Amortisation

Amortisation is the schedule of how a fixed-payment loan’s monthly payment is split between interest and principal across the loan’s life. The monthly payment is constant; the split is not. Early in the loan, most of each payment is interest; late in the loan, most is principal.

Concrete example: $400,000 mortgage at 7% over 30 years pays $2,661/month. In month one, $2,333 of that is interest (7% ÷ 12 × $400,000) and only $328 is principal. By month 180 (halfway through the loan), about $1,500 is interest and $1,150 is principal. Month 360 is almost entirely principal.

Why front-loaded? Interest is computed on the remaining principal balance. The balance is highest early; therefore interest is highest early. As principal shrinks, the interest portion of each payment shrinks proportionally and the principal portion grows. The math falls out of the constant-payment formula.

Practical implications: paying off a mortgage in year 5 returns very little equity beyond the down payment. Extra principal payments in the early years compound powerfully (each dollar paid early avoids years of compounding interest). The standard amortisation table is what every mortgage calculator generates.

The closed-form amortisation formula: for a loan of principal P, monthly rate r (annual / 12), and term n months, the constant monthly payment is M = P · r · (1 + r)ⁿ / ((1 + r)ⁿ − 1). This is the discounted-cash-flow inverse: the payment that makes the present value of all future payments equal the loan amount. Every entry in an amortisation table follows from this — each month, interest = (remaining balance) × r, principal = M − interest, new balance = old balance − principal. Spreadsheet functions PMT(), IPMT(), and PPMT() implement these directly.

Worked example

Take a $300,000 loan at 6% APR for 30 years. The monthly payment from the closed-form formula is exactly $1,798.65. Month 1 breaks down as: interest = $300,000 × (0.06 ÷ 12) = $1,500.00; principal = $1,798.65 − $1,500.00 = $298.65; new balance = $299,701.35. Month 120 (start of year 11), the balance is roughly $250,953; interest = $1,254.77, principal = $543.88 — principal has nearly doubled but is still less than a third of the payment. Crossover (where principal exceeds interest in a single payment) happens around month 222 — 18.5 years in. By month 360, the final payment is roughly $1,789 principal and $9 interest. Total interest paid over the life of the loan: $347,514, more than the original principal.

The crossover month — where principal first exceeds interest in a single payment — is a useful intuition pump. For a 30-year loan at 7%, crossover is roughly month 222 (year 18.5). For the same loan at 4%, crossover moves earlier to month 180 (year 15). For a 15-year loan at 7%, crossover is month 50 (year 4). Shorter terms reach crossover faster because the higher monthly payment chips away at principal sooner.

When and why it matters

Amortisation explains why a 30-year mortgage refinanced into another 30-year mortgage at year 5 is almost always a worse deal than the lower rate suggests — the homeowner is restarting the front-loaded interest schedule. It explains why making one extra principal payment per year (the common “26 biweekly payments” trick) shaves roughly 4-5 years off a 30-year term: each extra dollar paid in year 2 avoids 28 years of compound interest. It explains why bond traders care about the difference between coupon yield and yield-to-maturity, and why pension actuaries discount cash flows to present value. Anyone signing a mortgage, car loan, or amortising business loan should run the schedule once — most borrowers are surprised how little equity has accrued by year 5. Reference: CFPB — Loan amortisation.

Negative amortisation happens when the monthly payment is less than the interest accrued — the unpaid interest is added to the principal balance, which then grows. Some adjustable-rate mortgages, certain US student-loan income-driven repayment plans, and the pay-option ARMs sold during the 2000s housing bubble used this structure. It is generally a sign of stress in the borrower’s position and was a leading indicator of the 2008 mortgage crisis. Related: principal, APR, and the 15 vs 30-year mortgage comparison.