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An asymptotic characterization of hidden tail credit risk with actuarial applications

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Abstract

In this paper we study the tail risk of a properly diversified credit portfolio under a latent risk factor model. The usual perception is that if the diversification leads to asymptotic independence among the risk factors, then, because of the relatively low probability of simultaneous defaults, the tail risk of the entire portfolio is negligible. However, we point out that in fact there may be substantial tail risk hidden in this situation. We use a conditional tail probability of the portfolio loss to quantify the hidden tail risk, and then provide an asymptotic characterization for the risk under a hidden regular variation structure assumed for the risk factors. We also propose applications of the characterization to the determination and allocation of related insurance risk capital, based on the Conditional Tail Expectation risk measure. To understand the impact of dependence on the quantities of interest, we study two special cases where the risk factors have a Gaussian copula or an Archimedean copula. Numerical examples are provided to illustrate the results.

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Notes

  1. See NAIC archived reports “Year-End 2014 Insurance Industry Investment Portfolio Asset Allocations” and “U.S. Insurance Industry Cash and Invested Assets at Year-End 2015” at http://www.naic.org/capital_markets_archive_index.htm.

  2. If the stock price can be used as a proxy for the latent risk factor, then there has been empirical evidence showing that the risks of some technology firms and financial firms are asymptotically independent; see, e.g., [34].

  3. See NAIC archived report “Analysis of the U.S. Insurance Industrys Exposure to Below-Investment Grade Investments” at http://www.naic.org/capital_markets_archive_index.htm.

References

  1. Asimit AV, Furman E, Tang Q, Vernic R (2011) Asymptotics for risk capital allocations based on conditional tail expectation. Insur Math Econ 49:310–324

    Article  MathSciNet  MATH  Google Scholar 

  2. Asimit AV, Vernic R, Zitikis R (2013) Evaluating risk measures and capital allocations based on multi-losses driven by a heavy-tailed background risk: the multivariate Pareto-II model. Risks 1:14–33

    Article  Google Scholar 

  3. Charpentier A, Segers J (2009) Tails of multivariate Archimedean copulas. J Multivar Anal 100:1521–1537

    Article  MathSciNet  MATH  Google Scholar 

  4. Coles S, Heffernan J, Tawn J (1999) Dependence measures for extreme value analyses. Extremes 2:339–365

    Article  MATH  Google Scholar 

  5. Das B, Embrechts P, Fasen V (2013) Four theorems and a financial crisis. Int J Approx Reasoning 54:701–716

    Article  MathSciNet  MATH  Google Scholar 

  6. Das B, Mitra A, Resnick SI (2013) Living on the multidimensional edge: seeking hidden risks using regular variation. Adv Appl Probab 45:139–163

    Article  MathSciNet  MATH  Google Scholar 

  7. de Haan L, Ferreira A (2006) Extreme value theory: an introduction. Springer, New York

    Book  MATH  Google Scholar 

  8. Denuit M, Kiriliouk A, Segers J (2015) Max-factor individual risk models with application to credit portfolios. Insur Math Econ 62:162–172

    Article  MathSciNet  MATH  Google Scholar 

  9. Dhaene J, Tsanakas A, Valdez EA, Vanduffel S (2012) Optimal capital allocation principles. J Risk Insur 79:1–28

    Article  Google Scholar 

  10. Donnelly C, Embrechts P (2010) The devil is in the tails: actuarial mathematics and the subprime mortgage crisis. ASTIN Bull 40:1–33

    Article  MathSciNet  MATH  Google Scholar 

  11. Duffie D, Eckner A, Horel G, Saita L (2009) Frailty correlated default. J Financ 64:2089–2123

    Article  Google Scholar 

  12. Embrechts P, Klüppelberg C, Mikosch T (1997) Modelling extremal events for insurance and finance. Springer, Berlin

    Book  MATH  Google Scholar 

  13. Fougères A, Mercadier C (2012) Risk measures and multivariate extensions of Breiman’s theorem. J Appl Probab 49:364–384

