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\title{\vspace*{-2.5cm} Explicit Bounds and Approximations
for Polynomial Roots in Mathematics of Finance}
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\author{J\"urgen Herzberger\\
Fachbereich Mathematik\\
Carl von Ossietzky Universit\"at Oldenburg\\
26111 Oldenburg / GERMANY}
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{\it Keywords:\/} Polynomial roots, nonlinear equations\\
{\it MOS(AMS) Subject Classification:\/} 26C05, 65H05
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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.
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