Copula-based Modeling for IBNR Claim Loss Reserving

Document Type : Article


1 School of Mathematical and Statistical Sciences, Southern Illinois University Carbondale, IL 62901, USA

2 Department of Statistics, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran

3 Department of Statistics, Science and Research Islamic Azad University, Tehran, Iran

4 Department of Mathematical Sciences, Stevens Institute of Technology, Hoboken, NJ 07030, USA


There are growing concerns for reserves estimation of incurred but not reported (IBNR) claims in actuarial
sciences. In this paper, we propose a copula-based dependency model to capture the relationship between
two main IBNR reserve variables, i.e., the “time between two successive occurrences” and “delay time”.
A maximum likelihood estimation (MLE) method is used to estimate the parameters of the model. A
simulation study is conducted to evaluate the validity of the theoretical results. Moreover, the proposed
method is applied to predict the number of claims for the next years of a portfolio from a major automobile
insurer and is compared to the classical Chain Ladder model forecasting.



[1] Bornhuetter, R.L. and Ferguson, R.E. “November. The actuary and IBNR”, In Proceedings of the casualty actuarial society, 59(112), pp. 181-195 (1972).

[2] Benktander, G. “An Approach to Credibility in Calculating IBNR for Casualty Excess Reinsurance”, Actuarial Review, p7 (1976).

