Credit risk and solvency capital requirements

Abstract

Credit risk permeates the assets of most insurance companies. This article develops a framework for computing credit capital requirements under the constant position paradigm and taking into account recovery rates. Although this framework was originally derived under the Solvency 2 regulation, it also provides concepts that can be useful under other international regulations. After a brief survey of the existing technology on rating transitions and default probabilities, the paper provides new results on risk premium adjustment factors. Then, three different procedures for reconstructing constant position market-consistent histories of credit portfolios from quoted Merrill Lynch indices are given. The reconstructed historical credit values are modeled via mixed empirical-Generalized Pareto Distribution (GPD) dynamics and a detailed parameter estimation is performed. Several validations of the estimation are also provided. Finally, credit Solvency Capital Requirements are computed and an analysis of the results per rating class is given.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Notes

  1. 1.

    In structural approaches, it is possible to use the Girsanov theorem or the Wang [30] transform to construct the risk-neutral universe. Note that in the latter approach, a distorsion of decumulative historical probabilities, rather than of probabilities, is applied.

  2. 2.

    For AAA bonds, the models predict a null SCR because the recovery rate is \(100\%\) for all these bonds in our database.

References

  1. 1.

    Bauer D, Ha H (2016) A least-squares monte carlo approach to the calculation of capital requirements, Working Paper

  2. 2.

    Bauer D, Reuss A, Singer D (2012) On the calculation of the solvency capital requirement based on nested simulations. ASTIN Bull 42:453–499

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Bielecki TR, Rutkowski M (2002) Credit risk: modelling, valuation and hedging. Springer, Berlin

    Google Scholar 

  4. 4.

    Bruyère R, Cont R, Copinot R, Fery L, Jaeck C, Spitz T (2006) Credit derivatives and structured credit: a guide for investors. Wiley, Hoboken

    Google Scholar 

  5. 5.

    Cohen A, Costanzino N (2015) Bond and CDS pricing with recovery risk I: The stochastic recovery Merton model, Working Paper

  6. 6.

    Cohen A, Costanzino N (2017a) Bond and CDS Pricing via the stochastic recovery Black–Cox model. Risks 5(2):26

    Article  Google Scholar 

  7. 7.

    Cohen A, Costanzino N (2017b) A general framework for incorporating stochastic recovery in structural models of credit risk. Risks 5(4):65

    Article  Google Scholar 

  8. 8.

    Collin-Dufresne P, Goldstein R (2001) Do credit spreads reflect stationary leverage ratios? J Financ 56(5):1929–1957

    Article  Google Scholar 

  9. 9.

    Crouhy M, Galai D, Mark R (2000) A comparative analysis of current credit risk models. J Bank Financ 24:59–117

    Article  Google Scholar 

  10. 10.

    Duffie D, Lando D (2001) Term structures of credit spreads with incomplete accounting information. Econometrica 69(3):633–664

    MathSciNet  Article  Google Scholar 

  11. 11.

    Elliott RJ, Jeanblanc M, Yor M (2000) On models of default risk. Math Financ 10:179–195

    MathSciNet  Article  Google Scholar 

  12. 12.

    Embrechts P, Klüppelberg C, Mikosch T (1999) Modelling extremal events in insurance and finance. Springer, Berlin

    Google Scholar 

  13. 13.

    Floryszczak A, Le Courtois O, Majri M (2016) Inside the solvency 2 Black Box: net asset values and solvency capital requirements with a least-squares Monte–Carlo approach. Insur Math Econ 71:15–26

    MathSciNet  Article  Google Scholar 

  14. 14.

    Gatzert N, Martin M (2012) Quantifying credit and market risk under Solvency II: standard approach versus internal model. Insur Math Econ 51:649–666

    MathSciNet  Article  Google Scholar 

  15. 15.