    Article  MathSciNet  MATH  Google Scholar 

  14. Frees EW, Valdez EA (1998) Understanding relationships using copulas. N Am Actuar J 2:1–25

    Article  MathSciNet  MATH  Google Scholar 

  15. Frey R, McNeil AJ (2003) Dependent defaults in models of portfolio credit risk. J Risk 6:59–92

    Article  Google Scholar 

  16. Furman E, Landsman Z (2005) Risk capital decomposition for a multivariate dependent gamma portfolio. Insur Math Econ 37:635–649

    Article  MathSciNet  MATH  Google Scholar 

  17. Furman E, Landsman Z (2008) Economic capital allocations for non-negative portfolios of dependent risks. ASTIN Bull 38:601–619

    Article  MathSciNet  MATH  Google Scholar 

  18. Furman E, Zitikis R (2008) Weighted risk capital allocations. Insur Math Econ 43:263–269

    Article  MathSciNet  MATH  Google Scholar 

  19. Gabaix X (2009) Power laws in economics and nance. Annu Rev Econ 1:255–293

    Article  Google Scholar 

  20. Genest C, Nešlehová J (2007) A primer on copulas for count data. ASTIN Bull 37:475–515

    Article  MathSciNet  MATH  Google Scholar 

  21. Giesecke K (2004) Correlated default with incomplete information. J Bank Financ 28:1521–1545

    Article  Google Scholar 

  22. Glasserman P, Li J (2005) Importance sampling for portfolio credit risk. Manag Sci 51:1643–1656

    Article  MATH  Google Scholar 

  23. Gupton GM, Finger CC, Bhatia M (1997) CreditMetrics\(^{\rm TM}\)—technical document. Technical report. J. P. Morgan & Co., New York

  24. Hao X, Li X (2015) Pricing credit default swaps with a random recovery rate by a double inverse Fourier transform. Insur Math Econ 65:103–110

    Article  MathSciNet  MATH  Google Scholar 

  25. Hashorva E (2005) Asymptotics and bounds for multivariate Gaussian tails. J Theor Probab 18:79–97

    Article  MathSciNet  MATH  Google Scholar 

  26. Hashorva E, Hüsler J (2002) On asymptotics of multivariate integrals with applications to records. Stochas Models 18:41–69

    Article  MathSciNet  MATH  Google Scholar 

  27. Hashorva E, Hüsler J (2003) On multivariate Gaussian tails. Ann Inst Stat Math 55:507–522

    Article  MathSciNet  MATH  Google Scholar 

  28. Heffernan JE (2000) A directory of coefficients of tail dependence. Extremes 3:279–290

    Article  MathSciNet  MATH  Google Scholar 

  29. Hua L, Joe H (2011) Tail order and intermediate tail dependence of multivariate copulas. J Multivar Anal 102:1454–1471

    Article  MathSciNet  MATH  Google Scholar 

  30. Hua L, Joe H, Li H (2014) Relations between hidden regular variation and tail order of copulas. J Appl Probab 51:37–57

    Article  MathSciNet  MATH  Google Scholar 

  31. Joe H (2015) Dependence modeling with copulas. CRC Press, Boca Raton, FL

    MATH  Google Scholar 

  32. Kim JHT, Hardy MR (2009) A capital allocation based on a solvency exchange option. Insur Math Econ 44:357–366

    Article  MathSciNet  MATH  Google Scholar 

  33. Kimberling CH (1974) A probabilistic interpretation of complete monotonicity. Aequ Math 10:152–164

    Article  MathSciNet  MATH  Google Scholar 

  34. Kiriliouk A, Segers J, Warchoł M (2015) Nonparametric estimation of extremal dependence. In: Dey DK, Yan J (eds) Extreme value modeling and risk analysis: methods and applications, Chapman & Hall/CRC, Boca Raton, FL, pp 367–389

