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On unbalanced data and common shock models in stochastic loss reserving

Published online by Cambridge University Press:  27 July 2020

Benjamin Avanzi ,
Greg Taylor ,
Phuong Anh Vu  and
Bernard Wong
Benjamin Avanzi
Affiliation:
Centre for Actuarial Studies, Department of Economics, University of Melbourne, Melbourne, VIC3010, Australia
Greg Taylor
Affiliation:
School of Risk and Actuarial Studies, UNSW Business School, UNSW Sydney, NSW2052, Australia
Phuong Anh Vu*
Affiliation:
Taylor Fry, Level 22, 45 Clarence St, Sydney, NSW2000, Australia
Bernard Wong
Affiliation:
School of Risk and Actuarial Studies, UNSW Business School, UNSW Sydney, NSW2052, Australia
*
*Corresponding author. E-mail: [email protected]
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Abstract

Introducing common shocks is a popular dependence modelling approach, with some recent applications in loss reserving. The main advantage of this approach is the ability to capture structural dependence coming from known relationships. In addition, it helps with the parsimonious construction of correlation matrices of large dimensions. However, complications arise in the presence of “unbalanced data”, that is, when (expected) magnitude of observations over a single triangle, or between triangles, can vary substantially. Specifically, if a single common shock is applied to all of these cells, it can contribute insignificantly to the larger values and/or swamp the smaller ones, unless careful adjustments are made. This problem is further complicated in applications involving negative claim amounts. In this paper, we address this problem in the loss reserving context using a common shock Tweedie approach for unbalanced data. We show that the solution not only provides a much better balance of the common shock proportions relative to the unbalanced data, but it is also parsimonious. Finally, the common shock Tweedie model also provides distributional tractability.

Keywords

Stochastic loss reservingCommon shockUnbalanced dataNegative claimsMultivariate Tweedie distribution
Type
Review
Information
Annals of Actuarial Science , Volume 15 , Issue 1 , March 2021 , pp. 173 - 203
Copyright
© Institute and Faculty of Actuaries 2020

