Abstract
In this paper, it is analyzed whether a Fourier-based approach can be an efficient tool for calculating risk measures in the context of a credit portfolio model with integrated market risk factors. For this purpose, this technique is applied to a version of the well-known credit portfolio model CreditMetrics, extended by correlated interest rate and credit spread risk. While Fourier-based methods are reported to be superior to full Monte Carlo simulations for default mode models, this result cannot be confirmed for the integrated market and credit portfolio model used here. The application of standard importance sampling techniques for improving the performance of the Fourier-based approach is problematic, too. However, combining the full Monte Carlo simulation with an importance sampling technique indeed yields superior results, even for the integrated market and credit portfolio model.
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Notes
An exception is the approach Algo Credit developed by the risk management firm Algorithmics (see Iscoe et al. 1999).
Furthermore, an application of the Fourier-based approach to the analysis of asset-backed securities and collateralized debt obligations can be found in Debuysscher and Szegö (2003).
Using moment generating functions, Finger (1999) describes a similar approach for the original CreditMetrics framework without market risk.
For details concerning this procedure, see Gupton et al. (1997, p 85).
In the Vasicek model, the short rate at the risk horizon is normally distributed and can be represented by \(r{\left( t \right)} = \theta + {\left( {r{\left( 0 \right)} - \theta } \right)}e^{{ - \kappa t}} + {\sqrt {{\sigma ^{2}_{r} {\left( {1 - e^{{ - 2\kappa t}} } \right)}} \mathord{\left/ {\vphantom {{\sigma ^{2}_{r} {\left( {1 - e^{{ - 2\kappa t}} } \right)}} {{\left( {2\kappa } \right)}X_{r} }}} \right. \kern-\nulldelimiterspace} {{\left( {2\kappa } \right)}X_{r} }} }\,\,{\left( {X_{r} \; \sim \;N{\left( {0,1} \right)};\;\kappa ,\theta ,\sigma \in \mathbb{R}_{ + } } \right)}.\)
For simplicity, it is assumed that all defaultable zero coupon bonds have the same time to maturity T n =T.
For example, Barnhill and Maxwell (2002) estimate a short rate volatility of 0.007, whereas Lehrbaß (1997) finds σ r =0.029 and Huang and Huang (2003) even work with σ r =0.0468. With regard to the mean reversion parameter and the market price of interest rate risk, Lehrbaß finds κ=1.169 and absolute values of 0.59, 0.808, and 1.232 for the parameter λ, whereas Huang and Huang choose κ=0.226 and an absolute value of 0.248 for λ.
The grid points and weights of the Gauss–Legendre integration rule for n=96 are taken from Abramowitz and Stegun (1984, p 397). The length of the intervals on which the Gaussian integration is applied increases because, for rising values of t, the absolute value of the oscillating integrand decreases rapidly. Alternatively, the trapezoidal rule has been tried for computing Eq. 3. However, due to the larger number of grid points needed by this numerical integration rule, the performance of the Fourier-based approach gets worse.
However, in general, the number of exposure buckets will also increase when the number of obligors increases (see the comment later in this section). In Merino and Nyfeler’s approach, the number N of obligors has an influence on the dimension of the discrete Fourier transforms of the conditional loss distribution of each exposure bucket and, hence, on the number of component-wise complex multiplications needed to determine the discrete Fourier transform of the conditional portfolio loss distribution.
Binnenhei (2004) proposes an analytic extension of the CreditRisk+ model, which incorporates rating transitions. In his approach, all proper transition probabilities are approximated by a Poisson distribution. Aside from the problem that, with this approximation, multiple credit events of single obligors can occur in all possible combinations, one necessary condition to apply his approach is that there are no value changes when the obligor does not change its initial rating. However, this prerequisite is not fulfilled within the integrated market and credit portfolio model because, even given that an obligor maintains his initial rating, the exposures are stochastic due to the market risk.
The meaning of “similar” is concretized by the definition of a so-called basic loss unit. The losses of all exposure buckets are an integer multiple of this basic loss unit, and the exposure buckets only differ in the value of the multiple.
See Merino and Nyfeler (2002, p 83).
For the ease of exposition, it is assumed that the random vector (Z,X) is multivariate normally distributed.
For a more detailed discussion of this approach when applied to integrated market and credit portfolio models, see Grundke (2006).
Alternatively, to save computation time, one set of IS means could be used for the estimation of all percentiles corresponding to “higher” confidence levels. Glasserman and Li (2005) report that the variance reduction effect is relatively insensitive with respect to the IS means. For the simulation results reported in Tables 7 and 8, the exact percentile estimators for each confidence level are used for determining the IS means for the systematic risk factors Z and X r .
Other papers, which apply variance reduction techniques to Monte Carlo simulations in a credit risk context, are from Joshi (2004), Joshi and Kainth (2004), Kalkbrener et al. (2004), Merino and Nyfeler (2004), Morokoff (2004), Tchistiakov et al. (2004), Egloff et al. (2005), Rott and Fries (2005), Bassamboo et al. (2006), and Glasserman (2006).
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Acknowledgements
I wish to thank Rosalind Bennett, Marc Gürtler, and two anonymous referees for their helpful comments. For stimulating discussions, I also thank the participants of the Global Finance Conference in Dublin 2005, the Annual Meeting of the European Financial Management Association in Milan 2005, the International Conference on Finance in Copenhagen 2005, the International Scientific Annual Conference Operations Research in Bremen 2005, the Annual Meeting of the German Finance Association in Augsburg 2005, and the Annual Meeting of the Financial Management Association in Chicago 2005.
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A previous version of this paper was entitled “Application of Fourier inversion methods to credit portfolio models with integrated interest rate and credit spread risk”.
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Grundke, P. Computational aspects of integrated market and credit portfolio models. OR Spectrum 29, 259–294 (2007). https://doi.org/10.1007/s00291-006-0050-7
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DOI: https://doi.org/10.1007/s00291-006-0050-7
Keywords
- Credit risk
- Interest rate risk
- Credit spread risk
- Credit portfolio model
- Value at risk
- Characteristic function
- Fourier transforms