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MULTIVARIATE DISTRIBUTIONS WITH TIME AND CROSS-DEPENDENCE: AGGREGATION AND CAPITAL ALLOCATION

Published online by Cambridge University Press:  27 April 2022

Xiang Hu  and
Lianzeng Zhang
Xiang Hu*
Affiliation:
School of Finance Zhongnan University of Economics and Law Nanhu Road, Wuhan, 430073, P.R. China
Lianzeng Zhang
Affiliation:
Department of Actuarial Science, School of Finance Nankai University Tongyan Road, Tianjin, 300350, P.R. China E-mail: zhlz@nankai.edu.cn
*
E-mail: xianghu@zuel.edu.cn
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Abstract

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.

Keywords

Multivariate INAR(1) processestime dependencecross-dependencerisk aggregationcapital allocation
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 52 , Issue 2 , May 2022 , pp. 669 - 706
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The International Actuarial Association

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References

Acerbi, C. and Tasche, D. (2002) On the coherence of expected shortfall. Journal of Banking and Finance, 26(7), 1487–1503.CrossRefGoogle Scholar
Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D. (1999) Coherent measures of risk. Mathematical Finance, 9(3), 203–228.CrossRefGoogle Scholar
Bargès, M., Cossette, H. and Marceau, E. (2009) TVaR-based capital allocation with copulas. Insurance: Mathematics and Economics, 45(3), 348–361.Google Scholar
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.Google Scholar
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.CrossRefGoogle Scholar
Cai, J. and Li, H. (2010) Conditional tail expectations for multivariate phase-type distributions. Journal of Applied Probability, 42(3), 810–825.CrossRefGoogle Scholar
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.Google Scholar
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.Google Scholar
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.CrossRefGoogle Scholar
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.Google Scholar
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.Google Scholar
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.Google Scholar
Cummins, J.D. (2000) Allocation of capital in the insurance industry. Risk Management and Insurance Review, 3(1), 7–27.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Dhaene, J., Goovaerts, M.J. and Kaas, R. (2003) Economic capital allocation derived from risk measures. North American Actuarial Journal, 7(2), 44–59.CrossRefGoogle Scholar
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.Google Scholar
Dhaene, J., Tsanakas, A., Valdez, E.A. and Vanduffel, S. (2012) Optimal capital allocation principles. Journal of Risk and Insurance, 79(1), 1–28.CrossRefGoogle Scholar
Furman, E. and Landsman, Z. (2010) Multivariate Tweedie distributions and some related capital-at-risk analyses. Insurance: Mathematics and Economics, 46(2), 351–361.Google Scholar
Gourieroux, C. and Jasiak, J. (2004) Heterogeneous INAR(1) model with application to car insurance. Insurance: Mathematics and Economics, 34(2), 177–192.Google Scholar
Johnson, N.L., Kotz, S. and Balakrishnan, N. (1997) Discrete Multivariate Distributions. New York: Wiley.Google Scholar
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.CrossRefGoogle Scholar
Lindskog, F. and McNeil, A. (2003) Common Poisson shock models: Applications to insurance and credit risk modelling. ASTIN Bulletin, 33(2), 209–238.CrossRefGoogle Scholar
McNeil, A.J., Frey, R. and Embrechts, P. (2015) Quantitative Risk Management: Concepts, Techniques and Tools. New Jersey: Princeton University Press.Google Scholar
Myers, S.C. and Read, J.A. (2001) Capital allocation for insurance companies. Journal of Risk and Insurance, 68(4), 545–580.CrossRefGoogle Scholar
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.Google Scholar
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.CrossRefGoogle Scholar
Pedeli, X. and Karlis, D. (2013b) Some properties of multivariate INAR(1) processes. Computational Statistics and Data Analysis, 67(11), 213–225.CrossRefGoogle Scholar
Ratovomirija, G. (2016) On mixed Erlang reinsurance risk: aggregation, capital allocation and default risk. European Actuarial Journal, 6(1), 149–175.CrossRefGoogle Scholar
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.Google Scholar
Tsanakas, A. (2009) To split or not to split: capital allocation with convex risk measures. Insurance: Mathematics and Economics, 44(2), 268–277.Google Scholar
Vernic, R. (2006) Multivariate skew-normal distributions with applications in insurance. Insurance: Mathematics and Economics, 38(2), 413–26.Google Scholar
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.CrossRefGoogle Scholar
Vernic, R. (2017) Capital allocation for Sarmanov’s class of distributions. Methodology and Computing in Applied Probability, 19(1), 311–330.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.Google Scholar
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