Stochastic Simulations underpinning Compartmental Models for P&C Reserving
Compartmental models constitute a semi-deterministic method for estimating reserves. As explained in Morris (2016) and Gesmann and Morris (2020) they consist of systems of differential equations with unknown process parameters that usually depend on time. These parameters are determined using Bayesian methods that compare and update solutions of the differential equations against observed data (such as loss ratios of open and paid claims). As an alternative we borrow ideas from chemical kinetics and interpret the claim process as a stochastic process of chemical reactions. To simulate this stochastic process, variants of Gillespie’s stochastic simulation algorithm (SSA) are applied. Since its time-dependent parameters (the propensities) are closely related to the paid and reported loss ratios, these available data are incorporated into the stochastic model from the beginning. We employ optimal transport techniques to derive a time-varying probability distribution of the ratios between observation times (e.g. development years). With this probability distribution we construct the propensities for each run of SSA, thereby simulating one realization of the claim process. Repeatedly drawing a propensity and running SSA allows us to synthesize the distribution and confidence intervals for the reserves at a preset time horizon. An advantage of this method is its ability to adapt to changing conditions and its potential to include expert knowledge and further available information about the claims process – once we have a mathematical description of these additions.
The validation of our stochastic model is done using claims triangles where one part of each triangle is needed to calibrate the propensities and the other to check the predicted claims development. Back-testing is implemented in the R environment but can be transferred easily to other environments.
Find the Q&A here: Q&A on 'General Insurance Reserving'