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Loan payment calculator and formula

A level loan payment is M = P times i divided by 1 minus (1 + i) to the power of minus n. Borrow $250,000 at 6.5 percent over 30 years and the payment is $1,580.17 a month, of which $1,354.17 is interest in month one.

Monthly payment

$1,580.17

Over 360 payments you repay $568,861.22 in total.

Total interest
$318,861.22
Total repaid
$568,861.22
First payment: interest
$1,354.17
First payment: principal
$226.00

Amortisation schedule, first year

#InterestPrincipalBalance
1$1,354.17$226.00$249,774.00
2$1,352.94$227.23$249,546.77
3$1,351.71$228.46$249,318.31
4$1,350.47$229.70$249,088.61
5$1,349.23$230.94$248,857.67
6$1,347.98$232.19$248,625.48
7$1,346.72$233.45$248,392.04
8$1,345.46$234.71$248,157.32
9$1,344.19$235.98$247,921.34
10$1,342.91$237.26$247,684.07
11$1,341.62$238.55$247,445.53
12$1,340.33$239.84$247,205.69
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The formula

M=P×i1−(1+i)−nM = P \times \frac{i}{1 - (1 + i)^{-n}}

MM is the payment, PP the amount borrowed, ii the rate for one period (the annual rate divided by 12 for a monthly loan), and nn the total number of payments.

What this calculator works out

Enter the amount borrowed, the annual rate and the term. The calculator returns the level monthly payment, the total interest over the life of the loan, and a full payment-by-payment schedule you can scroll through.

The schedule is the part worth looking at. A level payment stays the same every month, but what it does changes completely: early payments are almost all interest, and late payments are almost all principal. The table shows the exact split for every month.

The loan payment formula

The payment is whatever amount, repeated nn times, exactly clears the balance:

M=P×i1−(1+i)−nM = P \times \frac{i}{1 - (1 + i)^{-n}}

For a monthly loan, ii is the annual rate divided by 12 and nn is the number of years times 12. A 30 year loan at 6.5 percent has i=0.065/12=0.00541667i = 0.065/12 = 0.00541667 and n=360n = 360.

The formula comes from setting the present value of all the payments equal to the amount borrowed. You do not need that derivation to use it, but it explains why the answer is exact rather than approximate: the payment is defined as the one that lands the balance on zero at payment nn.

How each payment splits

Every payment is applied in the same order. Interest is charged on the balance outstanding, that interest is taken out of the payment, and whatever is left reduces the balance.

  • Interest for the month is the current balance times ii.
  • Principal is the payment minus that interest.
  • The new balance is the old balance minus the principal.

Because the balance falls, the interest portion falls with it and the principal portion grows. The change is slow at the start. On the 30 year loan above, the principal part of the payment does not overtake the interest part until payment 233, which is year 20 of 30.

Term against total cost

A longer term buys a smaller payment by charging interest for longer. The two move in opposite directions, and only one of them is visible when you are deciding what you can afford each month.

The worked examples below run the same $250,000 at the same rate over 30 years and over 15 years. The shorter term raises the payment substantially and cuts the total interest by more than half.

The rate itself is worth checking carefully too, because a quoted rate and the rate you actually pay are not always the same number. The APR against APY calculator covers that gap. Running money the other way, into an account rather than out of one, is the compound interest calculator.

Worked examples

A 30 year loan at 6.5 percent

You borrow $250,000 over 30 years at 6.5 percent, paid monthly. What is the payment and what does the loan cost?

  1. Find the period rate: i=0.065/12=0.00541667i = 0.065/12 = 0.00541667.
  2. Count the payments: n=30×12=360n = 30 \times 12 = 360.
  3. Work out the discount term: (1+i)−360=0.143025(1 + i)^{-360} = 0.143025, so 1−0.143025=0.8569751 - 0.143025 = 0.856975.
  4. Apply the formula: M=250000×0.005416670.856975M = 250000 \times \frac{0.00541667}{0.856975}, which is $1,580.17.
  5. Multiply by the number of payments: 1580.17×360=1580.17 \times 360 = $568,861.22.
  6. Subtract the amount borrowed to isolate the interest: $568,861.22 minus $250,000.

The payment is $1,580.17 a month. Over 360 payments you repay $568,861.22 in total, so the interest alone is $318,861.22, which is more than the $250,000 you borrowed.

Where the first payment actually goes

Using the same loan, how much of the first $1,580.17 payment reduces what you owe?

  1. Interest for month one is the full balance times the period rate: 250000×0.00541667=250000 \times 0.00541667 = $1,354.17.
  2. Principal is what is left of the payment: $1,580.17 minus $1,354.17.
  3. The new balance is $250,000 minus that principal.

Only $226.00 of the first payment reduces the debt. The balance after one month is $249,774.00, so after paying $1,580.17 you owe just $226.00 less than you did.

The same loan over 15 years

Keep $250,000 at 6.5 percent but halve the term to 15 years. What happens to the payment and to the total interest?

  1. The period rate is unchanged at i=0.00541667i = 0.00541667, but now n=180n = 180.
  2. M=250000×0.005416671−(1.00541667)−180M = 250000 \times \frac{0.00541667}{1 - (1.00541667)^{-180}}, which is $2,177.77.
  3. Total interest is 2177.77×1802177.77 \times 180 minus the $250,000 borrowed.

The payment rises to $2,177.77, about 38 percent more each month, and the total interest falls to $141,998.31. Halving the term cuts the interest by well over half, because you are borrowing less money for less time at every point in the schedule.

The mistake that costs the most

Treating the monthly payment as the price of the loan.

The payment is what you can afford. The total interest is what the loan costs, and the two are pulled apart by the term. On the numbers above, the 30 year version costs $318,861.22 in interest and the 15 year version costs $141,998.31. Same amount borrowed, same rate, and the shorter term costs less than half as much to run.

A lender quotes the payment because it is the number that decides whether you say yes. Before comparing two offers, work out the total repaid for each, and compare those instead.

Common questions

Does this include property tax and insurance?

No. It returns principal and interest only, which is the part the formula governs. A mortgage servicer often collects tax and insurance in the same monthly bill, so the amount leaving your account can be noticeably higher than the payment shown here.

What happens if I pay extra each month?

Every extra dollar goes straight to principal, so it removes all the future interest that dollar would have carried. Early extra payments do far more than late ones, because they cut the balance that all the remaining interest is charged on.

Why is so much of an early payment interest?

Interest is charged on what you still owe, and at the start you still owe almost everything. The payment is level, so the interest share falls only as the balance falls. On a 30 year loan the principal part does not overtake the interest part until payment 233.

Keep reading

This page is educational material, not financial advice. The figures come from the formula shown and assume the inputs you enter hold for the whole term. Your own rate, fees, taxes and timing will differ, so treat the output as arithmetic to check a decision against, not as a recommendation.