The Math Behind Loan Amortization (With a Worked Example)

A loan schedule has one property that surprises almost everyone the first time they see it: every payment is identical, but what each payment does changes completely from the first month to the last. Understanding why takes about ten minutes of arithmetic, and it changes how you read every credit offer afterwards.

The problem the formula solves

You borrow an amount P at a periodic interest rate i, and you want to repay it in n equal instalments. What should each instalment be?

Too small and the debt never clears — the interest each period outruns what you pay. Too large and you overpay. There is exactly one value that lands the balance on zero at instalment n.

Deriving it

Think of each payment as a separate amount of money delivered at a different point in the future. A payment of PMT made k periods from now is worth PMT / (1 + i)^k today, because that smaller sum, left to compound at i for k periods, would grow into PMT.

The loan balances when the present value of all n payments equals the amount borrowed:

P = PMT/(1+i)^1 + PMT/(1+i)^2 + ... + PMT/(1+i)^n

Factor out PMT and you have a geometric series with ratio 1/(1+i). Its sum collapses to a closed form:

P = PMT * (1 - (1+i)^-n) / i

Rearranged for the payment:

PMT = P * i / (1 - (1+i)^-n)

That is the whole thing. Every amortisation table in the world is this line plus bookkeeping.

A worked example

Borrow 10,000 at 1% per month for 12 months.

PMT = 10000 * 0.01 / (1 - 1.01^-12)
    = 100 / (1 - 0.887449)
    = 100 / 0.112551
    = 888.49

Twelve payments of 888.49 come to 10,661.85, so the loan costs 661.85 in interest.

Now watch what each payment does. Interest is always charged on the balance that is currently outstanding, so it is recomputed every month:

month  payment  interest  principal  balance
   1    888.49    100.00     788.49   9211.51
   2    888.49     92.12     796.37   8415.14
   3    888.49     84.15     804.34   7610.80
  ...
  11    888.49     17.51     870.98    879.69
  12    888.49      8.80     879.69      0.00

The payment never changes. The split does, continuously: interest falls because the balance falls, and the principal portion absorbs exactly what interest gives up. That is the mechanism behind the whole schedule.

Why long loans feel like paying for nothing

In the twelve-month example above, interest is 11.3% of the first payment. Over a short term at a modest rate, the split is never dramatic.

Stretch the term and it becomes dramatic. On a 30-year mortgage at 9% a year, the first payment is overwhelmingly interest, and the crossover — the month where the principal portion finally exceeds the interest portion — arrives past the halfway point of the loan. Leave in year three and you have paid a lot of money and reduced the debt very little. Nothing improper has happened; it is arithmetic. Interest is charged on what you still owe, and in year three you still owe almost all of it.

This is also why the total cost climbs so steeply with term. Same 200,000 at 9% a year:

• 10 years — 2,534 a month, 104,022 in interest
• 20 years — 1,799 a month, 231,868 in interest
• 30 years — 1,609 a month, 379,328 in interest

Going from twenty years to thirty saves about 190 a month and costs about 147,000. That can still be the right decision for a household's cash flow. It should be a decision, though, not a default chosen because the monthly figure fit the budget. Running the two side by side takes seconds and the difference is rarely what people expect.

What overpaying actually does

An extra payment can be applied two ways, and they are not equivalent:

• Against the principal. The balance drops immediately. Every subsequent interest charge is computed on a smaller number, so the saving compounds for the rest of the term.
• Against future instalments. You have paid next month's bill early. The balance follows the original schedule, and you save almost nothing.

Lenders do not always default to the first. Whenever you overpay, state explicitly that it reduces the principal, and confirm afterwards that the term was shortened or the payment recalculated. The difference over a long loan runs into five figures.

Nominal versus effective

One more trap, because it decides which offer is actually cheaper.

A rate quoted as 12% a year, compounded monthly, is not 12% a year. Each month charges 1%, and twelve months of that compound to:

(1 + 0.01)^12 - 1 = 0.1268 = 12.68%

That 12.68% is the effective annual rate. The more often interest is capitalised, the further the effective rate drifts above the nominal one on the poster. When comparing offers, compare the same thing — either both effective, or both nominal with the same compounding frequency. The same arithmetic works in your favour when you are saving rather than borrowing, which is the entire point of a compound interest projection.

Reading a schedule critically

Three questions worth asking of any amortisation table:

1. What is the total paid? Not the monthly figure. The sum.
2. What is in the payment besides principal and interest? Arrangement fees and mandatory insurance are commonly financed into the balance, which means you pay interest on them too. A loan with a lower rate and higher fees can cost more.
3. When does the crossover happen? If the principal portion does not exceed the interest portion until year 17 of a 30-year loan, you now know what leaving early costs.

None of this requires financial expertise. It requires one formula, and the willingness to look at the total rather than the instalment.