A loan repayment can look deceptively simple: one amount leaves an account every month, so it is tempting to think that every payment is doing the same job.
It is not.
In an amortising loan, each repayment is split between two different things. One part pays the interest charged for using the lender’s money. The other part reduces the outstanding principal. Because the principal changes after every repayment, the next interest charge may change too.
That changing relationship is the mathematics of loan amortisation.
Understanding it matters because a monthly repayment is not merely a number to be accepted. It is the visible output of a system involving principal, rate, time, payment frequency and the rule used to calculate interest. Change any one of those inputs and the repayment pattern can change.
The one-sentence idea
Loan amortisation is the process of reducing a loan balance through scheduled repayments in which each payment covers interest and also repays part of the principal.
Start with four quantities
- Principal, P: the amount borrowed.
- Interest rate: the rate used to calculate the cost of borrowing.
- Term: how long the loan is scheduled to run.
- Repayment frequency: how often payments are made, such as monthly.
These quantities interact. A larger principal usually means larger repayments if the other conditions remain unchanged. A higher interest rate increases the cost of borrowing. A longer term often lowers each scheduled repayment but may increase the total interest paid because the debt remains outstanding for longer.
Singapore’s MoneySense explains the same important distinction in its current guidance on borrowing costs: a longer loan period can lead to more interest being paid, and interest may be calculated differently depending on whether a loan uses a flat-rate method or a reducing-balance method. See MoneySense: Costs of borrowing.
Amortisation is easiest to see one payment at a time
Suppose a borrower owes $120,000. For a simplified fixed-rate mathematical example, suppose the nominal annual rate is 4.8% and the loan is repaid monthly over 60 months.
The monthly rate in this simplified model is:
r = 0.048 ÷ 12 = 0.004
So the monthly rate is 0.4%.
If the loan is structured as a standard fixed-payment reducing-balance loan, the monthly repayment can be modelled by:
M = P × r(1 + r)^n ÷ ((1 + r)^n − 1)
where M is the regular payment, P is the starting principal, r is the interest rate per payment period and n is the number of payments.
For this example:
P = 120,000 r = 0.004 n = 60 M ≈ 2,253.57
The regular repayment is therefore about $2,253.57 under these assumptions.
The first repayment is not $2,253.57 of principal
The first month’s interest is calculated from the outstanding balance:
Interest = 120,000 × 0.004 = 480.00
So only the remainder of the payment reduces principal:
Principal repaid = 2,253.57 − 480.00
= 1,773.57
The new balance is approximately:
120,000 − 1,773.57 = 118,226.43
Now the mechanism becomes visible. The second month does not begin from $120,000. It begins from about $118,226.43. Therefore, if the rate remains unchanged, the interest charged for the next month is slightly smaller.
| Payment | Interest portion | Principal portion | Balance after payment |
|---|---|---|---|
| 1 | $480.00 | $1,773.57 | $118,226.43 |
| 2 | $472.91 | $1,780.66 | $116,445.77 |
| 3 | $465.78 | $1,787.79 | $114,657.98 |
The payment stays approximately constant in this simplified fixed-payment model, but its internal composition changes. Interest falls. Principal repayment rises.
Why does the principal portion grow?
Because interest is charged on a shrinking balance.
At the beginning, the borrower owes a large amount, so the interest portion is relatively large. Once part of the principal has been repaid, the balance becomes smaller. If the rate is unchanged, a smaller balance produces a smaller interest charge. Since the total scheduled payment remains the same, more of that payment is left to reduce principal.
This is why an amortisation schedule is not a row of identical transactions. It is a sequence of changing states.
The balance is the state variable
A useful mathematical way to think about the loan is to treat the outstanding balance as the quantity that carries information from one month to the next.
new balance = old balance + interest charged − repayment made
In a fixed-rate monthly model:
B(k+1) = B(k)(1 + r) − M
This recurrence relation explains the loan dynamically. Each new balance depends on the previous balance. That makes amortisation a useful example of how sequences, exponential growth and repeated subtraction can operate together.
Why the term changes more than the monthly payment
Learners often notice that extending a loan term can reduce the regular repayment. That is true in many standard fixed-payment models, but the smaller payment can hide a larger total cost.
Imagine repaying the same principal at the same rate over five years instead of three. Each payment can be spread over more months. That reduces the amount demanded at each payment date. But the lender’s money remains outstanding for longer, so interest has more time to accumulate.
Therefore:
A lower monthly repayment does not automatically mean a cheaper loan.
That is one of the most important transfer ideas in financial mathematics: cash-flow affordability and total borrowing cost are related, but they are not the same quantity.
