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2018 | OriginalPaper | Buchkapitel

10. Default and Asset Correlation

verfasst von : David Jamieson Bolder

Erschienen in: Credit-Risk Modelling

Verlag: Springer International Publishing

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Abstract

If default dependence is the heart of credit-risk modelling, then an empirical estimate of its magnitude is of primordial importance. Unlike estimation of default probabilities, addressed in the previous chapter, the characterization of default dependence is model dependent rendering this task more difficult. Since different models incorporate the relationship between obligor defaults in alternative ways, dependence is governed by some subset of a model’s parameters. As usual, a variety of techniques are presented, examined, and concretely implemented. The first, based on the method of moments, applies quite generally and is conceptually similar to the calibration techniques employed in previous chapters. A second approach, using observed default outcomes, exploits conditional independence to build a likelihood function and applies in both mixture and threshold settings. This permits use of the maximum-likelihood framework for the production of both point and interval estimates. The final, somewhat complex and fragile, approach is only applicable to the family of threshold models. It enjoys the advantage of using all transition data, but simultaneously requires inference of the unobservable global state variable values. The robustness of the final two techniques are assessed within separate simulation studies.

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Fußnoten
1
Practically speaking, we use a uniform distribution to assign counterparties to these categories using these weights. This implies that, given the randomness, the final portfolio weights can deviate slightly.
 
2
The inverse normal distribution is employed since we find ourselves in the Gaussian threshold setting. The inverse cumulative distribution function is, however, determined by the latent state variable’s underlying characteristics.
 
3
Any sufficiently large positive number will, of course, work.
 
4
It will, of course, vary depending on the actual default probability of the individual credit counterparties.
 
5
Although we do not provide the figures, if we estimate the default probabilities, the values are equivalent to those found in Chap. 9. Transition probabilities are thus preserved under the imposition of default correlation.
 
6
See Magnello (2009) for an illuminating review of Karl Pearson’s path-breaking contributions to the field of statistics.
 
7
Hansen (1982) offers a full-fledged framework, referred to as the generalized method of moments, where the moment equations are replaced with orthogonality conditions inferred from one’s model. While a useful and powerful approach, it seems a bit excessive in this context.
 
8
The mixture-model default correlation expression can also be found in Chap. 3.
 
9
This is, incidentally, the only model in this group with a single parameter value.
 
10
The Gaussian model, which takes only a single state variable, is performed separately.
 
11
It is, of course, possible to use Efron (1979)’s bootstrap technique to generate standard errors.
 
12
Normalized, in this context, means dividing all reported log-likelihood values by its maximum.
 
13
In this case, of course, we are minimizing the negative of the log-likelihood function, which amounts to maximization.
 
14
See Press et al. (1992, Chapter 10) for more detail on non-linear optimization algorithms.
 
15
Chernick (1999) provides an excellent overview of this technique with a wealth of useful references.
 
16
By extension, 1 − ω describes the relative importance of the idiosyncratic element.
 
17
Typically, the gamma distribution requires two parameters, but a parameter is lost due to the necessity of constraining the expectation of this random variable to unity. See Chap. 3 for much more detail.
 
18
See, for example, Judge et al. (1985, Chapter 17) or Dhrymes (1994, Chapter 6) for a more rigorous discussion of identification.
 
19
Not all systems of n equations and n unknowns have solutions, of course, but this is the basic idea.
 
20
The reader might be thinking that, since this is a simulated dataset, there is no empirical reality. While true, the actual dataset was generated from an alternative model, which is not unlike the situation faced by real-life statisticians.
 
21
If, as in Gordy and Heitfield (2002), we used the credit-rating, this would not be necessary.
 
22
As R increases, however, it will become increasingly difficult and time-consuming to maximize the likelihood kernel.
 
23
This follows directly from equation 10.9.
 
24
Some useful studies that address these issues, to a certain extent, include Hashimoto (2009) and Martin (2013).
 
25
It is, of course, implicit in the calculations.
 
26
It is clear that there is some simplification if F̆n ≡ F n for n = 1, …, N; we will address this point shortly.
 
27
For much more technical rigour on the notion of absolute continuity, see Billingsley (1995)
 
28
This is a technical constraint associated with the form of multivariate t distribution employed in this analysis. See Kotz and Nadarajah (2004) for more details.
 
29
For more detail, see Held and Bové (2014, Chapter 2).
 
30
There are a variety of approaches—both with and without numerical derivatives—to solve this problem. Beck (2014) is an excellent reference in this area.
 
31
This discussion can be framed more formally in terms of injective and surjective functions, but this is perhaps too abstract for this setting. See Royden (1988, Chapter 1) or Bartle and Sherbert (1982, Chapter 1) for more background on these ideas.
 
32
This approach is also described in detail in Berger et al. (1999).
 
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Metadaten
Titel
Default and Asset Correlation
verfasst von
David Jamieson Bolder
Copyright-Jahr
2018
DOI
https://doi.org/10.1007/978-3-319-94688-7_10