Dependence bounds for the difference of stop-loss payoffs on the difference of two random variables

TL;DR

This paper investigates bounds for the difference of stop-loss payoffs based on two random variables, analyzing how comonotonic and countermonotonic modifications can provide bounds despite the payoff's non-convexity, with applications to longevity bonds.

Contribution

It introduces a method to construct bounds for stop-loss payoff differences using dependence structures, supported by numerical analysis of mortality-linked securities.

Findings

Crossing points of cdfs are generally unique for longevity bonds.

Symmetric copulas allow median-based approximations of crossing points.

Extreme dependence structures can provide bounds for tail risk layers.

Abstract

This paper considers the difference of stop-loss payoffs where the underlying is a difference of two random variables. The goal is to study whether the comonotonic and countermonotonic modifications of those two random variables can be used to construct upper and lower bounds for the expected payoff, despite the fact that the payoff function is neither convex nor concave. The answer to the central question of the paper requires studying the crossing points of the cdf of the original difference with the cdfs of its comonotonic and countermonotonic transforms. The analysis is supplemented with a numerical study of longevity trend bonds, using different mortality models and population data. The numerical study reveals that for these mortality-linked securities the three pairs of cdfs generally have unique pairwise crossing points. Under symmetric copulas, all crossing points can reasonably be approximated by the difference of the marginal medians, but this approximation is not necessarily valid for asymmetric copulas. Nevertheless, extreme dependence structures can give rise to bounds if the layers of the bond are selected to hedge tail risk. Further, the dependence uncertainty spread can be low if the layers are selected to hedge median risk, and, subject to a trade-off, to hedge tail risk as well.