CreditRisk+ in the Banking Industry

Frontmatter

1. Introduction

Summary

We give a brief description of the history, possibilities and applications of CreditRisk⁺ as well as an overview of the book.

Volker Matthias Gundlach, Frank Berthold Lehrbass

2. Basics of CreditRisk+

Summary

We present the fundamental ideas of CreditRisk⁺ and give an introduction to the main notions of this credit risk model. In particular we set up the notations for the book and describe the original version of CSFP.

Volker Matthias Gundlach

3. Capital Allocation with CreditRisk+

Summary

Capital allocation for credit portfolios has two meanings. First, at portfolio level it means to determine capital as a buffer against an unexpected negative cash-flow resulting from credit losses. In this case, the allocation method can be specified by means of a risk measure. Its result is called economic capital of the portfolio. Second, at subportfolio or transaction level, capital allocation means breaking down the economic capital of the portfolio into its sub-units. The resulting capital assignments are called risk contributions. We discuss several current concepts for economic capital and risk contributions in a general setting. Then we derive formulas and algorithms for these concepts in the special case of the CreditRisk⁺ methodology with individual independent potential exposure distributions.

Dirk Tasche

4. Risk Factor Transformations Relating CreditRisk+ and CreditMetrics

Summary

CreditRisk⁺ and CreditMetrics furnish special cases of general credit risk factor models. On a respective model space, there is a symmetry of factor transformations that relates identical though differently represented models. In the simplest case of homogeneous one-factor one-band-models, there is an approximate symmetry between consistently parametrized CreditRisk⁺ and CreditMetrics. This can be viewed as evidence that there exists in general a consistent parametrization of both models that results in the same loss distribution.

Christian Wieczerkowski

5. Numerically Stable Computation of CreditRisk+

Summary

We present an alternative numerical recursion scheme for CreditRisk⁺, equivalent to an algorithm recently proposed by Giese, based on well-known expansions of the logarithm and the exponential of a power series. We show that it is advantageous to the Panjer recursion advocated in the original CreditRisk⁺ document, in that it is numerically stable. The crucial stability arguments are explained in detail. Furthermore, the computational complexity of the resulting algorithm is stated.

Hermann Haaf, Oliver Reiß, John Schoenmakers

6. Enhanced CreditRisk+

Summary

In this chapter we discuss the link between the CreditRisk⁺ loss distribution and the moment-generating function (MGF) of the risk factors. We show that the probability-generating function (PGF) of the loss variable is the MGF of the factors, evaluated at a particular “point”. This approach has two major advantages: it leads to a new recursion formula for the portfolio loss distribution that is faster and more accurate than the standard approach. It also allows us to extend the modelling framework to a wider class of factor distributions incorporating sector correlations. At the end of this chapter we show that risk contributions are related to the partial derivatives of the MGF. We derive the formula for exact risk contributions in this generalized modelling framework and highlight the differences from the corresponding result obtained in the saddlepoint approximation.

Götz Giese

7. Saddlepoint Approximation

Summary

Saddlepoint approximation offers a robust and extremely fast alternative to Panjer recursion for the solution of the CreditRisk⁺ loss distribution. This chapter shows how saddlepoint approximation can be applied to an extended version of CreditRisk⁺ that incorporates idiosyncratic severity risk. Regardless of the number of sectors and without any need for discretizing loss exposures, both value-at-risk and expected shortfall are easily calculated.

Michael B. Gordy

8. Fourier Inversion Techniques for CreditRisk+

Summary

The CreditRisk⁺ model is described in terms of characteristic functions, and two methods to determine the distribution of the credit loss based on Fourier inversion are presented. For the convenience of the reader, a short introduction to the theory of characteristic functions and the Fourier transformation is given. Then two general results are stated how to obtain the distribution of a random variable from its characteristic function. These general techniques, which are based on Fourier inversion, will be applied to the CreditRisk⁺ model and yield efficient and numerically stable algorithms, which provide the loss distribution in the CreditRisk⁺ framework. Advantages of this approach are that the algorithms are easy to implement and that a basic loss unit is not required.

Oliver Reiß

9. Incorporating Default Correlations and Severity Variations

Summary

The original CreditRisk⁺ methodology does not allow for incorporation of industry, geographical or other segment correlations in modelling default events. We provide an extension that enables modelling of default correlations among segments while preserving the analytical solution for the loss distribution. Moreover, the proposed methodology can consistently be extended to independently (of default events) model stochastic severities in collateral devaluation. This extension imposes further distributional assumptions on the model. Nevertheless, it is shown that in the limit of a large portfolio the loss distribution is determined solely by the systematic components of default and severity risk. Finally, the relevance of this observation to portfolio management and stress loss analysis is motivated.

Nese Akkaya, Alexandre Kurth, Armin Wagner

10. Dependent Risk Factors

Summary

As an extension to the standard CreditRisk⁺ model we discuss two multivariate factor distributions, which include factor correlations. The moment-generating functions (MGFs) of both distributions have a simple analytical form, which fits into the framework of Chapter 6 so that the nested evaluation recursion scheme can be applied. We show how the parameters of the new distributions can be fitted to an externally given covariance matrix for the risk factors. With the example of a test portfolio we compare the new models with a single-factor approach to correlation, which has been proposed in [1].

Götz Giese

11. Integrating Rating Migrations

Summary

While CreditRisk⁺ is generally regarded as a powerful, fast and easyto-use credit portfolio model, there is often criticism that its definition of credit loss according to the default mode approach is inferior to the more comprehensive markto-market approach used in other credit portfolio models like CreditMetrics. In this chapter we present a practical, “easy to implement” procedure that allows us to integrate the rating migration concept — an important advantage of CreditMetrics — into CreditRisk⁺. Rating-driven changes in market value that are characteristic of liquid portfolios are included in this model without losing the benefits of CreditRisk⁺.

