Sarmanov family of bivariate distributions for multivariate loss reserving analysis. (English) Zbl 1414.91154

Summary: The correlation among multiple lines of business plays a critical role in aggregating claims and thus determining loss reserves for an insurance portfolio. We show that the Sarmanov family of bivariate distributions is a convenient choice to capture the dependencies introduced by various sources, including the common calendar year, accident year, and development period effects. The density of the bivariate Sarmanov distributions with different marginals can be expressed as a linear combination of products of independent marginal densities. This pseudo-conjugate property greatly reduces the complexity of posterior computations. In a case study, we analyze an insurance portfolio of personal and commercial auto lines from a major U.S. property-casualty insurer.


91B30 Risk theory, insurance (MSC2010)
62P05 Applications of statistics to actuarial sciences and financial mathematics
62H05 Characterization and structure theory for multivariate probability distributions; copulas
Full Text: DOI Link


[1] Abdallah, A.; Boucher, J. P.; Cossette, H., Modeling Dependence between Loss Triangles with Hierarchical Archimedean Copulas, ASTIN Bulletin, 45, 3, 577-599, (2015) · Zbl 1390.91154
[2] Barnett, G.; Zehnwirth, B., Best Estimates for Reserves, Casualty Actuarial Society Forum, 1-54, (1998)
[3] Boucher, J. P.; Denuit, M.; Guillen, M., Risk Classification for Claim Counts: A Comparative Analysis of Various Zero-Inflated Mixed Poisson and Hurdle Models, North American Actuarial Journal, 11, 4, 110-131, (2007)
[4] Braun, C., The Prediction Error of the Chain Ladder Method Applied to Correlated Run-off Triangles, ASTIN Bulletin, 34, 2, 399-434, (2004) · Zbl 1274.62689
[5] Brehm, P., Correlation and the Aggregation of Unpaid Loss Distributions, Casualty Actuarial Society Forum, 1-23, (2002)
[6] Cohen, L., Probability Distributions with Given Multivariate Marginals, Journal of Mathematical Physics, 25, 2402-2403, (1984) · Zbl 0566.60013
[7] Danaher, P.; Smith, M., Modeling Multivariate Distributions Using Copulas: Applications in Marketing, Marketing Science, 30, 1, 4-21, (2011)
[8] De Jong, P., Forecasting Runoff Triangles, North American Actuarial Journal, 10, 2, 28-38, (2006)
[9] De Jong, P., Modeling Dependence between Loss Triangles, North American Actuarial Journal, 16, 1, 74-86, (2012)
[10] Hernández-Bastida, A.; Fernández-Sánchez, M., A Sarmanov Family with Beta and Gamma Marginal Distributions: An Application to the Bayes Premium in a Collective Risk Model, Statistical Methods and Applications, 21, 4, 91-409, (2012) · Zbl 1332.62042
[11] Hernández-Bastida, A.; Fernández-Sánchez, M.; Gómez-Déniz, E., The Net Bayes Premium with Dependence between the Risk Profiles, Insurance: Mathematics and Economics, 45, 2, 247-254, (2009) · Zbl 1231.91198
[12] Kirschner, G.; Kerley, C.; Isaacs, B., Two Approaches to Calculating Correlated Reserve Indications Across Multiple Lines of Business, Variance, 2, 1, 15-38, (2008)
[13] Kuang, D.; Nielsen, B.; Nielsen, J., Forecasting with the Age-Period-Cohort Model and the Extended Chain-Ladder Model, Biometrika, 95, 4, 987-991, (2008) · Zbl 1437.62516
[14] Lee, M.-L. T., Properties and Applications of the Sarmanov Family of Bivariate Distributions, Comm. Statist. Theory Methods, 25, 6, 1207-1222, (1996) · Zbl 0875.62205
[15] Lowe, J., A Practical Guide to Measuring Reserve Variability Using Bootstrapping, Operational Times and a Distribution-Free Approach, General Insurance Convention, Institute of Actuaries and Faculty of Actuaries, (1994)
[16] Miravete, E., Multivariate Sarmanov Count Data Models, (2009)
[17] Sarmanov, O., Generalized Normal Correlation and Two-Dimensional Fréchet Classes, Doklady (Soviet Mathematics), 168, 596-599, (1966) · Zbl 0203.20001
[18] Schmidt, K., Optimal and Additive Loss Reserving for Dependent Lines of Business, Casualty Actuarial Society Forum, 319-351, (2006)
[19] Schweidel, D.; Fader, P.; Bradlow, E., A Bivariate Timing Model of Customer Acquisition and Retention, Marketing Science, 27, 5, 829-843, (2008)
[20] Shi, P., A Copula Regression for Modeling Multivariate Loss Triangles and Quantifying Reserving Variability, ASTIN Bulletin, 44, 1, 85-102, (2014) · Zbl 1284.62644
[21] Shi, P.; Frees, E., Dependent Loss Reserving Using Copulas, ASTIN Bulletin, 41, 2, 449-486, (2011)
[22] Shi, P.; Basu, S.; Meyers, G., A Bayesian Log-Normal Model for Multivariate Loss Reserving, North American Actuarial Journal, 16, 1, 29-51, (2012) · Zbl 1291.91126
[23] Taylor, G.; McGuire, G., A Synchronous Bootstrap to Account for Dependencies between Lines of Business in the Estimation of Loss Reserve Prediction Error, North American Actuarial Journal, 11, 3, 70-88, (2007)
[24] Wüthrich, M., Accounting Year Effects Modeling in the Stochastic Chain Ladder Reserving Method, North American Actuarial Journal, 14, 2, 235-255, (2010) · Zbl 1219.91074
[25] Wüthrich, M.; Merz, M.; Hashorva, E., Dependence Modelling in Multivariate Claims Run-off Triangles, Annals of Actuarial Science, 7, 1, 3-25, (2013)
[26] Wüthrich, M.; Salzmann, R., Modeling Accounting Year Dependence in Runoff Triangles, European Actuarial Journal, 2, 2, 227-242, (2012) · Zbl 1256.91034
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.