Loan Amortization Curve: Payments and Balance

A loan amortization curve shows how a loan’s outstanding balance changes over time as scheduled payments are applied to interest and principal. Enter the loan amount, APR, and term below to calculate the fixed monthly payment and total interest, then use the explanation to understand the balance curve.

Formula

For a fixed-rate loan with equal monthly payments, the level-payment formula is:

Payment = round[P × r / (1 − (1 + r)⁻ⁿ), 2]

• P is the starting loan principal, in dollars.
• r is the monthly interest rate as a decimal: APR ÷ 1,200.
• n is the number of monthly payments: years × 12.
• Payment is the scheduled monthly amount, rounded to cents.

The calculator’s second output is estimated total interest: monthly payment × n − P. The formula assumes the rounded payment is made for every month. It does not draw the entire curve; rather, the payment and interest figures provide the inputs for understanding how the balance changes through time.

Worked Example

Using the default inputs—$20,000 principal, 8% APR, and a five-year term:

1. Convert the APR to a monthly decimal rate: 8 ÷ 1,200 = 0.0066667.
2. Calculate the number of monthly payments: 5 × 12 = 60.
3. Substitute into the payment formula: $20,000 × 0.0066667 ÷ [1 − (1.0066667)⁻⁶⁰] = $405.53 after rounding to cents.
4. Calculate total interest using the rounded payment: $405.53 × 60 − $20,000 = $4,331.80.

The displayed estimates are a monthly payment of $405.53 and total interest of $4,331.80. Actual final payments can differ if the lender uses different day-count or rounding conventions.

Assumptions

• The interest rate remains fixed for the entire term.
• Payments are equal, made monthly, and begin one month after the loan starts.
• Interest is modelled with monthly compounding at APR ÷ 12.
• The calculation excludes fees, taxes, insurance, PMI, and escrow, as well as any separately financed charges.

How an Amortization Curve Works

Each payment is divided between interest and principal. In a standard fixed-payment loan, interest for a period is calculated on the balance then outstanding. At the start, that balance is largest, so the interest share is generally larger. The remainder of the payment reduces principal. As the balance falls, the interest portion tends to shrink and the principal portion tends to grow.

For example, at the default rate, the first month’s approximate interest is $20,000 × 0.0066667, or $133.33. From a $405.53 payment, about $272.20 goes toward principal, leaving roughly $19,727.80. The next month’s interest is calculated on that lower balance, subject to the lender’s exact accrual and rounding method. A balance-over-time graph therefore usually slopes downward, with the balance declining more quickly later in the schedule.

The curve is not necessarily a straight line. Nor does “amortization” mean that each payment reduces principal by the same amount: the principal portion varies as the balance and interest charge change.

Reading the Curve and Schedule

A full amortization schedule lists each payment date or period, the opening balance, interest charged, principal repaid, and closing balance. Plotting closing balance against elapsed months gives the amortization curve. Useful points include the initial balance, balances after one year or halfway through the term, and the final scheduled balance.

• Steeper downward slope: the balance is being reduced more quickly over that interval.
• Flatter early section: a greater share of early payments is going to interest under many long-term, level-payment arrangements.
• End point: under the model, the balance reaches zero after the scheduled payments, allowing for rounding and the lender’s final-payment calculation.

To calculate a balance at a particular payment number, a standard fixed-rate formula is: balance after k payments = P(1 + r)^k − Payment × [(1 + r)^k − 1] ÷ r. Here k is the number of completed monthly payments. This is useful for examining one point on the curve, but it is not an additional output in this calculator.

What Changes the Curve

The starting principal, APR, and term shape the schedule together. A larger principal raises the payment and generally increases total interest if the rate and term stay the same. A higher APR increases the interest charged each month. A longer term usually lowers the required level payment but allows interest to accrue for more periods; comparing total interest also depends on the payment rounding and the particular loan terms.

Extra principal payments, missed or late payments, variable rates, payment holidays, and fees added to the balance can make the actual curve depart from the standard schedule. Extra payments may reduce later interest if the loan terms apply them to principal, but rules and timing vary. Check the loan documents or lender’s figures rather than assuming how an additional payment is allocated.

Common Calculation Limits and Mistakes

• Confusing APR and monthly rate: this calculator accepts an annual percentage rate and converts it to a monthly rate by dividing by 1,200. Do not enter 8 as though it were a monthly percentage.
• Mixing term units: the input is years; the formula multiplies by 12 to obtain months.
• Rounding too early: displayed payments are rounded to cents. Repeatedly rounding intermediate rates or balances can create small differences from a lender’s schedule.
• Treating APR as every borrowing cost: this simplified calculation does not add fees or other charges to the principal, and it cannot show how a lender’s APR calculation treats them.
• Assuming every loan is level-payment: interest-only periods, balloon payments, variable rates, irregular payment dates, and some repayment structures require different calculations.

Use the result as an arithmetic estimate, not as a promise of a specific payoff balance. To see the exact scheduled allocation, request the lender’s amortization schedule and compare its assumptions with the loan agreement.

Frequently Asked Questions

What is a loan amortization curve?
It is a graph or description of how the outstanding loan balance changes over the repayment period as payments cover interest and reduce principal.

Why does the balance fall slowly at first?
When the starting balance is high, the period’s interest charge is high relative to the fixed payment. Less of that payment is left to reduce principal. As the balance declines, the interest share usually declines too.

Is an amortization curve the same as an amortization schedule?
No. A schedule is a period-by-period table of balances and payment allocations; a curve is a visual representation of balance over time, often plotted from that table.

Does every loan have a fixed amortization curve?
No. The curve depends on the loan’s rate, payment pattern, fees, and terms. Variable rates or irregular payments can change the path, and some loans do not fully amortize through regular payments.

Does paying extra change the curve?
It can, if the extra amount is applied to principal and the loan terms allow it. The resulting schedule depends on timing and how the lender handles the payment, so confirm those details with the lender.

This result is an estimate; the lender’s disclosure gives the exact figures.

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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 | $ | round(p * (apr / 1200) / (1 - (1 + apr / 1200) ^ -(yrs * 12)), 2) | 2
out int | Total interest | $ | pmt * yrs * 12 - p | 2