## Sequential Monte Carlo samplers for capital allocation under copula-dependent risk models.(English)Zbl 1314.91241

Summary: In this paper we assume a multivariate risk model has been developed for a portfolio and its capital derived as a homogeneous risk measure. The Euler (or gradient) principle, then, states that the capital to be allocated to each component of the portfolio has to be calculated as an expectation conditional to a rare event, which can be challenging to evaluate in practice. We exploit the copula-dependence within the portfolio risks to design a Sequential Monte Carlo Samplers based estimate to the marginal conditional expectations involved in the problem, showing its efficiency through a series of computational examples.

### MSC:

 91G60 Numerical methods (including Monte Carlo methods) 91B30 Risk theory, insurance (MSC2010) 65C05 Monte Carlo methods 65C40 Numerical analysis or methods applied to Markov chains 62P05 Applications of statistics to actuarial sciences and financial mathematics

### Software:

copula; HAC; QRM; LibBi; copula
Full Text:

### References:

 [1] Arbenz, P., Cambou, M., Hofert, M., 2014. An importance sampling algorithm for copula models in insurance. ArXiv Preprint arXiv:1403.4291. · Zbl 1405.65005 [2] Artzner, P.; Delbaen, F.; Eber, J. M.; Heath, D., Coherent measures of risk, Math. Finance, 9, 3, 203-228, (1999) · Zbl 0980.91042 [3] Asimit, A. V.; Vernic, R.; Zitikis, R., Evaluating risk measures and capital allocations based on multi-losses driven by a heavy-tailed background risk: the multivariate Pareto-II model, Risks, 1, 1, 14-33, (2013) [4] Asmussen, S.; Glynn, P. W., Stochastic simulation: algorithms and analysis: algorithms and analysis, vol. 57, (2007), Springer · Zbl 1126.65001 [5] International convergence of capital measurement and capital standards—a revised framework (comprehensive version). tech. rep., (2006), Bank for International Settlements, URL: http://www.bis.org/publ/bcbs128.pdf [6] Böcker, K.; Klüppelberg, C., Modelling and measuring multivariate operational risk with Lévy copulas, J. Oper. Risk, 3, 2, 3-27, (2008) [7] Brechmann, E.; Czado, C.; Paterlini, S., Flexible dependence modeling of operational risk losses and its impact on total capital requirements, J. Bank. Finance, 40, 271-285, (2014) [8] Buch, A.; Dorfleitner, G., Coherent risk measures, coherent capital allocations and the gradient allocation principle, Insurance Math. Econom., 42, 1, 235-242, (2008) · Zbl 1141.91490 [9] Cai, J.; Einmahl, J. H.; De Haan, L.; Zhou, C., Estimation of the marginal expected shortfall: the mean when a related variable is extreme, J. R. Stat. Soc. Ser. B Stat. Methodol., (2015), forthcoming · Zbl 1414.91433 [10] Carmona, R.; Fouque, J. P.; Vestal, D., Interacting particle systems for the computation of rare credit portfolio losses, Finance Stoch., 13, 4, 613-633, (2009) · Zbl 1199.91248 [11] Chopin, N., Central limit theorem for sequential Monte Carlo methods and its application to Bayesian inference, Ann. Statist., 32, 6, 2385-2411, (2004) · Zbl 1079.65006 [12] Creal, D., A survey of sequential Monte Carlo methods for economics and finance, Econometric Rev., 31, 3, 245-296, (2012) [13] Crisan, D.; Doucet, A., A survey of convergence results on particle filtering methods for practitioners, IEEE Trans. Signal Process., 50, 3, 736-746, (2002) · Zbl 1369.60015 [14] Cruz, M. G.; Peters, G. W.; Shevchenko, P. V., Fundamental aspects of operational risk modeling and insurance analytics: A handbook of operational risk, (2015), John Wiley & Sons [15] De Luca, G.; Rivieccio, G., Multivariate tail dependence coefficients for Archimedean copulae, (Advanced Statistical Methods for the Analysis of Large Data-Sets, (2012), Springer), 287-296 [16] Del Moral, P., Feynman-Kac formulae, (2004), Springer · Zbl 1130.60003 [17] Del Moral, P.; Doucet, A.; Jasra, A., Sequential Monte Carlo samplers, J. R. Stat. Soc. Ser. B Stat. Methodol., 68, 3, 411-436, (2006) · Zbl 1105.62034 [18] Del Moral, P.; Peters, G. W.; Vergé, C., An introduction to