How we calculate

Every number on loan.tools comes from the formulas below. No black boxes.

Monthly payment (principal & interest)

The standard fixed-rate amortization formula:

M = P · r / (1 − (1 + r)⁻ⁿ)

P = amount borrowed · r = annual rate ÷ 1200 (monthly rate as a decimal) · n = term in months. If the rate is 0%, the payment is simply P ÷ n.

The amortization schedule

Each month, interest accrues on the remaining balance first; the rest of the payment reduces principal:

interest_m = balance_m−1 · r  principal_m = M − interest_m  balance_m = balance_m−1 − principal_m

This is why early payments are mostly interest: the balance is largest at the start. The crossover marked on our charts is the first month where principal_m ≥ interest_m.

Extra payments

Monthly, yearly, and one-time extra payments are added to principal_m in the month they occur. The payment M stays the same, so the loan ends early; the interest saved is the difference between the two schedules' total interest.

PMI (private mortgage insurance)

Applied when the down payment is under 20%. Monthly PMI is the annual PMI rate times the original loan amount, divided by 12. It's charged until the balance reaches 80% of the purchase price:

PMI/month = loan · rate_PMI / 12, charged while balance > 0.80 · price

Lenders differ on exact removal rules (78% automatic termination, appraisal-based removal). We use the 80%-of-original-price convention.

Property tax & insurance estimates

The "typical estimates" button uses 1.10% of the home price per year for property tax and 0.35% for homeowners insurance. ZIP lookups map to state-level averages (California uses 1.15%, the effective rate for a new purchase under Prop 13, because the state average misleads buyers). County rates vary — verify yours.

Rent vs Buy

A cumulative 30-year comparison, whatever the loan term. The cost of renting grows your rent by the chosen rate each year; if you invest the down payment instead of buying, those gains offset rent. The cost of home ownership counts every dollar that doesn't come back — interest, property tax, insurance, HOA, PMI, upkeep (a percentage of home value per year), plus 3% of the price at closing — and credits back what a sale would return: down payment, principal repaid, and appreciation, less 6% selling costs.

ownership_m = interest + taxes · ins · fees + upkeep + 3% · price − (equity_m − 6% · value_m)

Under appreciation, upkeep scales with home value and property tax grows with it, capped at 2%/yr (California's assessment cap, the strictest common one). Break-even is the last month the lead changes; with two compounding forces it can flip more than once. Not modeled: investing monthly cash-flow differences, and tax effects (deductions, capital gains).

Refinance break-even

break-even = first month m where closing costs + Σ new payments_1..m ≤ Σ old payments_1..m

Each sum stops when its loan is paid off, so the savings stream ends when the loans do. If the refinance never catches up before both loans end, the answer is "never", even when the new payment is lower. (Dividing closing costs by the monthly saving assumes the saving lasts forever; with two years left on a loan it can report a break-even month long after both loans are gone.) With no closing costs, a lower payment is ahead from the first month. Lifetime comparison uses total interest remaining on the current loan versus total interest on the new loan plus closing costs.

Adjustable-rate mortgages (ARMs)

An ARM holds its starting rate for a fixed period (7 years on a 7/6), then resets: every 6 months on a 7/6, every year on a 7/1. At each reset the new rate is the index plus the margin, rounded to the nearest 0.125 point, then held inside the caps:

fully indexed = round_1/8(index + margin)  rate_k = clamp(fully indexed, rate_k−1 ± cap_k) within [margin, start + lifetime cap]

index = 30-day average SOFR · cap_k = first-reset cap at the first reset, later-reset cap after that · start + lifetime cap = the rate ceiling. The floor is assumed at the margin; some notes name another.

On every reset date the payment is recalculated on the balance and the months left, whether or not the rate moved, so extra payments lower it there. We draw two paths. Index holds: the index stays at today's value, and the rate steps toward the fully indexed rate as fast as the caps allow. Rates rise to the caps: the rate rises by the full cap at every reset until it reaches the ceiling. It is the most the note allows, not a forecast.

Against a fixed loan we compare the same loan, term, and extra payment, month by month: the ARM costs more from the first month its running interest passes the fixed loan's. For an ARM you already hold, refinancing comes out ahead from the first month the new loan's running interest plus closing costs drops below the ARM's. Refinancing now instead of at the reset costs the interest gap over the months before the reset, assuming the same fixed rate and closing costs would be on offer then.

Default caps are 2/1/5 (first reset, later resets, lifetime) on the 6-month types and 2/2/5 on the 1-year types; many 7/6 and 10/6 notes allow up to 5 points at the first reset. Your Loan Estimate's Adjustable Interest Rate table lists yours. The mortgage page's starting rate defaults to the fixed rate less 0.66 points, the 2005 to 2022 average 5/1-ARM discount (Freddie Mac ended that series in 2022), so no ARM quote here is live. Not modeled: interest-only periods, negative amortization, payment caps, index lookback, conversion options, prepayment penalties.

Auto loans

Most states tax the price minus your trade-in; manufacturer incentives reduce the amount financed but are usually still taxed:

tax = max(price − trade-in value, 0) · rate_tax  financed = price + tax − down − (trade-in value − owed on trade-in) − incentives

Only your equity in the trade reduces the loan: a $20,000 trade with $15,000 still owed takes $5,000 off. If you owe more than the trade is worth, the difference (negative equity) is added to the new loan. The sales-tax credit still uses the full trade-in value, because that is what states discount.

A few states tax the full price or cap the trade-in credit (CA, HI, VA, and DC give no credit; MI caps it at $12,000). The auto calculator applies each state's rule.

Auto leases & money factor

A lease payment is depreciation plus a finance charge:

payment = (cap − residual)/n  +  (cap + residual) · MF

cap = negotiated (capitalized) cost after reductions · residual = MSRP × residual % · MF = money factor. Approximate APR = MF × 2400. Dealers often quote "2.5" to mean 0.00250. Working backwards from a quoted payment: MF = (payment − depreciation) / (cap + residual).

The lease-quote card on the auto page does exactly that with a dealer's quote, then prices the markup: the dealer's implied money factor minus the bank's buy rate, times (cap + residual), times the number of months. Residual is held to 0–100% of MSRP and the term to at least one month. A quoted payment below pure depreciation is shown as a subsidized lease rather than a negative rate.

Rates

Live rates are the Freddie Mac Primary Mortgage Market Survey weekly averages (30-year and 15-year fixed), fetched from FRED. They're national survey averages, not offers. The ARM index is the 30-day average SOFR (FRED SOFR30DAYAVG), refreshed every 12 hours.

How we keep it honest

An automated test suite checks the exact code these pages run against published example loans and closed-form results — payment amounts, total interest, PMI drop-off months, extra-payment payoffs, lease money factors — on every change.

Rounding: we display dollars rounded to the nearest dollar; schedules compute in full precision. Your lender's numbers may differ slightly due to payment dates, escrow, and fees. This is math, not financial advice.