  35. Ledford AW, Tawn JA (1996) Statistics for near independence in multivariate extreme values. Biometrika 83:169–187

    Article  MathSciNet  MATH  Google Scholar 

  36. Ledford AW, Tawn JA (1997) Modelling dependence within joint tail regions. J R Stat Soc B 59:475–499

    Article  MathSciNet  MATH  Google Scholar 

  37. Manner H, Segers J (2011) Tails of correlation mixtures of elliptical copulas. Insur Math Econ 48:153–160

    Article  MathSciNet  MATH  Google Scholar 

  38. McNeil AJ, Frey R, Embrechts P (2015) Quantitative risk management. Princeton University Press, Princeton

    MATH  Google Scholar 

  39. McNeil AJ, Nešlehová J (2009) Multivariate Archimedean copulas, \(d\)-monotone functions and \(l_{1}\)-norm symmetric distributions. Ann Stat 37:3059–3097

    Article  MATH  Google Scholar 

  40. Mitra A, Resnick SI (2011) Hidden regular variation and detection of hidden risks. Stochas Models 27:591–614

    Article  MathSciNet  MATH  Google Scholar 

  41. Malov SV (2001) On finite-dimensional Archimedean copulas. In: Balakrishnan N, Ibragimov IA, Nevzorov VB (eds), Asymptotic methods in probability and statistics with applications. Birkhäuser, Boston, MA, pp 19–35

  42. Myers SC, Read Jr JA (2001) Capital allocation for insurance companies. J Risk Insur 68:545–580

    Article  Google Scholar 

  43. Qi M, Zhao X (2011) Comparison of modeling methods for loss given default. J Bank Financ 35:2842–2855

    Article  Google Scholar 

  44. Ratovomirija G (2016) On mixed Erlang reinsurance risk: aggregation, capital allocation and default risk. Eur Actuar J 6:149–175

    Article  MathSciNet  MATH  Google Scholar 

  45. Resnick SI (1987) Extreme values, regular variation, and point processes. Springer, New York

    Book  MATH  Google Scholar 

  46. Resnick SI (2002) Hidden regular variation, second order regular variation and asymptotic independence. Extremes 5:303–336

    Article  MathSciNet  MATH  Google Scholar 

  47. Resnick SI (2007) Heavy-tail phenomena, probabilistic and statistical modeling. Springer, New York

    MATH  Google Scholar 

  48. Savage IR (1962) Mill’s ratio for multivariate normal distributions. J Res Natl Bureau Stand Sect B Math Sci 66:93–96

    Article  MATH  Google Scholar 

  49. Scherer M, Schmid L, Schmidt T (2012) Shot-noise driven multivariate default models. Eur Actuar J 2:161–186

    Article  MathSciNet  MATH  Google Scholar 

  50. Scott A, Metzler A (2015) A general importance sampling algorithm for estimating portfolio loss probabilities in linear factor models. Insur Math Econ 64:279–293

    Article  MathSciNet  MATH  Google Scholar 

  51. Shi X, Tang Q, Yuan Z (2017) A limit distribution of credit portfolio losses with low default probabilities. Insur Math Econ 73:156–167

    Article  MathSciNet  Google Scholar 

  52. Tang Q, Yuan Z (2013) Asymptotic analysis of the loss given default in the presence of multivariate regular variation. N Am Actuar J 17:253–271

    Article  MathSciNet  Google Scholar 

  53. Tankov P (2016) Tails of weakly dependent random vectors. J Multivar Anal 145:73–86

    Article  MathSciNet  MATH  Google Scholar 

  54. Wang R, Peng L, Yang J (2015) CreditRisk+ model with dependent risk factors. N Am Actuar J 19:24–40

    Article  MathSciNet  Google Scholar 

  55. Wei L, Yuan Z (2016) The loss given default of a low-default portfolio with weak contagion. Insur Math Econ 66:113–123

    Article  MathSciNet  MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Risk Management, Smeal College of Business, The Pennsylvania State University, 362 Business Building, University Park, PA, 16802, USA

    Zhongyi Yuan

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  1. Zhongyi Yuan
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Yuan, Z. An asymptotic characterization of hidden tail credit risk with actuarial applications. Eur. Actuar. J. 7, 165–192 (2017). https://doi.org/10.1007/s13385-017-0150-6

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  • DOI: https://doi.org/10.1007/s13385-017-0150-6

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