Finding Interest Rate without approximation or root-finding? - sci.math

Message from discussion Finding Interest Rate without approximation or root-finding?

David W. Cantrell

View profile

More options Sep 27 2007, 1:38 pm

Newsgroups: sci.math

From: David W. Cantrell <DWCantr...@sigmaxi.net>

Date: 27 Sep 2007 17:38:10 GMT

Local: Thurs, Sep 27 2007 1:38 pm

Subject: Re: Finding Interest Rate without approximation or root-finding?

  Reply to author
  |
  Forward
  | Print
  | View thread
  | Show original
  | Report this message
  | Find messages by this author
  

UKP <rockm...@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/e5a502cd7bd...>
and
<http://groups.google.com/group/sci.math/browse_frm/thread/8cd003a0ddd...>.
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

Reply to author Forward