The debt-to-income math lenders use
US underwriters start with two ratios. The front-end ratio is your housing payment divided by gross monthly income, traditionally capped near 28%. The back-end ratio adds every other monthly debt the credit report shows - car loans, student loans, credit card minimums, personal loans, child support - and is the one that usually binds. The old rule of thumb is 36%; the Consumer Financial Protection Bureau's Qualified Mortgage standard generally caps it at 43%, and Fannie Mae and Freddie Mac automated systems will approve up to about 50% when credit, reserves and down payment are strong.
The housing payment in the ratio is PITI: principal, interest, property taxes and insurance, plus HOA dues and mortgage insurance if they apply. This calculator works backwards from that. It takes your income times the DTI limit, subtracts your other debts, subtracts taxes and insurance, and converts what is left into a loan balance using the amortization factor r x (1 + r)^n / [(1 + r)^n - 1].
A worked example
Take $8,000 of gross monthly income, $500 in other debt payments, a 36% limit, a 6.5% rate on a 30-year fixed, 20% down, $300 a month in property tax and $125 for insurance.
Total debt allowed is $8,000 x 0.36 = $2,880. Subtract the $500 of car and card payments and $2,380 is left for housing. Take out $425 of taxes and insurance and $1,955 remains for principal and interest. The payment factor at 6.5% over 360 months is 0.0063207, so the maximum loan is $1,955 / 0.0063207 = about $309,300. With 20% down, the price that loan supports is $309,300 / 0.80 = roughly $386,600, needing about $77,300 in cash for the down payment.
Raise the limit to 43% and the same income supports about $497,000 in price. Clearing the $500 car payment is worth roughly $98,900 of additional price at 36%, which is often the fastest way to move the number.
What the approval number leaves out
Lenders do not look at your life. The ratio ignores income taxes, retirement contributions, childcare, health premiums, commuting and groceries, and it uses gross income, which for most households is 25% to 35% above take-home pay. A payment at the top of the approved range can be technically affordable and practically miserable.
Budget separately for costs the calculator only approximates. Maintenance runs roughly 1% to 2% of the home's value per year. Closing costs typically add 2% to 5% of the price on top of the down payment. Property tax is reassessed after a sale in many states, so the seller's tax bill may understate yours. And if you put less than 20% down, private mortgage insurance adds roughly 0.3% to 1.5% of the loan per year until you reach 20% equity - enter it in the insurance box to see the effect.