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Collateralized Debt Obligations have sparked interest in portfolio default models over multiple horizons. For these, in contrast to single period models, there is little understanding of the impact of model assumptions. We investigate four multiple horizon models, each calibrated to the same set of input data. Our results show a significant disparity, showing that the issue of model choice is more consequential here than in the single period case.
Introduction
In recent years, models of defaults in a portfolio context have been well studied. Three separate approaches (CreditMetrics, CreditRisk+, and CreditPortfolioView 1 ) were made public in 1997. Subsequently, researchers 2 have examined the mathematical structure of the various models. Each of these studies has revealed it is possible to calibrate the models to each other and the differences between the models lie in subtle choices of the driving distributions and in the data sources one would naturally use to feed the models. Common to all of these models and the subsequent examinations thereof is the fact that the models describe only a single period. In other words, the models describe, for a specific risk horizon, whether each asset of interest defaults within the horizon. The timing of defaults within the risk horizon is not considered, nor is the possibility of defaults beyond the horizon. This is not a flaw of the current models, but rather an indication of their genesis as approaches to risk management and capital allocation for a fixed portfolio.
Not entirely by chance, the development of portfolio models for credit risk management has coincided with an explosion in issuance of Collateralized Debt Obligations (CDO's). The performance of a CDO structure depends on the default behavior of a pool of assets. Significantly, the dependence is not just on whether the assets default over the life of the structure, but also on when the defaults occur. Thus, while an application of the existing models can give a cursory view of the structure (by describing, for instance, the distribution of the number of assets that will default over the structure's life), a more rigorous analysis requires a model of the timing of defaults.
In this paper, we will survey a number of extensions of the standard single-period models that allow for a treatment of default timing over longer horizons. We will examine two extensions of the CreditMetrics approach: one that models only defaults over time and a second that effectively accounts for rating migrations. In addition, we will examine the copula function approach introduced by Li (1999) and Li (2000), as well as a simple version of the stochastic intensity model applied by Duffie and Garleanu (1998).
We will investigate the differences in the four approaches that arise from model – rather than data – differences. Thus, we will suppose that we begin with satisfactory estimates of expected default rates over time and of the correlation of default events over one period. Higher order information, such as the correlation of defaults in subsequent periods or the joint behavior of three or more assets, will be driven by the structure of the models. The analysis of the models will then illuminate the range of results that can arise given the same initial data. Nagpal and Bahar (1999) adopt a similar approach in the single horizon context, investigating the range of possible full distributions that can be calibrated to first and second order default statistics.
In the following section, we present terminology and notation to be used throughout. We proceed to detail the four models. Finally, we present two comparison exercises: in the first, we use closed- form results to analyze default rate volatilities and conditional default probabilities, while in the second, we implement Monte Carlo simulations in order to investigate the full distribution of realized default rates.
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