Mortgage Solver: Find the Rate, Term, Amount or Payment
Input three loan components to calculate the missing value.
| Debt Component | Input | Output |
|---|---|---|
| Loan Amount The principal advanced at closing. | | - |
| Interest Rate The nominal annual rate. Quoted to three decimals because lenders price in eighths and sixteenths. | | - |
| Amortization (Years) The schedule the payment is calculated on, which is often longer than the term of the loan itself. | | - |
| Monthly P&I Principal and interest only — no taxes, insurance or reserves. Leave one box empty and it is the one solved. | | - |
Monthly P&I is the missing component, solved from the three above it.
How it is calculated
Payment = P × [ r(1 + r)ⁿ ] ÷ [ (1 + r)ⁿ − 1 ]
One relationship ties four numbers together — the principal, the periodic rate, the number of payments and the payment itself — so any three of them fix the fourth. Three of those rearrangements are ordinary algebra. The fourth, solving for the rate, has no closed form at all: the rate appears both in the numerator and inside an exponent, and no amount of rearranging isolates it. That one is found by search rather than by formula.
- P
- Principal — the loan amount advanced at closing.
- r
- Periodic interest rate: the annual rate divided by twelve.
- n
- Total payments — amortization years multiplied by twelve.
- P&I
- Principal and interest. Taxes, insurance and reserves are not included.
A worked deal example
A $2,450,000 loan at 6.750% on a 30-year amortization, paying $15,890.65 a month. Here the monthly payment was left blank, so it is the one solved.
- Loan Amount
- $2,450,000
- Interest Rate
- 6.750%
- Amortization
- 30 years
- Monthly P&I
- $15,890.65
- Solved For
- the monthly payment
The three given components fix the payment exactly: $15,890.65 a month retires $2,450,000 over 30 years. Across the full schedule the borrower pays $5,720,635, of which $3,270,635 is interest — 57% of everything paid. More than half of every dollar goes to interest rather than principal, which is what a long amortization at this rate costs.
Frequently asked questions
How do I find the interest rate from a loan payment?
How much can I borrow for a given monthly payment?
How long will it take to pay off a loan with a fixed payment?
The solver says my payment never retires the principal. What does that mean?
Amortization or term — which do I enter?
The four components, and which way each one solves
A level-payment loan is one relationship between four quantities. Fix any three and the fourth is determined — but the four are not equally easy to extract, and the difference is worth understanding because it explains why one of them occasionally refuses to answer.
- Payment is the direct form. Multiply the principal by the annuity factor and you are done.
- Loan amount is the same identity divided rather than multiplied. Also exact, also instant.
- Amortization comes out through logarithms. Exact, with one real precondition: the payment has to exceed one period's interest, or the balance grows and no term exists.
- Interest rate has no formula at all: the rate appears both as a multiplier and inside an exponent, so no rearrangement isolates it. The tool finds it by narrowing a range from 0% to 100% until the payment matches to the cent, which always lands on the answer if one exists.
Related calculators
Estimates only. This calculator is an educational tool, not financial advice, a loan offer, or a commitment to lend. Actual payments, loan sizing and closing costs depend on your lender's underwriting, the day-count convention in your loan documents, and fees not modeled here. Confirm any figure with your lender or advisor before acting on it. For an official reference, see Buying a House: tools and resources (Consumer Financial Protection Bureau).