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Applying state space models to stochastic claims reserving. (English) Zbl 1471.91462

Summary: The paper solves the loss reserving problem using Kalman recursions in linear state space models. In particular, if one orders claims data from run-off triangles to time series with missing observations, then state space formulation can be applied for projections or interpolations of IBNR (incurred but not reported) reserves. Namely, outputs of the corresponding Kalman recursion algorithms for missing or future observations can be taken as the IBNR projections. In particular, by means of such recursive procedures one can perform effectively simulations in order to estimate numerically the distribution of IBNR claims which may be very useful in terms of setting and/or monitoring of prudency level of loss reserves. Moreover, one can generalize this approach to the multivariate case of several dependent run-off triangles for correlated business lines and the outliers in claims data can be also treated effectively in this way. Results of a numerical study for several sets of claims data (univariate and multivariate ones) are presented.

MSC:

91G05 Actuarial mathematics
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[1] Alpuim, T. and Ribeiro, I. (2003) A state space model for run-off triangles. Applied Stochastic Models in Business and Industry, 19, 105-120. · Zbl 1052.91056
[2] Atherino, R., Pizzinga, A. and Fernandes, C. (2010) A row-wise stacking of the runoff triangle: State space alternatives for IBNR reserve prediction. ASTIN Bulletin, 40(2), 917-946. · Zbl 1232.91338
[3] Avanzi, B., Lavender, M., Taylor, G. and Wong, B. (2016) On the impact detection and treatment of outliers in robust loss reserving. In Proceedings of the Actuaries Institute, General Insurance Seminar, pp. 13-15, Melbourne.
[4] 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, 27-37. · Zbl 1218.91069
[5] Braun, C. (2004) The prediction error of the chain ladder method applied to correlated run-off triangles. ASTIN Bulletin, 34(2), 399-423. · Zbl 1274.62689
[6] Brockwell, P.J. and Davis, R.A. (2016) Introduction to Time series and Forecasting, 3^rd edition. Springer. · Zbl 1355.62001
[7] Carrato, A., Concina, F., Gesmann, M., Murphy, D., Wuthrich, M. and Zhang, W. (2019) Claims reserving with R: ChainLadder-0.2.10 Package Vignette. https://rdrr.io/cran/ChainLadder/.
[8] Chukhrova, N. and Johannssen, A. (2017) State space models and the Kalman-filter in stochastic claims reserving: Forecasting, filtering and smoothing. Risks5, 30. doi: doi:10.3390/risks5020030.
[9] Cipra, T. (2010) Financial and Insurance Formulas. Springer. · Zbl 1200.91001
[10] Cipra, T. and Romera, R. (1997) Kalman filter with outliers and missing observations. Test6, 379-395. · Zbl 0893.62094
[11] 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. · Zbl 1514.62500
[12] De Jong, P. (2005) State space models in actuarial science. Research Paper No. 2005/02, Macquarie University, Sydney.
[13] De Jong, P. (2006) Forecasting runoff triangles. North American Actuarial Journal, 10(2), 28-38. · Zbl 1479.91317
[14] De Jong, P. (2012) Modeling dependence between loss triangles. North American Actuarial Journal, 16(1), 74-86.
[15] De Jong, P. and Zehnwirth, B. (1983) Claim reserving, state-space models and the Kalman filter. Journal of the Institute of Actuaries, 110, 157-181.
[16] Durbin, J. and Koopman, S.J. (2002) A simple and efficient simulation smoother for state space time series analysis. Biometrika, 89(3), 603-615. · Zbl 1036.62071
[17] Durbin, J. and Koopman, S.J. (2012) Time Series Analysis by State Space Methods, 2^nd edition. Oxford University Press. · Zbl 1270.62120
[18] England, P.D. and Verrall, R.J. (2002) Stochastic claims reserving in general insurance. British Actuarial Journal, 8(3), 443-518.
[19] Hamilton, J.D. (1994) Time Series Analysis. Princeton University Press. · Zbl 0831.62061
[20] Harvey, A.C. (1989) Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge University Press.
[21] Helske, J. (2017) KFAS: Exponential family state space models in R. Journal of Statistical Software, 78(10), 1-38. doi: doi:10.18637/jss.v078.i10.
[22] Johannssen, A. (2016) Stochastische Schadenreservierung unter Verwendung vonZustands-raummodellen und des Kalman-Filters. Hamburg: Verlag Dr. Kovac.
[23] Kloek, T. (1998) Loss development forecasting models: An econometrician’s view. Insurance: Mathematics and Economics, 23, 251-261. · Zbl 0917.62090
[24] Koopman, S.J., Harvey, A.C., Doornik, J.A. and Shephard, N. (2009) STAMP 8.2: Structural Time Series Analyser, Modeller and Predictor. London: Timberlake Consultants.
