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Articles Referenced in
Loan or Investment Formulas

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Summary: My article Loan or Investment Formulas references several articles from the Usenet newsgroup sci.math. Finding Usenet articles in Google is unreasonably, and don’t even get me started on the requirement to enable Javascript. To make matters worse, links that used to work will unaccountably stop working. To save effort and frustration for myself and my readers, I’ve just captured the referenced articles here.

D. Cantrell, 26 Aug 2004

Subject: A Series of Interest
From: David W. Cantrell <DWCan...@sigmaxi.org>
Date: 26 Aug 2004 04:39:50 GMT
Newsgroups: sci.math
Organization: NewsReader.Com Subscriber
Message-ID: <20040826003950.596$yG@newsreader.com>

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

Retrieved 2016-01-03 from
https://groups.google.com/forum/#!msg/sci.math/5aUCzXvU4KQ/IVz7uXGmpmUJ.

D. Cantrell, 09 Oct 2004

Subject: A Series of Interest
From: David W. Cantrell <DWCan...@sigmaxi.org>
Date: 09 Oct 2004 22:17:15 GMT
Newsgroups: sci.math
Organization: NewsReader.Com Subscriber
Message-ID: <20041009181715.799$E7@newsreader.com>

David W. Cantrell <DWCan...@sigmaxi.org> wrote:
> 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

Retrieved 2016-01-03 from
https://groups.google.com/d/msg/sci.math/jNADoN3QgoE/v3hs8T6b9mEJ.

D. Cantrell, 13 Oct 2004

Subject: Re: A Series of Interest
From: David W. Cantrell <DWCan...@sigmaxi.org>
Date: 13 Oct 2004 13:40:57 GMT
Newsgroups: sci.math
Organization: NewsReader.Com Subscriber
Message-ID: <20041013094057.830$RY@newsreader.com>

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

Retrieved 2016-01-03 from
https://groups.google.com/d/msg/sci.math/jNADoN3QgoE/J84zVDo2uycJ.
(Quoted material suppressed.)

D. Cantrell, 27 Sept 2007

Subject: Re: Finding Interest Rate without approximation or root-finding?
From: David W. Cantrell <DWCan...@sigmaxi.net>
Date: 27 Sep 2007 17:38:10 GMT
Newsgroups: sci.math
Organization: NewsReader.Com Subscriber
Message-ID: <20070927133811.945$vG@newsreader.com>
References: <1190790526.315104.83600@57g2000hsv.googlegroups.com>

UKP <rock...@gmail.com> wrote:
> How would you find Interest Rate *without approximation* from the
> formula below..if the values for rest variables are given?
>
> Is there any direct formula to calculate Interest Rate for such loan
> calculations - without using approximation or root-finding? I'm having
> troubles programming in Java with approximation or root-finding.

Based on your sentence above, I'm guessing that you might be happy with a
simple formula for the interest rate, even if it is not precise, because
that would avoid your "troubles programming". So I will mention a formula I
discovered a few years ago. I hadn't mentioned it in this newsgroup before
because it had hoped to write an article comparing it with other such
formulae in the literature... Anyway, for now, I'll just give the formula
below and make a few comments.

> The formula to find monthly payment is, MP = P*R / [(1-1/(1+R)^T))],
> where -
>
> Monthly Payment= MP
>
> P = principal (or loan amount)
> R, Rate = monthly interest rate (not annual interest rate)
> T, Term = number of months (not number of years
>
> Example to find Monthly Payment -
>
> Principal = $250,000
> Rate = 5.5% (0.055) annual interest, or 5.5 / 12 = 0.458% (0.00458)
> monthly interest
> Term = 30 years, or 30 X 12 = 360 months
> Payment = 250000 X 0.00458 / (1 - 1/(1 + 0.00458) ^ 360 )) = 1419.47 =
> $1,419.47

Often, in practice, a way to find the interest rate is to use a series
expansion. But unfortunately, in the example above, that method fails.
[For anyone interested, see the sci.math thread "A Series of Interest",
started in Aug. 2004, at
<http://groups.google.com/group/sci.math/browse_frm/thread/e5a502cd7bd4e0a4>
and
<http://groups.google.com/group/sci.math/browse_frm/thread/8cd003a0ddd08281>.
As mentioned in the latter part of the thread, the series converges if,
using the OP's notation, T * MP < 2 * P.
But that is not the case in the example above; the series diverges.]

But there is a simple formula, which AFAIK has never appeared in the
literature, for approximating the interest rate. Using the OP's notation:

R = ((MP/P + 1)^(1/q) - 1)^q - 1   approximately                  (*)

where q = lg(1 + 1/T), with lg denoting the binary logarithm.

Example: Suppose that MP, the monthly payment, is $1419.47; P, the
  principal, is $250000; and T, the term, is 360 months. We wish to find R,
  the rate of interest monthly.

Using (*), q = lg(1 + 1/360) = ln(1 + 1/360)/ln(2) = 0.0040019...

and then, approximately,

R = (($1419.47/$250000 + 1)^(1/q) - 1)^q - 1 = 0.004558... = 0.4558...%

For comparison, from the OP's example, we know that the precise interest
rate is actually 5.5%/12 = 0.4583...%. The relative error in our
approximation is then about -0.0055 .

Considering the simplicity of (*), it is reasonably accurate. More
importantly, it is reasonably accurate over a _wide_ range of values for
the variables, and I believe that is not the case for any of the
approximations which have appeared in the literature previously.

David W. Cantrell

Retrieved 2016-01-03 from
https://groups.google.com/d/msg/sci.math/hop5mBJX1TA/suyDwCZA1ZEJ.

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