Lesson 2 of 25 · 7 min

The installment formula, worked

A fixed monthly amount is the number that makes every period's interest and remaining principal add up.

A fixed installment is the single monthly amount that, at a constant rate, retires a stated principal over a stated number of periods. It is not a fraction of the cash price. It is the number that makes each month's interest on the outstanding balance, plus a slice of principal, add up to the same cash figure every time.

The formula is mechanical. Let P be the principal, n the number of months, and r the monthly rate, which is APR divided by 12. Then the installment is P × r × (1+r)^n / ((1+r)^n − 1). The (1+r)^n terms are the compounding of that monthly rate across the whole term. The division forces the present value of n equal payments, discounted at r, to equal P.

PMT = P × r × (1+r)^n / ((1+r)^n − 1), r = APR / 12

PMT: The fixed monthly installment.
P: The principal being financed.
r: The monthly rate, APR divided by 12.
n: The number of months in the term.

Work Tallow Yard Autos at P = $28,420, an illustrative 6.85 percent APR, and n = 60. Then r = 0.0685 ÷ 12 = 0.00570833. Substitute: $28,420 × 0.00570833 × (1.00570833)^60 / ((1.00570833)^60 − 1) = $560.74. Sixty of those installments total $33,644.46. Interest over the life is $33,644.46 − $28,420 = $5,224.46.

Track 1 lesson 14 is daily variable accrual: the charge changes whenever the balance or the rate changes. A car installment is the other shape. The rate is fixed for the term, the payment is fixed, and the split inside each payment — interest versus principal — is the only thing that moves. That split is a later lesson. This one is the formula that produces the payment.

The installment is a solved-for remainder. Change P, r, or n, and the remainder changes. The cash price of the car is none of those three.

Key terms

Installment (PMT): The fixed monthly amount that retires a principal over n months at a constant monthly rate.
Monthly rate (r): APR divided by 12. The rate applied to the outstanding balance each month.
Term (n): The number of months over which the principal is scheduled to be retired.

Worked example

One substitution, three outputs

Tallow Yard Autos finances $28,420 at an illustrative 6.85 percent APR for 60 months.

Monthly rate r: 0.00570833
Installment: $560.74
Total of payments: $33,644.46
Interest over the life: $5,224.46

Sixty equal cash figures of $560.74 are one principal plus $5,224.46 of time. The formula is how those two parts are forced into a single monthly number.

Check yourself

Principal $18,640, illustrative 7.2 percent APR, 48 months. What is the monthly installment, and what is the total of payments?

A fixed installment is the everyday version of a known accrual schedule, unlike the daily variable accrual Track 1 lesson 14 walks through. Open Daily accrual, and why a monthly statement hides it →

Saved in this browser only. No account, nothing sent anywhere.

Educational only. Nothing in this lesson is investment, legal or tax advice, or a recommendation to enter into any transaction. Figures are illustrative and computed from the inputs stated in the lesson.