Random Walk Model for Paid Loss Development: A

Traditional loss development techniques focus on estimating the expected ultimate loss but do not generally indicate the magnitude of possible deviation from this estimate. In a variety of circumstances, however, point reserve estimates are not sufficient. In particular, loss portfolio transfers, commutations, innovations, and reserve margin securitization all typically require an estimate of the range of possible loss outcomes. By adjusting a paid loss model described in Foundations of Casualty Actuarial Science to incorporate a random fluctuation component, a stochastic differential equation model is obtained, This model is analogous to the stock price model used to develop the Black-Scholes option pricing formula. Furthermore, this differential equation has an explicit solution that yields Lognormal distributed development factors similar to the Lognormal link-ratio model published by Roger Hayne. A slight modification to the model for undiscounted reserves provides a differential equation that accounts for variation in both the amount and timing of loss payments. This equation does not have an explicit solution but can be solved numerically to yield the distribution of the present value reserve.