# Finite-time ruin probabilities under large-claim reinsurance treaties for heavy-tailed claim sizes

## TL;DR

This paper derives explicit asymptotic formulas for the probability of ruin within a finite time for heavy-tailed claim sizes under large-claim reinsurance treaties, comparing different contracts and validating with simulations.

## Contribution

It provides the first explicit asymptotic expressions for finite-time ruin probabilities under large-claim reinsurance for heavy-tailed claims, using recent large deviations theory.

## Key findings

- Ruin probability is regularly varying with initial capital under heavy tails.
- Explicit asymptotic formulas for LCR and ECOMOR reinsurance contracts.
- Simulation confirms the accuracy of the asymptotic approximations.

## Abstract

We investigate the probability that an insurance portfolio gets ruined within a finite time period under the assumption that the r largest claims are (partly) reinsured. We show that for regularly varying claim sizes the probability of ruin after reinsurance is also regularly varying in terms of the initial capital, and derive an explicit asymptotic expression for the latter. We establish this result by leveraging recent developments on sample-path large deviations for heavy tails. Our results allow, on the asymptotic level, for an explicit comparison between two well-known large-claim reinsurance contracts, namely LCR and ECOMOR. We finally assess the accuracy of the resulting approximations using state-of-the-art rare event simulation techniques.