Quick read. A loan repayment schedule is not a mysterious banking table. It is an iterative mathematical system. Start with a balance, apply the periodic interest rate, subtract the payment, and repeat. The resulting recurrence relation explains why a fixed monthly instalment can contain mostly interest near the beginning of a loan and mostly principal near the end.

This article is an educational explanation of financial mathematics, not personal financial advice. Real loan contracts may include fees, penalties, changing rates, legal terms, insurance and other rules that must be read separately.

The one-sentence answer

For a standard reducing-balance loan, the next balance is the current balance plus one period of interest minus the payment: B_(t+1)=B_t(1+r)-P.

Why this belongs in a mathematics classroom

Loan amortisation connects percentage, exponential growth, algebra, sequences, functions, recurrence relations, present value and numerical checking. It is a strong example of what mathematics does after the examination: mathematics turns a changing real-world process into a model that can be computed, inspected and challenged.

The same mathematical habit appears across engineering, computing and science: define the state, define the update rule, run the rule, then verify whether the output behaves as expected. Bukit Timah Tutor’s broader How Mathematics Works page develops that habit beyond any one application.

1. Define the inputs before calculating anything

Principal, L: the starting amount borrowed.
Periodic interest rate, r: the interest rate for one repayment period, not automatically the advertised annual rate.
Number of payments, n: the number of repayment periods.
Payment, P: the amount paid each period.
Balance, B_t: the amount still outstanding immediately after payment t.

The first major failure mode is unit mismatch. An annual rate cannot simply be dropped into a monthly recurrence. If the contract compounds monthly and the nominal annual rate is 4.8%, a simplified monthly model may use 0.048/12 = 0.004 per month. Real contracts can define rate conversion differently, so the contract remains authoritative.

2. The algorithm is a recurrence relation

Suppose the balance just after this month’s payment is B_t. During the next month, interest is added. Then the next payment is removed:

B_(t+1)=B_t+rB_t-P=B_t(1+r)-P

This tiny equation is the core of a standard amortisation engine. A spreadsheet, calculator or banking system can iterate it month by month. The important point is that interest is recalculated from the remaining balance. As the balance falls, the interest component normally falls too.

3. Why the principal-interest split changes

At period t, the interest component is approximately I_t=rB_t. The principal component is therefore P-I_t. Early in the loan, B_t is large, so I_t is large. Later, B_t is smaller, so less of the same payment is required for interest and more is available to reduce principal.

This matches the consumer explanation from the U.S. Consumer Financial Protection Bureau: in a typical fixed-rate mortgage, the combined principal-and-interest payment can remain unchanged while the allocation between principal and interest changes over time.

4. From the recurrence to the equal-payment formula

Repeated substitution gives:

B_n=L(1+r)^n-P[(1+r)^(n-1)+(1+r)^(n-2)+...+1]

The bracket is a finite geometric series. Using the geometric-series formula:

B_n=L(1+r)^n-P[((1+r)^n-1)/r]

For a fully amortising loan, the target is B_n = 0. Solving for P gives:

P = Lr(1+r)^n / ((1+r)^n-1)

An equivalent form is P=Lr/[1-(1+r)^(-n)]. This is the familiar annuity-payment formula. The formula is not a separate piece of magic; it is the closed form of the month-by-month recurrence.

5. A worked example

Take a simplified educational loan of $100,000, repaid monthly over 10 years at a nominal 4.8% per year with monthly rate r = 0.004 and n = 120. The equal-payment formula gives a monthly payment of about $1,051. This is only an illustrative mathematical model, not a quote for any real loan.

Month 1 begins with $100,000. Interest is about $400. If the payment is about $1,051, roughly $651 reduces principal, leaving a balance near $99,349. In the next month, interest is calculated on the lower balance, so the interest portion is slightly smaller. The recurrence continues until rounding and the final contractual adjustment bring the balance to zero.

