The Math of Simple Debt Amortization - Business Math

Use Math to Determine the Payment Needed for a Loan

Simple debt amortization can help you figure out car and home payments to see how they affect your loan.
Simple debt amortization can help you figure out car and home payments to see how they affect your loan. Peter Dazeley, Getty Images

Incurring debt and making a series of payments to reduce this debt to nil is something you are very likely to do in your lifetime. Most people make purchases, such as a home or auto, that would only be feasible if we are given sufficient time to pay down the amount of the transaction.

This is referred to as amortizing a debt, a term that takes its root from the French term amortir, which is the act of providing death to something.

Amortizing a Debt

The basic definitions required for someone to understand the concept are:
1. Principal - the initial amount of the debt, usually the price of the item purchased.
2. Interest Rate - the amount one will pay for the use of someone else's money. Usually expressed as a percentage so that this amount can be expressed for any period of time.
3. Time - essentially the amount of time that will be taken to pay down (eliminate) the debt. Usually expressed in years, but best understood as the number of and interval of payments, i.e., 36 monthly payments.
Simple interest calculation follows the formula:​ I = PRT, where

  • I = Interest
  • P = Principal
  • R = Interest Rate
  • T = Time.

Example of Amortizing a Debt

John decides to buy a car. The dealer gives him a price and tells him he can pay on time as long as he makes 36 installments and agrees to pay six per cent interest. (6%). The facts are:

  • Agreed price 18,000 for the car, taxes included.
  • 3 years or 36 equal payments to pay out the debt.
  • Interest rate of 6%.
  • The first payment will occur 30 days after receiving the loan

To simplify the problem, we know the following:

1. The monthly payment will include at least 1/36th of the principal so we can pay off the original debt.
2. The monthly payment will also include an interest component that is equal to 1/36 of the total interest.


3. Total interest is calculated by looking at a series of varying amounts at a fixed interest rate.

Take a look at this chart reflecting our loan scenario.

Payment Number

Principle Outstanding

Interest

018000.0090.00
118090.0090.45
217587.5087.94
317085.0085.43
416582.5082.91
516080.0080.40
615577.5077.89
715075.0075.38
814572.5072.86
914070.0070.35
1013567.5067.84
1113065.0065.33
1212562.5062.81
1312060.0060.30
1411557.5057.79
1511055.0055.28
1610552.5052.76
1710050.0050.25
189547.5047.74
199045.0045.23
208542.5042.71
218040.0040.20
227537.5037.69
237035.0035.18
246532.5032.66

This table shows the calculation of interest for each month, reflecting the declining balance outstanding due to the principal pay down each month (1/36 of the balance outstanding at the time of the first payment. In our example 18,090/36 = 502.50)

By totaling the amount of interest and calculating the average, you can arrive at a simple estimation of the payment required to amortize this debt. Averaging will differ from exact because you are paying less than the actual calculated amount of interest for the early payments, which would change the amount of the outstanding balance and therefore the amount of interest calculated for the next period.



Understanding the simple effect of interest on an amount in terms of a given time period and realizing that amortization is nothing more then a progressive summary of a series of simple monthly debt calculations should provide a person with a better understanding of loans and mortgages. The math is both simple and complex; calculating the periodic interest is simple but finding the exact periodic payment to amortize the debt is complex. 

Edited by Anne Marie Helmenstine, Ph.D.