Stochastic Processes in Actuarial Surplus Modelling: A Validation Framework

This study develops a validation framework for actuarial surplus modelling using stochastic processes, focusing on Geometric Brownian Motion (GBM) for assets, the Vasicek model for interest rates, and the Ornstein-Uhlenbeck (OU) process for liabilities. Through Monte Carlo simulations (10,000 paths, 1-year horizon), we evaluate three scenarios: GBM with deterministic liabilities, GBM with OU liabilities, and GBM with Vasicek-discounted liabilities. Analytical derivations of the stochastic differential equations and their solutions provide the mathematical foundation, while Euler-Maruyama discretisation ensures numerical tractability. Results demonstrate that mean-reverting models (OU and Vasicek) reduce surplus volatility (standard deviation: €360,000–€375,000 vs. €400,000) and tail risk (99.5% VaR: -€60,000, –€75,000 vs. -€100,000; CVaR: -€100,000 -€120,000 vs. -€150,000) compared to the deterministic liability model. These findings align with Solvency II’s risk-based capital requirements, highlighting the Vasicek model’s superior stability for liability valuation and the OU model’s suitability for dynamic liability adjustments. The framework supports enhanced solvency assessment and product pricing, with implications for regulatory compliance and risk management in insurance.