Loan Payment Calculator

Fixed-rate annuity payment with full amortization breakdown.

Worked example — Loan Payment Calculator

Monthly payment	485.65
Total paid	29139
Total interest	4138.9
Interest share	14.204%

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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Lesson

The theory — Loan Payment Calculator

An amortising loan is repaid by a constant payment, but what that payment does changes every month: part settles the interest accrued since the last one, and whatever is left reduces the debt. Because the debt shrinks, next month’s interest is smaller and more of the same payment goes to principal.

What each symbol means

r: the monthly rate, not the annual one. The box asks for 6.2% a year; the formula uses 0.0051667, which is 0.062 / 12.
n: the number of payments. Five years of monthly payments is n = 60.
M: the payment itself — the one number held constant, 485.65 here.

Where the formula comes from

Ask what the lender needs: the present value of all your future payments must equal the sum lent today. A payment M arriving k months from now is worth M/(1+r)^k now.
So P = M/(1+r) + M/(1+r)² + ⋯ + M/(1+r)ⁿ — a geometric series again, with ratio 1/(1+r). Summing it gives P = M·(1 − (1+r)⁻ⁿ) / r.
Solve for M and tidy by multiplying top and bottom by (1+r)ⁿ: M = P·r(1+r)ⁿ / ((1+r)ⁿ − 1). That is the formula above, and it is the annuity formula run backwards — a loan is a savings plan seen from the other side.

Assumes: A fixed rate, equal payments, and interest compounding at exactly the payment frequency. Nothing here models fees, insurance, an early repayment, or a rate that moves.
Breaks when: It also quietly refutes something you have probably been told. “Early payments are almost all interest” is true of a 30-year mortgage, not of loans in general — in month 1 of this 5-year loan, principal (356.48) already exceeds interest (129.17) nearly three to one. The claim is about the ratio of rate to term, not about loans as such.

why you pay so much interest up front 🖖

With a fixed monthly payment, the interest portion is charged on the outstanding balance each month. How much that comes to depends entirely on the term. On the thirty-year mortgage preset the first payment is 82.4% interest and principal does not overtake it until month 218 of 360; on the five-year car loan the very first payment is already mostly principal. Front-loaded interest is not a bank trick, it is what a long balance does to arithmetic.

why the lower rate costs more 🖖

The monthly payment is not chosen — it is the one amount that, paid every month, drives the balance to exactly zero on the final payment, and the principal, the rate and the term fix it completely. Which makes the two presets worth pressing back to back. The car loan carries the higher rate, 6.2%, and its interest share comes out at about a seventh of everything paid. The mortgage is cheaper per year, 5.8%, and its interest share is over half — the tool puts its total interest above the sum it lent. Six times the term, not the rate, is what did that. The rate is the number borrowers negotiate; the term is the number that settles the bill.

amortization literally means killing the loan 🖖

The word amortization comes from the Latin ad mortem, "to death," via the Old French amortir, "to kill." An amortizing loan is a debt you slowly put to death, payment by payment, until the balance is dead at zero. The same root gives us mortgage — literally a "death pledge," because the pledge dies the moment the debt is paid off.

Practice

Check yourself

Predict the answer first, then use the controls above to find out. Reveal only after you have committed to a guess — that is what makes it practice.

Press the mortgage preset — 420,000 at 5.8% over 30 years. Month 1 sends 2,030.00 of the 2,464.36 payment straight to interest. Before you open the schedule: which year is it before the principal share overtakes the interest share?

Show answer

Year 19, and the note under the chart says so outright. Row 218 of the schedule is where it happens — 1,236.68 of principal against 1,227.68 of interest, with 254,003.03 still owed. Nineteen of the thirty years pass before the payment is doing more repaying than renting. Change nothing but the term and the crossing moves: the 5-year default is already past it in month 1.
Same mortgage. Set Term in years to 15. The monthly payment has to rise — predict what the total interest does.

Show answer

It more than halves, 4.672 × 10⁵ down to 2.098 × 10⁵, and the interest share of everything paid falls from 52.658% to 33.314%. The payment only went from 2464.4 to 3499, about 42% more. You are not being charged a different rate; you are being charged for half as long, on a balance that falls twice as fast.
Back to 30 years, and open the schedule. You are exactly halfway through the term — month 180, fifteen years of payments made. How much of the 420,000 is still owed?

Show answer

295,809.96: seven tenths of it. Fifteen years of payments have removed under a third of the debt, while the Cum. Interest column has already reached 319,395.26. The balance does not drop below half until month 250, five sixths of the way through the term. Question 1 dated the front-loading; this prices it.

Problem solved in full

The monthly payment for borrowing 25 000 at 6.2% over 5 years 5 steps
Borrow 25 000 at 6.2% over 5 years. Derive the monthly payment from scratch, then find what doubling the term does to the total cost.
1. A loan is a promise to make n equal payments. Its value today is the sum of those payments discounted back, and that sum is a geometric series — which is the only mathematics in the whole of consumer finance.
  
  $P = k = 1 \sum n \frac{A}{( 1 + i ) ^{k}} = A \cdot \frac{1 - ( 1 + i ) ^{- n}}{i}$
2. Collapse the series and invert it for the payment. The monthly rate is the annual rate divided by twelve, and the term is in months, not years — the commonest error in this calculation is mixing the two units.
  
  $A = \frac{P i}{1 - ( 1 + i ) ^{- n}}, i = \frac{0.062}{12} = 0.0051 \overline{6}, n = 60$
3. Substitute. The calculator above prints exactly this figure, and it does so by evaluating this expression rather than by iterating a schedule.
  
  $A = \frac{25 000 \times 0.0051 6}{1 - ( 1.0051 6 ) ^{- 60}} = 485.65$
4. Total paid is just payment times count, and whatever exceeds the principal is interest. The tool prints both, along with the interest share of 14.204%.
  
  $total = 485.65 \times 60 = 29139 ⟹ interest = 4 138.9$
5. Now double the term to ten years, which halves the payment — almost. Run the same formula again.
  
  $n = 120 : A = 280.07, total = 33608, interest = 8608$
Answer

485.65 a month. Doubling the term drops the payment to 280.07 — not to 242.83, because you also owe interest for twice as long — and the interest more than doubles, from 4139 to 8608. This is the shape of every "lower monthly payment" offer: the payment falls by 42% and the cost rises by 108%. The series is where that asymmetry lives, and it is visible in the formula before any number is put in.

$A = \frac{P i}{1 - ( 1 + i ) ^{- n}} = 485.65 per month$

Learning path

Money over time

Leads to inflation-adjuster

References (1)

The amortisation schedule the lesson derives, and how the split moves month to month: S. Garrett, "Loan Repayment Schedules," in Introduction to the Mathematics of Finance, 2nd ed., 79–93. Butterworth-Heinemann, 2013.

By the 42Math team · Updated 2026-07-29

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

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Example problems

car loan - 25,000 at 6.2% over five years: 485.65 a month and 4,138.90 of interest, about a seventh of everything paid. Principal beats interest from the very first payment.
mortgage - 420,000 at 5.8% over thirty years: 2,464.36 a month and 467,170.59 of interest - more than the sum borrowed. The first payment is 82.4% interest, and principal does not overtake it until month 218 of 360.
student loan - 60,000 at 4.9% over ten years: 633.46 a month and 16,015.72 of interest, 21.1% of everything paid. Squarely between the car loan's seventh and the mortgage's half.