On the Computation of A New Formula for the Duration of A Bond that Yields Precise Results Without the Need for Convexity and Other Devices

Abstract

The time value of money (TVM) equation is a core equation in finance. It is often differentiated to obtain the interest rate sensitivity of whatever is being valued. In fixed income analysis the result is incorporated into the concept known as duration. It is well known, however, that the various versions of duration yield only approximate answers. Even after the considerable extra effort necessary to improve the approximations by adding concepts such as convexity, or creating a duration vector, the results remain approximations. This paper summarizes an entirely new approach to the TVM equation based on the fact that it is a polynomial, and a polynomial does not have only one root. It has many, distributed around the complex plane. The result of taking the multiple solutions into account is a solution to the problem of inaccuracy that has existed in the duration literature since the time of Macauley [1938]. In this paper, a new equation for duration is given that provides precise results for the measure of interest rate sensitivity, no matter what the change in the rate. No extra devices, such as convexity, are needed. The contribution of the paper is to summarize work to date, then to go on to describe the computational issues presented by the new approach, and, finally, to suggest ways to deal with the issues. More important, perhaps, is that the analysis provides a new, more general, perspective to the TVM equation that may be a launch pad for further research into the many other financial concepts that depend on it.