The LaPlace Transform as Stochastic Present Value

In finance, the LaPlace transform is used to calculate the distribution of stochastic present value. There are several practical impediments to the use of the LaPlace transform in actuarial science: we lack a physical interpretation of the transform, it requires a change in perspective to a frame of reference that we seldom use, and it involves complex arithmetic. This paper will define terms and provide interpretations and visualizations of the transform, the alternate frame of reference as well as the meaning of complex values in an actuarial setting. The dual purposes of this paper are to give actuaries a formula for computing the stochastic present value distribution and to attach a meaning to the LaPlace transform in an actuarial context.