Mortgage formula reference
Every formula this site computes with, stated so you can check our arithmetic against your own. Each one carries a worked example in real numbers and — the part most references skip — a sentence on where it stops being the right tool.
The payment itself
What you send the servicer each month, and what it is made of.
Monthly principal and interest
- M
- Monthly payment of principal and interest
- P
- Principal — the amount borrowed, not the purchase price
- r
- Monthly interest rate: the annual rate divided by 12, as a decimal
- n
- Total number of payments: years × 12
The standard amortizing payment. It is the constant monthly amount that exactly retires the principal over n payments while paying interest on the balance outstanding in each one. Every fixed-rate mortgage payment in the United States comes from this expression.
Worked example
$320,000 borrowed at 6.5% for 30 years
M = $2,022.62 per month
Where it stops applying
This is principal and interest only. Property tax, homeowners insurance, mortgage insurance and HOA dues are added on top and are usually collected into escrow, which is why a quoted 'payment' is often far below what leaves your account.
Payment at a zero rate
- M
- Monthly payment
- P
- Principal
- n
- Number of payments
The general formula divides by zero when r = 0, because (1 + 0)ⁿ − 1 = 0. At a zero rate there is no interest to compound, so the principal simply spreads evenly across the term. Worth stating because 0% promotional financing exists, and a calculator that returns NaN for it is broken.
Worked example
$120,000 at 0% for 10 years
M = $1,000.00 per month
Where it stops applying
A genuinely zero-rate mortgage is rare outside employer and nonprofit programmes. If a quoted rate is 0% but there are large fees, the APR is not zero — see the APR formula below.
How the balance moves
Amortization: why early payments barely dent the principal.
Splitting a payment into interest and principal
- B
- Balance at the start of the month
- r
- Monthly interest rate
- M
- Monthly payment
- B′
- Balance carried into the next month
Interest is charged on what you still owe, so it falls a little every month as the balance drops. The payment is fixed, so the principal portion rises by exactly the same amount. Repeating this for n months is the amortization schedule.
Worked example
First payment on $320,000 at 6.5%, M = $2,022.62
86% of the first payment is interest
Where it stops applying
This is why extra principal is worth so much more early than late: every dollar you take off the balance now avoids interest in all 359 remaining months. The same dollar in the final year avoids almost nothing.
Balance remaining after k payments
- B(k)
- Balance after k payments have been made
- P
- Original principal
- r
- Monthly interest rate
- n
- Total payments in the term
- k
- Payments made so far
A closed form for the balance at any point, without stepping through the schedule month by month. Useful for finding when a loan crosses a loan-to-value threshold — which is exactly how PMI cancellation dates are computed.
Worked example
$320,000 at 6.5% for 30 years, after 5 years (k = 60)
B(60) = $299,555 — after five years, 6.4% of the principal is gone
Where it stops applying
Assumes every payment was made exactly on schedule with no extra principal. Any overpayment makes the real balance lower than this predicts.
Ratios lenders underwrite against
The arithmetic behind an approval or a decline.
Loan-to-value
- B
- Current loan balance
- V₀
- Original value: the lesser of purchase price and original appraised value
The proportion of the property still financed. It governs pricing, whether mortgage insurance is required, and when that insurance ends.
Worked example
Balance of $299,555 on a home bought for $400,000
LTV = 74.9% — below 78%, so PMI has already terminated automatically
Where it stops applying
For PMI cancellation the denominator is the ORIGINAL value, fixed by statute, not what the home is worth today. An appreciating market does not move the statutory thresholds — though many servicers will cancel on a current appraisal under their own policy.
Debt-to-income, both ratios
- H
- Total monthly housing payment: principal, interest, tax, insurance, mortgage insurance, HOA
- D
- All other monthly debt payments on your credit report
- I
- Gross monthly income, before tax
The two ratios an underwriter computes. The classic guideline is 28% and 36%; agency-backed conventional loans commonly stretch the back-end ratio to 45%, and to 50% with compensating factors.
Worked example
$8,500 gross monthly income, $2,400 housing, $650 other debt
Back-end inside the 36% guideline; front-end just over 28%
Where it stops applying
Income means documented, stable income. Bonus, commission and self-employment earnings normally require a two-year history and are averaged. Childcare, utilities and groceries do not count as debt, however large they are.
What it really costs
Fees restated as rates, and insurance that ends.
Annual percentage rate
- i
- Monthly APR, the value being solved for
- M
- Monthly payment, fixed by the note rate
- P
- Loan amount
- F
- Finance charges: origination, points, underwriting, mortgage insurance
- n
- Number of payments
APR restates the cost of borrowing as a single rate by treating the fees you paid to obtain the loan as though they were extra interest. It is the discount rate at which the payment stream equals the money that actually reached you. There is no closed form; it is solved numerically.
Worked example
$320,000 at 6.5% for 30 years with $4,400 of finance charges
APR = 6.634% — the fees add 0.134 points
Where it stops applying
APR assumes you keep the loan for the full term. Most borrowers do not, and leaving early means the fee was spread over far fewer payments than the disclosure assumed. It also excludes third-party costs you may shop for, so two lenders can quote the same APR and charge different amounts at closing.
When PMI reaches the cancellation threshold
- B(k)
- Scheduled balance after k payments
- V₀
- Original value
- k
- The month being solved for
The Homeowners Protection Act gives two rights against the original value: you may request cancellation at 80% LTV, and the servicer must cancel automatically at 78%. Both are found by stepping the balance formula until it crosses the threshold.
Worked example
$400,000 home, 10% down, 6.5%, 30 years
Month 95 to request cancellation, month 109 for automatic termination
Where it stops applying
Applies to conventional loans only. FHA MIP on a loan with less than 10% down has no cancellation right at all and lasts the life of the loan — escaping it requires refinancing.
Deciding between options
Break-even points, where the answer is a date rather than a number.
Refinance break-even
- C
- Total closing costs of the refinance
- M_old
- Current monthly payment
- M_new
- Payment on the new loan
How long the monthly saving takes to repay what the refinance cost. Below that horizon you have lost money; beyond it you are ahead.
Worked example
$6,500 closing costs, payment falls from $2,332 to $2,098
28 months to break even
Where it stops applying
This is necessary but not sufficient. A refinance that restarts a 30-year term can lower the payment and still raise total interest — compare lifetime cost as well as the break-even month, particularly when refinancing a loan you are years into.
Discount point break-even
- points
- Points paid, as a decimal — one point is 1% of the loan
- P
- Loan amount
- M_base
- Payment at the quoted rate
- M_bought
- Payment at the bought-down rate
Buying the rate down costs cash now for a lower payment later. The break-even is when the accumulated saving repays the points.
Worked example
2 points on $320,000 to drop 6.5% to 6.0%
62 months — a little over five years
Where it stops applying
Only worth it if you will hold the loan past the break-even. Points are also the reason APR and note rate diverge, so a lender quoting an unusually low rate may simply have assumed you are buying it down.
Checking these yourself
Every figure in the worked examples above is produced by the engine that runs the calculators, and pinned by a test — so an example here cannot quietly drift from what the site actually computes. The constants each formula consumes are published with their sources on the methodology page, downloadable as CSV and JSON, and callable through the REST API if you would rather compare against a running implementation than a table.
Free to reproduce, including in teaching material. Attribution to usmortgagecalc.com is appreciated.