stochastic particle integration methods: with applications to risk and insurance, (Monte Carlo and Quasi-Monte Carlo Methods 2012, (2013), Springer), 39-81 · Zbl 1302.65034 [19] Denault, M., Coherent allocation of risk capital, J. Risk, 4, 1, 1-34, (2001) [20] Doucet, A.; De Freitas, N.; Gordon, N., Sequential Monte Carlo methods in practice, (2001), Springer · Zbl 0967.00022 [21] Doucet, A.; Johansen, A. M., A tutorial on particle filtering and smoothing: fifteen years later, (Handbook of Nonlinear Filtering, vol. 12, (2009)), 656-704 · Zbl 05919872 [22] Gamerman, D.; Lopes, H. F., Markov chain Monte Carlo: stochastic simulation for Bayesian inference, (2006), CRC Press · Zbl 1137.62011 [23] Giacometti, R.; Rachev, S.; Chernobai, A.; Bertocchi, M., Aggregation issues in operational risk, J. Oper. Risk, 3, 3, 3-23, (2008) [24] Gilks, W. R.; Berzuini, C., Following a moving target Monte Carlo inference for dynamic Bayesian models, J. R. Stat. Soc. Ser. B Stat. Methodol., 63, 1, 127-146, (2001) · Zbl 0976.62021 [25] Glasserman, P., Measuring marginal risk contributions in credit portfolios, J. Comput. Finance, 9, 2, 1, (2005) [26] Gouriéroux, C.; Laurent, J. P.; Scaillet, O., Sensitivity analysis of values at risk, J. Empir. Finance, 7, 3, 225-245, (2000) [27] Hofert, J. M., Sampling nested Archimedean copulas: with applications to CDO pricing, (2010), Suedwestdeutscher Verlag fuer Hochschulschriften [28] Hofert, M., Kojadinovic, I., Maechler, M., Yan, J., 2014. copula: multivariate dependence with copulas. URL: http://CRAN.R-project.org/package=copula. R package version 0.999-9. [29] Kalkbrener, M., An axiomatic approach to capital allocation, Math. Finance, 15, 3, 425-437, (2005) · Zbl 1102.91049 [30] Künsch, H. R., Recursive Monte Carlo filters: algorithms and theoretical analysis, Ann. Statist., 33, 5, 1983-2021, (2005) · Zbl 1086.62106 [31] Liu, J. S.; Chen, R., Sequential Monte Carlo methods for dynamic systems, J. Amer. Statist. Assoc., 93, 443, 1032-1044, (1998) · Zbl 1064.65500 [32] McLeish, D. L., Bounded relative error importance sampling and rare event simulation, Astin Bull., 40, 1, 377-398, (2010) · Zbl 1191.65007 [33] McNeil, A. J.; Frey, R.; Embrechts, P., Quantitative risk management: concepts, techniques, and tools, (2010), Princeton University Press · Zbl 1089.91037 [34] Murray, L.M., 2013. Bayesian state-space modelling on high-performance hardware using LibBi. ArXiv Preprint arXiv:1306.3277. [35] Neal, R. M., Slice sampling, Ann. Statist., 31, 3, 705-741, (2003) · Zbl 1051.65007 [36] Okhrin, O.; Ristig, A., Hierarchical Archimedean copulae: the HAC package, J. Stat. Softw., 58, 4, 1-20, (2014), URL: http://www.jstatsoft.org/v58/i04 [37] Peters, G. W., Topics in sequential Monte Carlo samplers, (2005), University of Cambridge, Department of Engineering, M.Sc. [38] Peters, G. W.; Shevchenko, P. V., Advances in heavy tailed risk modeling: A handbook of operational risk, (2015), John Wiley & Sons [39] Peters, G. W.; Shevchenko, P. V.; Wüthrich, M. V., Dynamic operational risk: modeling dependence and combining different sources of information, J. Oper. Risk, 4, 2, 69-104, (2009) [40] Ristic, B.; Arulampalam, S.; Gordon, N., Beyond the Kalman filter: particle filters for tracking applications, (2004), Artech House · Zbl 1092.93041 [41] Rosen, D.; Saunders, D., Risk factor contributions in portfolio credit risk models, J. Bank. Finance, 34, 2, 336-349, (2010) [42] Shevchenko, P. V., Modelling operational risk using Bayesian inference, (2011), Springer · Zbl 1213.91011 [43] Siller, T., Measuring marginal risk contributions in credit portfolios, Quant. Finance, 13, 12, 1915-1923, (2013) · Zbl 1282.91379 [44] Tasche, D., Risk contributions and performance measurement. report of the lehrstuhl für mathematische statistik, (1999), TU München [45] Tasche, D., Capital allocation to business units and sub-portfolios: the Euler principle, (Pillar II in the New Basel Accord: The Challenge of Economic Capital, (2008), Risk Books), 423-453
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