A Series of Interest
David W. Cantrell
> David W. Cantrell <DWCan...@sigmaxi.org> wrote:
> > 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.)
>
> Perhaps it's not well known. But in any event, I found out just today
> that the series had been already published:
>
> H. E. Stelson, "Note on finding the interest rate" _Amer. Math.
> Monthly_ 60:10 (Dec. 1963) 703-705.
>
> As I did, he obtained the series by reversion, and did not discuss
> convergence
Another comment:
It seems that the series converges if NP < 2A. In other words, the infinite
series gives the interest rate precisely as long as the total payment is
less than twice the amount of the loan.
In the previously mentioned example from Stan Brown's page, we had
P = $200,000, A = $2,800,000 and N = 19, and so NP = 1.36 A approximately;
in the previously mentioned example at the end of the sci.math FAQ entry,
we had P = $50, A = $10,000 and N = 260, and so NP = 1.3 A. This leads me
to suspect that, _for most situations arising in practice_, the series will
converge.
This comment will perhaps conclude this thread.
> or mention the form of the general term. He then expressed
> the interest rate as a continued fraction and, by considering
> convergents, obtained "some very excellent approximations". I may
> discuss his and various other approximations in another thread soon.
I will do that, but I don't know how soon.
David Cantrell
David W. Cantrell
Nope. One more comment:
I see now that the series was known before Stelson. It was used by
Ralph W. Snyder, C.P.A. in "Schurig's and Baily's Formulae for Finding
the Interest Rate" _The American Accountant_ 17 (Dec. 1932) 362-365.
Perhaps it was also known before Snyder.
BTW, I found Snyder's article to be "amusing". For example, after
mentioning that he was going to use reversion of series, and thinking that
some of the accountants reading his article might not be familiar with
reversion, he says parenthetically
for an example of reversion in detail, see Wentworth's _College Algebra_,
rev. ed., p. 340
Hmm. Just how many "college algebra" texts cover reversion of series
nowadays?
David