nach oben

Erschienen in:

2018 | OriginalPaper | Buchkapitel

4. Threshold Models

verfasst von : David Jamieson Bolder

Erschienen in: Credit-Risk Modelling

Verlag: Springer International Publishing

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

The binomial- and Poisson-mixture models offer a useful range of possible credit-risk implementations. Not only do we actively seek alternative approaches, but the mixture models are silent on the ultimate reason for default. In other words, the actuarial or mixture methodology is reduced form. A competing structural modelling family, referred to as the set of threshold models, is offered in this chapter. This approach is, in fact, a clever combination of a pragmatic, latent-variable approach and the classic Merton (1974, Journal of Finance, 29, 449–470) model. Default, therefore, occurs when a statistical proxy of the firm’s asset value falls below a predetermined threshold. The eponymous threshold is inferred from the obligor’s unconditional default probability. The basic structure and convergence properties of this technique are first examined in the Gaussian setting. This initial logic is then generalized—allowing for both thicker tails and tail dependence—through the introduction of the class of normal-variance mixture models. Parameter-calibration techniques are also reviewed and all models are exhaustively applied to our ongoing portfolio example.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Vorheriges Kapitel Mixture or Actuarial Models

Nächstes Kapitel The Genesis of Credit-Risk Modelling

Merton (1974) is sufficiently important and fundamental that we assign the entirety of Chap. 5 to its consideration.

It also acts, as we’ll see in Chap. 6, as the foundation for the Basel bank-capital requirements. See Fok et al. (2014) and BIS (2006a).

Techniques to estimate these important probabilities are addressed in Chap. 9.

This will, we assure the reader, be relaxed in latter sections.

The upper limits of integration in the classic description are typically standard normal inverse cumulative distribution functions of standard normal variates. That is, \(u_i\sim \mathcal {U}(0,1)\). Since the domain of each Φ ⁻¹(p _i) is also [0, 1], the form in equation 4.28 is mathematically equivalent to the Gaussian copula.

In this simple setting, the correlation matrix, \(\varOmega \in \mathbb {R}^{N\times N}\), has ones along the main diagonal and the parameter ρ populates all off-diagonal elements; as in equation 4.10.

This can be slightly relaxed as long as none of the counterparties dominates the portfolio.

Gordy (2002) offers concentration adjustment for such, real-world, non-infinitely grained portfolios. This interesting and pertinent issue will be addressed in Chap. 6.

We’ve restricted the support of this density, because inspection of equation 4.55 reveals that the density is undefined for ρ = 0 and takes the value of zero when ρ = 1.

This fact presumably explains, at least in part, its popularity in regulatory circles.

An intuitive, rough-and-ready justification from Schmitz (2003, Section 2.2) involves,

https://static-content.springer.com/image/chp%3A10.1007%2F978-3-319-94688-7_4/MediaObjects/458636_1_En_4_Equ62_HTML.png

(4.62)

This is neither precise, nor probably even exactly true, but it is a useful bit of motivation for the preceding result.

See McNeil et al. (2015, Chapter 7) for more details.

With the exception of perfect correlation—that is, the uninteresting case when ρ = 1.

See Kuhn (2005) and Kostadinov (2005) for more detail on the t-distributed case.

See Appendix A for a detailed description of the construction of the t-distribution.

This logic could have saved us the effort, although we would have missed the edification, in equation 4.72. Naturally, this is only really sensible for values of ν > 2.

Indeed, when \(\varLambda _{\mathcal {D}}\approx 0.00\), we have ρ = 0.29 and ν = 71. This is still, at the extreme tails, quite far from the Gaussian distribution.

Additional details, for the interested reader, on the generation of multivariate t-distributed random variables are found in Appendix A.

A more general framework was provided a few years later by Eberlein and Keller (1995).

Tail dependence is, in general, a fairly complex area. See Schmidt (2003) for a useful description of the concept in very general setting of elliptical copulae.