6. Flat-rate interest and reducing-balance interest are not the same model

MoneySense Singapore distinguishes flat-rate borrowing from monthly-rest borrowing. Under a flat-rate method, interest may be calculated from the original principal even as the borrower repays the loan. Under a monthly-rest or reducing-balance method, interest is calculated from the outstanding balance. Two loans displaying similar headline rates can therefore have materially different effective costs.

This is a useful mathematical warning: the percentage printed in an advertisement is not enough to identify the algorithm. You need the state variable on which the percentage acts.

7. Effective interest rate is a comparison problem

MoneySense describes the Effective Interest Rate, or EIR, as a way to compare the true cost of borrowing across repayment structures. Payment frequency, fees and the way interest is applied can change effective cost. A mathematically literate comparison therefore asks not only “What is the rate?” but also “Applied to what balance, at what times, with what cash flows?”

8. Where the simple algorithm can fail

Variable rates: r changes over time, so one closed-form payment formula may no longer describe the entire loan.
Fees: fees alter effective cost even if they do not appear in the recurrence for principal.
Prepayments: an extra principal payment changes the future state and therefore future interest.
Late or missed payments: the contract may add charges or interest according to separate rules.
Balloon structures: the scheduled payments may be intentionally insufficient to reduce the ending balance to zero.
Negative amortisation: if a payment is smaller than the interest due, the balance can grow rather than shrink.
Rounding: systems usually work in currency units such as cents, so repeated rounding can create a small final adjustment.
Day-count conventions: some products calculate interest from actual days rather than a simple monthly fraction of an annual rate.

9. Diagnostics: how to tell whether an amortisation schedule is plausible

At every step, does principal reduction equal payment minus interest?
For a fixed positive payment on a standard reducing-balance loan, does interest generally fall as the balance falls?
If the payment is intended to fully amortise the loan, is the final balance near zero?
If the interest rate rises while everything else is unchanged, does the required payment rise?
If the term becomes longer, does the monthly payment normally fall while total interest rises?
Does an extra principal payment reduce future interest in the model?
Are all rates and time units consistent?

These are invariants and directional checks. They are often more valuable than trusting a single calculator output.

10. A falsifier is more useful than confidence

A model should state what would prove it unsuitable. The recurrence B_(t+1)=B_t(1+r)-P is falsified as a full description if the real contract applies flat interest to original principal, changes rates according to an index, adds fees to principal, uses irregular payment dates, or has a contractual rule not represented in the update equation.

This distinction matters for computational reasoning. A correct calculation inside the wrong model is still the wrong answer.

11. Connection to Secondary and Additional Mathematics

Percentages: convert and interpret periodic rates.
Indices: understand repeated multiplicative change.
Sequences: follow a balance through time.
Algebra: rearrange the payment formula.
Functions: examine payment as a function of rate, term or principal.
Graphs: compare balance trajectories under different scenarios.
Numerical methods: reproduce a schedule iteratively and test it.

Students who want to trace the algebraic dependencies behind this kind of model can also use Bukit Timah Tutor’s public Algebra and Functions HELP Tasks.

12. What a robust computational implementation should store

A dependable implementation should preserve the original inputs, rate definition, timing convention, payment dates, balance before interest, interest charged, payment received, principal reduction, balance after payment and any exceptional adjustment. This makes the schedule auditable. If two systems disagree, the disagreement can be located instead of guessed at.

13. Update triggers

This educational model should be revisited whenever the interest-rate convention changes, payments become irregular, fees are capitalised, a prepayment occurs, the rate is reset, or a contract uses a different method of calculating interest. The algorithm is only as current as its assumptions.

Sources and further reading

Final idea. The useful lesson is larger than loans. Whenever a quantity changes step by step, ask what the state is, what rule updates it, what assumptions the rule hides, and what observation would show that the rule is no longer adequate. That is mathematics becoming an algorithm.

Small Group Tutorials

How Banks Calculate Loan Repayments: Amortisation, Compound Interest and Recurrence Relations