Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T17:11:43.293Z Has data issue: false hasContentIssue false
  • English
  • Français

NEW LOSS RESERVE MODELS WITH PERSISTENCE EFFECTS TO FORECAST TRAPEZOIDAL LOSSES IN RUN-OFF TRIANGLES

Published online by Cambridge University Press:  21 September 2022

Farha Usman  and
Jennifer S.K. Chan
Farha Usman*
Affiliation:
School of Mathematics and Statistics University of Sydney Sydney, NSW 2006, Australia
Jennifer S.K. Chan*
Affiliation:
School of Mathematics and Statistics University of Sydney Sydney, NSW 2006, Australia
*
E-Mail: [email protected]
E-Mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Modelling loss reserve data in run-off triangles is challenging due to the complex but unknown dynamics in the claim/loss process. Popular loss reserve models describe the mean process through development year, accident year, and calendar year effects using the analysis of variance and covariance (ANCOVA) models. We propose to include in the mean function the persistence terms in the conditional autoregressive range model for modelling the persistence of claim across development years. In the ANCOVA model, we adopt linear trends for the accident and calendar year effects and a quadratic trend for the development year effect. We investigate linear or log-transformed mean functions and four distributions, namely generalised beta type 2, generalised gamma, Weibull, and exponential extension, with positive support to enhance the model flexibility. The proposed models are implemented using the Bayesian user-friendly package Stan running in the R environment. Results show that the models with log-transformed mean function and persistence terms provide better model fits. Lastly, the best model is applied to forecast partial loss reserve and calendar year reserve for three years.

Keywords

Loss reservingrun-off trianglepersistenceconditional autoregressive range modelBayesian analysisgeneralised beta type 2 distribution
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 52 , Issue 3 , September 2022 , pp. 877 - 920
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The International Actuarial Association

