Subject: Analysis - Loan Payments and Amortization

Last-Revised: 16 Feb 2003
Contributed-By: Hugh Chou

This article presents the formula for computing monthly payments on loans. A listing of the full series of payments (principal and interest) that show how a loan is paid off is known as a loan amortization table. This article will explain how these tables are generated for the U.S. system in which interest is compounded monthly.

First you must define some variables to make it easier to set up:

  • P = principal, the initial amount of the loan
  • I = the annual interest rate (from 1 to 100 percent)
  • L = length, the length (in years) of the loan, or at least the length over which the loan is amortized.

The following assumes a typical conventional loan where the interest is compounded monthly. But first, just two more variables to make the calculations easier:

  • J = monthly interest in decimal form. This is I / (12 x 100).
  • N = number of months over which loan is amortized. This is L x 12.

Finally here is for the big monthly payment (M) formula:

                              J
         M  =  P  x ------------------------
                      1  - ( 1 + J ) ^ -N

Please note "1" is just the number one (it does not appear too clearly on some browsers) and "^" is the exponentiation operator (2 ^ 3 = 8).

So to calculate M, you would first calculate 1 + J, then take that to the -N (negative N) power, subtract that from the number 1. Now take the inverse of that (if you have a 1/X button on your calculator push that). Then multiply the result times J and then times P. Sorry for the long way of explaining it, but I just wanted to be clear for everybody.

Here's the one-liner for a program (adjust for your favorite language):

         M = P * ( J / (1 - (1 + J) ** -N))

So now you should be able to calculate the monthly payment, M. To calculate the amortization table you need to do some iteration (i.e. a simple loop). I will tell you the simple steps:

  1. Calculate H = P x J, this is your current monthly interest
  2. Calculate C = M - H, this is your monthly payment minus your monthly interest, so it is the amount of principal you pay for that month
  3. Calculate Q = P - C, this is the new balance of your principal of your loan.
  4. Set P equal to Q and go back to Step 1: You thusly loop around until the value Q (and hence P) goes to zero.

Programmers will see how this makes a trivial little loop to code, but I have found that many people now surfing on the Internet are NOT programmers and still want to calculate their mortgages!

Note that just about every PC or Mac has a spreadsheet of some sort on it, and they are very good tools for doing mortgage analysis. Most of them have a built-in PMT type function that will calculate your monthly payment given a loan balance, interest rate, and the number of terms. Check the help text for your spreadsheet.

Please visit Hugh Chou's web site for a calculator that will generate amortization tables according to the formulas discussed here. He also offers many other calculators:
http://www.hughchou.org/calc/

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