Help - Periodic Payment for a Loan (loan)
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Periodic Payment for a Loan
Purpose
This calculator computes the fixed payment required each period to fully repay a loan, along with the total amount paid over the life of the loan. It is useful for quickly estimating a monthly (or other periodic) installment for a mortgage, auto loan, or any loan repaid in equal instalments.
Background
Amortizing a loan to zero
A standard loan is repaid with equal payments at the end of every period until the balance reaches exactly zero. Each payment covers the interest owed on the remaining balance for that period, with the rest reducing the principal. This calculator finds the single fixed payment amount that achieves that outcome over the stated Number of Periods, given the Loan Amount and Interest Rate.
Inputs
Loan Amount
The total amount borrowed.
Interest Rate
The loan's nominal interest rate, entered with a time basis (for example,
6.0 pct/yr). It is converted proportionally to match the time unit of
Number of Periods — for example, an annual rate is divided by 12 to
get a monthly rate. This is a simple proportional conversion, not a
compounding conversion, so entering the rate directly in the same time
unit as Number of Periods (for example, a monthly rate when periods
are months) gives the same result and avoids any ambiguity.
Number of Periods
The total number of equal payments over which the loan is repaid (for
example, 12.0 mo). Its time unit also determines how Interest Rate
is converted, as described above.
Results
Payment Per Period
The fixed amount to pay at the end of every period so the loan is fully repaid, with interest, by the end of Number of Periods.
Total Payment
Payment Per Period multiplied by Number of Periods — the total of all payments over the life of the loan. It is always more than the Loan Amount, with the difference being the total interest paid.
Understanding the Calculation
The interest rate is first expressed as a rate per period matching the time unit of Number of Periods. The required periodic payment is then:
where is the interest rate per period and is the Number of Periods. This is the standard loan amortization formula — the fixed payment whose repeated application, with interest charged on the declining balance, brings the balance to exactly zero after payments.
Example
Using the default inputs — a Loan Amount of 100,000, an Interest Rate of 6.0% per year, and 12.0 monthly periods — the calculator converts 6% per year to 0.5% per month, and returns a Payment Per Period of approximately 8,606.64 (per month) and a Total Payment of approximately 103,279.72. The gap between the Total Payment and the 100,000 borrowed — about 3,279.72 — is the total interest paid over the year.
Important Assumptions and Interpretation
- Payment Per Period is assumed to be made at the end of each period, a fixed and evenly spaced amount, with no missed or extra payments.
- Interest Rate is assumed constant across every period; this is a fixed-rate loan calculation, not an adjustable-rate one.
- Because the rate conversion to match Number of Periods is a simple proportional split rather than a compounding-equivalent conversion, for the clearest results enter Interest Rate already in the same time unit as Number of Periods (for example, a monthly rate for monthly periods) if you want to avoid any approximation from the conversion step.
- The result does not include fees, insurance, taxes, or prepayments — it reflects only principal and interest on the stated terms.