See, for example, Embrechts et al. (1999, Chapter 1) for vastly more detail on the notion of heavy tails. See also , Kotz et al. (2001) or Barndorff-Nielsen et al. (1982) for more information on variance-mixture distributions and tail dependence.

The cubic in equation 4.90 gives rise to many more terms than shown. Any term including G or 𝜖 _n not raised to a power, however, is equal to zero and can be ignored.

See Hu and Kercheval (2007) for more details.

Another possible choice is to set V to an exponentially-distributed random variable, which gives rise to a Laplace distribution. Kotz et al. (2001, Chapter 2) provides more details.

More specifically, if X follows a three-parameter generalized inverse Gaussian distribution, it is written X ∼GIG(a, b, p). In the two-parameter version, X ∼GIG(a, p) ≡GIG(a, a, p).

These specialized functions arise naturally as the solution to various differential equations. The interested reader is referred to Abramovitz and Stegun (1965, Chapter 9) for more details.

We could, of course, require that var(V ) = 1 as in the variance-gamma case. This would, however, require an additional parameter, which we would prefer to avoid.

There are, incidentally, many useful R packages that might be employed. The variance-gamma and generalized-hyperbolic distributions are, for example, handled in Scott (2015a,b).

This package was developed by Laurent Gautier. It does not seem possible to find any formal paper written on this package. There are, however, numerous useful blogs and online documentation available on the Internet for the interested reader.

The very convenient notation used in this section has been borrowed, naturally without implication, from Glasserman (2006).

To resolve equation 4.107, it helps to recall that for independent random variables, x ₁, …, x _N, each with \(\mathbb {E}(x_i)=0\), their product \(\mathbb {E}(x_n x_m)=\mathbb {E}(x_n)\cdot \mathbb {E}(x_m)\ne 0\) only when n = m.

It can, of course, get significantly more involved. See Daul et al. (2005) for additional extensions in this area with a particular focus on the t-distribution.

This excludes an obligors’s correlation with itself.

The idea is to induce heightened correlation among regions and were each credit counterpart to form its own region, the effect would be indistinguishable from the idiosyncratic factor.

One could naturally add an additional index to permit varying loadings to the global variable.

The measure used to describe the distance between the two matrices is the Frobenius norm; see Golub and Loan (2012, Chapter 2) for more details.

The ρ parameter is approximately 0.25.

In this case, the ρ parameter is about 0.07 and ν = 30.

The reader is also encouraged to compare the results with the multivariate CreditRisk+ implementation in Table 3.15 on page 147.

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

Barndorff-Nielsen, O., Kent, J., & Sørensen, M. (1982). Normal variance-mean mixtures and z distributions. International Statistical Review, 50(2), 145–159.CrossRef

BIS. (2006a). International convergence of capital measurement and capital standards: A revised framework comprehensive version. Technical report. Bank for International Settlements.

Daul, S., de Giorgi, E., Lindskog, F., & McNeil, A. (2005). The grouped t-copula with an application to credit risk. ETH Zürich.

de Kort, J. (2007). Modeling tail dependence using copulas—literature review. University of Amsterdam.

Devroye, L. (2012). Random variate generation for the generalized inverse Gaussian distribution. Statistical Computing, 24(1), 239–246.

Eberlein, E., & Keller, U. (1995). Hyperbolic distributions in finance. Bernoulli, 3(1), 281–299.CrossRef

Embrechts, P., Klüppelberg, C., & Mikosch, T. (1999). Modelling extremal events for insurance and finance (1st edn.). New York, NY: Springer-Verlag. Stochastic modelling and applied probability.

Fok, P.-W., Yan, X., & Yao, G. (2014). Analyis of credit portfolio risk using hierarchical multi-factor models. University of Delaware.

Glasserman, P. (2006). Measuring marginal risk contributions in credit portfolios. Risk Measurement Research Program of the FDIC Center for Financial Research.

