Debt
Loan Payment Formula
The fixed monthly payment required to fully amortize a loan over its term.
Formula
M = P × [r(1+r)^n] / [(1+r)^n − 1]
Variables
M- Monthly payment
P- Loan principal
r- Monthly interest rate
n- Number of payments
Worked Example
Scenario: $300,000 mortgage, 6% annual rate, 30 years: monthly rate = 0.06/12 = 0.005, n = 360; payment ≈ $1,799.
Steps:
- Convert annual rate to monthly: 6% ÷ 12 = 0.5% = 0.005
- Calculate number of payments: 30 × 12 = 360
- Calculate (1 + r)^n = 1.005^360 ≈ 6.023
- Apply numerator: 0.005 × 6.023 = 0.03012
- Apply denominator: 6.023 − 1 = 5.023
- Divide and multiply by principal: $300,000 × (0.03012 ÷ 5.023) ≈ $1,799
When to Use This Formula
- Calculate mortgage or loan payment
- Compare loan offers with different terms
- Plan debt repayment budget
Limitations & Assumptions
- Assumes fixed rate and regular payments
- Does not include taxes, insurance, or maintenance
- Ignores early payoff or additional payments
Common Questions
Why does extending the term reduce the payment?
The principal is spread over more periods, so each payment is smaller (but total interest increases).
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- APR formulaFormula
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- How expensive is this loan really?Decision
Reference note: This is an educational mathematical reference. The formula's applicability depends on assumptions, market conditions, and use case. Always verify results independently and consult domain experts for decisions.