Frank Bröker, Stefan Schweizer

12. An Analytic Approach to Rating Transitions

Summary

Extending CreditRisk⁺ to a multi-state model allows one to incorporate credit quality changes into the calculation of portfolio credit risk. Several modifica­tions to the original methodology are proposed to make the extension to a mark­to-market model tractable. The distribution of portfolio value changes is obtained analytically by a two-dimensional recursion algorithm.

Carsten Binnenhei

13. Dependent Sectors and an Extension to Incorporate Market Risk

Summary

In standard CreditRisk⁺ the risk factors are assumed to be independently gamma distributed and as a consequence the model can be computed analytically. If one extends the model such that the risk factors are dependently distributed with quite arbitrary distributions, one has to give up the existence of a closed-form solution. The advantage of this approach is that one gains interesting generalizations and the computational effort to determine the loss distribution still remains quite small.

In the first step the model will be generalized such that the risk factors are dependently distributed with quite arbitrary distributions and as a suitable choice in practice, the dependent lognormal distribution is suggested. This model can then be transferred to a time continuous model and the risk factors become processes, more precisely geometric Brownian motions. Having a time continuous credit risk model is an important step to combining this model with market risk. Additionally a portfolio model will be presented where the changes of the spreads are driven by the risk factors. Using a linear expansion of the market risk, the distribution of this portfolio can be determined. In the special case that there is no credit risk, this model yields the well-known delta normal approach for market risk, hence a link between credit risk and market risk has been established.

Oliver Reiß

14. Econometric Methods for Sector Analysis

Summary

Default event correlation and dependency plays a key role in determining portfolio credit risk. We present two econometric methods that can be employed to forecast default rates and describe correlations of the forecasts. The first approach is company-specific, based on a Merton-style threshold model. The second focuses on the systematic risk factors influencing the obligors of a specific industry or country. Both approaches are particularly well suited to estimating default rate development and correlation in a conditional independence framework such as CreditRisk⁺. Based on panel data covering a variety of countries and industries we present an implementation example for the second method of seemingly unrelated regression and illustrate the implications of dynamic and correlated default rates on the risk of a large portfolio of credits.

Leif Boegelein, Alfred Hamerle, Michael Knapp, Daniel Rösch

15. Estimation of Sector Weights from Real-World Data

Summary

We discuss four different approaches to the estimation of sector weights for the CreditRisk⁺ model from German real-world data. Using a sample loan portfolio, we compare these approaches in terms of the resulting unexpected loss risk figures.

Michael Lesko, Frank Schlottmann, Stephan Vorgrimler

16. Risk-Return Analysis of Credit Portfolios

Summary

We consider a problem of real-world risk-return analysis of credit portfolios in a multi-objective function setting with respect to additional constraints. For the approximation of a set of feasible, risk-return-efficient portfolio structures in this setting we discuss a flexible approach that incorporates multi-objective evolutionary and local search methods as well as specific features of the CreditRisk⁺ model. We apply the hybrid approach to a sample loan portfolio to illustrate its working principle.

Frank Schlottmann, Detlef Seese, Michael Lesko, Stephan Vorgrimler

17. Numerical Techniques for Determining Portfolio Credit Risk

Summary

Two numerical algorithms for the risk analysis of credit portfolios are presented. The first one determines the distribution of credit losses and is based on the fast Fourier transform. The algorithm has a strong analogy to the CreditRisk⁺ approach, since both the Poisson approximation as well as the technique of forming homogeneous exposure bands are being used. An application to the analysis of collateralized debt obligations is also given. The second algorithm makes use of an importance sampling technique for allocating credit risk contributions according to the risk measure expected shortfall. The coherent risk spectrum that is obtained by varying the loss exceedance level is introduced and its properties are discussed.

Sandro Merino, Mark Nyfeler

18. Some Remarks on the Analysis of Asset-Backed Securities

Summary

In this chapter we discuss the analysis of asset-backed securities (ABS) in the environment of competitive risk-based pricing in the banking industry. We will cover the relevant aspects that need to be considered before investing in an ABS structure. When it comes to model-based pricing approaches a portfolio model is needed. The practitioner may either choose a simulation-based approach or an analytical model, where both have their advantages and shortcomings. We will focus on the usage of CreditRisk⁺ in the context of ABS pricing, outline the prerequisites for running the model in practice and finally discuss the pricing of a simple ABS structure with CreditRisk⁺.

Daniel Kluge, Frank B. Lehrbass

19. Pricing and Hedging of Structured Credit Derivatives

Summary

Closed-form solutions are the Holy Grail in derivative pricing. We provide such pricing formulas for structured credit derivatives such as first- and k-th-to-default baskets as well as all kinds of tranches of collateralized debt obligations. First, we employ the conditional independence framework to derive semi-explicit pricing formulas for basket credit default swaps. Second, by introducing a linear factor model for individual hazard rates we obtain pricing formulas in terms of the moment-generating functions of the risk factors. We show how to calibrate this factor model to market data, especially to time series of credit spreads. Thus we distinguish between exogenous and endogenous modelling. For the cases of explicitly given moment-generating functions the pricing formulas become explicit.

This approach is characterized by its great flexibility and easy calibration to observed market data and leads to an efficient pricing by analytical calculation. In particular, our approach eases the calculation of hedge ratios for dynamic hedges of basket credit default swaps (also known as single-tranche technology or correlation trading) and other complex trading strategies.

Martin Hellmich, Oliver Steinkamp

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