Consistent Iterated Simulation of Multivariate Defaults: Markov Indicators, Lack of Memory, Extreme-Value Copulas, and the Marshall–Olkin Distribution

A current market-practice to incorporate multivariate defaults in global risk-factor simulations is the iteration of (multiplicative) i.i.d. survival indicator increments along a given time-grid, where the indicator distribution is based on a copula ansatz. The underlying assumption is that the behavior of the resulting iterated default distribution is similar to the one-shot distribution. It is shown that in most cases this assumption is not fulfilled and furthermore numerical analysis is presented that shows sizable differences in probabilities assigned to both “survival-of-all” and “mixed default/survival” events. Moreover, the classes of distributions for which probabilities from the “terminal one-shot” and “terminal iterated” distribution coincide are derived for problems considering “survival-of-all” events as well as “mixed default/survival” events. For the former problem, distributions must fulfill a lack-of-memory type property, which is, e.g., fulfilled by min-stable multivariate exponential distributions. These correspond in a copula-framework to exponential margins coupled via extreme-value copulas. For the latter problem, while looping default inspired multivariate Freund distributions and more generally multivariate phase-type distributions could be a solution, under practically relevant and reasonable additional assumptions on portfolio rebalancing and nested distributions, the unique solution is the Marshall–Olkin class.