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2018 | OriginalPaper | Buchkapitel

3. Mixture or Actuarial Models

verfasst von : David Jamieson Bolder

Erschienen in: Credit-Risk Modelling

Verlag: Springer International Publishing

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Abstract

The independent-default model is deeply flawed. Not only is it fair to argue that dependence is the single most important aspect of credit-risk modelling, but the tails of the associated loss distribution are overly thin and its asymptotic behaviour is simply too well behaved. This chapter offers a family of approaches—generally referred to as mixture or actuarial models—to address each of these shortcomings. The principle idea behind this new methodology is the randomization of the default probability. Practically, a common state variable is introduced, which induces default dependence among all obligors. Conditionally, default events remain independent, but unconditionally they are related through the realization of the systematic state variable. The structure of the default-loss distribution thus depends on the statistical properties of one’s state-variable choice mixed with underlying binomial-default structure. A variety of possible choices are investigated, convergence properties are explored, and our portfolio example is examined from both analytic and numerical perspectives. Through the law of rare events, a separate class of Poisson-mixture models are explored ultimately leading to the celebrated CreditRisk+ model used widely in practice and first suggested by Wilde (1997, CreditRisk+: A credit risk management framework, Credit Suisse First Boston).

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Vorheriges Kapitel A Natural First Step

Nächstes Kapitel Threshold Models

See Abramovitz and Stegun (1965, Chapter 6) for a detailed description of the gamma function.

See Casella and Berger (1990, Chapter 3) and Fishman (1995, Chapter 3) for more information on the beta distribution.

The associated standard deviation of Z is about 0.02, which implies that it can deviate fairly importantly from the average of 1%.

See Fishman (1995, Chapter 3) for more detail and a number of more efficient algorithms.

The generateGamma function merely exploits the relationship from equation 3.30.

Mathematically, it is possible for σ ₁ < 0, but it then takes an inverted S-form—for this reason, we restrict our attention to $\mathbb {R}_+$.

This, in turn, is based on the Fortran 77 library function QUADPACK, which employs the Gauss-Kronrod quadrature formulae. For more information on this approach, in particular or numerical integration in general, see Ralston and Rabinowitz (1978, Chapter 4) or Press et al. (1992, Chapter 4).

See Press et al. (1992, Chapter 9) for a detailed discussion of numerically solving non-linear systems of equations.

Although we are basically consumers of this algorithm, a few details are nonetheless useful. hybrd determines the roots of a system of N non-linear functions in N unknowns using the Powell-hybrid, or dogleg, method—this approach tries to minimize the sum of the squared function values using Newton’s method. The Jacobian is calculated numerically, employing user-provided function routines, with a forward finite-difference approximation.

The gamma density function can be written in alternative formats; we have opted for the shape-rate parametrization. Ultimately, although the moments are different, the choice is immaterial.

The Python implementation uses the shape-scale representation of the gamma distribution. This means that we need to send the reciprocal of the b parameter to their random-number engine.

We have dispensed with showing the poissonMixtureSimulation algorithm, because it is almost identical to Algorithm 3.10 and this chapter is already sufficiently long and repetitive. The interested reader can easily locate it, however, in the mixtureModels library. See Appendix D for more details.

Or, in an equivalent manner, as a becomes small.

It also helps, as we see in the next chapter, to create a closer conceptual link to the family of threshold models.

As we’ve stressed so far, one could, of course, work in the opposite direction.

The reader is naturally free to disagree; for those that do, Gundlach and Lehrbass (2004) is the place to start.

An important step in the previous development stems from our constraint in equation 3.94, which implies that,

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \underbrace{\sum_{k=0}^K \sum_{j=0}^K}_{k\ne j} \omega_{n,k} \omega_{m,j} + \sum_{k=0}^K \omega_{n,k} \omega_{m,k} &\displaystyle =&\displaystyle \sum_{k=0}^K \sum_{j=0}^K \omega_{n,k} \omega_{m,j},\\ &\displaystyle =&\displaystyle \sum_{k=0}^K \omega_{n,k} \underbrace{\sum_{j=0}^K \omega_{m,j}}_{=1},\\ &\displaystyle =&\displaystyle 1. \end{array} \end{aligned} $$

(3.98)

Abramovitz, M., & Stegun, I. A. (1965). Handbook of mathematical functions. New York: Dover Publications.

Billingsley, P. (1995). Probability and measure (3rd edn.). Third Avenue, New York, NY: Wiley.

Casella, G., & Berger, R. L. (1990). Statistical inference. Belmont, CA: Duxbury Press.

Davis, P. J. (1959). Leonhard euler’s integral: A historical profile of the gamma function. The American Mathematical Monthly, 66(10), 849–869.

Fishman, G. S. (1995). Monte Carlo: Concepts, algorithms, and applications. 175 Fifth Avenue, New York, NY: Springer-Verlag. Springer series in operations research.

Frey, R., & McNeil, A. (2003). Dependent defaults in model of portfolio credit risk. The Journal of Risk, 6(1), 59–92.CrossRef

Gundlach, M., & Lehrbass, F. (2004). CreditRisk+ in the banking industry (1st edn.). Berlin: Springer-Verlag.CrossRef

Panjer, H. H. (1981). Recursive evaluation of a family of compound distributions. Astin Bulletin, (12), 22–26.CrossRef

Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. (1992). Numerical recipes in C: The art of scientific computing (2nd edn.). Trumpington Street, Cambridge: Cambridge University Press.

Ralston, A., & Rabinowitz, P. (1978). A first course in numerical analysis (2nd edn.). Mineola, NY: Dover Publications.

Sharpe, W. F. (1963). A simplified model for portfolio analysis. Management Science, 9(2), 277–293.CrossRef

Sharpe, W. F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk. The Journal of Finance, 19(3), 425–442.

Stuart, A., & Ord, J. K. (1987). Kendall’s advanced theory of statistics. New York: Oxford University Press.

Wilde, T. (1997). CreditRisk+: A credit risk management framework. Credit Suisse First Boston.

Titel: Mixture or Actuarial Models
verfasst von: David Jamieson Bolder
Verlag: Springer International Publishing
Buch: Credit-Risk Modelling
Print ISBN: 978-3-319-94687-0

Electronic ISBN: 978-3-319-94688-7

Copyright-Jahr: 2018
DOI: https://doi.org/10.1007/978-3-319-94688-7_3