Skip to main content
SpringerLink
Log in

Calculation of Compound Distribution

  • Chapter
  • First Online:
Modelling Operational Risk Using Bayesian Inference

  • 2229 Accesses

Abstract

Estimation of the capital under the LDA requires evaluation of compound loss distributions. Closed-form solutions are not available for the distributions typically used in operational risk and numerical evaluation is required. This chapter describes numerical algorithms that can be successfully used for this problem. In particular Monte Carlo, Panjer recursion and Fourier transformation methods are presented. Also, several closed-form approximations are reviewed.

Science never solves a problem without creating ten more.

Bernard Shaw

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Computing time quoted in this chapter is for a standard Dell laptop Latitude D820 with Intel(R) CPU T2600 @ 2.16 GHz and 3.25 GB of RAM.

  2. 2.

    Underflow/overflow are the cases when the computer calculations produce a number outside the range of representable numbers leading 0 or \(\pm\infty\) outputs respectively.

  3. 3.

    Note that often, in the relevant literature, notation “∽” is used to indicate that the ratio of the left- and righthand sides converge to 1; here we use “→” to avoid confusion with notation used to indicate that a random variable is distributed from a distribution.

References

  1. Abate, J., Whitt, W.: Numerical inversion of Laplace transforms of probability distributions. ORSA Journal of Computing 7, 36–43 (1992)

    Google Scholar 

  2. Abate, J., Whitt, W.: Numerical inversion of probability generating functions. Operations Research Letters 12, 245–251 (1995)

    Article  MathSciNet  Google Scholar 

  3. Acerbi, C., Tasche, D.: On the coherence of expected shortfall. Journal of Banking and Finance 26, 1487–1503 (2002)

    Article  Google Scholar 

  4. Bladt, M.: A review of phase-type distributions and their use in risk theory. ASTIN Bulletin 35(1), 145–167 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Böcker, K., Klüppelberg, C.: Operational VAR: a closed-form approximation. Risk Magazine 12, 90–93 (2005)

    Google Scholar 

  6. Böcker, K., Sprittulla, J.: Operational VAR: meaningful means. Risk Magazine 12, 96–98 (2006)

    Google Scholar 

  7. Bohman, H.: Numerical inversion of characteristic functions. Scandinavian Actuarial Journal pp. 121–124 (1975)

    Google Scholar 

  8. Brass, H., Förster, K.J.: On the estimation of linear functionals. Analysis 7, 237–258 (1987)

    MATH  MathSciNet  Google Scholar 

  9. Brigham, E.O.: The Fast Fourier Transform. Prentice-Hall, Englewood Cliffs, NJ (1974)

    MATH  Google Scholar 

  10. Bühlmann, H.: Numerical evaluation of the compound Poisson distribution: recursion or Fast Fourier Transform? Scandinavian Actuarial Journal 2, 116–126 (1984)

    Google Scholar 

  11. Clenshaw, C.W., Curtis, A.R.: A method for numerical integration on an automatic computer. Num. Math 2, 197–205 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  12. Craddock, M., Heath, D., Platen, E.: Numerical inversion of Laplace transforms: a survey of techniques with applications to derivative pricing. Computational Finance 4(1), 57–81 (2000)

    Google Scholar 

  13. Den Iseger, P.W.: Numerical Laplace inversion using Gaussian quadrature. Probability in the Engineering and Informational Sciences 20, 1–44 (2006)

    MATH  MathSciNet  Google Scholar 

  14. Embrechts, P., Frei, M.: Panjer recursion versus FFT for compound distributions. Mathematical Methods of Operations Research 69(3), 497–508 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  15. Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling Extremal Events for Insurance and Finance. Springer, Berlin (1997). Corrected fourth printing 2003

    MATH  Google Scholar 

  16. Gerhold, S., Schmock, U., Warnung, R.: A generalization of Panjer’s recursion and numerically stable risk aggregation. Finance and Stochastics 14(1), 81–128 (2010)

    Article  MathSciNet  Google Scholar 

  17. Glasserman, P.: Monte Carlo Methods in Financial Engineering. Springer, New York, NY (2004)

    MATH  Google Scholar 

  18. Glasserman, P.: Measuring marginal risk contributions in credit portfolios. Journal Computational Finance 9(2), 1–41 (2005)

    Google Scholar 

  19. Golub, G.H., Welsch, J.H.: Calculation of Gaussian quadrature rules. Mathematics of Computation 23, 221–230 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  20. Grubel, R., Hermesmeier, R.: Computation of compound distributions I: aliasing errors and exponential tilting. ASTIN Bulletin 29(2), 197–214 (1999)

    Article  MathSciNet  Google Scholar 

  21. Heckman, P.E., Meyers, G.N.: The calculation of aggregate loss distributions from claim severity and claim count distributions. Proceedings of the Casualty Actuarial Society LXX, 22–61 (1983)

    Google Scholar 

  22. Hess, K.T., Liewald, A., Schmidt, K.D.: An extension of Panjer’s recursion. ASTIN Bulletin 32(2), 283–297 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  23. Hesselager, O.: Recursions for certain bivariate counting distributions and their compound distributions. ASTIN Bulletin 26(1), 35–52 (1996)

    Article  MathSciNet  Google Scholar 

  24. Hipp, C.: Speedy convolution algorithms and Panjer recursions for phase-type distributions. Insurance: Mathematics and Economics 38(1), 176–188 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  25. Kahaner, D., Moler, C., Nash, S.: Numerical Methods and Software. Prentice-Hall, Englewood Cliffs, NJ (1989)

