On distributions with fixed marginals maximizing the joint or the prior default probability, estimation, and related results

TL;DR

This paper investigates the maximum probability of certain joint or sequential default events between two components with fixed marginals, using copula theory, and provides estimators and characterizations for these probabilities.

Contribution

It introduces copula-based formulas for maximizing default probabilities under fixed marginals and generalizes results to non-monotonic transformations, with estimators and illustrative examples.

Findings

Derived explicit formulas for maximum joint default probabilities.

Characterized transformations where maximum probabilities coincide.

Proposed a strongly consistent estimator with asymptotic normality.

Abstract

We study the problem of maximizing the probability that (i) an electric component or financial institution $X$ does not default before another component or institution $Y$ and (ii) that $X$ and $Y$ default jointly within the class of all random variables $X,Y$ with given univariate continuous distribution functions $F$ and $G$, respectively, and show that the maximization problems correspond to finding copulas maximizing the mass of the endograph $\Gamma^\leq(T)$ and the graph $\Gamma(T)$ of $T=G \circ F^-$, respectively. After providing simple, copula-based proofs for the existence of copulas attaining the two maxima $\overline{m}_T$ and $\overline{w}_T$ we generalize the obtained results to the case of general (not necessarily monotonic) transformations $T:[0,1] \rightarrow [0,1]$ and derive simple and easily calculable formulas for $\overline{m}_T$ and $\overline{w}_T$ involving the distribution function $F_T$ of $T$ (interpreted as random variable on $[0,1]$). The latter are then used to charac\-terize all non-decreasing transformations $T:[0,1] \rightarrow [0,1]$ for which $\overline{m}_T$ and $\overline{w}_T$ coincide. A strongly consistent estimator for the maximum probability that $X$ does not default before $Y$ is derived and proven to be asymptotically normal under very mild regularity conditions. Several examples and graphics illustrate the main results and falsify some seemingly natural conjectures.