Asymptotics of Random Contractions

TL;DR

This paper investigates the asymptotic behavior of random contractions, which are products of a positive random variable and a bounded random variable, with applications in insurance and finance, focusing on tail asymptotics and risk measures.

Contribution

It derives the tail asymptotics of random contractions under specific distributional assumptions and applies these results to risk models and tail risk measures.

Findings

Derived tail asymptotics for random contractions with max-domain of attraction distributions.

Quantified the impact of random scaling on Conditional Tail Expectations.

Established joint asymptotic distributions of linear combinations of contractions.

Abstract

In this paper we discuss the asymptotic behaviour of random contractions $X=RS$, where $R$, with distribution function $F$, is a positive random variable independent of $S\in (0,1)$. Random contractions appear naturally in insurance and finance. Our principal contribution is the derivation of the tail asymptotics of $X$ assuming that $F$ is in the max-domain of attraction of an extreme value distribution and the distribution function of $S$ satisfies a regular variation property. We apply our result to derive the asymptotics of the probability of ruin for a particular discrete-time risk model. Further we quantify in our asymptotic setting the effect of the random scaling on the Conditional Tail Expectations, risk aggregation, and derive the joint asymptotic distribution of linear combinations of random contractions.