How to Calculate a Monthly Loan Payment by Hand

The payment formula

For a loan with a fixed interest rate and equal monthly payments, the payment depends on three inputs: the amount borrowed, the monthly interest rate and the number of payments.

payment = P × r ÷ (1 − (1 + r)^−n)P is the amount borrowed, r is the annual rate ÷ 12 as a decimal, and n is the number of monthly payments.

The monthly rate is the annual rate divided by 12, so 6% a year becomes 0.06 ÷ 12 = 0.005. The number of payments is the term in years multiplied by 12. If the rate is zero, the formula cannot be used because it would divide by zero, but the answer is simply the amount borrowed divided by the number of payments.

The structure is easier to trust once you see what it does. The top, P × r, is the interest due in the first month. The bottom scales that figure up so that the payment is just large enough to clear the whole balance by the final month, no more and no less.

A worked example, step by step

Example: borrow 20,000 at 6% a year, repaid over 5 years.

1. Monthly rate: r = 0.06 ÷ 12 = 0.005.
2. Number of payments: n = 5 × 12 = 60.
3. Growth factor: (1.005)^60 ≈ 1.34885, so (1.005)^−60 ≈ 0.74137.
4. Denominator: 1 − 0.74137 = 0.25863.
5. Numerator: 20,000 × 0.005 = 100.
6. Payment: 100 ÷ 0.25863 ≈ 386.66 per month.

Over 60 payments you repay about 23,199, so the interest cost is about 3,199. The power in step 3 is the only awkward part; the exponent calculator handles it, and the loan calculator gives the whole answer in one go, so you can use it to confirm your hand result.

How term and rate change the cost

Stretching a loan lowers each payment, but you pay interest for longer. Here is the same 20,000 at 6% over three different terms:

The rate matters as well. Keeping the 5-year term, a 7% loan costs 396.02 a month and an 8% loan costs 405.53. Each extra percentage point adds roughly 9 to 10 to the payment here, and about 560 to 570 to the total cost over the life of the loan.

When comparing offers, look at the total repaid as well as the payment. A lower payment is only an advantage if you can also afford the extra interest that a longer term brings.

Using the formula to check a quote

If a lender quotes a payment, you can test it in a minute. Ask for the amount financed, the annual rate and the number of monthly payments, then run the formula.

• If your answer matches within a few cents, the quote is a plain fixed-rate loan.
• If the quote is higher, the amount financed may include fees or add-ons rolled into the loan, or the payment may include insurance or other charges.
• If the quote is lower, check that the term and rate you were given are the ones used, and whether the rate is only an introductory one.

Asking why the numbers differ is reasonable and costs nothing. A clear answer usually points to a specific fee or product that you can then decide on separately.

What the formula leaves out

• Fees. Origination fees, closing costs and insurance are not part of the formula, so the real cost of a loan can be higher than the payment suggests.
• Variable rates. If the rate can change, the payment can change too. The formula applies to each period using the then-current rate and remaining balance.
• Daily interest. Some loans accrue interest daily or use different payment dates, which can shift the figures by small amounts.
• Rounding. Lenders round the payment to the nearest cent, so the final payment is often a few cents different from the rest.

A useful sanity check is to compare the payment with P × r, the first month's interest. In the example that is 100, and the payment of 386.66 is well above it. If a payment were smaller than the monthly interest, the balance would never fall.

This article is general education about loan arithmetic, not financial advice. Check the actual terms, fees and your own circumstances with the lender or a qualified adviser before borrowing.