferguson revisited, Scandinavian Actuarial Journal 2 pp 151– (1992) · Zbl 0770.62092
[18] Mack, Credible claims reserves: the Benktander method, Astin Bulletin 30 (2) pp 333– (2000) · Zbl 1087.91514
[19] Hess, Convergence of Bayes and Credibility Premiums in the Bühlmann-Straub Model (1994)
[20] Bühlmann, Glaubwürdigkeit für Schadensätze, Bulletin of Swiss Association of Actuaries 70 (1) pp 111– (1970)
[21] Dubey, On parameter estimators in credibility, Bulletin of Swiss Association of Actuaries 81 (1) pp 187– (1981) · Zbl 0482.62092
[22] Norberg, On optimal parameter estimation in credibility, Insurance: Mathematics and Economics 1 (2) pp 73– (1982) · Zbl 0504.62090
[23] Vylder De, Estimation of the heterogeneity parameter in the Bühlmann-Straub credibility theory model, Insurance: Mathematics and Economics 10 (4) pp 233– (1991) · Zbl 0745.62097
[24] Vylder De, Optimal parameter estimation under zero-excess assumptions in the Bühlmann-Straub model, Insurance: Mathematics and Economics 11 (3) pp 167– (1992) · Zbl 0783.62088
[25] Norberg, Empirical Bayes credibility, Scandinavian Actuarial Journal 4 pp 177– (1979) · Zbl 0447.62107
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.