\documentclass[12pt]{article} \textheight24cm \textwidth17cm \oddsidemargin-0.5cm \title{\vspace*{-2.5cm} Explicit Bounds and Approximations for Polynomial Roots in Mathematics of Finance} % \author{J\"urgen Herzberger\\ Fachbereich Mathematik\\ Carl von Ossietzky Universit\"at Oldenburg\\ 26111 Oldenburg / GERMANY} % \date{} \begin{document} \large \maketitle \thispagestyle{empty} \noindent {\it Keywords:\/} Polynomial roots, nonlinear equations\\ {\it MOS(AMS) Subject Classification:\/} 26C05, 65H05 \vspace*{0.5cm} \noindent Considering several financial problems leads to polynomial equations in mathematics of finance. Those are, for example the equations for the effective rate-of-return of a bond, of an installment loan or of an annuity. The calculation of the roots in question -- usually a positive root of the equation -- is an easy task with a programmable computer. This is due to the fact that the polynomial is always convex in the range under consideration. Thus NEWTON's method is monotone and NEWTON-FOURIER's method calculates bounds for the root. However, there are more general aspects which do not allow to use such kind of methods. Firstly, the equation may contain some parameters which make it impossible to do explicit numerical calculations. Secondly, one needs some simple formulas in order to use more primitive desk calculators allowing only the four basic operations. In all this cases one needs in some sense simple expressions for the roots only requiring a sequence of a modest number of basic operations. The talk gives examples for such polynomial equations and brings new formulas for approximations and for bounds of the roots. It turns out that some results of estimating polynomial roots which were considered in the calculation of the order of convergence of iterative numerical processes can be useful here since the structure of the polynomials in both fields is very similar. It is shown that the monotonicity principle as well as a very special approach considered by TRAUB, HERZBERGER and KJURKCHIEV in earlier papers can be used here. Numerical examples for concrete cases are presented and a comparison with some results in the literature is made. \end{document} 