[25] Kremer, E. (1982) IBNR claims and the two way model of ANOVA. Scandinavian Actuarial Journal, 1982(1), 47-55. · Zbl 0495.62092
[26] Li, J. (2006) Comparison of stochastic reserving methods. Australian Actuarial Journal, 12, 489-569.
[27] Mack, T. (1993) Distribution-free calculation of the standard error of chain ladder reserve estimates. ASTIN Bulletin, 23(2), 213-225.
[28] Merz, M. and Wüthrich, M.V. (2007) Prediction error of the chain ladder reserving method applied to correlated run-off triangles. Annals of Actuarial Science, 2(1), 25-50.
[29] Merz, M. and Wüthrich, M.V. (2008) Prediction error of the multivariate chain ladder reserving method. North American Actuarial Journal, 12(2), 175-197. · Zbl 1481.91180
[30] Merz, M., Wüthrich, M.V. and Hashorva, E. (2012) Dependence modelling in multivariate claims run-off triangles. Annals of Actuarial Science, 7(1), 3-25.
[31] Ntzoufras, I. and Dellaportas, P. (2002) Bayesian modelling of outstanding liabilities incorpo-rating claim count uncertainty. North American Actuarial Journal, 6(1), 13-28. · Zbl 1084.62544
[32] Pang, L. and He, S. (2012) The application of state-space model in outstanding claims reserve. In Conference Paper, International Conference on Information Management, Innovation Management and Industrial Engineering 2012. Sanya.
[33] Peremans, K., Van Aelst, S. and Verdonck, T. (2018) A robust general multivariate chain ladder method. Risks, 6, 108. doi: doi:10.3390risks6040108.
[34] Pitselis, G., Grigoriadou, V. and Badounas, I. (2015) Robust loss reserving in a log-linear model. Insurance: Mathematics and Economics, 64, 14-27. · Zbl 1348.62244
[35] Pröhl, C. and Schmidt, K.D. (2005) Multivariate chain-ladder. Dresdner Schriften zur Versi-cherungsmathematik. Technische Universität Dresden.
[36] (1991): Historical Loss Development Study. Reinsurance Association of America, Washington D.C.
[37] Renshaw, A.E. (1989) Chain-ladder and interactive modelling (claims reserving and GLIM). Journal of the Institute of Actuaries, 116, 559-587.
[38] Renshaw, A.E. and Verrall, R.J. (1998) A stochastic model underlying the chain-ladder technique. British Actuarial Journal, 4(4), 903-923.
[39] Shi, P., Basu, S. and Meyers, G.G. (2012) A Bayesian log-normal model for multivariate loss reserving. North American Actuarial Journal, 16(1), 29-51. · Zbl 1291.91126
[40] Shumway, R.H. and Stoffer, D.S. (1982) An approach to time series smoothing and forecasting using the EM algorithm. Journal of Time Series Analysis, 3(4), 253-264. · Zbl 0502.62085
[41] Shumway, R.H. and Stoffer, D.S. (2017) Time Series Analysis and Its Applications (With R Examples), 4th edition. New York: Springer. · Zbl 1367.62004
[42] Stoffer, D.S. and Wall, K.D. (1991) Bootstrapping state-space models: Gaussian maximum likelihood estimation of the Kalman filter. Journal of the American Statistical Association, 86(416), 1024-1033. · Zbl 0850.62693
[43] Taylor, G.C. and Ashe, F.R. (1983) Second moments of estimates of outstanding claims. Journal of Econometrics, 23, 37-61.
[44] Taylor, G.C., Mcguire, G. and Greenfield, A. (2003) Loss reserving: Past, present and future. Research Paper No. 109. University of Melbourne 2003 (an invited lecture to the 34th ASTIN Colloquium, Berlin 2003).
[45] Verdonck, T. and Van Wouwe, M. (2011) Detection and correction of outliers in the bivariate chain-ladder method. Insurance: Mathematics and Economics, 49, 188-193. · Zbl 1218.91098
[46] Verdonck, T., Van Wouwe, M. and Dhaene, J. (2009) A robustification of the chain-ladder method. North American Actuarial Journal, 13(2), 280-298. · Zbl 1483.91210
[47] Verrall, R. (1989) A state space representation of the chain ladder linear model. Journal of the Institute of Actuaries, 116, 589-610.
[48] Verrall, R. (1991) On the estimation of reserves from loglinear models. Insurance: Mathematics and Economics, 10, 75-80. · Zbl 0723.62070
[49] Verrall, R. (1994) A method for modelling varying run-off evolutions in claims reserving. ASTINBulletin, 24(2), 325-332.
[50] Wright, T. (1990) A stochastic method for claims reserving in general insurance. Journal of the Institute of Actuaries, 117, 677-731.
[51] Wüthrich, M.V. and Merz, M. (2008) Stochastic Claims Reserving Methods in Insurance. Chichester: Wiley. · Zbl 1273.91011
[52] Zehnwirth, B. (1996) Kalman filters with applications to loss reserving. Research Paper 27, University of Melbourne, Department of Economics, Centre for Actuarial Studies.
[53] Zhang, Y. (2010) A general multivariate chain ladder model. Insurance: Mathematics and Economics, 46, 588-599. · Zbl 1231.91258
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