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/
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