Formula & Calculator
Loan/Mortgage Monthly Payment
Calculates the fixed monthly payment required to fully pay off a loan over its term, given principal, rate, and number of payments.
Interpretation
M = P · [r(1+r)^n] / [(1+r)^n − 1]. Monthly payment for a fixed‑rate loan or mortgage. P principal, r monthly rate, n number of payments. Used in real estate and personal finance.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| M | Monthly payment | currency |
| P | Loan principal | currency |
| r | Monthly interest rate (annual rate/12) | |
| n | Total number of monthly payments |
What it means
This formula calculates the fixed monthly payment for a fully amortizing loan (e.g., mortgage, car loan) where the interest rate is constant and the loan is paid off over a fixed term. The payment is derived from the present value of an annuity. It ensures that each payment covers interest and a portion of principal, resulting in a zero balance after n payments. This calculation is essential for borrowers to understand affordability and for lenders to structure loans. It is also used in reverse to determine the maximum loan amount for a given payment. Understanding this formula is fundamental for personal financial planning and for real estate investment analysis.
Worked example
Loan Monthly Payment – Two Detailed Examples
Real‑World| Parameter | Value |
|---|---|
| P | 300000 |
| Annual Rate | 6.5% |
| n (months) | 360 |
| Parameter | Value |
|---|---|
| P | 25000 |
| Annual Rate | 7% |
| n | 60 |
Common mistakes
- Monthly payment M: The fixed payment amount for each period.
- Principal P: The loan amount.
- Monthly interest rate r: Annual rate divided by 12 – in decimal form.
- Number of payments n: Total number of monthly payments (loan term in years × 12).
- Sign: Payments are typically outflows – but the formula gives the positive payment amount.
- Ensure r and n are consistent (both monthly).
Applications
The loan/mortgage monthly payment formula, M = P·[r(1+r)^n]/[(1+r)^n − 1], determines the fixed periodic payment required to fully amortise a loan over a given term. This is the basis for most home mortgages, auto loans, and student loans. Banks, brokers, and consumers use it to understand affordability, to compare loan offers, and to plan budgets. By adjusting the interest rate, term, or principal, borrowers can see the effect on monthly payments. This formula also helps in refinancing decisions and in evaluating the total cost of borrowing. Understanding it is essential for personal financial management and for professionals in banking, real estate, and financial planning.
- Mortgage payment calculation for home buyers
- Auto loan and personal loan budgeting
- Loan comparison and refinancing analysis
- Amortisation schedule preparation
- Financial literacy and consumer protection education
Frequently Asked Questions
M = P * (r*(1+r)^n) / ((1+r)^n - 1). This calculates the equal monthly payment required to fully amortize a loan over a fixed term, given the principal P, monthly interest rate r, and number of payments n.
Divide the annual percentage rate (APR) by 12: r = (APR / 100) / 12. For example, 6% APR → monthly rate = 0.06/12 = 0.005.
Amortization refers to the process of paying off a loan with regular payments over time. Each payment covers both interest and a portion of the principal, gradually reducing the balance to zero by the end of the term.
A longer term (larger n) reduces the monthly payment but increases the total interest paid over the life of the loan. A shorter term increases the payment but saves on total interest.
Total interest = (M × n) − P. This is the sum of all payments minus the original principal. It can be significantly more than the principal for long‑term loans.
Use the outstanding balance formula: B = P*(1+r)^p - M*(((1+r)^p - 1)/r), where p is the number of payments made. This tells you how much principal is still owed.
- Using the annual rate directly instead of dividing by 12.
- Forgetting to convert the number of years to months (n = years × 12).
- Ignoring that the payment is due at the end of each period (ordinary annuity).
It is derived from the present value of an annuity formula: PV = PMT × (1 − (1+r)^-n)/r. Setting PV = loan amount and solving for PMT gives the payment formula.
For variable‑rate loans, the monthly payment may change when the rate resets. You would need to recalculate M using the new rate and the remaining balance and term to determine the new payment.
Making an extra payment directly reduces the principal balance, which reduces future interest and can shorten the loan term. The savings depend on when the extra payment is made; earlier payments have a larger impact.