Calendar Year Dependence Modeling in Run-Off Triangles

Transcription

1 Calendar Year Dependence Modeling in Run-Off Triangles Mario V. Wüthrich RiskLab, ETH Zurich May 21-24, 2013 ASTIN Colloquium The Hague wueth

2 Claims reserves at time I = 2010 accident year development year j i to be predicted X i,j denotes the payments for accident year i in development year j. Thus, X i,j is paid in calendar year k = i + j. Observations at time I = 2010: D I = {X i,j ; i + j I}. Best-estimate reserves: R I = i+j>i E [X i,j D I ]. 1

3 Independence between accident years i accident year development year j i Classical stochastic claims reserving methods assume independence between payments X i,j of different accident years i. 2

4 Claims inflation and calendar year effects accident year development year j i Payments in different accident years i are not independent if, for instance, they are subject to claims inflation and (external) calendar year effects. de Jong (2006, 2012), Kuang-Nielsen-Nielsen (2008), Shi-Basu-Meyers (2012), W. (2010, 2012), Donnelly-W. (2012), Merz-W.-Hashorva (2013). 3

5 Log-link chain-ladder model Assume cumulative payments C i,j = X i, X i,j > 0. We consider the individual log-link ratios C i,j = C i,j 1 exp (ξ i,j ) ξ i,j = log (C i,j /C i,j 1 ). Set d = (J + 1)I and define the random vectors ξ i = (ξ i,0,..., ξ i,j ) R J+1 and ξ = (ξ 1,..., ξ I) R d. = Random vector ξ contains all individual log-link ratios. 4

6 Bayesian multivariate Gauss chain-ladder model Model assumptions. Conditionally, given Θ R J+1, ξ {Θ} is multivariate Gauss with covariance matrix Σ R d d, and mean E [ξ i Θ] = Θ for all i {1,..., I}. The parameter has multivariate prior distribution Θ (d) N (µ, T ). 5

7 Bayesian multivariate Gauss chain-ladder model Model assumptions. Conditionally, given Θ R J+1, ξ {Θ} is multivariate Gauss with covariance matrix Σ R d d, and mean E [ξ i Θ] = Θ for all i {1,..., I}. The parameter has multivariate prior distribution Θ (d) N (µ, T ). 6

8 Consequences of the model assumptions The mean E [ξ i Θ] = Θ does not depend on i {1,..., I}: E [ξ i,j Θ] = Θ j, with Θ j reflecting the chain-ladder factor for development year j. Covariance matrix Σ allows for any correlation structure between (all) individual log-link ratios in ξ, for example, for (i, j) (l, m) Corr (ξ i,j, ξ l,m Θ) = ρ > 0 for all i + j = l + m. (1) These are exactly claims inflation and calendar year effects. See also Shi-Basu-Meyers (NAAJ 2012) and W. (NAAJ 2012). The choice of the prior distribution allows to implement parameter uncertainty and expert knowledge about chain-ladder parameters in a natural way Θ (d) N (µ, T ). 7

9 Possible choices of covariance matrix Σ i i j j i j top, lhs: independent ξ i,j s (= independent accident years) top, rhs: calendar year correlation (1), i.e. for ξ i,j and ξ l,m with i + j = l + m bottom, rhs: random walk on calendar year axis further models: AR(p) process on calendar year axis, etc. 8

10 Predictive distribution in lower triangles Denote by ξ DI the observations in the upper triangle D I ; and by ξ D c I the unobserved components in the lower triangle D c I. Theorem 1 (predictive distribution). We have posterior distribution ξ D c I { ξ DI } (d) N (µ post, S post ), with explicit (credibility) formulas for µ post and S post. S post contains both covariance matrix Σ and parameter uncertainty T and reflects volatilities in individual log-link ratios ξ i,j s, dependence between individual log-link ratios ξ i,j and ξ l,m, prior uncertainty in chain-ladder parameters Θ. 9

11 Summary We can choose any covariance structure Σ between individual log-link ratios ξ i,j s, and any prior information µ and T on the parameter space Θ, and then Theorem 1 provides the predictive distribution of ξ D c I, given the observations ξ DI, in closed form. Moreover, we have closed form solutions for the ultimate claim predictor and the claims reserves, the MSEP of the total uncertainty and the one-year CDR uncertainty. Our model allows in solvency models for a bottom-up calibration of correlation by specifying Σ. 10

12 Case study in correlation model (1) 3000, , , , , ,000,0 0% 20% 40% 60% 80% 100% claims reserves confidence ultimate confidence CDR 100% 80% 60% 40% 20% 0% 0% 20% 40% 60% 80% 100% ultimate msep^(1/2) CDR msep^(1/2) linear function (a) lhs: prediction and 1-std.dev. confidence bounds of total uncertainty and one-year CDR uncertainty as a function of ρ [0, 1) in model (1). (b) rhs: relative increase of uncertainty as a function of ρ [0, 1) in model (1). For more details see conference paper. 11

13 Conclusions Dependencies and correlation have a huge effect on confidence bounds and quantiles: more case studies are needed! Effects of correlations are often counter-intuitive, see Conclusions 1-5 in Merz-W.-Hashorva [1]. Our model allows in solvency models for a bottom-up calibration of correlation. Issue: choices of reasonable correlation structures: choose, for instance, an AR(1) process on the calendar year axis. 12

14 References [1] Merz, M., Wüthrich, M.V., Hashorva, E. (2013). Dependence modelling in multivariate claims run-off triangles. Annals Actuarial Science 7/1, [2] Shi, P., Basu, S., Meyers, G.G. (2012). A Bayesian log-normal model for multivariate loss reserving. North American Actuarial Journal 16/1, [3] Wüthrich, M.V. (2012). Discussion of A Bayesian lognormal model for multivariate loss reserving by Shi-Basu- Meyers. North American Actuarial Journal 16/3, [4] Wüthrich, M.V., Merz, M. (2013). Financial Modeling, Actuarial Valuation and Solvency in Insurance. Springer. Jakob Bernoulli, Law of Large Numbers, Ars Contectandi,