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: On October 30, 2006, the UK government's Stern Review on the Economics of Climate Change made startling claims about the present value today of climate change effects predicted to emerge primarily 150 years or more in the future. The ensuing debate included challenges to and defenses of the particular methodology used and parameter values selected to perform the discounting of future values. Noticing an overlooked faux pas in the Stern Review, this presentation uncovers the likely motivation for choosing the most mystifying of the parameters values, and provides a lesson on the underpinnings of the basic tool of actuarial science.

Description: The behavior of extreme paths in the usual stochastic interest rate models is not nearly so plausible as the behavior of their expected values and variances. This paper proposes a new class of models that deliver more plausible extreme paths while preserving the usual expected value and variance behavior. Along the way we derive the closed form solution for the traditional mean-reverting lognormal process, including the drift compensation.