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

AbuJarad, M.H. and Khan, A.A. (2018) Exponential model: A Bayesian study with Stan. International Journal of Recent Scientific Research, 9(8), 28495–28506.Google Scholar
Aiuppa, T.A. (1988) Evaluation of Pearson curves as an approximation of the maximum probable annual aggregate loss. The Journal of Risk and Insurance, 55(3), 425–441.CrossRefGoogle Scholar
Anderson, D.R. (1971) Effects of under and over-evaluations in loss reserves. Journal of Risk and Insurance, 585–600.CrossRefGoogle Scholar
Annis, J., Miller, B. and Palmeri, T. (2017) Bayesian inference with Stan: A tutorial on adding custom distributions. Behavior Research Methods, 49(3), 863–886.CrossRefGoogle ScholarPubMed
Antonio, K. and Plat, R. (2014) Micro-level stochastic loss reserving for general insurance. Scandinavian Actuarial Journal, 2014(7), 649–669.CrossRefGoogle Scholar
Atherino, R., Pizzinga, A. and Fernandes, C. (2010) A row-wise stacking of the run-off triangle: State space alternatives for IBNR reserve prediction. ASTIN Bulletin: The Journal of the IAA, 40(2), 917–946.Google Scholar
Bakar, S.A., Hamzah, N.A., Maghsoudi, M. and Nadarajah, S. (2015) Modeling loss data using composite models. Insurance: Mathematics and Economics, 61, 146–154.Google Scholar
Berger, J. (2006) The case for objective Bayesian analysis. Bayesian Analysis, 1(3), 385–402.CrossRefGoogle Scholar
Björkwall, S., Hössjer, O., Ohlsson, E. and Verrall, R. (2011) A generalized linear model with smoothing effects for claims reserving. Insurance: Mathematics and Economics, 49(1), 27–37.Google Scholar
Carpenter, B., Gelman, A., Hoffman, M.D., Lee, D., Goodrich, B., Betancourt, M., Brubaker, M., Guo, J., Li, P. and Riddell, A. (2017) Stan: A probabilistic programming language. Journal of statistical software, 76(1).CrossRefGoogle Scholar
Chan, J. (2015). Predicting loss reserves using quantile regression. Journal of Data Science, 13(1), 127–155.Google Scholar
Chan, J., Choy, S. and Makov, U. (2008) Robust Bayesian analysis of loss reserves data using the generalized-t distribution. ASTIN Bulletin: The Journal of the IAA, 38(1), 207–230.CrossRefGoogle Scholar
Chan, J., Choy, S., Makov, U. and Landsman, Z. (2018) Modelling insurance losses using contaminated generalised beta type-II distribution. ASTIN Bulletin: The Journal of the IAA, 48(2), 871–904.CrossRefGoogle Scholar
Chou, R.Y., Chou, H. and Liu, N. (2015) Range volatility: A review of models and empirical studies. In Handbook of Financial Econometrics and Statistics, pp. 2029–2050.CrossRefGoogle Scholar
Costa, L. and Pizzinga, A. (2020) State-space models for predicting IBNR reserve in row-wise ordered run-off triangles: Calendar year IBNR reserves & tail effects. Journal of Forecasting, 39(3), 438–448.CrossRefGoogle Scholar
Costa, L., Pizzinga, A. and Atherino, R. (2016) Modeling and predicting IBNR reserve: Extended chain ladder and heteroscedastic regression analysis. Journal of Applied Statistics, 43(5), 847–870.CrossRefGoogle Scholar
Cummins, D., Lewis, C. and Phillips, R. (1999) Pricing excess-of-loss reinsurance contracts against cat as trophic loss. In The Financing of Catastrophe Risk, pp. 93–148. University of Chicago Press.Google Scholar
Cummins, J., Mcdonald, J. and Merrill, C. (2007) Risky loss distributions and modeling the loss reserve pay-out tail. Review of Applied Economics, 3, 1–23.Google Scholar
Cummins, J.D., Dionne, G., McDonald, J.B. and Pritchett, B.M. (1990) Applications of the GB2 family of distributions in modeling insurance loss processes. Insurance: Mathematics and Economics, 9(4), 257–272.Google Scholar
De Jong, P. (2006) Forecasting runoff triangles. North American Actuarial Journal, 10(2), 28–38.Google Scholar
De Jong, P., Heller, G.Z., et al. (2008). Generalized Linear Models for Insurance Data. Cambridge University Press: Cambridge Books.CrossRefGoogle Scholar
De Jong, P. and Zehnwirth, B. (1983) Claims reserving, state-space models and the Kalman filter. Journal of the Institute of Actuaries (1886-1994), 110(1), 157–181.CrossRefGoogle Scholar
De Myttenaere, A., Golden, B., Le Grand, B. and Rossi, F. (2016) Mean absolute percentage error for regression models. Neurocomputing, 192, 38–48.CrossRefGoogle Scholar
Dong, A.X. and Chan, J. (2013). Bayesian analysis of loss reserving using dynamic models with generalized beta distribution. Insurance: Mathematics and Economics, 53(2), 355–365.Google Scholar
Duane, S., Kennedy, A.D., Pendleton, B.J. and Roweth, D. (1987) Hybrid Monte Carlo. Physics Letters B, 195(2), 216–222.CrossRefGoogle Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (2013) Modelling Extremal Events: For Insurance and Finance, vol. 33. Springer Science & Business Media.Google Scholar
England, P.D. and Verrall, R.J. (2001) A flexible framework for stochastic claims reserving. In Proceedings of the Casualty Actuarial Society, 88, 1–38.Google Scholar
Fernandes, M. and Grammig, J. (2006). A family of autoregressive conditional duration models. Journal of Econometrics, 130(1), 1–23.CrossRefGoogle Scholar
