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Sharp Large Deviations and Gibbs Conditioning for Threshold Models in Portfolio Credit Risk

Fengnan Deng, Anand N. Vidyashankar, Jeffrey F. Collamore

TL;DR

The paper studies sharp large deviations for exceedance probabilities in dependent threshold models for portfolio credit risk using a diverging-factor triangular array. It develops a Laplace Olver endpoint analysis together with conditional Bahadur Rao bounds to obtain regime-specific nonlogarithmic prefactors that reflect latent-factor tail geometry. A key finding is a Gibbs conditioning principle in total variation, yielding asymptotically i.i.d. tilted defaults under large-loss events, with explicit tilts for both unbounded and boundary cases. The framework unifies GN and RV tail behaviors through log-smooth conditions and provides second-order VaR and ES expansions, offering transferable techniques for sharp rare-event analysis in dependent threshold systems.

Abstract

We obtain sharp large deviation estimates for exceedance probabilities in dependent triangular array threshold models with a diverging number of latent factors. The prefactors quantify how latent-factor dependence and tail geometry enter at leading order, yielding three regimes: Gaussian or exponential-power tails produce polylogarithmic refinements of the Bahadur-Rao $n^{-1/2}$ law; regularly varying tails yield index-driven polynomial scaling; and bounded-support (endpoint) cases lead to an $n^{-3/2}$ prefactor. We derive these results through Laplace-Olver asymptotics for exponential integrals and conditional Bahadur-Rao estimates for the triangular arrays. Using these estimates, we establish a Gibbs conditioning principle in total variation: conditioned on a large exceedance event, the default indicators become asymptotically i.i.d., and the loss-given-default distribution is exponentially tilted (with the boundary case handled by an endpoint analysis). As illustrations, we obtain second-order approximations for Value-at-Risk and Expected Shortfall, clarifying when portfolios operate in the genuine large-deviation regime. The results provide a transferable set of techniques-localization, curvature, and tilt identification-for sharp rare-event analysis in dependent threshold systems.

Sharp Large Deviations and Gibbs Conditioning for Threshold Models in Portfolio Credit Risk

TL;DR

The paper studies sharp large deviations for exceedance probabilities in dependent threshold models for portfolio credit risk using a diverging-factor triangular array. It develops a Laplace Olver endpoint analysis together with conditional Bahadur Rao bounds to obtain regime-specific nonlogarithmic prefactors that reflect latent-factor tail geometry. A key finding is a Gibbs conditioning principle in total variation, yielding asymptotically i.i.d. tilted defaults under large-loss events, with explicit tilts for both unbounded and boundary cases. The framework unifies GN and RV tail behaviors through log-smooth conditions and provides second-order VaR and ES expansions, offering transferable techniques for sharp rare-event analysis in dependent threshold systems.

Abstract

We obtain sharp large deviation estimates for exceedance probabilities in dependent triangular array threshold models with a diverging number of latent factors. The prefactors quantify how latent-factor dependence and tail geometry enter at leading order, yielding three regimes: Gaussian or exponential-power tails produce polylogarithmic refinements of the Bahadur-Rao law; regularly varying tails yield index-driven polynomial scaling; and bounded-support (endpoint) cases lead to an prefactor. We derive these results through Laplace-Olver asymptotics for exponential integrals and conditional Bahadur-Rao estimates for the triangular arrays. Using these estimates, we establish a Gibbs conditioning principle in total variation: conditioned on a large exceedance event, the default indicators become asymptotically i.i.d., and the loss-given-default distribution is exponentially tilted (with the boundary case handled by an endpoint analysis). As illustrations, we obtain second-order approximations for Value-at-Risk and Expected Shortfall, clarifying when portfolios operate in the genuine large-deviation regime. The results provide a transferable set of techniques-localization, curvature, and tilt identification-for sharp rare-event analysis in dependent threshold systems.

Paper Structure

This paper contains 37 sections, 27 theorems, 256 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Main Results
  3. Model description and assumptions
  4. Gibbs conditioning in total variation
  5. Preliminary results
  6. Proofs
  7. Proof of Theorem \ref{['thm:LDP_ndZ']} and Proposition \ref{['prop:LDP_ndZ']}
  8. Proof of Theorem \ref{['thm:LDP_kappa']}
  9. Proof of the lower bounded support case
  10. Proof of the degenerate case
  11. Proof of Theorem \ref{['thm:LDP_gen_ndZ']}
  12. Proof of Theorem \ref{['thm:gibbs']}
  13. Concluding Remarks
  14. Appendix
  15. Proof of Lemma \ref{['pf:lem_Gen_1']}
  16. ...and 22 more sections

Key Result

Theorem 2.1

Assume that conditions (assumption1)-(assumption3), asp:L2-asp:L3 hold. There exists a sequence of constants $M_n \nearrow \infty$ and a collection of functions $\{ \phi_n \}$ such that where $\exp(n \phi_n(-\tilde{M}_n))$ converges to a constant depending only on the distribution of $\epsilon$ and $\mathcal{Z}_n$, and $H_n(-\tilde{M}_n)$ diverges to infinity as $n \to \infty$.

Theorems & Definitions (59)

  • Theorem 2.1
  • Proposition 2.1
  • Remark 2.1
  • Remark 2.2: Multi-factor regularly varying tails
  • Theorem 2.2
  • Remark 2.3
  • Theorem 2.3
  • Remark 2.4: Scope and reduction to $\mathcal{C}$
  • Remark 2.4: Scope and reduction to $\mathcal{C}$
  • Theorem 2.4
  • ...and 49 more