Finding Interest Rate without approximation or root-finding?
UKP
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.
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
Proginoskes
> 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?
No. The formula, by its very nature, requires root-finding or root-
approximating.
You owe the Internet Oracle the billionth digit of the Euler-
Mascheroni constant.
--- Christopher Heckman
UKP
MP = P*R / [(1-1/(1+R)^T))]
be represented in terms of exp() and log()?
say the values of MP, P and T is given..How would then I take log to
find the value?
Thanks :)
Han de Bruijn
hagman
> 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.
You should rather attack this problem.
A simple binary search will do for everyday problems.
Virgil
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.
>
> The formula to find monthly payment is, MP = P*R / [(1-1/(1+R)^T))],
The formula may be resolved into a polynomial equation in R of degree
T+1.
Except for very small values of T, approximation is by far the easiest
method unless exact solutions are required, and there will be, in
general, no method for finding exact solutions for such T.
David W. Cantrell
> 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
nib...@yahoo.com
for the present value (PV). Now, show us your equation, on
interest rate, for the future value (FV)?
Luiz
May, 14, 2017.