A new approach to interest rate sensitivity using the complex plane

The time value of money equation is often differentiated to obtain the interest rate sensitivity of whatever is being valued. The process is so common in finance that it could be said to be automatic it is never questioned. Two recent examples, chosen at random from the literature, are Skinner [1998] and Crack and Nawalkha [2000]. It is well known that it yields approximate answers. Considerable extra effort is necessary to improve the approximations by adding the extra terms of a Taylor series expansion. But the results remain approximations. This paper summarises an entirely new approach based on the fact that the time value of money is a polynomial, and a polynomial does not have one root, it has many, distributed around the complex plane. The result of taking the multiple solutions into account is a new equation that provides precise results for the measure of interest rate sensitivity. The paper describes the computational issues presented by the new approach and suggests a way to deal with them. The analysis provides a new perspective that may be a launch pad for further research.