Time dependent stop-loss reinsurance and exposure curves

Abstract

Stop-loss contracts are the most commonly used reinsurance agreements in insurance whose important factors are the retention and the maximum (cap) values attained on the random loss, which may occur within the policy period. Therefore, determining and forecasting the loss amounts is an important issue for both the insurer and the reinsurer. Along with many approaches in actuarial literature, we propose a geometric Brownian motion (BM) with the time-varying parameters to capture the time-dependent loss amounts. We implement the time-influence on stop-loss contract in the frame of the stochastic model and find the analytical derivations of costs associated with reinsurance contract for reinsurer and insurer with constraints on time, loss amount, retention, and both retention and cap levels. Additionally, the analytical forms of exposure curves are depicted to determine the premium share between reinsurer and insurer under time, loss, retention, and both retention and cap constraints. An application of the proposed methodology on real-life data and the calibration of time-varying parameters using dynamic maximum likelihood estimator and simulations on the proposed model are performed. Finally, we forecast the claim amounts, expected costs, and exposure curves on time-varying parameters using the cubic spline extrapolation and the dynamic ARIMA with trend search. It is shown that the time-varying approach using the stochastic model copes with the behavior of the claims and assures fair share between insurer and reinsurer.