    Gatzert N, Wesker H (2011) A comparative assessment of basel II/III and solvency II. Geneva Papers Risk Insur Issues Pract 37:539–570

    Article  Google Scholar 

  16. 16.

    Gini C (1921) Measurement of inequality of income. Econ J 31:22–43

    Article  Google Scholar 

  17. 17.

    Hill BM (1975) A simple general approach to inference about the tail of a distribution. Ann Stat 3(5):1163–1174

    MathSciNet  Article  Google Scholar 

  18. 18.

    Hosking JRM, Wallis JR (1987) Parameter and quantile estimation for the generalized Pareto distribution. Technometrics 29(3):339–349

    MathSciNet  Article  Google Scholar 

  19. 19.

    Israel RB, Rosenthal JS, Wei JZ (2001) Finding generators for Markov chains via empirical transition matrices with applications to credit ratings. Math Financ 11:245–265

    MathSciNet  Article  Google Scholar 

  20. 20.

    Jarrow RA, Lando D, Turnbull SM (1997) A Markov model for the term structure of credit risk spreads. Rev Financ Stud 10:481–523

    Article  Google Scholar 

  21. 21.

    Jarrow RA, Protter P (2004) Structural versus reduced form models: a new information based perspective. J Invest Manag 2:1–10

    Google Scholar 

  22. 22.

    Leadbetter MR (1991) On a basis for peaks over threshold modeling. Stat Probab Lett 12:357–362

    MathSciNet  Article  Google Scholar 

  23. 23.

    Le Courtois O, Walter C (2014) Extreme financial risks and asset allocation. Imperial College Press, London

    Google Scholar 

  24. 24.

    Leland H (1994) Corporate debt value, bond covenants, and optimal capital structure. J Financ 49(4):1213–1252

    Article  Google Scholar 

  25. 25.

    Longstaff FA, Schwartz ES (1995) A simple approach to valuing risky fixed and floating rate debt. J Financ 50(3):789–820

    Article  Google Scholar 

  26. 26.

    Lorenz MO (1905) Methods of measuring the concentration of wealth. Publ Am Stat Assoc 9:209–219

    Google Scholar 

  27. 27.

    Official Journal of the European Union (2015) Commission delegated regulation (EU) 2015/35 of 10 October 2014 supplementing Directive 2009/138/EC of the European Parliament and of the Council on the taking-up and pursuit of the business of Insurance and Reinsurance (Solvency II), L12, 58

  28. 28.

    Schönbucher PJ (2003) Credit derivatives pricing models. Wiley finance, Hoboken

    Google Scholar 

  29. 29.

    Vasicek OA (2002) The distribution of loan portfolio value. RISK 15(12):160–162

    Google Scholar 

  30. 30.

    Wang SS (2000) A class of distortion operators for pricing financial and insurance risks. J Risk Insur 67(1):15–36

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Philippe Desurmont, Anthony Floryszczak, Donatien Hainaut, François-Xavier de Lauzon, Jacques Lévy-Véhel, Hubert Rodarie, Li Shen, Xia Xu, and Christian Walter for their useful comments.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Olivier Le Courtois.

Appendices

Appendix 1

See Table 6.

Table 6 Bond dataset as of 31 / 12 / 2014

Appendix 2

See Table 7.

Table 7 Average 1-year corporate transition rates (%). Source: Standard and Poor’s

Appendix 3

See Tables 8, 9 and 10.

Table 8 Parameter a of the standard formula
Table 9 Parameter b of the standard formula
Table 10 Parameter c of the standard formula

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Allali, J., Le Courtois, O. & Majri, M. Credit risk and solvency capital requirements. Eur. Actuar. J. 8, 487–515 (2018). https://doi.org/10.1007/s13385-018-0183-5

Download citation

Keywords

  • Credit spread
  • Risk premium adjustment factor
  • Solvency capital requirement
  • General pareto distribution
  • Market consistency
  • Rating transition
  • Credit benchmarking
  • Constant position