contingent functions

With two lives, you can promise a benefit not just on a death, but on a death in a particular order. A contingent benefit pays only if a specified life dies first (or second, or before some other event). Picture a policy that pays a widow a sum only if her husband dies before her — if she dies first, nothing is paid even though a death has occurred. The benefit is 'contingent' on whose death comes first, not merely on a death happening at all.

Contingent functions value exactly these order-dependent promises. The classic one is a contingent insurance written A^1_{xy}: it pays 1 at the moment life x dies, but only if x dies before life y. The little 1 over the x marks 'this is the life whose death triggers payment, and it must die first'. To compute it you sum, over each future moment, the benefit times the probability that x dies right then while y is still alive. A natural and tidy relationship falls out: the contingent insurance where x dies first, plus the contingent insurance where y dies first, equals the joint-life insurance paid on the first death of either, A^1_{xy} + A^1_{yx} = A_{xy} (first-death). The two order-specific pieces partition the first-death benefit between them.

Contingent functions are how you price reversionary annuities (income that starts for one person only after another dies), survivor pensions, and order-of-death clauses in family and business policies. They are the most error-prone corner of multi-life work, for two reasons. First, the order matters and is easy to mislabel — A^1_{xy} and A^1_{yx} are different benefits. Second, they lean even harder on the independence assumption than joint-life functions do, because correlated mortality distorts not just whether two people die but the likelihood of one dying before the other. Treat them carefully and check which life the contingency is on.