How an installment loan is amortized
Car loans, personal loans, student loans and equipment loans are usually repaid in equal monthly installments over a fixed number of months. The lender charges interest each month on whatever you still owe, the installment covers that interest first, and the rest of the installment repays part of the amount borrowed. Spreading the repayment this way is called amortization. The calculator finds the installment that brings the balance to exactly zero on the last payment and lists every month of the loan.
The installment is:
M = P × r ÷ (1 − (1 + r)^−n)
Pis the amount you borrow, including any fees the lender adds to the loan.ris the monthly rate: the yearly interest rate divided by 100 and then by 12. A 9% loan charges 0.0075 a month.nis the term in months, because installment loans are usually quoted as 36, 48, 60 or 72 months rather than in years.Mis the monthly installment.
Each row of the schedule repeats the same three steps: interest = balance × r, principal = M − interest, new balance = balance − principal. Interest is rounded to the cent every month, and the last installment pays whatever is left, so the final balance is exactly zero and the final installment can be a few cents off. At a 0% rate the installment is simply P ÷ n.
Worked example: the Truth in Lending APR example
Appendix J of Regulation Z, the U.S. Truth in Lending rule, explains how the annual percentage rate (APR) of a loan is computed and works through examples. Its first example is a $5,000.00 advance repaid in 24 monthly payments of $230, which the rule shows has an APR of 9.69%. Running that loan through the formula shows why: the APR is the rate at which the payments exactly repay the amount advanced.
- Monthly rate: 9.69 ÷ 100 ÷ 12 = 0.00807500.
- Number of payments: 24.
- Installment: $5,000.00 × 0.00807500 ÷ (1 − (1 + 0.00807500)^−24) = $230.01, which is the published $230 to within a cent of rounding.
- Month 1: interest $40.38, principal $189.63, balance $4,810.37.
- Month 12: interest has fallen to $22.83, principal has grown to $207.18, and $2,620.53 is still owed.
- Over all 24 payments you repay $5,520.21, of which $520.21 is interest.
The same check works in reverse for any offer you hold: enter the amount, the stated APR and the term, and the installment should match the one on the offer. If the offer's installment is higher, the loan includes costs that the APR you entered does not capture, or the amount financed is larger than you assumed.
What moves the installment and the total cost
The calculator opens with $25,000.00 at 7.5% over 60 months: an installment of $500.95 and $5,056.96 of interest. Changing only the term pulls the two numbers in opposite directions:
- 36 months: $777.66 a month, $2,995.58 of interest in total.
- 48 months: $604.47 a month, $4,014.68 of interest in total.
- 60 months: $500.95 a month, $5,056.96 of interest in total.
- 72 months: $432.25 a month, $6,122.18 of interest in total.
- 84 months: $383.46 a month, $7,210.24 of interest in total.
The rate
Three percentage points more (10.5%) on the same loan raises the installment to $537.35 and interest to $7,240.78. A rate difference adds less interest on a five-year loan than on a thirty-year mortgage, because the balance is outstanding for fewer months, but it still shows up in every installment.
Fees added to the loan
An origination or documentation fee that is added to the balance is borrowed money too. Financing a $500 fee on top of $25,000.00 raises the installment to $510.97 and interest to $5,158.04, because you pay interest on the fee for the whole term. Enter the amount actually financed, not the price of what you are buying.
Car loans versus personal loans
The arithmetic is identical. What differs is what goes into P: for a vehicle it is the price minus the down payment and any trade-in, plus taxes and fees that you finance; for a personal loan it is the amount paid out to you plus any fee deducted or added. Some loans compute interest on a daily basis rather than once a month, and the timing of each payment then shifts the split slightly; this calculator assumes payments exactly one month apart.
Frequently asked questions
Should I enter the interest rate or the APR?
Either, as long as you know which one you used. The APR folds certain finance charges into a yearly rate, so entering the APR with the amount financed shown on a Truth in Lending disclosure reproduces the installment, as in the worked example. Entering the note rate with the full balance you owe, fees included, gives the installment the lender bills when fees are added to the loan.
Why does the installment stay the same while the interest falls?
The installment is fixed when the loan is made. Each month the interest part is computed on a smaller balance, so a larger part of the same installment goes to principal. In the worked example the principal part grows from $189.63 in month 1 to $207.18 in month 12.
What happens if I pay the loan off early?
On a simple-interest installment loan, interest stops accruing on the principal you repay, so paying early reduces the total interest; the schedule shows the balance you would need to clear at any month. Check your agreement for a prepayment charge before relying on that saving. To model a regular extra amount on a home loan, use the mortgage payoff calculator.
How do I compare two offers with different terms?
Put each offer into the calculator and compare the total of payments as well as the installment. A longer term lowers the installment and raises the total; the table above shows how much for one loan. The interest share in the result tells you what part of everything you repay is interest.
Does changing the currency change the result?
No. It only changes the symbol and number format. Enter amounts in the currency of your loan; the calculation is the same in every currency, and your choice is kept in this browser only.
Sources
- Regulation Z, Appendix J to Part 1026: annual percentage rate computations for closed-end credit transactions — the definition of the APR used in the worked example.
- Regulation Z, Appendix J, paragraph (c)(1), example (i) — $5,000.00 advanced, 24 monthly payments of $230, annual percentage rate 9.69%.
