Bounds for joint default probability

This question considers two risky bonds and asks for the range of possible values for the probability that at least one of them defaults over a horizon, given only their individual default probabilities. The candidate must reason about all admissible joint distributions of two binary default indicators that are consistent with the specified marginals. The focus is not on any particular credit model, but on understanding how far dependence can push systemic default risk, from the most favorable to the most adverse co-movement between the two bonds' default events.

The solution leans on basic probability inequalities for unions and intersections, along with the Fréchet-Hoeffding bounds for joint distributions with fixed marginals. It also requires familiarity with indicator random variables and how correlation is derived from their joint and marginal probabilities. An interviewer is watching for clear identification of feasible joint probabilities, correct normalization when translating joint probabilities into correlation, and an understanding that correlation is constrained by the marginal default rates. Logical consistency checks, such as ensuring probabilities stay within the unit interval, are also important.

What it tests

When dealing with two binary events with fixed marginal probabilities, the joint probability structure—and thus any function of it, like the probability of at least one event or their correlation—is constrained by the possible ways the events can overlap. The minimum and maximum probabilities for unions and intersections arise from the Fréchet-Hoeffding bounds, which reflect the limits imposed by the marginals: the joint probability cannot be less than the sum minus one, nor more than the smaller marginal. This is because probabilities must remain between zero and one, and the overlap cannot exceed what the marginals allow. The correlation between indicators is then determined by how much the joint probability deviates from the product of the marginals, normalized by the product of their standard deviations. This structure holds for any pair of binary events with given marginals, regardless of the context.