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Modelling time-dependent interest rates by mathematical formula. (English) Zbl 0785.90011
Summary: The modelling of time-dependent interest rates by means of a mathematical formula containing a finite number of parameters is considered. It is shown that Stoodley’s formula, and its generalization due to McCutcheon, may both the regarded as special cases of a more general formula for the time-dependent force of interest function which permits exact analytic calculation of the corresponding valuation function. This is achieved by considering a change of variables which transforms the differential equation satisfied by the valuation function to a first-order equation of the Fuchsian class. Three specific functional forms for the change of variables are considered, and examples of the corresponding force of interest functions are displayed graphically. The present value of a continuous cash flow of constant rate under the action of such a general time-dependent force of interest function is also considered, and is shown to be expressible in closed form in terms of hypergeometric functions of several variables.
91B28 Finance etc. (MSC2000)
91B62 Economic growth models