Description: The series expansion of a probability density function, known to actuaries by Esscher's name and to statisticians as the saddlepoint approximation, typically gets integrated to approximate probability values under the density, with a location parameter in the expansion chosen to optimize convergence of the integrated series. We review the general form, the accuracy and the derivation of the Esscher approximation. Then we propose using related series expansions for the derivative of a probability density function to approximate the value of the point of maximum likelihood for the density. The key is to make the value of the location parameter an unknown and assume it to be the point of maximum likelihood, which leads to a tractable optimization problem to find its value.

Description: This paper discusses an example showing how simulation has been used to develop financial projections for variable life insurance with a guaranteed minimum death benefit and how recourse to mathematical analysis based on risk-theoretic considerations can enable us to see the impact of stock price changes more clearly. These projections reveal the effects of stock price fluctuations interacting with the more common elements, such as mortality and persistency, on the financial results of a variable life insurance operation.