Application of the Laplace Transform for the Evaluation of Consols‘ Present Value

The aim of the article is to demonstrate the use of the Laplace transform to the evaluation of consols’ present value under different streams of returns (dividends). Consols (or perpetual bonds, perpetuities) are bonds with no maturity, which means that interest is paid to a bondholder perpetually (usually annually) forever. The present value of a consol, when constant interest is paid, is simply a ratio of this interest and interest rate. However, when interest payments (stream of returns to a bondholder) change in time the evaluation of consol’s present value is more complicated. In the first part of this article it is shown that the Laplace transform of a stream of returns can be used to the evaluation present value of a consol under assumption of continuous compounding; and moreover, with the inverse Laplace transform an unknown stream of returns can be reconstructed from the known consol’s present value. In the second part of the paper, the use of the Laplace transform for the present value evaluation is illustrated by examples. Introduction The Laplace transform, named after French mathematician and astronomer Pierre-Simon Laplace (1749-1827), is the integral transformation widely used in many areas of physics and engineering. It belongs into broader class of integral transformations including Fourier, Hartley, Hilbert, Poisson, Weierstrass and other transforms. The main area of the application of the Laplace transform is the solution of ordinary and partial differential equations (see e.g. [3], [8], [9] or [10]), because the Laplace transform converts differential equations into algebraic ones, which are easier to handle with. Another area of application includes signal and image processing, the theory of electric circuits, probability theory or impuls-response analysis. In economics, the Laplace transform can be applied to the analysis of dynamics and shocks in time series of macroeconomic indicators [4], for pricing barrier options [7], analysis