Conditional expectations given the sum of independent random variables with regularly varying densities

Abstract The conditional expectation mₗ (s) =EX|S=s, where X and Y are two independent random variables with S=X+Y, plays a key role in various actuarial applications. For instance, considering the conditional mean risk-sharing rule, mX (s) determines the contribution of the agent holding the risk X to a risk-sharing pool. It is also a relevant function in the context of risk management, for example, when considering natural capital allocation principles. The monotonicity of mX (\!\!) is particularly significant under these frameworks, and it has been linked to log-concave densities since Efron (1965). However, the log-concavity assumption may not be realistic in some applications because it excludes heavy-tailed distributions. We consider random variables with regularly varying densities to illustrate how heavy tails can lead to a nonmonotonic behavior for mX (\!\!). This paper first aims to identify situations where mX (\!\!) could fail to be increasing according to the tail heaviness of X and Y. Second, the paper aims to study the asymptotic behavior of mX (s) as the value s of the sum gets large. The analysis is then extended to zero-augmented probability distributions, commonly encountered in applications to insurance, and to sums of more than two random variables and to two random variables with a Farlie–Gumbel–Morgenstern copula. Consequences for risk sharing and capital allocation are discussed. Many numerical examples illustrate the results.