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References

Abdallah, A., Boucher, J.-P. & Cossette, H. (2015). Modeling dependence between loss triangles with hierarchical Archimedean copulas. ASTIN Bulletin, 45, 577–599.CrossRefGoogle Scholar
Ajne, B. (1994). Additivity of chain-ladder projections. ASTIN Bulletin, 24(2), 311–318.CrossRefGoogle Scholar
Alai, D.H., Landsman, Z. & Sherris, M. (2013). Lifetime dependence modelling using a truncated multivariate gamma distribution. Insurance: Mathematics and Economics, 52(3), 542–549.Google Scholar
Alai, D.H., Landsman, Z. & Sherris, M. (2015). A multivariate Tweedie lifetime model: censoring and truncation. Insurance: Mathematics and Economics, 64, 203–213.Google Scholar
Alai, D.H., Landsman, Z. & Sherris, M. (2016). Multivariate Tweedie lifetimes: the impact of dependence, Scandinavian Actuarial Journal, 2016(8), 692–712.CrossRefGoogle Scholar
Alai, D.H. & Wüthrich, M.V. (2009). Taylor approximations for model uncertainty within the Tweedie exponential dispersion family. ASTIN Bulletin, 39(2), 453.CrossRefGoogle Scholar
Avanzi, B., Taylor, G., Vu, P.A. & Wong, B. (2016 b). Stochastic loss reserving with dependence: a flexible multivariate Tweedie approach. Insurance: Mathematics and Economics, 71, 63–78.Google Scholar
Avanzi, B., Taylor, G., Vu, P.A. & Wong, B. (2020). A multivariate evolutionary generalised linear model framework with adaptive estimation for claims reserving. Insurance: Mathematics and Economics, 93, 50–71.Google Scholar
Avanzi, B., Taylor, G. & Wong, B. (2016 a). Correlations between insurance lines of business: an illusion or a real phenomenon? Some methodological considerations. ASTIN Bulletin, 46(2), 225–263.CrossRefGoogle Scholar
Avanzi, B., Taylor, G. & Wong, B. (2018). Common shock models for claim arrays. ASTIN Bulletin, 48(3), 1–28.CrossRefGoogle Scholar
Avanzi, B., Wong, B. & Yang, X. (2016). A micro-level claim count model with overdispersion and reporting delays. Insurance: Mathematics and Economics, 71, 1–14.Google Scholar
Boucher, J.P. & Davidov, D. (2011). On the importance of dispersion modeling for claims reserving: an application with the Tweedie distribution. Variance, 5(2), 158.Google Scholar
Braun, C. (2004). The prediction error of the chain ladder method applied to correlated run-off triangles. ASTIN Bulletin, 34, 399–424.CrossRefGoogle Scholar
Congdon, P.D. (2010). Applied Bayesian Hierarchical Methods. Chapman & Hall, Boca Raton.CrossRefGoogle Scholar
Côté, M.-P., Genest, C. & Abdallah, A. (2016). Rank-based methods for modeling dependence between loss triangles. European Actuarial Journal, 6(2), 377–408.CrossRefGoogle ScholarPubMed
De Alba, E. (2006). Claims reserving when there are negative values in the runoff triangle: Bayesian analysis using the three-parameter log-normal distribution. North American Actuarial Journal, 10(3), 45–59.CrossRefGoogle 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
England, P.D. & Verrall, R.J. (2002). Stochastic claims reserving in general insurance. British Actuarial Journal, 8(3), 443–518.CrossRefGoogle Scholar
England, P.D., Verrall, R.J. & Wüthrich, M.V. (2012). Bayesian over-dispersed Poisson model and the Bornhuetter– Ferguson claims reserving method. Annals of Actuarial Science, 6(2), 258–283.CrossRefGoogle Scholar
Furman, E. & Landsman, Z. (2010). Multivariate Tweedie distributions and some related capital-at-risk analyses. Insurance: Mathematics and Economics, 46(2), 351–361.Google Scholar
Gismondi, F., Janssen, J. & Manca, R. (2012). The construction of the claims reserve distribution by means of a semi-Markov backward simulation model. Annals of Actuarial Science, 6(1), 23–64.CrossRefGoogle Scholar
Haario, H., Saksman, E. & Tamminen, J. (2001). An adaptive Metropolis algorithm. Bernoulli, 7(2), 223–242.CrossRefGoogle Scholar
Heberle, J. & Thomas, A. (2016). The fuzzy Bornhuetter–Ferguson method: an approach with fuzzy numbers. Annals of Actuarial Science, 10(2), 303–321.CrossRefGoogle Scholar
Hess, K.T., Schmidt, K.D. & Zocher, M. (2006). Multivariate loss prediction in the multivariate additive model. Insurance: Mathematics and Economics, 39(2), 185–191.Google Scholar
