How loan amortization works, with a worked schedule

Amortization is repaying a loan in equal installments, where each payment first covers the interest charged on the balance since the last payment and the rest reduces the balance. The payment stays the same, but its split changes: as the balance falls, each period’s interest gets a little smaller and the principal part a little larger. A formula sets the payment so the last one brings the balance to exactly $0.00. Because payments are made in whole cents, that last payment is usually a few cents different from the others.

The examples use assumed rates, not current ones: a $10,000 loan at 6% repaid in 12 monthly payments, and later a $300,000 loan at 6.5% over 30 years, where interest is the larger part of every payment until payment 233, in year 20.

How is a loan payment calculated?

For a fixed-rate loan of $P$ repaid in $n$ equal payments at a rate of $i$ per period, the payment is

• $P$ is the amount borrowed.
• $i$ is the interest rate per payment period: the annual rate ÷ 12 for monthly payments.
• $n$ is the number of payments: years × 12 for monthly payments.

In the example, $i$ = 6% ÷ 12 = 0.005 and $n$ = 12. Then $(1.005)^{12}$ = 1.0616778…, so $1 - (1.005)^{-12}$ = 0.0580947…, and $M$ = $10,000 × 0.005 ÷ 0.0580947… = $860.6643…, which rounds to $860.66.

The formula comes from one line of bookkeeping. Each period the balance grows by its interest and falls by the payment, $B_k = B_{k-1}(1 + i) - M$. Require the balance after payment $n$ to be zero, solve for $M$, and you get the formula above. Put another way, the amount borrowed equals the present value of all the payments at the loan’s rate.

Use the loan’s interest rate, not its APR. The APR also counts points and fees, so it is usually higher and would overstate the payment; APR, APY and the interest rate explains the difference.

How is each payment split between interest and principal?

Interest comes first. Each period’s interest is the balance owed times the periodic rate, rounded to the cent, and whatever is left of the payment reduces the balance.

• Payment 1: interest is $10,000 × 0.005 = $50.00, so $860.66 − $50.00 = $810.66 goes to principal. The balance falls to $9,189.34.
• Payment 2: interest is $9,189.34 × 0.005 = $45.9467, rounded to $45.95, so $814.71 goes to principal. The balance falls to $8,374.63.

Notice that $810.66 × 1.005 = $814.71. Each payment’s principal is the previous one times $(1 + i)$, give or take a cent of rounding, because the interest saved on last period’s principal moves to this period’s principal. Following that back from the last payment gives the principal part of payment $k$ directly:

In words, it is the payment discounted at the loan’s rate over the number of payments left, counting this one. For payment 1: $860.6643 ÷ 1.0616778 = $810.66.

What does a full amortization schedule look like?

Every row repeats those two steps.

Three things to check in any schedule:

• The principal column adds up to the amount borrowed, to the cent. Here it is $10,000.00, and everything else paid, $327.96, is interest.
• The interest share falls every month, from 5.8% of payment 1 to 0.5% of payment 12. On a one-year loan it starts small; on a 30-year loan the same pattern starts from most of the payment.
• The last payment differs: $860.70, not $860.66.

Why is the last payment different?

Because payments are made in whole cents. The exact payment is $860.6643…, so each $860.66 payment is about 0.43 of a cent short. With each month’s interest also rounded to the cent, twelve payments of $860.66 would leave 4 cents owing, so the last payment is $860.70: its $4.28 of interest plus the whole remaining balance of $856.42. Regulation Z’s official interpretation recognizes this: because payments can’t be collected in fractions of a cent, the last one may need adjusting.

Not every schedule rounds the same way. The OpenStax Contemporary Mathematics text rounds both the payment and each month’s interest up to the next cent. On the same loan, that gives eleven payments of $860.67, a last payment of $860.66 and $328.03 of interest, 7 cents more. The balances drift a few cents apart along the way: $5,074.79 after payment 6, against $5,074.83.

Two other things can make a lender’s schedule differ from a formula-built one:

• Daily interest. Some lenders charge interest for the actual days since the last payment, at 1/365 of the annual rate per day, instead of 1/12 of the annual rate per month. On $10,000 at 6%, a 31-day month then costs $50.96, a 30-day month $49.32 and a 28-day month $46.03, against $50.00 every month under the monthly convention. Under a daily method, paying a few days late means more interest and less principal that month. Regulation Z lets the disclosed schedule assume on-time payments and months of equal length, so for these loans it is an approximation.
• An odd first period. If the first payment is due more or less than one period after the loan starts, the first interest charge is larger or smaller. Regulation Z lets the disclosure ignore a first period that is only moderately long or short.

How do you find the balance without building the schedule?

The balance after $k$ payments is what the loan would have grown to with no payments, minus what the payments would have grown to at the same rate:

After 6 payments: $10,000 × 1.005⁶ = $10,303.775…, and $860.66 × 6.075502… = $5,228.941…, leaving $5,074.83, the same as row 6 of the schedule. Use the rounded payment you actually make; the unrounded $860.6643… gives $5,074.81. Over a long loan, rounding each month’s interest moves the schedule slightly off the formula: by 21 cents near the end of the 30-year loan below.

