A Series of Interest

From: David W. Cantrell (DWCantrell_at_sigmaxi.org)
Date: 08/26/04 

Date: 26 Aug 2004 04:39:50 GMT

A common financial formula is P = iA/(1 - (1+i)^(-N)), where P denotes the
payment, i the interest rate, A the loan amount, and N the number of
payments. A series is presented here for the interest rate.

In the aus.mathematics newsgroup recently, someone asked about solving such
an equation for i. Ken Pledger mentioned some appropriate links, such as
Stan Brown's <http://oakroadsystems.com/math/loan.htm> (see formula (2)
there) and the sci.math FAQ entry
<http://db.uwaterloo.ca/~alopez-o/math-faq/node76.html>. The formula cannot
be solved for the interest rate in closed form in terms of elementary
functions. Apparently the most common approach to determining i is to use a
numerical technique, such as Newton's method. But an alternative technique
is to use a series:

Letting u = (PN/A - 1)/(N + 1),

i = 2( u - (N-1) u^2/3 + (N-1) (2N+1) u^3/9 - (N-1) (2N+1) (11N+7) u^4/135
         + (N-1) (2N+1)^2 (13N+11) u^5/405 -+...)

which I obtained by reversion of series. Surely this series must be well
known to those who deal with such matters often; I would greatly appreciate
references to it. (Of course, more terms of the series could be given here,
but references should make doing so unnecessary.)

Examples:

1. In example 11 from Stan Brown's page, in which P = $200,000,
A = $2,800,000 and N = 19, using just the five terms of the series shown
above, we obtain i = 3.2611% as the approximate interest rate. (For
comparison: The approximation given by Stan was 3.26%. Using an accurate
numerical technique, we find that i = 3.2596...%)

2. In the example at the end of the sci.math FAQ entry, in which P = $50,
A = $10,000 and N = 260, using just the five terms of the series shown
above, we obtain i = 10.9648% as the approximate interest rate. (For
comparison: The final approximation given in the FAQ entry was 10.997%. But
using an accurate numerical technique, we find that i = 10.9624...%)

David W. Cantrell