International Actuarial Association (2004). A global framework for insurer solvency assessment, Website. Last accessed: 9/12/2018. https://www.actuaries.org/LIBRARY/Papers/Global_Framework_Insurer_Solvency_Assessment-public.pdf Google Scholar
Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman & Hall, New York.Google Scholar
Jørgensen, B. (1997). The Theory of Dispersion Models. Chapman & Hall, London.Google Scholar
Kunkler, M. (2006). Modelling negatives in stochastic reserving models. Insurance: Mathematics and Economics, 38(3), 540–555.Google Scholar
Merz, M. & 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. & Wüthrich, M.V. (2009). Prediction error of the multivariate additive loss reserving method for dependent lines of business. Variance, 3(1), 131–151.Google Scholar
Meyers, G.G. (2007). The common shock model for correlated insurance losses. Variance, 1(1), 40–52.Google Scholar
Peters, G.W., Shevchenko, P. & Wüthrich, M.V. (2009). Model uncertainty in claims reserving within Tweedie’s compound Poisson models. ASTIN Bulletin, 39(1), 1–33.CrossRefGoogle Scholar
Pinheiro, P.J.R., Andrade e Silva, J.M. & de Lourdes Centeno, M. (2003). Bootstrap methodology in claim reserving. Journal of Risk and Insurance, 70(4), 701–714.CrossRefGoogle Scholar
Pröhl, C. & Schmidt, K.D. (2005). Multivariate Chain-Ladder, Technology University, Institute für Mathematische Stochastik.Google Scholar
Renshaw, A.E. & Verrall, R.J. (1998). A stochastic model underlying the chain-ladder technique. British Actuarial Journal, 4(04), 903–923.CrossRefGoogle Scholar
Saluz, A. & Gisler, A. (2014). Best estimate reserves and the claims development results in consecutive calendar years. Annals of Actuarial Science, 8(2), 351–373.CrossRefGoogle Scholar
Schmidt, K.D. (2006). Optimal and additive loss reserving for dependent lines of business. Casualty Actuarial Society Forum (Fall), pp. 319–351.Google Scholar
Shi, P. (2014). A copula regression for modeling multivariate loss triangles and quantifying reserving variability. ASTIN Bulletin, 44(1), 85–102.CrossRefGoogle Scholar
Shi, P., Basu, S. & Meyers, G.G. (2012). A Bayesian log-normal model for multivariate loss reserving. North American Actuarial Journal, 16(1), 29–51.CrossRefGoogle Scholar
Shi, P. & Frees, E.W. (2011). Dependent loss reserving using copulas. ASTIN Bulletin, 41(2), 449–486.Google Scholar
Taylor, G. (2009). The chain ladder and Tweedie distributed claims data. Variance, 3(1), 96–104.Google Scholar
Taylor, G. (2015). Bayesian chain ladder models. ASTIN Bulletin, 45(1), 75–99.CrossRefGoogle Scholar
Taylor, G.C. (2000). Loss Reserving: An Actuarial Perspective. Kluwer Academic Publishers, Boston.CrossRefGoogle Scholar
Verrall, R., Hössjer, O. & Björkwall, S. (2012). Modelling claims run-off with reversible jump Markov chain Monte Carlo methods. ASTIN Bulletin, 42(1), 35–58.Google Scholar
Vihola, M. (2012). Robust adaptive metropolis algorithm with coerced acceptance rate. Statistics and Computing, 22(5), 997–1008.CrossRefGoogle Scholar
Wüthrich, M.V. (2003). Claim reserving using Tweedie’s compound poisson model. ASTIN Bulletin, 33(3), 331–346.CrossRefGoogle Scholar
Wüthrich, M.V. (2018). Machine learning in individual claims reserving. Scandinavian Actuarial Journal, 2018(6), 465–480.CrossRefGoogle Scholar
Wüthrich, M.V. & Merz, M. (2008). Stochastic Claims Reserving Methods in Insurance. John Wiley & Sons, Chichester.Google Scholar
Zhang, Y. (2010). A general multivariate chain ladder model. Insurance: Mathematics and Economics, 46(3), 588–599.Google Scholar
Zhang, Y. & Dukic, V. (2013). Predicting multivariate insurance loss payments under the Bayesian copula framework. The Journal of Risk and Insurance, 80(4), 891–919.CrossRefGoogle Scholar
Zhang, Y., Dukic, V. & Guszcza, J. (2012). A Bayesian non-linear model for forecasting insurance loss payments. Journal of Royal Statistical Society, 175(2), 637–656.CrossRefGoogle Scholar
Zhao, X.B., Zhou, X. & Wang, J.L. (2009). Semiparametric model for prediction of individual claim loss reserving. Insurance: Mathematics and Economics, 45(1), 1–8.Google Scholar