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Modeling portfolio loss distribution under infectious defaults and immunization

Gabriele Torri, Rosella Giacometti, Gianluca Farina

TL;DR

The paper introduces a tractable contagion-based framework for portfolio loss distributions in credit portfolios that blends idiosyncratic defaults with systemic infection and an immunization mechanism to curb feedback. It develops a recursive algorithm to compute the loss distribution and extends the model with two factor-based specifications: a conditional contagion model tied to a Gaussian factor and a two-state mixture contagion-default regime. The framework is applied to pricing synthetic iTraxx CDO tranches, demonstrating that the mixture (MIX) specification delivers the best fit across multiple dates and offers clear economic interpretation via parameters like contagion share $\omega$ and mixing probability $\pi$, while remaining computationally efficient. The approach provides a versatile tool for pricing, scenario analysis, and systemic risk monitoring, enabling regime-switching views of default dependence and contagion dynamics in credit markets.

Abstract

We introduce a model for the loss distribution of a credit portfolio considering a contagion mechanism for the default of names which is the result of two independent components: an infection attempt generated by defaulting entities and a failed defence from healthy ones. We then propose an efficient recursive algorithm for the loss distribution. Then we extend the framework with more flexible distributions that integrate a contagion component and a systematic factor to better fit real-world data. Finally, we propose an empirical application in which we price synthetic CDO tranches of the iTraxx index, finding a good fit for multiple tranches.

Modeling portfolio loss distribution under infectious defaults and immunization

TL;DR

The paper introduces a tractable contagion-based framework for portfolio loss distributions in credit portfolios that blends idiosyncratic defaults with systemic infection and an immunization mechanism to curb feedback. It develops a recursive algorithm to compute the loss distribution and extends the model with two factor-based specifications: a conditional contagion model tied to a Gaussian factor and a two-state mixture contagion-default regime. The framework is applied to pricing synthetic iTraxx CDO tranches, demonstrating that the mixture (MIX) specification delivers the best fit across multiple dates and offers clear economic interpretation via parameters like contagion share and mixing probability , while remaining computationally efficient. The approach provides a versatile tool for pricing, scenario analysis, and systemic risk monitoring, enabling regime-switching views of default dependence and contagion dynamics in credit markets.

Abstract

We introduce a model for the loss distribution of a credit portfolio considering a contagion mechanism for the default of names which is the result of two independent components: an infection attempt generated by defaulting entities and a failed defence from healthy ones. We then propose an efficient recursive algorithm for the loss distribution. Then we extend the framework with more flexible distributions that integrate a contagion component and a systematic factor to better fit real-world data. Finally, we propose an empirical application in which we price synthetic CDO tranches of the iTraxx index, finding a good fit for multiple tranches.

Paper Structure

This paper contains 20 sections, 9 theorems, 68 equations, 4 figures, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Literature review
  3. The proposed infection model
  4. Portfolio loss distribution at time $t$
  5. Portfolio loss distribution with contagion and correlations
  6. Conditional contagion loss distribution
  7. Mixture contagion loss distribution
  8. Default correlations
  9. An application to CDO pricing
  10. CDO pricing
  11. Specifications and calibration of the loss distribution
  12. Choice of the parameters $\mu_i$
  13. Simulation study
  14. Empirical analysis
  15. Dataset
  16. ...and 5 more sections

Key Result

Proposition 1

Let $X_i,V_i,U_i$, $i=1,...,n$ be mutually independent Bernoulli variables with probabilities $p_i= P\{X_i=1\}, \,\, u_i=P\{U_i=1\} , \,\, v_i=P\{V_i=1\}$ on a time horizon $[0,t]$. Let $Z_i$ be defined according to Eq. eq:MainModDesc and let $\tilde{p}_i := P\{Z_i=1\}$. We have: where

Figures (4)

  • Figure 1: Sensitivity of the loss distribution to parameters $\tilde{p}$, $\omega$, $\mu$. The baseline parameters for the distribution are $\tilde{p} = 0.05$, $\omega=0.5$, $\mu = 0.1$, and there are 125 names in the portfolio. The horizontal axis represents the number of defaults and, although the maximum possible number is 125, only the range up to 40 is displayed for clarity. Parameters are homogeneous across all names.
  • Figure 2: Loss distribution for different models with the parameters in Table \ref{['tab:loss_distr_parameters']}. The horizontal axis represents the number of defaults and, although the maximum possible number is 125, only the range up to 40 is displayed for clarity.
  • Figure 3: Evolution over time of the average Mean Absolute Error across the priced tranches with spread of the untranched index shaded in the background (upper panels) and optimal parameters (bottom panels) for four selected models (OFG, CON-FIN, COND-FIN, and MIX-FIN models from October 31, 2019 to May 30, 2025.
  • Figure 4: 10 Monte Carlo estimates of the loss distribution based on 1000, 5000, and 10000 simulations.

Theorems & Definitions (14)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • proof : Proof
  • Proposition : 1
  • proof
  • Proposition : 2
  • proof
  • ...and 4 more