Flat-rate interest is a different model
Not every advertised loan rate is used in the same way. MoneySense distinguishes a flat-rate calculation from a monthly-rest or reducing-balance calculation. Under a flat-rate method, interest is calculated from the original principal even while the outstanding balance is being repaid. Under monthly rest, interest is calculated from the outstanding balance as it reduces.
This means two loans can display similar-looking percentage rates yet produce different effective borrowing costs. For real financial decisions, the contract terms, fees, rate basis and effective interest rate matter. This article explains the mathematics; it is not personalised financial advice.
A common mistake: dividing total interest evenly
A learner may calculate total interest, divide it by the number of months, and assume that this gives the interest charged in every period.
That may resemble a flat-rate model, but it does not describe a standard reducing-balance amortisation schedule. In a reducing-balance model, the interest portion depends on the outstanding balance at that point in time. The balance changes after every repayment, so the interest portion changes too.
Another mistake: applying an annual rate directly to each month
If a model states a nominal annual rate of 4.8% with monthly calculation, using 4.8% every month would be a major unit error. The rate and the time period must match.
annual rate ↔ annual period monthly rate ↔ monthly period
This is the same dimensional discipline used elsewhere in mathematics. A rate cannot be separated from the interval over which it applies.
What happens when the interest rate changes?
The elegant fixed-payment formula assumes the relevant rate remains fixed for the modelled period. Real loans may have floating or repriced rates. When the rate changes, the repayment, the remaining term or both may need to change according to the loan contract.
This is an important model limitation. A formula is not reality itself. It is a representation built from assumptions. When the assumptions change, the model must be updated.
MoneySense’s current home-loan guidance similarly notes that floating rates can move and that higher rates increase interest expense. See MoneySense: How home loans work.
What happens when an extra repayment is made?
Mathematically, an extra principal repayment reduces the balance sooner than the original schedule expected. If later interest is calculated from the reduced balance, future interest charges can become smaller.
But real loan agreements may include restrictions, lock-in periods, prepayment rules or fees. So the mathematical statement “smaller balance means less future interest under a reducing-balance rule” should not be confused with the practical statement “every borrower should prepay immediately.” The second question depends on the actual contract and the borrower’s circumstances.
How to read an amortisation table
- Check the starting balance.
- Check the rate for the payment period.
- Calculate the interest from the correct balance.
- Subtract interest from the payment to find principal repaid.
- Subtract principal repaid from the old balance.
- Use that new balance as the next row’s starting point.
If a spreadsheet or calculator gives a strange answer, these six checks often reveal the problem. The most common failures are mismatched time units, an incorrect rate per period, an off-by-one payment count, rounding too early, or using the original principal when the model requires the outstanding balance.
Diagnostic check: does the learner understand the mechanism?
A learner who genuinely understands amortisation should be able to answer questions such as these without relying on a memorised formula alone:
- If the payment stays constant, why can the interest portion still change?
- Why does a longer term often reduce the payment but increase total interest?
- Why must the interest rate period match the repayment period?
- What quantity changes after every payment?
- What assumption breaks if the rate changes halfway through the loan?
- Why are a flat-rate loan and a reducing-balance loan not mathematically identical?
The deeper mathematics
Loan amortisation sits at the intersection of several mathematical ideas.
- Percentages describe the periodic interest rate.
- Sequences describe the changing balance over time.
- Exponential growth appears because interest repeatedly acts on a balance.
- Algebra rearranges the balance equation to solve for payment, rate, term or principal.
- Graphs can show the declining balance and the changing interest-principal split.
- Modelling forces us to state assumptions about rates, timing and payment rules.
This is why financial mathematics becomes powerful when it is taught as a connected system rather than as a calculator recipe.
A useful transfer question
Suppose two loans have the same principal and the same stated annual rate, but one requires monthly repayments and the other has a different payment structure.
Can we immediately conclude that they cost the same?
No. We must know how the rate is applied, how often interest is calculated, when repayments occur, whether fees are included, and whether the stated rate is directly comparable. This is a modelling question before it is a calculation question.
For students, parents and teachers
For a student, the main goal is to understand why the balance changes and how each payment is decomposed.
For a parent, the useful lesson is that a monthly repayment alone does not describe the full cost structure of a loan.
For a teacher or tutor, amortisation is an excellent context for testing whether a learner can coordinate percentages, algebra, sequences, units and assumptions without losing the meaning of the quantities.
Final idea
A loan is not merely a principal plus an interest rate.
It is a changing balance moving through time under a repayment rule.
Once that mechanism is visible, amortisation stops looking like a mysterious bank calculation. It becomes a sequence that can be followed, checked, modelled and explained.
That is the mathematical habit worth keeping: do not only ask, “What is the repayment?” Ask, “What system produced it?”