Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T17:03:59.041Z Has data issue: false hasContentIssue false
  • English
  • Français

APPLYING STATE SPACE MODELS TO STOCHASTIC CLAIMS RESERVING

Published online by Cambridge University Press:  24 November 2020

Radek Hendrych Open the ORCID record for Radek Hendrych [Opens in a new window]  and
Tomas Cipra
Radek Hendrych*
Affiliation:
Faculty of Mathematics and Physics Department of Probability and Mathematical Statistics Charles University Prague, Czech Republic E-Mail: hendrych@karlin.mff.cuni.cz
Tomas Cipra
Affiliation:
Faculty of Mathematics and Physics Department of Probability and Mathematical Statistics Charles University Prague, Czech Republic E-Mail: cipra@karlin.mff.cuni.cz
*
E-Mail: hendrych@karlin.mff.cuni.cz
Get access Rights & Permissions [Opens in a new window]

Abstract

The paper solves the loss reserving problem using Kalman recursions in linear statespace 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.

Keywords

Dependent run-off trianglesIBNR projectionsmissing observationsstate space modelstochastic claims reservingC22C32G22
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 51 , Issue 1 , January 2021 , pp. 267 - 301
Copyright
© 2020 by Astin Bulletin. All rights reserved

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

Alpuim, T. and Ribeiro, I. (2003) A state space model for run-off triangles. Applied Stochastic Models in Business and Industry, 19, 105–120.CrossRefGoogle Scholar
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.Google Scholar
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.Google 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, 27–37.Google Scholar
Braun, C. (2004) The prediction error of the chain ladder method applied to correlated run-off triangles. ASTIN Bulletin, 34(2), 399–423.CrossRefGoogle Scholar
Brockwell, P.J. and Davis, R.A. (2016) Introduction to Time series and Forecasting, 3rd edition. Springer.CrossRefGoogle Scholar
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/.Google Scholar
Chukhrova, N. and Johannssen, A. (2017) State space models and the Kalman-filter in stochastic claims reserving: Forecasting, filtering and smoothing. Risks 5, 30. doi: 10.3390/risks5020030.CrossRefGoogle Scholar
Cipra, T. (2010) Financial and Insurance Formulas. Springer.CrossRefGoogle Scholar
Cipra, T. and Romera, R. (1997) Kalman filter with outliers and missing observations. Test 6, 379–395.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
de Jong, P. (2005) State space models in actuarial science. Research Paper No. 2005/02, Macquarie University, Sydney.Google Scholar
de Jong, P. (2006) Forecasting runoff triangles. North American Actuarial Journal, 10(2), 28–38.CrossRefGoogle Scholar
de Jong, P. (2012) Modeling dependence between loss triangles. North American Actuarial Journal, 16(1), 74–86.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Durbin, J. and Koopman, S.J. (2002) A simple and efficient simulation smoother for state space time series analysis. Biometrika, 89(3), 603–615.CrossRefGoogle Scholar
Durbin, J. and Koopman, S.J. (2012) Time Series Analysis by State Space Methods, 2nd edition. Oxford University Press.CrossRefGoogle Scholar
England, P.D. and Verrall, R.J. (2002) Stochastic claims reserving in general insurance. British Actuarial Journal, 8(3), 443–518.CrossRefGoogle Scholar
Hamilton, J.D. (1994) Time Series Analysis. Princeton University Press.CrossRefGoogle Scholar
Harvey, A.C. (1989) Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge University Press.Google Scholar
Helske, J. (2017) KFAS: Exponential family state space models in R. Journal of Statistical Software, 78(10), 1–38. doi: 10.18637/jss.v078.i10.CrossRefGoogle Scholar
Johannssen, A. (2016) Stochastische Schadenreservierung unter Verwendung vonZustands-raummodellen und des Kalman-Filters. Hamburg: Verlag Dr. Kovac.Google Scholar
Kloek, T. (1998) Loss development forecasting models: An econometrician’s view. Insurance: Mathematics and Economics, 23, 251–261.Google Scholar
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.Google Scholar
Kremer, E. (1982) IBNR claims and the two way model of ANOVA. Scandinavian Actuarial Journal, 1982(1), 47–55.CrossRefGoogle Scholar
Li, J. (2006) Comparison of stochastic reserving methods. Australian Actuarial Journal, 12, 489–569.Google Scholar
Mack, T. (1993) Distribution-free calculation of the standard error of chain ladder reserve estimates. ASTIN Bulletin, 23(2), 213–225.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Merz, M. and Wüthrich, M.V. (2008) Prediction error of the multivariate chain ladder reserving method. North American Actuarial Journal, 12(2), 175–197.Google Scholar
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.CrossRefGoogle Scholar
Ntzoufras, I. and Dellaportas, P. (2002) Bayesian modelling of outstanding liabilities incorpo-rating claim count uncertainty. North American Actuarial Journal, 6(1), 13–28.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Peremans, K., Van Aelst, S. and Verdonck, T. (2018) A robust general multivariate chain ladder method. Risks, 6, 108. doi: 10.3390risks6040108.CrossRefGoogle Scholar
Pitselis, G., Grigoriadou, V. and Badounas, I. (2015) Robust loss reserving in a log-linear model. Insurance: Mathematics and Economics, 64, 14–27.Google Scholar
Pröhl, C. and Schmidt, K.D. (2005) Multivariate chain-ladder. Dresdner Schriften zur Versi-cherungsmathematik. Technische Universität Dresden.Google Scholar
RAA (1991): Historical Loss Development Study. Reinsurance Association of America, Washington D.C.Google Scholar
Renshaw, A.E. (1989) Chain-ladder and interactive modelling (claims reserving and GLIM) . Journal of the Institute of Actuaries, 116, 559–587.Google 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
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Shumway, R.H. and Stoffer, D.S. (2017) Time Series Analysis and Its Applications (With R Examples), 4th edition. New York: Springer.CrossRefGoogle Scholar
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.Google Scholar
Taylor, G.C. and Ashe, F.R. (1983) Second moments of estimates of outstanding claims. Journal of Econometrics, 23, 37–61.CrossRefGoogle Scholar
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).Google Scholar
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.Google Scholar
Verdonck, T., Van Wouwe, M. and Dhaene, J. (2009) A robustification of the chain-ladder method. North American Actuarial Journal, 13(2), 280–298.CrossRefGoogle Scholar
Verrall, R. (1989) A state space representation of the chain ladder linear model. Journal of the Institute of Actuaries, 116, 589–610.CrossRefGoogle Scholar
Verrall, R. (1991) On the estimation of reserves from loglinear models. Insurance: Mathematics and Economics, 10, 75–80.Google Scholar
Verrall, R. (1994) A method for modelling varying run-off evolutions in claims reserving. ASTINBulletin, 24(2), 325–332.Google Scholar
Wright, T. (1990) A stochastic method for claims reserving in general insurance. Journal of the Institute of Actuaries, 117, 677–731.CrossRefGoogle Scholar
Wüthrich, M.V. and Merz, M. (2008) Stochastic Claims Reserving Methods in Insurance. Chichester: Wiley.Google Scholar
Zehnwirth, B. (1996) Kalman filters with applications to loss reserving. Research Paper 27, University of Melbourne, Department of Economics, Centre for Actuarial Studies.Google Scholar
Zhang, Y. (2010) A general multivariate chain ladder model. Insurance: Mathematics and Economics, 46, 588–599.Google Scholar