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01 Loan payments

Interest Loan Equation: Monthly Payment and Total Interest

Calculator No. 01-8111 · Updated · Formula and assumptions shown below

Monthly payment $405.53
Total interest
$4,331.67
= Monthly payment × loan term × 12 − loan amount
Loan amount
$20,000
Input
Interest rate (APR)
8 %
Input
You enter
Loan amount ($), Interest rate (APR) (%), Loan term (years)
You get
Monthly payment ($), Total interest ($)
Rounding
2 dp

Estimate for education, not financial advice. The result follows from the numbers you enter and the formula shown. It is not a quote or an offer, and it does not include fees, taxes or insurance unless the calculator asks for them. Ask your lender for the exact figures.

An interest loan equation calculates a fixed monthly payment from the amount borrowed, annual percentage rate (APR), and loan term. Enter those three figures to estimate the monthly payment and total interest paid over the scheduled term.

Formula

For a fixed-rate loan repaid in equal monthly installments, the payment is calculated with:

Payment = P × r ÷ [1 − (1 + r)−n]

  • P is the starting principal, or amount borrowed, in dollars.
  • r is the monthly interest rate as a decimal. With APR entered as a percentage per year, r = APR ÷ 1200.
  • n is the total number of monthly payments: loan term in years × 12.

The total interest estimate is monthly payment × n − P. The calculator rounds the monthly payment to the nearest cent, then uses that rounded payment to calculate total interest. For the defaults, r is 0.08 ÷ 12, or about 0.0066667, and n is 5 × 12, or 60.

Worked Example

Using the calculator’s default values—$20,000 principal, 8% APR, and a 5-year term—the calculation proceeds as follows:

  1. Convert the annual percentage rate to a monthly decimal: 8 ÷ 1200 = 0.0066667.
  2. Count the monthly payments: 5 × 12 = 60.
  3. Apply the payment formula: $20,000 × 0.0066667 ÷ [1 − (1 + 0.0066667)−60] = about $405.53 per month, rounded to cents.
  4. Calculate total interest using the rounded payment: $405.53 × 60 − $20,000 = $4,331.80.

The scheduled total of the payments is $24,331.80: $20,000 repays principal and the remaining $4,331.80 is estimated interest. Actual lender figures can differ if the payment schedule, rate, or amount financed differs from these inputs.

How the Equation Works

Each installment covers interest accrued since the previous payment and reduces principal by whatever remains. Early in a level-payment loan, the outstanding balance is larger, so the interest portion is larger. As principal falls, less interest accrues per period and more of each later payment goes toward principal. The payment formula accounts for that changing balance while keeping the scheduled payment level.

This is different from simple interest, which is often represented as I = P × r × t for a single period or a situation where interest is calculated only on the original principal. Applying that simple-interest equation directly to a loan with monthly payments will not generally give its amortizing payment. The loan equation must account for interest on the remaining balance over repeated periods.

When to Use This Calculator

Use the estimate to understand the payment and scheduled interest implied by a fixed APR, principal, and term. It can also help illustrate how the inputs affect the result: holding the other two inputs constant, a larger principal or rate raises the calculated payment, while a longer term generally lowers the payment but extends repayment and can increase total interest.

Use the rate and amount that apply to the actual loan offer if you have them. The calculator does not find an available rate, determine eligibility, or establish what payment fits a personal budget. For a variable-rate loan, the payment may change when the rate changes, so one fixed-rate calculation cannot represent the full repayment path.

How to Read the Results

Monthly payment is the scheduled amount under the model, rounded to cents. Total interest is the estimated sum of interest across the full term, based on that rounded payment and the assumption that all scheduled payments occur as specified. It is not the total amount repaid; add the original principal to total interest to obtain the modeled payment total.

A small change in the inputs can change the result. A quoted rate may be expressed as APR or as a periodic rate; enter an annual APR here, not a monthly rate. If a lender quotes a different amount financed because of included charges, use that financed principal only when you intend to model those charges as part of the balance.

Common Mistakes and Formula Limits

  • Mixing rate units: Enter 8 for 8% APR, not 0.08. The formula divides the percentage by 1200 to get a monthly decimal.
  • Mixing months and years: This calculator expects years and converts them to months. A 5-year term means 60 payments, not 5.
  • Rounding too early: The displayed payment is rounded to cents. Repeating the calculation with a rounded monthly rate or a different payment-rounding convention can create small differences in totals.
  • Assuming every loan is level-payment: Interest-only periods, balloon payments, irregular payment dates, variable rates, and changing installments require different calculations.
  • Confusing interest with all borrowing costs: Fees and other charges are not included unless they are represented in the APR or principal supplied, and the calculator does not separately model them.

The result is an estimate and the lender’s disclosure gives the exact figures.

Assumptions

  • The interest rate is fixed for the full modeled term.
  • Payments are equal, made monthly, and treated as occurring at the end of each monthly period.
  • Interest accrues monthly at APR divided by 12, with APR entered as a percentage.
  • Fees, taxes, insurance, PMI, and escrow are not separate inputs and are not included.

Frequently Asked Questions

What is the interest loan equation?
The standard fixed-payment equation is Payment = P × r ÷ [1 − (1 + r)−n], where P is principal, r is the periodic interest rate as a decimal, and n is the number of payments.

How do I calculate monthly interest from APR?
For this monthly-payment model, divide APR entered as a percentage by 1200. For example, 8% ÷ 1200 gives a monthly decimal rate of about 0.0066667.

Does the monthly payment include all interest?
It includes the scheduled principal and interest under the stated assumptions. It does not necessarily include separate fees, taxes, insurance, or other charges.

Why does a longer loan term usually increase total interest?
A longer term spreads principal repayment across more months. The scheduled payment may be lower, but interest accrues for more periods, so the modeled total interest can be higher.

Can I use this equation for a variable-rate loan?
It can estimate payments for a period if you assume a particular fixed rate, but it cannot predict future payments when the rate changes. Recalculate with each applicable rate to make separate estimates.

Related

Formula used
in p | Loan amount | $ | 20000 | 1 | 100000000
in apr | Interest rate (APR) | % | 8 | 0.01 | 100
in yrs | Loan term | years | 5 | 0.0833 | 50
out pmt | Monthly payment | $ | p * (apr / 1200) / (1 - (1 + apr / 1200) ^ -(yrs * 12)) | 2
out int | Total interest | $ | pmt * yrs * 12 - p | 2

Each line is one input (in, sel, list) or one result (out: label, unit, formula, decimals). How the calculators work

Method: the standard loan formula for this calculation, fixed rate and monthly compounding unless stated, rounded as each result states. Not financial advice; no lender, rate or product is recommended. See formulas and assumptions, the disclaimer or report an error.

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