Gelman, A. and Rubin, D.B. (1992). Inference from iterative simulation using multiple sequences. Statistical Science, 7(4), 457–472.CrossRefGoogle Scholar
Geman, S. and Geman, D. (1984) Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-6(6), 721–741.CrossRefGoogle Scholar
Hoffman, M.D. and Gelman, A. (2014) The No-U-Turn sampler: Adaptively setting path lengths in Hamiltonian Monte Carlo. Journal of Machine Learning Research, 15(1), 1593–1623.Google Scholar
Kremer, E. (1984) A class of autoregressive models for predicting the final claims amount. Insurance: Mathematics and Economics, 3(2), 111–119.Google Scholar
Lee, M.D. and Wagenmakers, E.-J. (2014) Bayesian Cognitive Modeling: A Practical Course. Cambridge University Press.CrossRefGoogle Scholar
Mack, T. (1993) Distribution-free calculation of the standard error of chain ladder reserve estimates. ASTIN Bulletin: The Journal of the IAA, 23(2), 213–225.CrossRefGoogle Scholar
Mack, T. and Venter, G. (2000) A comparison of stochastic models that reproduce chain ladder reserve estimates. Insurance: Mathematics and Economics, 26(1), 101–107.Google Scholar
McKenzie, J. (2011) Mean absolute percentage error and bias in economic forecasting. Economics Letters, 113(3), 259–262.Google Scholar
Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H. and Teller, E. (1953) Equation of state calculations by fast computing machines. The Journal of Chemical Physics, 21(6), 1087–1092.CrossRefGoogle Scholar
Meyers, G. (2015) Stochastic loss reserving with Bayesian MCMC models. Geophysical Monograph Series, 1, 1.Google Scholar
Morteza, K. and Ahmadabadi, A. (2010) Some properties of generalized gamma distribution. Mathematical Sciences Quarterly Journal, 4, 9-28.Google Scholar
Nadarajah, S. and Haghighi, F. (2011) An extension of the exponential distribution. Statistics, 45, 543–558.Google Scholar
Neal, R.M. (1994) An improved acceptance procedure for the hybrid Monte Carlo algorithm. Journal of Computational Physics, 111(1), 194–203.CrossRefGoogle Scholar
Ntzoufras, I. and Dellaportas, P. (2002) Bayesian modelling of outstanding liabilities incorporating claim count uncertainty. North American Actuarial Journal, 6(1), 113–125.CrossRefGoogle Scholar
Paulson, A.S. and Faris, N. (1985) A practical approach to measuring the distribution of total annual claims. In Strategic Planning and Modeling in Property-Liability Insurance, pp. 205–223. Springer.CrossRefGoogle Scholar
Peters, G.W., Wüthrich, M.V. and Shevchenko, P.V. (2010) Chain ladder method: Bayesian bootstrap versus classical bootstrap. Insurance: Mathematics and Economics, 47(1), 36–51.Google Scholar
Ramlau-Hansen, H. (1988) A solvency study in non-life insurance. Scandinavian Actuarial Journal, 1988(1–3), 35–59.CrossRefGoogle Scholar
Renshaw, A.E. (1989) Chain ladder and interactive modelling (Claims reserving and GLIM). Journal of the Institute of Actuaries (1886-1994), 116(3), 559–587.CrossRefGoogle Scholar
Renshaw, A.E. and Verrall, R.J. (1998) A stochastic model underlying the chain-ladder technique. British Actuarial Journal, 4(4), 903–923.CrossRefGoogle Scholar
Taylor, G. (2012) Loss Reserving: An Actuarial Perspective, vol. 21. Springer Science & Business Media.Google Scholar
Taylor, G. and McGuire, G. (2016) Stochastic loss reserving using generalized linear models. Arlington: Casualty Actuarial Society.Google Scholar
Vehtari, A., Gelman, A., Simpson, D., Carpenter, B., Bürkner, P.-C. (2021) Rank-normalization, folding, and localization: An improved $\widehat{R}$ for assessing convergence of MCMC. Bayesian Analysis, 16(2):667–718.Google Scholar
Venter, G. (2007) Generalized linear models beyond the exponential family with loss reserve applications. ASTIN Bulletin: The Journal of the IAA, 37(2), 345–364.Google Scholar
Venter, G. (2018) Loss reserving using estimation methods designed for error reduction. Available at SSRN 3096178.CrossRefGoogle Scholar
Venter, G., Gutkovich, R. and Gao, Q. (2017) Parameter reduction in actuarial triangle models. Variance, Forthcoming, UNSW Business School Research Paper Forthcoming.CrossRefGoogle Scholar
Venter, G. and Sahin, Ş. (2018) Parsimonious parameterization of age-period-cohort models by Bayesian shrinkage. ASTIN Bulletin: The Journal of the IAA, 48(1), 89–110.CrossRefGoogle Scholar
Verrall, R. (1996) Claims reserving and generalised additive models. Insurance: Mathematics and Economics, 19(1), 31–43.Google Scholar
Verrall, R. (2000) An investigation into stochastic claims reserving models and the chain-ladder technique. Insurance: Mathematics and Economics, 26, 91–99.Google Scholar
Zhang, C. and Browne, M.J. (2013) Loss reserve errors, income smoothing and firm risk of property and casualty insurance companies. In Annual Meeting of the American Risk and Insurance Association, Working Pa, pp. 1–55.Google Scholar
Zhang, Z.J., Milovanovic, D.J., Tomita, M. and Zicarreli, D.J. (2018) Using generalized linear models to develop loss triangles in reserving. Actuarial Review, 37–40.Google Scholar
Supplementary material: PDF

Usman and Chan supplementary material

Usman and Chan supplementary material

Download Usman and Chan supplementary material(PDF)
PDF 144.4 KB