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A dimension reduction approach for loss valuation in credit risk modelling

Jian He, Asma Khedher, Peter Spreij

TL;DR

This work tackles the curse of dimensionality in loss valuation for credit risk by introducing a dimension reduction method based on the Bayesian filter and smoother. The approach projects high-dimensional transition-factor models onto a lower-dimensional latent space, achieving an MMSE-optimal, interpretable projection that feeds a valuation grid used for loss distribution calculations. The authors derive a closed-form Black-Scholes-type formula for the expected LGD (ELGD) and demonstrate through numerical experiments that the Bayesian projection outperforms PCA in estimating expected losses and Value-at-Risk across several risk metrics and applications (CREC, IRC, IFRS 9 ECL, market risk, IRRBB). The results indicate faster, more accurate loss valuations with broad applicability to risk valuation beyond credit losses, offering a practical, implementable framework for financial institutions.

Abstract

This paper addresses the ``curse of dimensionality'' in the loss valuation of credit risk models. A dimension reduction methodology based on the Bayesian filter and smoother is proposed. This methodology is designed to achieve a fast and accurate loss valuation algorithm in credit risk modelling, but it can also be extended to valuation models of other risk types. The proposed methodology is generic, robust and can easily be implemented. Moreover, the accuracy of the proposed methodology in the estimation of expected loss and value-at-risk is illustrated by numerical experiments. The results suggest that, compared to the currently most used PCA approach, the proposed methodology provides more accurate estimation of expected loss and value-at-risk of a loss distribution.

A dimension reduction approach for loss valuation in credit risk modelling

TL;DR

This work tackles the curse of dimensionality in loss valuation for credit risk by introducing a dimension reduction method based on the Bayesian filter and smoother. The approach projects high-dimensional transition-factor models onto a lower-dimensional latent space, achieving an MMSE-optimal, interpretable projection that feeds a valuation grid used for loss distribution calculations. The authors derive a closed-form Black-Scholes-type formula for the expected LGD (ELGD) and demonstrate through numerical experiments that the Bayesian projection outperforms PCA in estimating expected losses and Value-at-Risk across several risk metrics and applications (CREC, IRC, IFRS 9 ECL, market risk, IRRBB). The results indicate faster, more accurate loss valuations with broad applicability to risk valuation beyond credit losses, offering a practical, implementable framework for financial institutions.

Abstract

This paper addresses the ``curse of dimensionality'' in the loss valuation of credit risk models. A dimension reduction methodology based on the Bayesian filter and smoother is proposed. This methodology is designed to achieve a fast and accurate loss valuation algorithm in credit risk modelling, but it can also be extended to valuation models of other risk types. The proposed methodology is generic, robust and can easily be implemented. Moreover, the accuracy of the proposed methodology in the estimation of expected loss and value-at-risk is illustrated by numerical experiments. The results suggest that, compared to the currently most used PCA approach, the proposed methodology provides more accurate estimation of expected loss and value-at-risk of a loss distribution.
Paper Structure (32 sections, 2 theorems, 43 equations, 13 figures, 1 table, 2 algorithms)

This paper contains 32 sections, 2 theorems, 43 equations, 13 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem description and background
  3. Contribution
  4. Organization of the paper
  5. Set up and background
  6. Overview of loss calculations in credit risk
  7. Valuation model for the credit losses
  8. Valuation grid for losses valuation
  9. valuation grid pre-processing and loss simulation
  10. A dimension reduction approach: Bayesian filter projection
  11. State space model
  12. Bayesian filter and smoother
  13. Bayesian filter recursion
  14. Bayesian smoother recursion
  15. Bayesian filter/smoother projection
  16. ...and 17 more sections

Key Result

Proposition A.1

For given $\mathrm{LC}_0$, it holds that with

Figures (13)

  • Figure 2.1: An illustration to determine the loss distribution
  • Figure 2.2: An illustration to determine the loss distribution using valuation grids
  • Figure 3.1: High level description of projection algorithm
  • Figure 3.2: High level description of the algorithm
  • Figure 4.1: Relative difference of the transition probabilities: the red line is for the proposed Bayesian projection approach and the blue line is for the PCA. On the $x$-axis are the 100 simulated scenarios.
  • ...and 8 more figures

Theorems & Definitions (5)

  • Remark 3.1
  • Proposition A.1
  • proof
  • Corollary A.2
  • proof