Threshold Models and the Gaussian Copula for Portfolio Default

A single-obligor PD of 1% looks manageable. A portfolio of 1,000 obligors each with 1% PD looks diversified. But the aggregate loss distribution is qualitatively different depending on whether defaults are independent or correlated: with even a 0.5% default correlation, the probability of 30 or more simultaneous defaults in that portfolio is roughly five times higher than under independence (Figure 11.1). This is the portfolio credit risk problem.

The threshold model provides a natural extension of Merton's firm-value model to a portfolio: each obligor

i

defaults if a latent "credit quality" variable

X_i

falls below a threshold

d_i

. The joint default distribution is then entirely controlled by the copula of the vector

X = (X_1,\ldots,X_m)

. Choosing the Gaussian copula recovers the industry-standard Gaussian threshold model — equivalently, the multivariate version of Merton's model. Choosing a

t

-copula or Clayton copula produces models with heavier joint tails and more tail dependence.

Key result (Lemma 11.2). Two threshold models produce the same joint default distribution if and only if their critical variables have (i) the same marginal default probabilities and (ii) the same copula. The copula is therefore the sole determiner of default dependence structure for a given set of marginal PDs.

Learning objectives. By the end of this note you should be able to:

define a threshold model and state the default probability in terms of the marginal CDF of $X_i$ ;
define default correlation and distinguish it from asset correlation;
state Lemma 11.2 and use it to compare models with the same marginals but different copulas;
set up the Gaussian threshold model with one-factor structure and compute asset correlations;
describe how alternative copulas ( $t$ , Clayton) affect the tail of the default count distribution;
explain why the copula choice is a major source of model risk in portfolio credit.

In this lesson

1. The one-period portfolio credit model
1. Threshold models and Sklar's theorem
1. The Gaussian threshold model
1. Alternative copulas and model risk

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