Loan Payment Formula with Irregular First Period

Date: 05/16/2003 at 09:57:20
From: Art Norman
Subject: Loan payment formula with irregular first period

What is the mathematical formula for calculating a level payment 
payment on a simple interest loan with an irregular (short or long) 
first period?

I have searched the Internet and textbooks but only find derivations 
on the formula for simple interest loans where all the time periods 
are assumed equal: PMT = LOAN AMT * r / (1-(1+r)^-t) where r = rate 
per period, t = number of periods, and the ^ means 'to the power of'. 
I know most lenders request the interest of the irregular period be 
paid up front to avoid this, but I need the formula. 

Date: 05/16/2003 at 10:22:02
From: Doctor Mitteldorf
Subject: Re: Loan payment formula with irregular first period

Dear Art,

Do you want the first payment to be the same as all the others, even 
if the period it covers is much longer or shorter than a full period?

Are you looking for a theoretical understanding or a practical
answer?  

Several years back, I wrote a commercial software application that
does this and many other calculations - now available for free 
download from http://www.persense.org . In that program, go to the
"computational settings" and set "prepaid interest" to "no," and then
the Amortization Screen will do just what you want.

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 05/18/2003 at 23:55:38
From: Art Norman
Subject: Loan payment formula with irregular first period

Thank you for your response. Yes, I want the first payment to be the 
same as all the others even if the period it covers is much longer or 
shorter then a full period. Although I would like theoretical 
understanding, I need the practical answer that includes a formula 
that I can solve for the equal payment amount; if I have the loan 
amount, interest rate per period, number of equal periods, and the 
"irregular" length (short or long) or fraction of the first period.  
My task once I have the formula is to convert it into a computer 
program.  

Thanks again for the dialog. 
Art

Date: 05/19/2003 at 10:23:14
From: Doctor Mitteldorf
Subject: Re: Loan payment formula with irregular first period

Dear Art,

Here's how to do it:  

First, distinguish between true rate and loan rate. Loan rate, or
"monthly rate," is the one quoted by lending institutions, and which
corresponds to APR. True rate is the one that goes into the formulas
below. If the true rate is r and the loan rate is i, then they are
related by  i = 12*(exp(rt/12)-1), or i = 12*ln(1+i/12)

Second is the fundamental principle behind all financial calculations:
equalizing present values. The principle of the loan on the day the
loan is issued is equal to the sum of the present values of all
payments. The present value of a payment is exp(-rt) times the amount
of that payment, where t is time (in years after the present) that the
payment is made.

Third, the implementation: We are going to divide the original loan
principle by the sum of N exponential factors in the form exp(-rt) to
get the payment amount.  If these are all one payment period (one
month) apart, then they can be summed using the formula for a
geometric series; see the Dr. Math FAQ:

   Loans and Interest    
   http://mathforum.org/dr.math/faq/faq.interest.html 

The exponential factors differ from one month to the next by a factor
f=exp(-rt/12)=1+i/12.  The sum of N such factors is 

             (1-f^N)
             -------
               1-f

This needs to be multiplied by the value of the first payment, which
is exp(-rT), where T is the time between the loan date and the first
payment.

Fourth is the bad news: the above will bring you very close to the
answer for most practical cases, but if you're writing a general- 
purpose computer program, that isn't good enough. The reason that it 
is not exact is that loan calculations are ruled by convention as well 
as mathematics, and convention dictates that interest is not 
compounded WITHIN a given loan period. So if the first period is long, 
it offers the borrower a small discount, and if it is short, it gives 
the lender a small bonus.  

So here's what I do in my financial calculator: After finding an
approximation to the payment amount using the above calculation, I run
it through the amortization table generator, which adds interest for
each payment period, subtracts the payment amount, and repeats this
calculation for as many payment periods as the loan specifies. If at
the end there is principal left to be paid, the payment amount is
slightly increased; if the loan is overpaid, then the payment amount
is slightly decreased. The amount of this correction is calculated
with a Newton-Raphson algorithm:

   Inventing an Operation to Solve x^x = y
   http://mathforum.org/library/drmath/view/54586.html 

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/