Three extensions to the visualisation of financial concepts in the complex plane

In a previous edition of this journal Osborne [2000] contains new expressions for two financial concepts from the bond markets: modified duration and the yield to maturity. The concepts are given in terms of the distances between the roots of the time value of money equation and other salient points in the complex plane. This note offers three extensions. First, another new expression is offered, this time a simple and elegant equation for the price of a bond. Secondly, since its introduction by Macauley [1938], the concept of duration has suffered from several shortcomings. One of these shortcomings is the fact that the traditional formulas for duration, Macauley or modified, give estimates of the interest elasticity of the bond price that do not allow for the curvature of the link between price and the interest rate. The orthodox formula can be made more accurate by supplementing it with convexity; however, the result is still an approximation. The problem can be found illustrated in any finance text. It is shown here that the new approach to duration outlined in Osborne [2000] can be adjusted to give a formula that yields precise results for the change in the price of a bond in response to a change in the interest rate. This result, and the methodology surrounding it, is important because it demonstrates that the new perspective from the complex plane not only gives an alternative view of an existing concept, but also improves on it. In addition, it explains why the orthodox formula does not provide precise results, and why it cannot be adjusted to do so. Thirdly, in order to make the theory operational in the bond markets, a question must be answered. How can the theory be adjusted to cope with bonds priced part way through a coupon period? The required level of detail in time in the bond markets is down to the individual day, and the required level of accuracy down to $1 in a million. In the context of these requirements the new approach poses questions about computation. The questions are stated here, but not answered.