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Multivariate distributions with time and cross-dependence: aggregation and capital allocation. (English) Zbl 1492.91297

Summary: This paper investigates risk aggregation and capital allocation problems for an insurance portfolio consisting of several lines of business. The class of multivariate INAR(1) processes is proposed to model different sources of dependence between the number of claims of the portfolio. The total capital required for the whole portfolio is evaluated under the TVaR risk measure, and the contribution of each line of business is derived under the TVaR-based allocation rule. We provide the risk aggregation and capital allocation formulas in the general case of continuous and strictly positive claim sizes and then in the case of mixed Erlang claim sizes. The impact of both time dependence and cross-dependence on the behavior of risk aggregation and capital allocation is numerically illustrated.

MSC:

91G05 Actuarial mathematics
62P05 Applications of statistics to actuarial sciences and financial mathematics
62H10 Multivariate distribution of statistics

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[1] Acerbi, C. and Tasche, D. (2002) On the coherence of expected shortfall. Journal of Banking and Finance, 26(7), 1487-1503.
[2] Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D. (1999) Coherent measures of risk. Mathematical Finance, 9(3), 203-228. · Zbl 0980.91042
[3] Bargès, M., Cossette, H. and Marceau, E. (2009) TVaR-based capital allocation with copulas. Insurance: Mathematics and Economics, 45(3), 348-361. · Zbl 1231.91141
[4] Bermúdez, L., Guillén, M. and Karlis, D. (2018) Allowing for time and cross dependence assumptions between claim counts in ratemaking models. Insurance: Mathematics and Economics, 83(6), 161-169. · Zbl 1417.91262
[5] Cai, J., Landriault, D., Shi, T. and Wei, W. (2017) Joint insolvency analysis of a shared MAP risk process: a capital allocation application. North American Actuarial Journal, 21(2), 178-192. · Zbl 1414.91168
[6] Cai, J. and Li, H. (2010) Conditional tail expectations for multivariate phase-type distributions. Journal of Applied Probability, 42(3), 810-825. · Zbl 1079.62022
[7] Cossette, H., Côté, M.-P., Marceau, E. and Moutanabbir, K. (2013) Multivariate distribution defined with Farlie-Gumbel-Morgenstern copula and mixed Erlang marginals: aggregation and capital allocation. Insurance: Mathematics and Economics, 52(3), 560-572. · Zbl 1284.60027
[8] Cossette, H., Mailhot, M. and Marceau, E. (2012) TVaR-based capital allocation for multivariate compound distributions with positive continuous claim amounts. Insurance: Mathematics and Economics, 50(2), 247-256. · Zbl 1235.91086
[9] Cossette, H., Marceau, E. and Maume, D.V. (2010) Discrete-time risk models based on time series for count random variables. ASTIN Bulletin, 40(1), 123-150. · Zbl 1230.91071
[10] Cossette, H., Marceau, E. and Perreault, S. (2015) On two families of bivariate distributions with exponential marginals: aggregation and capital allocation. Insurance: Mathematics and Economics, 64(5), 214-224. · Zbl 1348.91137
[11] Cossette, H., Marceau, E. and Toureille, F. (2011) Risk models based on time series for count random variables. Insurance: Mathematics and Economics, 48(1), 19-28. · Zbl 1218.91074
[12] Cossette, H., Marceau, E., Trufin, J. and Zuyderhoff, P. (2020). Ruin-based risk measures in discrete-time risk models. Insurance: Mathematics and Economics, 93(4), 246-261. · Zbl 1447.91132
[13] Cummins, J.D. (2000) Allocation of capital in the insurance industry. Risk Management and Insurance Review, 3(1), 7-27.
[14] Darolles, S., Le Fol, G., Lu, Y. and Sun, R. (2019) Bivariate integer-autoregressive process with an application to mutual fund flows. Journal of Multivariate Analysis, 173, 181-203. · Zbl 1451.60038