When does principal overtake interest on a long loan?

When the payments left, counting the current one, are fewer than the number of months money takes to double at the loan’s rate. For $300,000 at 6.5% for 30 years, a payment of $1,896.20, that is payment 233, in year 20.

Until then, most of each payment is interest. Payment 1 is $1,625.00 of interest and only $271.20 of principal. After five years, $113,772.00 of payments have brought the balance down to $280,833.26, so 6.4% of the loan is repaid. Payment 233 puts $949.69 toward principal against $946.51 of interest, and by then $314,659.48 of the loan’s $382,636.71 of lifetime interest (82%) has already been paid. The balance drops below half the original loan at payment 257, in year 22.

The principal formula above says why it happens so late. Principal exceeds interest once it is more than half the payment, which happens when $(1 + i)^{n - k + 1} < 2$. At 6.5% ÷ 12 per month, money doubles in 128.3 months, so principal overtakes interest with 128 payments left, about 10.7 years before the end. The rule of 72 gives a quick check: 72 ÷ 6.5 ≈ 11 years.

The loan amount doesn’t enter into it. At 6.5% over 30 years, the crossover is payment 233 whether you borrow $100,000 or $1,000,000. Rate and term decide it:

A loan with fewer payments than the doubling time puts more toward principal than interest from the first payment: a 15-year loan at 4%, a 10-year loan at 6.5% or less, and a 5-year loan at any rate in the table. That is why the “mostly interest” pattern is a feature of long loans at meaningful rates, not of amortization itself. For what the full 30-year pattern means for a mortgage payment, see what’s included in a mortgage payment.

What happens to the schedule when you pay extra?

Extra principal cuts the balance directly, so every later interest charge is smaller. Suppose $1,000 extra goes with payment 4 of the 12-month loan:

The $1,000 is no longer charged 0.5% a month, so payment 5’s interest is exactly $5.00 lower. Keep the payment the same and the saving grows a little each month ($5.03 in payment 6, $5.15 by payment 11), and the loan ends one payment early. A recast, if your lender offers one, re-amortizes the lower balance over the payments left and turns it into a lower payment instead. That saves less interest because the balance falls more slowly: the monthly saving shrinks from $5.00 to $0.63 by payment 12.

Some mortgages carry a prepayment penalty, a fee for paying off all or part of the loan early, so check your loan’s terms before a large prepayment. For mortgage-sized amounts, prepayment penalties and the trade-off against other uses of the money, see extra mortgage payments.

What is negative amortization?

Negative amortization means the balance grows even though you are paying, because the payment doesn’t cover the interest. The unpaid interest is added to the balance and is then charged interest itself. On $10,000 at 6%, a $40.00 payment against $50.00 of interest leaves $10,010.00 owing. The next month’s interest is $50.05, and the balance becomes $10,020.05, then $10,030.15.

Two other structures postpone amortization:

• An interest-only period. The payment equals the interest, so the balance doesn’t move: $50.00 a month on $10,000 at 6% leaves $10,000 owing indefinitely. Once principal payments start, the payment rises. Borrow $300,000 at 6.5% with 10 years interest-only: the payment is $1,625.00 for 10 years, then $2,236.72 to repay the $300,000 over the remaining 20. Total interest is $431,812.49, against $382,636.71 for the same loan amortized over the full 30 years.
• A balloon loan. The payments are too small to repay the loan within its term, so a large final payment is due at the end.

What a schedule doesn’t show

• Taxes and insurance. Many mortgage payments also include amounts for property tax and insurance. A schedule covers principal and interest only.
• Fees. Points and other charges show up in the APR, not in the schedule.
• Rate changes. On an adjustable-rate loan, a rate change typically (though not always) resets the payment from the new rate and the term left, which starts a new schedule from the balance at that point.

These are educational estimates, not financial or lending advice. Your loan’s note and disclosures show its actual terms.

Try it

Loan payment calculator, with monthly payments:

• Loan amount 10,000, annual interest rate 6, loan term 12 months. The payment is $860.66, the final payment $860.70 and total interest $327.96. Row 6 of the schedule shows a balance of $5,074.83.
• Loan amount 300,000, rate 6.5, term 30 years. The payment is $1,896.20 and total interest $382,636.71. Row 233 is the first where principal ($949.69) exceeds interest ($946.51), and row 257 leaves $149,389.63. Change the amount to 100,000 and the crossover is still row 233.
• For the interest-only example, amount 300,000, rate 6.5, term 20 years gives $2,236.72.
• For the recast, amount 5,732.96, rate 6, term 8 months gives $732.84.

Mortgage extra payment calculator: current balance 10,000, interest rate 6, time left 12 months, “Monthly P&I” left blank so the computed $860.66 is used, and a one-time extra of 1,000 in month 4. Clear “Extra monthly”, which the page’s example fills in. The loan is repaid in 11 months instead of 12, and $39.81 of interest is saved. Its recast line shows the other choice: a payment of $732.84 and $22.62 saved.