Mortgage Calculator — monthly payment with PITI, PMI & amortization

A mortgage payment is built from four components — principal, interest, taxes, and insurance — known together as PITI. Lenders quote a single monthly figure to make budgeting easy, but each piece is computed separately and behaves differently over the life of the loan.

The principal-and-interest portion (P&I) is a fixed-rate, fully-amortising annuity. Every month the lender charges interest on the outstanding balance, and whatever is left of the monthly payment pays the balance down. The closed form drops out of the present-value-of-annuity identity in finance:

P&I = L × r × (1 + r)^n
      ─────────────────
        (1 + r)^n − 1

L is the loan amount (home price − down payment), r is the monthly periodic rate (annual rate ÷ 12), and n is the total number of monthly payments (term in years × 12). The same payment repeats every month, but the split between interest and principal shifts: early months are almost all interest, late months are almost all principal.

Property tax and homeowners insurance are added as flat amounts — the annual figure divided by 12 — because most U.S. lenders escrow both. PMI (Private Mortgage Insurance) is added when the down payment is less than 20% of the home price. The Homeowners Protection Act of 1998 (HPA), 12 U.S.C. § 4901, requires the lender to automatically drop PMI the month the projected balance falls to 78% of the original home price, and to drop it at borrower request at 80%. The calculator uses the 78% automatic threshold so the schedule reflects what the lender is legally required to do.

The amortization is then built one month at a time: interest portion = balance × r; principal portion = P&I − interest + any extra principal payment; closing balance = opening balance − principal. A sub-unit floating-point residue is snapped onto the final month so the loan closes at exactly zero.

Bi-weekly mode simulates 26 half-payments per year — equivalent to 13 monthly payments — by applying P&I ÷ 12 of extra principal every month. That produces the same payoff date as a true bi-weekly schedule held by a servicer that applies funds when the full monthly P&I has accumulated. The parallel “no-extras” simulation lets the calculator quote how many months and how much interest you save.

Every P&I is cross-checked against the present-value-of-annuity identity P = PMT × (1 − (1 + r)^−n) / r. If the re-derived principal does not match the loan amount to within a unit, the “PV-identity verified” badge above will not light up — a quick safety check against arithmetic drift.

• CFPB — When can I remove PMI from my loan? (78% LTV auto-termination)
• U.S. Homeowners Protection Act of 1998 — 12 U.S.C. § 4901 et seq.
• Investopedia — How to Calculate the Mortgage Payment
• U.S. Federal Reserve — Consumer Credit G.19 (periodic-rate amortization)

The amortization formula is universal across lenders. PMI rules referenced here are U.S.-specific (Homeowners Protection Act of 1998); other countries use different lender-insurance schemes. Cross-checked on 2026-05-11.