    MATH  Google Scholar 

  26. Kronrod, A.S.: Nodes and weights of quadrature formulas. Sixteen-place tables. New York: Consultants Bureau Authorized translation from Russian Doklady Akad. Nauk SSSR 154, 283–286 (1965)

    Google Scholar 

  27. Luo, X., Shevchenko, P.V.: Computing tails of compound distributions using direct numerical integration. The Journal of Computational Finance 13(2), 73–111 (2009)

    MATH  MathSciNet  Google Scholar 

  28. Luo, X., Shevchenko, P.V.: A short tale of long tail integration. Numerical Algorithms (2010). DOI: 10.1007/s11075-010-9406-9

    Google Scholar 

  29. Moscadelli, M.: The modelling of operational risk: experiences with the analysis of the data collected by the Basel Committee. Bank of Italy (2004). Working paper No. 517

    Google Scholar 

  30. Panjer, H., Willmot, G.: Insurance Risk Models. Society of Actuaries, Chicago, IL (1992)

    Google Scholar 

  31. Panjer, H.H.: Recursive evaluation of a family of compound distribution. ASTIN Bulletin 12(1), 22–26 (1981)

    MathSciNet  Google Scholar 

  32. Panjer, H.H.: Operational Risks: Modeling Analytics. Wiley, New York, NY (2006)

    MATH  Google Scholar 

  33. Panjer, H.H., Wang, S.: On the stability of recursive formulas. ASTIN Bulletin 23(2), 227–258 (1993)

    Article  Google Scholar 

  34. Peters, G.W., Johansen, A.M., Doucet, A.: Simulation of the annual loss distribution in operational risk via Panjer recursions and Volterra integral equations for value-at-risk and expected shortfall estimation. The Journal of Operational Risk 2(3), 29–58 (2007)

    Google Scholar 

  35. Piessens, R., Doncker-Kapenga, E.D., Überhuber, C.W., Kahaner, D.K.: QUADPACK – a Subroutine Package for Automatic Integration. Springer, New York, NY (1983)

    MATH  Google Scholar 

  36. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C. Cambridge University Press, New York, NY (2002)

    Google Scholar 

  37. Pugachev, V.S.: Theory of Random Functions and its applications to control problems, 1st edn. Pergamon Press, London (1965)

    Google Scholar 

  38. Robertson, J.: The computation of aggregate loss distributions. Proceedings of the Casuality Actuarial Society 79, 57–133 (1992)

    Google Scholar 

  39. Seal, H.L.: Numerical inversion of characteristic functions. Scandinavian Actuarial Journal pp. 48–53 (1977)

    Google Scholar 

  40. Shephard, N.G.: From characteristic function to distribution function: a simple framework for the theory. Econometric Theory 7, 519–529 (1991)

    Article  MathSciNet  Google Scholar 

  41. Sidi, A.: Extrapolation methods for oscillatory infinite integrals. Journal of the Institute of Mathematics and Its Applications 26, 1–20 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  42. Sidi, A.: A user friendly extrapolation method for oscillatory infinite integrals. Mathematics of Computation 51, 249–266 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  43. Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis, 3rd edn. Springer, New York, NY (2002)

    MATH  Google Scholar 

  44. Stuart, A., Ord, J.K.: Kendall’s Advanced Theory of Statistics: Volume 1, Distribution Theory, Sixth Edition. Edward Arnold, London/Melbourne/Auckland (1994)

    Google Scholar 

  45. Sundt, B.: On some extensions of Panjer’s class of counting distributions. ASTIN Bulletin 22(1), 61–80 (1992)

    Article  Google Scholar 

  46. Sundt, B.: On multivariate Panjer recursions. ASTIN Bulletin 29(1), 29–45 (1999)

    Article  Google Scholar 

  47. Sundt, B., Jewell, W.S.: Further results on recursive evaluation of compound distributions. ASTIN Bulletin 12(1), 27–39 (1981)

    MathSciNet  Google Scholar 

  48. Sundt, B., Vernic, R.: Recursions for Convolutions and Compound Distributions with Insurance Applications. Springer, Berlin (2009)

    MATH  Google Scholar 

  49. Szegö, G.: Orthogonal Polynomials, 4th edn. American Mathematical Society, Providence, RI (1975)

    MATH  Google Scholar 

  50. Vernic, R.: Recursive evaluation of some bivariate compound distributions. ASTIN Bulletin 29(2), 315–325 (1999)

    Article  Google Scholar 

  51. Waller, L.A., Turnbull, B.G., Hardin, J.M.: Obtaining distribution functions by numerical inversion of characteristic functions with applications. The American Statistician 49(4), 346–350 (1995)

    Article  Google Scholar 

  52. Wynn, P.: On a device for computing the \(e_m(s_n)\) tranformation. Mathematical Tables and Other Aids to Computation 10, 91–96 (1956)

    Article  MATH  MathSciNet  Google Scholar 

  53. Yamai, Y., Yoshiba, T.: Comparative analyses of expected shortfall and Value-at-Risk: Their estimation error, decomposition, and optimization. Monetary and Economic Studies pp. 87–121 (January 2002)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CSIRO, Mathematics, Informatics and Statistics, Locked Bag 17, North Ryde, NSW, 1670, Australia

    Dr. Pavel V. Shevchenko

Authors
  1. Dr. Pavel V. Shevchenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pavel V. Shevchenko .

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Shevchenko, P.V. (2011). Calculation of Compound Distribution. In: Modelling Operational Risk Using Bayesian Inference. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15923-7_3

Download citation

Publish with us

Policies and ethics

Search

Navigation