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Efficient estimation of large portfolio loss probabilities in \(t\)-copula models. (English) Zbl 1188.91231
Summary: We consider the problem of accurately measuring the credit risk of a portfolio consisting of loans, bonds and other financial assets. One particular performance measure of interest is the probability of large portfolio losses over a fixed time horizon. We revisit the so-called \(t\)-copula that generalizes the popular normal copula to allow for extremal dependence among defaults. By utilizing the asymptotic description of how the rare event occurs, we derive two simple simulation algorithms based on conditional Monte Carlo to estimate the probability that the portfolio incurs large losses under the \(t\)-copula. We further show that the less efficient estimator exhibits bounded relative error. An extensive simulation study demonstrates that both estimators outperform existing algorithms. We then discuss a generalization of the \(t\)-copula model that allows the multivariate defaults to have an asymmetric distribution. Lastly, we show how the estimators proposed for the \(t\)-copula can be modified to estimate the portfolio risk under the skew \(t\)-copula model.

MSC:
91G40 Credit risk
91G70 Statistical methods; risk measures
91G10 Portfolio theory
Software:
QRM
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[1] Asmussen, S.; Glynn, P.W., Stochastic simulation: algorithms and analysis, (2007), Springer New York · Zbl 1126.65001
[2] Asmussen, S.; Kroese, D.P., Improved algorithms for rare event simulation with heavy tails, Advances in applied probability, 38, 545-558, (2006) · Zbl 1097.65017
[3] Azzalini, A., A class of distributions which includes the normal ones, Scandinavian journal of statistics, 12, 171-178, (1985) · Zbl 0581.62014
[4] Azzalini, A.; Dalla Valle, A., The multivariate skew-normal distribution, Biometrika, 83, 715-726, (1996) · Zbl 0885.62062
[5] Bassamboo, A.; Juneja, S.; Zeevi, A., Portfolio credit risk with extremal dependence: asymptotic analysis and efficient simulation, Operations research, 56, 3, 593-606, (2008) · Zbl 1167.91362
[6] Bluhm, C.; Overbeck, L.; Wagner, C., An introduction to credit risk modeling, (2002), Chapman & Hall/CRC
[7] Chan, J.C.C., Kroese, D.P., in press. Rare-event probability estimation with Monte Carlo. Annals of Operations Research. doi:10.1007/s10479-009-0539-y. · Zbl 1279.60064
[8] Egloff, D., Leippold, M., Jöhri, S., Dalbert, C., 2005. Optimal importance sampling for credit portfolios with stochastic approximation SSRN working paper.
[9] Fernandez, C.; Steel, M.F.J., On Bayesian modeling of fat tails and skewness, Journal of the American statistical association, 93, 441, 359-371, (1998) · Zbl 0910.62024
[10] Franses, P.H.; van Dijk, D., Non-linear time series models in empirical finance, (2000), Cambridge University Press Cambridge
[11] Geweke, J., 1991. Efficient simulation from the multivariate normal and student-t distributions subject to linear constraints and the evaluation of constraint probabilities. In: Keramidas, E., Kaufman, S. (Eds.), Computing Science and Statistics: Proceedings of the 23rd Symposium on the Interface, pp. 571-578.
[12] Geweke, J., Bayesian treatment of the independent student-t linear model, Journal of applied econometrics, 8, S19-S40, (1993)
[13] Glasserman, P.; Li, J., Importance sampling for portfolio credit risk, Management science, 51, 11, 1643-1656, (2005) · Zbl 1232.91621
[14] Glasserman, P.; Kang, W.; Shahabuddin, P., Large deviations of multifactor portfolio credit risk, Mathematical finance, 17, 345-379, (2007) · Zbl 1186.91227
[15] Glasserman, P.; Kang, W.; Shahabuddin, P., Fast simulation of multifactor portfolio credit risk, Operations research, 56, 5, 1200-1217, (2008) · Zbl 1167.91369
[16] Grundke, P., Importance sampling for integrated market and credit portfolio models, European journal of operational research, 194, 1, 206-226, (2009) · Zbl 1158.91378
[17] Gupton, G., Finger, C., Bhatia, M., 1997. Creditmetrics technical document. Technical report, J.P. Morgan & Co., New York.
[18] Joshi, M.S., 2004. Applying importance sampling to pricing single tranches of CDOs in a one-factor Li model. Technical report, QUARC, Group Risk Management, Royal Bank of Scotland.
[19] Juneja, S.; Shahabuddin, P., Simulating heavy tailed processes using delayed hazard rate twisting, ACM transactions on modeling and computer simulation, 12, 2, 94-118, (2002) · Zbl 1390.65033
[20] Kalkbrener, M.; Lotter, H.; Overbeck, L., Sensible and efficient capital allocation for credit portfolios, Risk, 17, 1, S19-S24, (2004)
[21] Li, D., On default correlations: A copula function approach, Journal of fixed income, 9, 43-54, (2000)
[22] McNeil, A.; Frey, R.; Embrechts, P., Quantitative risk management: concepts, techniques and tools, (2005), Princeton University Press Princeton, New Jersey · Zbl 1089.91037
[23] Robert, C.P., Simulation of truncated normal variables, Statistics and computing, 5, 121-125, (1995)
[24] Rubinstein, R.Y.; Kroese, D.P., The cross-entropy method: A unified approach to combinatorial optimization Monte-Carlo simulation, and machine learning, (2004), Springer-Verlag New York · Zbl 1061.90032
[25] Rubinstein, R.Y.; Kroese, D.P., Simulation and the Monte Carlo method, (2007), John Wiley & Sons New York · Zbl 1061.90032
[26] van der Vert, A.W., Asymptotic statistics, (1998), Cambridge University Press · Zbl 0910.62001
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