Golub, G. H., & Loan, C. F. V. (2012). Matrix computations. Baltimore, Maryland: The John Hopkins University Press.

Gordy, M. B. (2002). A risk-factor model foundation for ratings-based bank capital rules. Board of Governors of the Federal Reserve System.

Gupton, G. M., Finger, C. C., & Bhatia, M. (2007). CreditMetrics — technical document. New York: Morgan Guaranty Trust Company.

Hu, W., & Kercheval, A. (2007). Risk management with generalized hyperbolic distributions. Proceedings of the fourth IASTED international conference on financial engineering and applications (pp. 19–24).

Joe, H. (1997). Multivariate models and dependence concepts. Boca Raton, FL: Chapman & Hall/CRC. Monographs on statistics and applied probability (vol. 73).

Johnson, N. L., Kotz, S., & Balakrishnan, N. (1994). Continuous univariate distributions: volume I (2nd edn.). New York, NY: John Wiley & Sons.

Kostadinov, K. (2005). Tail approximation for credit-risk portfolios with heavy-tailed risk factors. Zentrum Mathematik, Technische Universität München.

Kotz, S., Kozubowski, T., & Podgorski, K. (2001). The Laplace distribution and generalizations: A revisit with applications to communications, economics, engineering, and finance. Basel, Switzerland: Birkhäuser.CrossRef

Kuhn, G. (2005). Tails of credit default portfolios. Zentrum Mathematik, Technische Universität München.

Leydold, J., & Hörmann, W. (2013). Generating generalized inverse gaussian random variates. Institute for Statistics and Mathematics, Report 123, Wirstshafts Universität Wien.

Leydold, J., & Hörmann, W. (2017). Package ‘GIGrvg. R Package Documentation.

MacKenzie, D., & Spears, T. (2012). “The Formula That Killed Wall Street?” The Gaussian copula and the material cultures of modelling. University of Edinburgh.

Madan, D. B., & Seneta, E. (1990). The variance gamma (V.G.) model for share market returns. The Journal of Business, 63(4), 511–524.CrossRef

McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative risk management: Concepts, tools and techniques. Princeton, NJ: Princeton University Press.

Merton, R. (1974). On the pricing of corporate debt: The risk structure of interest rates. Journal of Finance, 29, 449–470.

Nelsen, R. B. (2006). An introduction to copulas. New York, NY: Springer.

Roth, M. (2013). On the multivariate t distribution. Division of Automatic Control, Department of Electric Engineering, Linköpings Universitet, Sweden.

Schmidt, R. (2003). Credit risk modelling and estimation via elliptical copulae. In G. Bohl, G. Nakhaeizadeh, S. Rachev, T. Ridder, & K. Vollmer (Eds.), Credit risk-measurement, evaluation and management (pp. 267–289). Physica-Verlag.

Schmitz, V. (2003). Copulas and stochastic processes. Rheinisch-Westfällisch Technischen Hochschule Aaachen.

Schönbucher, P. J. (2000b). Factor models for portfolio credit risk. Department of Statistics, University of Bonn.

Scott, D. (2015a). Package ‘GeneralizedHyperbolic. R Package Documentation.

Scott, D. (2015b). Package ‘VarianceGamma. R Package Documentation.

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.

Sklar, A. (1959). Fonctions de répartition à n dimensions et leurs marges. Publications de l’Institut de Statistique de L’Université de Paris, 8, 229–231.

Vasicek, O. A. (1987). Probability of loss on loan distribution. KMV Corporation.

Vasicek, O. A. (1991). Limiting loan loss probability distribution. KMV Corporation.

Vasicek, O. A. (2002). The distribution of loan portfolio value. Risk, (12), 160–162.

von Hammerstein, E. A. (2016). Tail behaviour and tail dependence of generalized hyperbolic distributions. In J. Kallsen, & A. Papapantoleon (Eds.), Advanced modelling in mathematical finance (pp. 3–40). Springer-Verlag,

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

Titel: Threshold 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_4