Amortizing Loan Payment (PMT) Equation, Derivation & Rearrangements

Core idea

Overview

The Amortizing Loan Payment formula determines the fixed periodic payment required to fully retire a debt over a specified duration at a constant interest rate. It mathematically balances the interest due on the remaining principal with a partial repayment of the principal itself in every installment.

When to use: This formula is applied when calculating monthly payments for fixed-rate installment products like mortgages, car loans, and personal loans. It assumes that interest is calculated on the declining balance and that the interest rate remains unchanged throughout the life of the loan.

Why it matters: It allows borrowers to budget with certainty by knowing the exact amount required to clear a debt by the end of its term. For financial institutions, it structures the recovery of principal while ensuring the time value of money is compensated via interest.

Symbols

Variables

PMT = Payment, PV = Principal (Present Value), r = Rate per Period, n = Number of Payments

P M T

Payment

$

P V

Principal (Present Value)

$

r

Rate per Period

V a r iab l e

n

Number of Payments

V a r iab l e

Walkthrough

Derivation

Derivation of the Amortizing Loan Payment Formula (PMT)

A fixed loan payment is found by treating repayments as an annuity whose present value equals the loan principal.

Payments are equal and made at the end of each period (ordinary annuity).
Interest rate per period is constant r.
Principal is PV and total number of payments is n.

1

Start from annuity present value:

The loan principal equals the present value of all future payments discounted at rate r.

P V = P M T \frac{1 - ( 1 + r ) ^{- n}}{r}

2

Multiply both sides by r:

Clear the fraction to isolate PMT.

P V r = P M T (1 - (1 + r)^{- n})

3

Divide by the bracket to make PMT the subject:

This is the standard fixed-payment (mortgage/car loan) formula. If r=0, it reduces to PMT=PV/n.

P M T = \frac{P V r}{1 - ( 1 + r ) ^{- n}}

Result

P M T = \frac{P V r}{1 - ( 1 + r ) ^{- n}}

Free formulas

Rearrangements

Solve for $P M T$

Make PMT the subject

P M T = \frac{P V r}{1 - ( 1 + r ) ^{- n}}

PMT is already the subject of the formula.

Difficulty: 1/5

Solve for $P V$

Make PV the subject

P V = P M T \frac{1 - ( 1 + r ) ^{- n}}{r}

Start with the Amortizing Loan Payment formula. To make PV the subject, multiply both sides by the denominator $1 - (1 + r)^{- n}$ , then divide by $r$ .

Difficulty: 3/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph is a straight line passing through the origin because the payment is directly proportional to the principal. For a finance student, this means that borrowing a larger principal requires a proportionally higher payment, while a smaller principal results in a smaller payment. The most important feature is that the linear relationship means doubling the principal always results in a doubling of the payment, provided the interest rate and number of periods remain constant.

Graph type: linear

Why it behaves this way

Intuition

A financial picture of a constant stream of payments, where early payments are mostly interest and later payments are mostly principal, gradually reducing the loan balance to zero over time.

PMT

The constant periodic payment required to fully repay the loan.

This is the fixed amount you pay each month (or period) that covers both the interest due and a portion of the principal, ensuring the loan balance eventually reaches zero.

PV

The initial principal amount of the loan (Present Value).

The total amount of money borrowed at the start, which needs to be amortized over the loan term.

r

The interest rate per payment period.

The cost of borrowing for each period, expressed as a decimal. A higher 'r' means more interest accrues, increasing the payment needed to clear the loan in the same time.

n

The total number of payment periods over the loan's life.

The total count of individual payments that will be made. A larger 'n' (longer loan term) generally leads to smaller individual payments but more total interest paid over time.

Signs and relationships

r: The interest rate 'r' in the numerator directly scales the payment. A higher 'r' means a higher cost of borrowing per period, thus requiring a proportionally larger payment to cover the interest and amortize the
1-(1+r)^{-n}: This entire term represents the present value interest factor of an ordinary annuity. It aggregates the discounted values of a series of future payments of $1.

Free study cues

Insight

Canonical usage

The payment (PMT) and present value (PV) must be in the same currency unit. The periodic interest rate (r) and the number of periods (n)

Common confusion

The most common mistake is using an annual interest rate (APR) directly for 'r' without converting it to the periodic rate that matches 'n', or using 'r' as a percentage instead of a decimal.

Dimension note

The terms 'r' and 'n' are dimensionless quantities representing a rate per period and a count of periods, respectively. The ratio PMT/PV is also dimensionless, ensuring that PMT and PV must be expressed in the same

Unit systems

$P M T$ Currency (e.g., USD, EUR) · The fixed periodic payment amount. Must be in the same currency as PV.

$P V$ Currency (e.g., USD, EUR) · The present value or principal amount of the loan. Must be in the same currency as PMT.

$r$ dimensionless · The periodic interest rate, expressed as a decimal (e.g., 0.005 for 0.5% per period). It must match the period of 'n'.

$n$ dimensionless · The total number of payment periods. It must match the period of 'r'.

One free problem

Practice Problem

A homeowner takes out a 30-year mortgage for 300,000 at an annual interest rate of 4.8%. If payments are made monthly, what is the monthly payment (PMT)?

Principal (Present Value)300000 $

Rate per Period0.004

Number of Payments360

Solve for: $P M T$

Hint: Convert the 4.8% annual rate to a monthly rate by dividing by 12, and the 30-year term to 360 months.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Monthly mortgage payment on a fixed-rate loan.

Study smarter

Tips

Ensure the interest rate (r) and number of periods (n) match the payment frequency (e.g., monthly).
Always convert percentage interest rates to decimals (e.g., 5% becomes 0.05) before dividing by periods.
The denominator 1 - (1 + r)⁻ⁿ represents the present value factor of an ordinary annuity.

Avoid these traps

Common Mistakes

Using APR as r without dividing by payment frequency.
Using years for n when payments are monthly.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

A fixed loan payment is found by treating repayments as an annuity whose present value equals the loan principal.

This formula is applied when calculating monthly payments for fixed-rate installment products like mortgages, car loans, and personal loans. It assumes that interest is calculated on the declining balance and that the interest rate remains unchanged throughout the life of the loan.

It allows borrowers to budget with certainty by knowing the exact amount required to clear a debt by the end of its term. For financial institutions, it structures the recovery of principal while ensuring the time value of money is compensated via interest.

Using APR as r without dividing by payment frequency. Using years for n when payments are monthly.