Loan Amortisation: The Maths Behind Monthly Payments
Amortisation explains how each payment splits between interest and principal over time. Early payments are interest-heavy; later payments reduce the balance faster. Here’s the maths, the intuition, and worked examples.
1) The Core Formula (PMT)
The fixed payment for a fully amortising loan is:
PMT = P · r · (1+r)n / ((1+r)n − 1)
- P = principal (amount borrowed)
- r = periodic interest rate (APR ÷ periods per year)
- n = total number of payments (years × periods per year)
This ensures the balance reaches exactly £0 after the final payment.
2) How Each Payment Splits
Each period:
- Interest = current balance × r
- Principal = PMT − Interest
- New balance = old balance − Principal
Because the balance shrinks, the interest portion falls over time, and the principal portion rises — the classic “amortisation curve”.
3) APR, EAR (AER) & Compounding
APR is the nominal annual rate (often ignoring compounding frequency for comparisons). EAR/AER reflects compounding:
EAR = (1 + APR/m)m − 1
where m is periods per year (12 for monthly). Amortisation uses the periodic rate r = APR/m (or a periodic rate implied by EAR/AER).
4) Worked Example
Loan: £250,000, APR: 5.0%, Term: 25 years, Monthly (m=12)
- r = 0.05 / 12 = 0.0041667
- n = 25 × 12 = 300
- PMT ≈ £1,462.00
Month 1: Interest = 250,000 × 0.0041667 ≈ £1,041.67; Principal = £1,462.00 − £1,041.67 ≈ £420.33; New balance ≈ £249,579.67.
5) Overpayments & Term Reduction
Paying extra each month increases the principal portion and shortens the term. Even small overpayments (e.g., £50–£100) can save thousands in interest. Many fixed-rate deals allow up to 10% overpayment per year without penalty.
6) Fees, APRC & True Cost
Arrangement fees increase the real cost. Some lenders allow fees to be added to the balance — which then accrue interest. For mortgages, the APRC (Annual Percentage Rate of Charge) is a standardised measure of total cost over the illustrative term.
7) Amortisation vs Interest-Only
With interest-only, PMT ≈ interest only; the balance doesn’t fall. With amortisation, PMT stays fixed but the interest share shrinks and the principal share grows — you steadily build equity until the loan is cleared.
8) Quick Reference Table
| Concept | Formula / Idea |
|---|---|
| Periodic rate | r = APR ÷ m |
| Payment (PMT) | P · r · (1+r)^n / ((1+r)^n − 1) |
| Interest per period | Balance × r |
| Principal per period | PMT − Interest |
| EAR/AER | (1 + APR/m)^m − 1 |
9) Try It Yourself
Use CalcFlow’s tools to model the maths with your own numbers: