Mortgage Math: Payments, Qualification, and Interest-Only Traps

A mortgage is the largest financial contract most people ever sign, and its math has three distinct layers: the payment formula, the qualification math that decides how much house you can buy, and the loan structures that can quietly undermine both. Understanding all three — especially the third — is what separates a mortgage you forget about from one that ambushes you.

The payment is PITI, not just principal and interest

The famous mortgage formula prices only the loan itself:

M = P × r(1 + r)^n / ((1 + r)^n − 1)

with r the monthly rate (annual ÷ 12) and n the number of monthly payments. But the check you actually write is PITI: principal, interest, property taxes, and insurance — homeowner's coverage, plus private mortgage insurance if your down payment is under 20%. Taxes and insurance are collected into escrow and ride on top of the formula; they do not amortize away. A quote given as principal-and-interest-only can look hundreds of dollars smaller than the payment that actually leaves your account. (The amortization machinery itself is covered in how loan interest works; making it vanish early is the subject of mortgage payoff explained.)

A worked example: $400,000 at 6.5% for 30 years

r = 0.065 / 12 = 0.0054167,   n = 360
M = 400,000 × 0.0054167 × (1.0054167)^360 / ((1.0054167)^360 − 1)
  = $2,528.27 per month (principal + interest)

360 payments of $2,528.27 total $910,178 — so the interest is about $510,178 on a $400,000 loan. You repay roughly 2.3 times what you borrowed; the interest alone exceeds the principal. That is not a scam, it is what a 6.5% rate on a slowly-falling 30-year balance produces. Small rate changes compound into large differences: at 7.5% the same loan costs $2,796.86 per month — about $269 more every month and roughly $97,000 more interest over the term.

Qualification: the debt-to-income gate

Lenders qualify you with debt-to-income (DTI) ratios computed on gross monthly income. The classic guideline is 28/36: housing costs (PITI) under about 28% of gross, and all monthly debt payments combined under about 36%. The binding constraint is the tighter of the two:

housing cap  = gross monthly × 0.28
all-debt cap = gross monthly × 0.36 − other debt payments
max PITI     = min(housing cap, all-debt cap)

Example: a $90,000 salary is $7,500 gross monthly. The housing cap is $2,100. The all-debt cap is $2,700 minus a $400 car payment and a $150 student loan, i.e. $2,150 — so the 28% rule binds at $2,100. Then invert the payment formula to find the loan that payment supports: set aside, say, $350 for taxes and insurance, leaving $1,750 of principal and interest — which at 6.5% for 30 years supports roughly a $277,000 loan. Modern programs sometimes tolerate DTIs into the 40s, but 28/36 remains the conservative planning baseline, and the Loan Qualifier Calculator runs this inversion for you.

The interest-only version looks cheaper

An interest-only (IO) loan requires only the interest accrual each month; the principal never moves. On the same $400,000 at 6.5%:

IO payment:      400,000 × 0.065 / 12 = $2,166.67
amortized:       $2,528.27
apparent saving: $361.60 per month

That $361 looks like free cash flow. The trap is the reset. IO periods are temporary — typically 5 to 10 years — after which the loan must amortize its full remaining balance over the shortened remainder of the term. After ten years of paying interest only, our $400,000 balance amortizes over just 20 years:

reset payment = 400,000 × r(1 + r)^240 / ((1 + r)^240 − 1) = $2,982.29

The payment jumps upward by $454 — about 18% — the moment the IO period ends, and a decade of payments built zero equity. If the property has fallen in value or your income has not grown, refinancing out of the reset may not be available. A nastier variant is negative amortization: when the allowed payment is below the interest accrual, the unpaid interest is added to the balance and the debt grows while you keep paying.

Who interest-only actually suits

IO is a legitimate tool for borrowers with lumpy incomes or a disciplined plan to retire principal somewhere else — but only if that plan is real. If the plan is "the property will appreciate," the loan has become a leveraged bet with a scheduled payment shock attached. The honest test: could you afford the reset payment today, on today's income? If not, the affordability is borrowed, not real.

Try it yourself

Model the full picture with the Mortgage Calculator, then check what income supports it with the Loan Qualifier Calculator. Put the IO and amortized structures side by side in the Interest-Only Loan Calculator to see the reset arrive on the schedule, and watch where every payment lands in the Amortization Schedule Generator.