[15] Dhaene, J., Goovaerts, M.J. and Kaas, R. (2003) Economic capital allocation derived from risk measures. North American Actuarial Journal, 7(2), 44-59. · Zbl 1084.91515
[16] Dhaene, J., Henrard, L., Landsman, Z., Vandendorpe, A. and Vanduffel, S. (2008) Some results on the CTE-based capital allocation rule. Insurance: Mathematics and Economics, 42(2), 855-863. · Zbl 1152.91577
[17] Dhaene, J., Tsanakas, A., Valdez, E.A. and Vanduffel, S. (2012) Optimal capital allocation principles. Journal of Risk and Insurance, 79(1), 1-28.
[18] Furman, E. and Landsman, Z. (2010) Multivariate Tweedie distributions and some related capital-at-risk analyses. Insurance: Mathematics and Economics, 46(2), 351-361. · Zbl 1231.91185
[19] Gourieroux, C. and Jasiak, J. (2004) Heterogeneous INAR(1) model with application to car insurance. Insurance: Mathematics and Economics, 34(2), 177-192. · Zbl 1107.62110
[20] Johnson, N.L., Kotz, S. and Balakrishnan, N. (1997) Discrete Multivariate Distributions. New York: Wiley. · Zbl 0868.62048
[21] Lee, S.C.K. and Lin, X.S. (2010) Modeling and evaluating insurance losses via mixtures of Erlang distributions. North American Actuarial Journal, 14(1), 107-130.
[22] Lindskog, F. and Mcneil, A. (2003) Common Poisson shock models: Applications to insurance and credit risk modelling. ASTIN Bulletin, 33(2), 209-238. · Zbl 1087.91030
[23] Mcneil, A.J., Frey, R. and Embrechts, P. (2015) Quantitative Risk Management: Concepts, Techniques and Tools. New Jersey: Princeton University Press. · Zbl 1337.91003
[24] Myers, S.C. and Read, J.A. (2001) Capital allocation for insurance companies. Journal of Risk and Insurance, 68(4), 545-580.
[25] Panjer, H.H. (2002) Measurement of Risk, Solvency Requirements, and Allocation of Capital within Financial Conglomerates. Institute of Insurance and Pension Research, University of Waterloo Research.
[26] Pedeli, X. and Karlis, D. (2013a) On composite likelihood estimation of a multivariate INAR(1) model. Journal of Time Series Analysis, 34(2), 206-220. · Zbl 1274.62376
[27] Pedeli, X. and Karlis, D. (2013b) Some properties of multivariate INAR(1) processes. Computational Statistics and Data Analysis, 67(11), 213-225. · Zbl 1471.62158
[28] Ratovomirija, G. (2016) On mixed Erlang reinsurance risk: aggregation, capital allocation and default risk. European Actuarial Journal, 6(1), 149-175. · Zbl 1415.91162
[29] Ratovomirija, G., Tamraz, M. and Vernic, R. (2017) On some multivariate Sarmanov mixed Erlang reinsurance risks: aggregation and capital allocation. Insurance: Mathematics and Economics, 74(3), 197-209. · Zbl 1394.62145
[30] Tsanakas, A. (2009) To split or not to split: capital allocation with convex risk measures. Insurance: Mathematics and Economics, 44(2), 268-277. · Zbl 1165.91423
[31] Vernic, R. (2006) Multivariate skew-normal distributions with applications in insurance. Insurance: Mathematics and Economics, 38(2), 413-26. · Zbl 1132.91501
[32] Vernic, R. (2011) Tail conditional expectation for the multivariate Pareto distribution of the second kind: another approach. Methodology and Computing in Applied Probability, 13(1), 121-137. · Zbl 1208.60014
[33] Vernic, R. (2017) Capital allocation for Sarmanov’s class of distributions. Methodology and Computing in Applied Probability, 19(1), 311-330. · Zbl 1358.60034
[34] Willmot, G.E. and Lin, X.S. (2011) Risk modeling with the mixed Erlang distribution. Applied Stochastic Models in Business and Industry, 27(1), 8-22.
[35] Zhang, L., Hu, X. and Duan, B. (2015) Optimal reinsurance under adjustment coefficient measure in a discrete risk model based on Poisson MA(1) process. Scandinavian Actuarial Journal, 2015(5), 455-467. · Zbl 1401.91213
[36] Zhou, M., Dhaene, J. and Yao, J. (2018) An approximation method for risk aggregations and capital allocation rules based on additive risk factor models. Insurance: Mathematics and Economics, 79(2), 92-100. · Zbl 1401.91218
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