Recursive Credibility Formula for Chain Ladder Factors and the Claims Development Result

Abstract

In recent Solvency II considerations much effort has been put into the development of appropriate models for the study of the one-year loss reserving uncertainty in non-life insurance. In this article we derive formulas for the conditional mean square error of prediction of the one-year claims development result in the context of the Bayes chain ladder model studied in Gisler-Wüthrich [9].The key to these formulas is a recursive representation for the results obtained in Gisler-Wüthrich [9]. KEYWORDSClaims reserving, chain ladder method, credibility chain ladder method, claims development result, year end expectation, loss experience prior accident years, liability at maturity, solvency, mean square error of prediction.% % % % % % % % % %

Chain Ladder and Error Propagation

“…For the development over the one-year horizon, our formula generalizes the formula given by Merz and Wüthrich (2008) and Bühlmann et al (2009) -for a proof, see the appendix. It also specializes to Result 4.5 (case H = I,q j =q j ,q j = 0,r j = 1 for all j ).…”

“…For a discussion of the split into process and parameter error, see also Merz and Wüthrich (2008) and, in the context of a Bayesian chain ladder model, Bühlmann et al (2009), Remarks 4.2 and 4.8.…”

“…Changing the summation order, we get Bühlmann et al (2009) considered the case of a one-period development and derived various formulae for the prediction error of the claims development result. We will show that their "linear approximation" (Bühlmann et al, 2009, Result 4.10, which goes back to Merz and Wüthrich, 2008), is a special case of our formula (36).…”

Reversible Jump Markov Chain Monte Carlo Method for Parameter Reduction in Claims Reserving

“…On the one hand it provides the chain-ladder reserves and on the other hand it is very easy to generate bootstrap samples from. For the calculation of predictive distributions of the lower triangle and for the study of prediction and model uncertainty it is often advantageous to introduce prior distributions for the parameters (including any available expert knowledge on them), see Gisler-Wüthrich [8] and Bühlmann et al [2]. A Bayesian ODP model was briefly covered in England-Verrall [5] and discussed in detail in England et al [6].…”

Market Value Margin calculations under the Cost of Capital approach within a Bayesian chain ladder framework

“…For the claims reserving problem, Bayesian methods are now well investigated (see e.g. [17], [2]) and they provide an interesting approach for a successive information update in each accounting year. Log-normal chain ladder models have been introduced by Hertig [8] and their Bayesian versions have been recently used by Merz and Wüthrich [12] to model paid-incurred chain claims.…”