[3] Hovinen, E. Additive and continuous IBNR, ASTIN Colloquium, Loen, Norway, (1981).
[4] Bühlmann, H., and Straub, E. “Estimation of IBNR reserves by the methods chain ladder, Cape Cod
and complementary loss ratio”, In International Summer School, (1983).
[5] Stanard, J. N. A simulation test of prediction errors of loss reserve estimation techniques (Doctoral
dissertation, New York University, Graduate School of Business Administration), (1985).
[6] Mack, T. “Distribution-free calculation of the standard error of chain ladder reserve estimates”, ASTIN
Bulletin: The Journal of the IAA, 23(2), pp. 213-225 (1993).
[7] Bettonville, C., d’Oultremont, L., Denuit, M., et al. “Matrix calculation for ultimate and 1-year risk
in the Semi-Markov individual loss reserving model”, Scandinavian Actuarial Journal, 5, pp. 380-407
[8] Wang, Z., Wu, X. and Qiu, C. “The impacts of individual information on loss reserving”, ASTIN
Bulletin: The Journal of the IAA, 51(1), pp. 303-347 (2021).
[9] Yanez, J.S. and Pigeon, M. “Micro-level parametric duration-frequency-severity modeling for outstanding
claim payments”, Insurance: Mathematics and Economics, 98, pp. 106-119 (2021).
[10] Barbiero, A. “A proposal for modeling and simulating correlated discrete Weibull variables”, Scientia
Iranica. Transaction E, Industrial Engineering 25(1), pp. 386-397 (2018).
[11] Rasaizadi, A. and Kermanshah, M. “Mode choice and number of non-work stops during the commute:
Application of a copula-based joint model”, Scientia Iranica, 25(3), pp.1039-1047 (2018).
[12] Purutcuoglu, V. and Farnoudkia, H. “Copula Gaussian graphical modelling of biological networks and
Bayesian inference of model parameters”, Scientia Iranica, 26(4), pp. 2495-2505 (2019).
[13] Pettere, G. and Kollo, T. “Modelling claim size in time via copulas”, In Transactions of 28th International
Congress of Actuaries, 206, (2006).
[14] Zhao, X., and Zhou, X. “Applying copula models to individual claim loss reserving methods”, Insurance:
Mathematics and Economics, 46(2), pp. 290-299 (2010).
[15] Zhao, X.B., Zhou, X. and Wang, J.L. “Semiparametric model for prediction of individual claim loss
reserving”, Insurance: Mathematics and Economics, 45(1), pp. 1-8 (2009).
[16] Shi, P. and Frees, E.W. “Dependent loss reserving using copulas”, ASTIN Bulletin: The Journal of the
IAA, 41(2), pp. 449-486 (2011).
[17] Badescu, A.L., Lin, X.S. and Tang, D. “A marked Cox model for the number of IBNR claims: Theory”,
Insurance: Mathematics and Economics, 69, pp. 29-37 (2016).
[18] Avanzi, B., Wong, B. and Yang, X. “A micro-level claim count model with overdispersion and reporting
delays”, Insurance: Mathematics and Economics, 71, pp. 1-14 (2016).
[19] Landriault, D., Willmot, G.E. and Xu, D. “Analysis of IBNR claims in renewal insurance models”,
Scandinavian Actuarial Journal, 7, pp. 628-650 (2017).
[20] Crevecoeur, J., Antonio, K. and Verbelen, R. “Modeling the number of hidden events subject to observation
delay”, European Journal of Operational Research, 277(3), pp. 930-944 (2019).
[21] Atatalab, F. and Payandeh Najafabadi, A.T. “Prediction of outstanding IBNR liabilities using delay
probability”, Journal of Mathematics and Modeling in Finance, (2021).
[22] Cheung, E.C., Rabehasaina, L., Woo, J.K., et al. “Asymptotic correlation structure of discounted
Incurred But Not Reported claims under fractional Poisson arrival process”, European Journal of Operational
Research, 276(2), pp. 582-601 (2019).
[23] Costa, L. and Pizzinga, A. “State-space models for predicting IBNR reserve in row-wise ordered runoff
triangles: Calendar year IBNR reserves & tail effects”, Journal of Forecasting, 39(3), pp. 438-448 (2020).
[24] Fung, T.C., Badescu, A.L. and Lin, X.S. “A new class of severity regression models with an application
to IBNR prediction”, North American Actuarial Journal, pp. 1-26 (2020).
[25] Hendrych, R. and Cipra, T. “Applying State Space Models to Stochastic Claims Reserving”, ASTIN
Bulletin: The Journal of the IAA, 51(1), pp. 267-301 (2021).
[26] Manski, S., Yang, K., Lee, G.Y., et al. “Extracting information from textual descriptions for actuarial
applications”, Annals of Actuarial Science, pp. 1-18 (2021).
[27] Weissner, E.W. “Estimation of the distribution of report lags by the method of maximum likelihood”,
In Proceedings of the Casualty Actuarial Society, 65, pp. 1-9 (1978).
[28] Clayton, D.G. “A model for association in bivariate life tables and its application in epidemiological
studies of familial tendency in chronic disease incidence”, Biometrika, 65(1), pp. 141-151 (1978).
[29] Oakes, D. “On the preservation of copula structure under truncation”, The Canadian Journal of Statistics,
33, pp. 465-468 (2005).
[30] Jewell, W. “Predicting IBNYR events and delays, part I continuous time”, Astin Bulletin, 19, , pp.
25-56 (1989).
[31] Jewell, W. “Predicting IBNYR events and delays, part II discrete time”, Astin Bulleti, 20, pp. 93-111
[32] Wüthrich, M.V. and Merz, M. Stochastic claims reserving methods in insurance 435, John Wiley and
Sons, (2008).
[33] Badounas, I., Bozikas, A. and Pitselis, G. “A robust random coefficient regression representation of the
chain-ladder method”, Annals of Actuarial Science, pp. 1-32 (2021).
[34] Delong, Ł., Lindholm, M. and Wüthrich, M.V. “Collective reserving using individual claims data”
Scandinavian Actuarial Journal, pp. 1-28 (2021).
[35] Fischinger, D. and Gach, F. “The 1-year premium risk”, European Actuarial Journal, pp. 1-21 (2021).
[36] Portugal, L., Pantelous, A.A. and Verrall, R. “Univariate and multivariate claims reserving with Generalized
Link Ratios”, Insurance: Mathematics and Economics, 97, pp. 57-67 (2021).
[37] Sklar, M. “Fonctions de répartition à n dimensions et leurs marges”, Université Paris 8, (1959).
[38] Nelsen, R. B. An Introduction to Copulas, New York: Springer Science Business Media, (2006).
[39] Liu, G., Long, W., Yang, B., et al. “Semiparametric estimation and model selection for conditional
mixture copula models”, Scandinavian Journal of Statistics, (2021).
[40] Cherubini, U., Luciano, E., and Vecchiato, W. Copula Methods in Finance, John Wiley and Sons,
Chichester, (2004).
[41] Katesari, H. S., and Vajargah, B. F. “Testing Adverse Selection Using Frank Copula Approach in Iran
Insurance Markets”, Mathematics and Computer Science, 15(2), pp. 154-158 (2015).
[42] Katesari, H.S. and Zarodi, S. “Effects of Coverage Choice by Predictive Modeling on Frequency of
Accidents”, Caspian Journal of Applied Sciences Research, 5(3), pp. 28-33 (2016).
[43] Safari-Katesari, H. and Zaroudi, S. “Count copula regression model using generalized beta distribution
of the second kind”, Statistics in Transition New Series, 21(2), pp. 1-12 (2020).
[44] Safari-Katesari, H. and Zaroudi, S. “Analysing the impact of dependency on conditional survival functions
using copulas”, Statistics in Transition New Series, 22(1), pp. 217-226 (2021).