A current market-practice to incorporate multivariate defaults in global riskfactor
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 sizeable 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.