Journal of Risk

Multi-factor default correlation model estimation: enhancement with bootstrapping

Zhihui Yang, Saikat Ray Majumder, Weiwei Shen, Stephane Karm, Douglas Cameron and James Gellert

  • We introduce a three-factor default correlation model for more accurate credit risk quantification.
  • Grid search estimates as initial values for the maximum likelihood estimators (MLE) are proposed.
  • The proposed method offers better MLE convergence with initial values from our grid search approach.
  • We demonstrate a bootstrap method to circumvent the sparsity of historical default data.

To assess the overall credit risk of a portfolio, it is important to consider the risk correlation between the counterparties (obligors) in addition to their individual credit risks. However, the estimation of the default correlations between obligors is challenging due to the scarcity of data and a lack of reporting default events as common practice. A constrained two-factor Merton correlation model with homogeneous risk classes has been proposed in the literature to achieve a balance between simplicity and usefulness. Yet, the assumption that assets are classified by two risk factors is practically oversimplified. Therefore, in this paper we extend the two-factor Merton correlation model into a three-factor Merton model to more realistically represent real-world practice. The maximum likelihood estimation and the grid search estimation methods for calibration are discussed in detail. For validation, we evaluate the robustness and stability of model parameters estimated from both methods on simulated and real-world default data. We also adopt the bootstrap approach to address the small sample size challenge of real-world data. The proposed model allows us to reveal the additional risk specification without sacrificing estimation accuracy. In addition, we present a convergence analysis to provide guidance on the number of observations to ensure stability of the calibrated maximum likelihood estimation.

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