Loan Amortization Payment

Core idea

Overview

The Loan Amortization Payment formula determines the constant payment amount needed to repay a loan, including both principal and interest, over a specified duration. It is fundamental in personal and corporate finance for structuring loans like mortgages, car loans, and student loans. This formula ensures that by the end of the loan term, the entire principal balance and all accrued interest are paid off.

When to use: Use this equation when you need to determine the regular payment amount for a fully amortizing loan, given the principal amount, the periodic interest rate, and the total number of payment periods. It's crucial for budgeting and understanding the financial commitment of a loan.

Why it matters: This formula is vital for financial planning, allowing borrowers to understand their monthly obligations and lenders to structure loan products. It underpins the calculation of mortgage payments, car loan installments, and other forms of debt, enabling individuals and businesses to manage their cash flow effectively and make informed borrowing decisions.

Symbols

Variables

P = Principal Loan Amount, r = Periodic Interest Rate, n = Total Number of Payments, PMT = Periodic Payment

P

Principal Loan Amount

$

r

Periodic Interest Rate

%

n

Total Number of Payments

payments

PMT

Periodic Payment

$

Walkthrough

Derivation

Formula: Loan Amortization Payment

The loan amortization payment formula calculates the constant periodic payment required to fully repay a loan over its term.

Payments are made at regular intervals (e.g., monthly, quarterly).
The interest rate is constant over the life of the loan.
Payments are made at the end of each period (ordinary annuity).
The loan is fully amortized, meaning the principal and interest are fully paid off by the end of the term.

1

Start with the Present Value of an Ordinary Annuity:

The principal amount of a loan (P) is equivalent to the present value of all future periodic payments (PMT), discounted at the periodic interest rate (r) over the total number of periods (n). This is the standard formula for the present value of an ordinary annuity.

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

2

Isolate PMT:

To find the periodic payment (PMT), we rearrange the present value of an annuity formula by multiplying both sides by 'r' and dividing by '(1 - (1+r)^-n)'. This isolates PMT on one side of the equation.

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

Result

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

Source: Brealey, Myers, and Allen, Principles of Corporate Finance, 13th Edition, McGraw-Hill Education

Free formulas

Rearrangements

Solve for $P$

Loan Amortization Payment: Make P the subject

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

To make $P$ (Principal Loan Amount) the subject, multiply both sides of the formula by the term representing the present value factor of an annuity.

Difficulty: 2/5

Solve for $r$

Loan Amortization Payment: Make r the subject

No direct algebraic solution for r

Making $r$ (Periodic Interest Rate) the subject of the loan amortization formula is not possible through direct algebraic manipulation and typically requires numerical methods.

Difficulty: 4/5

Solve for $n$

Loan Amortization Payment: Make n the subject

n = - \frac{ln ( 1 - \frac{P \cdot r}{P M T} )}{ln ( 1 + r )}

To make $n$ (Total Number of Payments) the subject, rearrange the formula to isolate the exponential term and then use logarithms.

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 principal amount is directly proportional to the periodic payment. For a student of finance, this means that borrowing a larger principal requires a proportionally higher payment, while a smaller principal results in a lower, more manageable commitment. The most important feature is that the linear relationship means doubling the principal amount exactly doubles the periodic payment required to amortize the loan.

Graph type: linear

Why it behaves this way

Intuition

Visualize a timeline where a series of identical, equally spaced payments are made, each payment comprising both interest and principal, such that by the final payment, the entire initial loan amount and all accrued

PMT

The fixed amount of money paid by the borrower to the lender at regular intervals.

This is the recurring financial obligation; it's what you budget for each period.

P

The initial sum of money borrowed or the principal amount of the loan.

This is the total debt you start with before any interest or payments.

r

The interest rate applied per payment period.

This is the cost of borrowing per period; a higher 'r' means more interest accrues, increasing the payment.

n

The total number of payment periods over the entire life of the loan.

This is how many times you'll make a payment; more periods generally mean smaller individual payments but more total interest paid.

Signs and relationships

(1+r)^-n: The negative exponent signifies that future payments are being discounted back to their present value. It represents the present value of a single dollar received 'n' periods in the future.
1 - (1+r)^{-n}: This entire term forms the present value interest factor of an annuity (PVIFA). It represents the present value of a series of 'n' future payments of $1, each discounted by 'r'.

Free study cues

Insight

Canonical usage

Ensures consistency in currency units for principal and payment, and in time periods for the periodic interest rate and total number of periods.

Common confusion

The most frequent error is using an annual interest rate (often given as a percentage) directly in the formula without converting it to the correct periodic decimal rate (e.g., monthly, quarterly)

Dimension note

The periodic interest rate 'r' and the number of payment periods 'n' are dimensionless quantities. 'r' is a ratio representing interest per principal per period, and 'n' is a count of periods.

