How this instrument works
A loan amount is, mathematically, the present value of every payment still owed, each one discounted back to today at the loan's periodic rate. The payment formula is that present-value-of-an-annuity relationship run in reverse: instead of asking what a stream of future payments is worth today, it asks what level payment, repeated N times at rate r, is worth exactly L today. Solve that equation for the payment and the familiar L · r(1+r)^N ⁄ ((1+r)^N − 1) shape falls out.
This version treats the loan amount as something to test rather than a fact already fixed by a specific transaction — useful for someone who has not chosen a home yet and wants to compare what a $250,000 loan costs monthly against a $300,000 one, or how a rate a half-point higher moves the number, before any purchase price or down payment enters the picture. Three fields, one answer, meant to be rerun rather than read once. The payment shown is principal and interest only; a PITI-style calculator is the place to fold in tax, insurance, and HOA dues once a specific loan shape is worth pricing out fully.
- Enter the loan amount, $ you want to test — a target figure, not necessarily tied to a specific home yet.
- Set the annual interest rate, % you are comparing — a lender's quote or a rate you are curious about.
- Choose the loan term, years, such as 30, 20, or 15.
- Read the monthly payment (principal & interest), $ — the level amount the formula produces for those three numbers.
- Change one field at a time and re-read the payment to see which variable moves the number the most.
Worked example — $300,000 at 7% over 30 years
Set the loan amount to $300,000, the annual interest rate to 7%, and the term to 30 years. The periodic rate works out to r = 7 ⁄ 1200 = 0.0058333, and the term becomes N = 30 × 12 = 360 payments. Running L, r, and N through the formula returns PMT = $1,995.91 a month — principal and interest only, with nothing added for tax, insurance, or PMI.
Hold the rate and term fixed and drop the loan amount to $250,000 instead, and the payment falls in the same proportion, to $1,663.26 — the formula is linear in L, so a one-sixth smaller loan produces a payment exactly one-sixth smaller too. Change the term to 15 years instead and that proportionality disappears: the payment rises to $2,696.48, more than simple scaling would suggest, because a shorter term also changes how many times the balance compounds. That contrast is why the three fields are worth testing one at a time rather than all at once.
Questions
Where does the payment formula actually come from?
It comes from solving the present-value-of-an-annuity equation for the payment instead of the value: the loan amount is set equal to the sum of every future payment discounted back at the periodic rate, and the algebra is rearranged so the payment sits by itself on one side. The result is the same formula used for retirement annuities and lease payments, just applied to a borrowed balance instead of a saved one.
Should I enter the interest rate or the APR my lender quoted?
Enter the interest rate — the figure this formula is built on. APR is a separate, usually slightly higher number that spreads certain closing costs and fees across the loan term to approximate a fuller borrowing cost; feeding an APR into this formula in place of the note rate overstates the actual monthly payment, sometimes by a noticeable amount on a loan with heavy up-front fees.
Do I enter the home price here, or the amount I am borrowing?
The amount borrowed — if a purchase price and down payment are already decided, subtract one from the other first. This calculator is deliberately agnostic about where that number comes from: it works the same whether it is a real balance from a signed contract or a round figure you are testing while deciding how much house to even look for.
Why does the payment not include property tax or insurance?
Because those figures depend on where the home is and which insurer quotes it, and mixing a guess into an otherwise exact number would make the whole result less reliable, not more useful. Keeping this page to principal and interest only also keeps it fast to rerun for several loan amounts or rates in a row; a companion PITI-style calculator holds the tax and insurance line items once a single loan shape has been chosen.
Does doubling the loan amount double the monthly payment?
Yes, exactly — the formula is linear in the loan amount, so twice the balance at the same rate and term produces exactly twice the payment. Changing the rate or the term does not behave this simply, because both sit inside the (1+r)^N part of the formula; equal-sized changes to either one move the payment by different, and usually larger, amounts than the same-sized change to the loan amount does.
What happens if I enter a 0% interest rate?
The calculator will not return a payment — the rate has to be greater than zero for this particular formula to work, since dividing by ((1+r)^N − 1) breaks down once r reaches zero. A true zero-interest loan simply divides the loan amount evenly across the number of payments, which is a different, much simpler calculation than the one this instrument runs.
References
- CFPB — Understand your mortgage loan options
- Federal Reserve — Consumer's guide to mortgage settlement costs
Read this first: This instrument shows arithmetic, not advice. Real offers add fees, taxes and terms that vary by lender and place — verify the figures against your actual paperwork before deciding anything.