Loan Amortization Monthly Payment
Loan Amortization Monthly Payment
Anyone taking out a mortgage, car loan, or personal loan runs into this formula, directly or indirectly, the moment they ask "what will my monthly payment be?" It converts a lump-sum borrowed amount into a level payment that, over the life of the loan, pays off both the principal and all the accrued interest exactly at the final payment.Small changes in the interest rate or the loan term move the monthly payment more than most borrowers expect, because the formula compounds the rate over every payment period — which is why shopping for even a fraction of a percentage point of interest rate, or choosing a 15-year over a 30-year term, makes a meaningful difference to both the monthly payment and the total interest paid.
The level monthly payment is M = P·r(1+r)^n/((1+r)^n − 1), the standard amortization formula. where P_loan is the loan principal, r_mo is the monthly interest rate (as a decimal), n_pay is the total number of monthly payments, and M_pay is the resulting fixed monthly payment, all in dollars (this engine has no currency unit, so amounts are entered as plain numbers).
Inputs
P_loan := 250000=250000
r_mo := 0.005=5e-03
n_pay := 360=360
Apply the standard amortization formula: the (1+r)^n growth factor compounds the rate over every payment, spreading both principal and interest into one level payment for the whole term.
M_pay := P_loan·(r_mo·(1+r_mo)^n_pay)/((1+r_mo)^n_pay - 1)=1498.88
Results
On a $250,000 loan at 6% annual interest over 30 years, the level payment comes out to roughly $1,500 per month — a useful sanity-check figure against typical mortgage payment size for a loan this size. Over the full 360 payments the borrower pays substantially more than the original principal in total, almost entirely because of compounding interest in the early years when the outstanding balance is still large; shortening the term or lowering the rate both reduce that total interest meaningfully.
Assumptions & limitations
- Fixed interest rate for the life of the loan.
- Level (fully amortizing) monthly payments, no balloon payment.
- No fees, points, taxes, or insurance included.
- Monetary amounts are plain numbers (dollars implied), since no currency unit exists in this engine.