Suppose that in a one-factor Gaussian copula model the 5...

Suppose that in a one-factor Gaussian copula model the 5 -year probability of default for each of 125 names is $3 \%$ and the pairwise copula correlation is 0.2 . Calculate, for factor values of $-2,-1,0,1$, and 2 : (a) the default probability conditional on the factor value and (b) the probability of more than 10 defaults conditional on the factor value.

Key Concepts

Gaussian Copula

A Gaussian copula is a function that links together marginal distributions into a joint multivariate distribution using the Gaussian (normal) dependence structure. In credit risk, it is used to model the joint default behavior of different entities by correlating their underlying latent variables using the multivariate normal distribution.

One-Factor Model

In a one?factor model, all individual default risks are driven by a single common systematic risk factor in addition to idiosyncratic risk components. This framework simplifies the dependence structure by assuming that the latent variable associated with each obligor is a linear combination of a common factor and an independent noise term.

Conditional Default Probability

This concept refers to the probability that an individual entity defaults given a specific realization of the common systematic risk factor. By conditioning on the factor's value, the model adjusts the default probability using the relationship between the underlying latent variable, the default threshold, and the specified factor, often via transformation using the inverse normal cumulative distribution function.

Binomial Distribution for Defaults

When considering a large portfolio of names, once the conditional default probability is determined for a given realization of the factor, the total number of defaults can be modeled using a binomial distribution. This approach assumes that, given the systematic factor, defaults occur independently across names, allowing for the computation of probabilities of observing more than a certain number of defaults.

Inverse Cumulative Distribution Function (Quantile Function)

The inverse cumulative distribution function of the standard normal distribution (often denoted as ??¹) is used to determine the default thresholds corresponding to a given default probability. In the context of the Gaussian copula model, it transforms unconditional probabilities into latent variable cutoffs that are then adjusted by the systematic factor in the computation of conditional default probabilities.

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