Unit systems

PMTCurrency (e.g., USD, EUR) - The calculated fixed periodic payment amount. Its currency unit will match that of the principal (P).

$P$ Currency (e.g., USD, EUR) - The principal loan amount. Its currency unit must be consistent with the expected PMT unit.

$r$ dimensionless - The periodic interest rate, expressed as a decimal. For example, a 6% annual rate paid monthly would be 0.06/12 = 0.005. The period (e.g., monthly, quarterly, annually) must match the period used for 'n'.

$n$ dimensionless - The total number of payment periods. This must correspond to the period used for 'r'. For example, a 30-year loan with monthly payments would have n = 30 * 12 = 360 periods.

One free problem

Practice Problem

A student takes out a loan of $20,000 to be repaid over 5 years with monthly payments. The annual interest rate is 6%, compounded monthly. What is the monthly payment amount?

Principal Loan Amount20000 $

Periodic Interest Rate0.005 %

Total Number of Payments60 payments

Solve for: PMT

Hint: Ensure the interest rate and number of periods are consistent with monthly payments.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

In the monthly payment for a 30-year fixed-rate mortgage, Loan Amortization Payment is used to calculate Periodic Payment from Principal Loan Amount, Periodic Interest Rate, and Total Number of Payments. The result matters because it helps compare incentives, policy effects, market outcomes, or financial decisions in context.

Study smarter

Tips

Ensure the interest rate 'r' and the number of periods 'n' are consistent with the payment frequency (e.g., if payments are monthly, 'r' should be the monthly rate and 'n' the total number of months).
Convert annual interest rates to periodic rates by dividing by the number of compounding periods per year (e.g., annual rate / 12 for monthly payments).
The formula assumes payments are made at the end of each period (ordinary annuity).
Be mindful of rounding intermediate calculations, as small errors can accumulate over many periods.

Avoid these traps

Common Mistakes

Using an annual interest rate for 'r' when payments are monthly or quarterly, instead of converting it to the periodic rate.
Incorrectly calculating 'n' (total number of periods) by not multiplying the loan term by the number of payments per year.
Confusing the formula for an ordinary annuity with an annuity due.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

The loan amortization payment formula calculates the constant periodic payment required to fully repay a loan over its term.

Use this equation when you need to determine the regular payment amount for a fully amortizing loan, given the principal amount, the periodic interest rate, and the total number of payment periods. It's crucial for budgeting and understanding the financial commitment of a loan.

This formula is vital for financial planning, allowing borrowers to understand their monthly obligations and lenders to structure loan products. It underpins the calculation of mortgage payments, car loan installments, and other forms of debt, enabling individuals and businesses to manage their cash flow effectively and make informed borrowing decisions.

Using an annual interest rate for 'r' when payments are monthly or quarterly, instead of converting it to the periodic rate. Incorrectly calculating 'n' (total number of periods) by not multiplying the loan term by the number of payments per year. Confusing the formula for an ordinary annuity with an annuity due.

In the monthly payment for a 30-year fixed-rate mortgage, Loan Amortization Payment is used to calculate Periodic Payment from Principal Loan Amount, Periodic Interest Rate, and Total Number of Payments. The result matters because it helps compare incentives, policy effects, market outcomes, or financial decisions in context.

Ensure the interest rate 'r' and the number of periods 'n' are consistent with the payment frequency (e.g., if payments are monthly, 'r' should be the monthly rate and 'n' the total number of months). Convert annual interest rates to periodic rates by dividing by the number of compounding periods per year (e.g., annual rate / 12 for monthly payments). The formula assumes payments are made at the end of each period (ordinary annuity). Be mindful of rounding intermediate calculations, as small errors can accumulate over many periods.

Yes. Open the Loan Amortization Payment equation in the Equation Encyclopedia app, then tap "Copy Excel Template" or "Copy Sheets Template".

References

Sources

Brealey, Richard A., Myers, Stewart C., and Allen, Franklin. Principles of Corporate Finance.
Brigham, Eugene F., and Ehrhardt, Michael C. Financial Management: Theory & Practice.
Wikipedia: Amortization (business)
Brealey, R. A., Myers, S. C., & Allen, F. (2020). Principles of Corporate Finance (13th ed.). McGraw-Hill Education.
Brealey, Myers, and Allen, Principles of Corporate Finance, 13th Edition, McGraw-Hill Education

Overview

Variables

Derivation

Start with the Present Value of an Ordinary Annuity:

Isolate PMT:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Present Value of an Ordinary Annuity

Future Value of a Lump Sum

Present Value of a Lump Sum

Frequently Asked Questions

Sources