How every number is worked out

Money in whole cents

Every amount is held as a whole number of cents, and every rate as a whole number of millionths of a percent. There are no decimal fractions anywhere money passes, so nothing drifts by a cent from one screen to the next.

What you type is read exactly. An amount with more than cents, or a rate with more than six decimals, is refused rather than rounded, so the engine never works with a number you didn’t type.

Rounding

Rounding happens in one place in the code, at a short list of points, and everything in between is kept exact. Each tax line, each fee line, each month’s interest and the monthly payment is rounded once, to the cent. Rounding to the nearest cent, a half cent goes up.

The monthly payment is rounded to the nearest cent unless you pick otherwise: lenders differ, so the loan details let you round it up or down instead. Here is what each choice does, worked by the engine’s own rounding:

In every state the calculator has rules for, each tax line and each fee line is rounded to the nearest cent on its own, never as one total, with ten exceptions the state’s law makes. New York and Washington round a deal’s whole tax once. Pennsylvania rounds its state sales tax by the Department of Revenue’s bracket schedule, and each local tax (Philadelphia’s and Allegheny County’s) to the cent with a half cent going down. Idaho rounds its sales tax by the State Tax Commission’s bracket schedule, six cents a whole dollar plus a set amount for the cents. Delaware rounds its motor vehicle document fee to the nearest whole dollar, and Oklahoma its motor vehicle excise tax. New Hampshire rounds its town permit fee, Iowa its registration fee, and Minnesota its registration tax, to the nearest whole dollar. Michigan rounds its registration tax to the nearest whole dollar with half a dollar going down, which matters when a first registration runs longer or shorter than a year.

The monthly payment

The sheet uses the standard monthly method. The monthly rate is the APR ÷ 12, kept exact. The payment is the one that repays the loan in equal months:

payment = A × i ÷ (1 − (1 + i)⁻ⁿ)

A is the amount financed, i the monthly rate and n the number of months. At a rate of zero it is simply A ÷ n.

Then the schedule is walked month by month. Each month is charged interest on what is still owed, rounded to the cent. The payment pays that interest first, and the rest pays down the loan. The last payment is whatever clears what is left, so it can be a few cents more or less than the others.

If a payment is ever less than the interest due, which can happen after a long wait for the first payment, the unpaid interest is set aside and paid first from the next payment. It is never added to the loan and never charged interest of its own.

An example

A new car at $32,700.00, $5,000.00 down, financed at 5.90% for 60 months, bought from a dealer and registered in Harris County. The engine finances $30,159.75 at 5.90% for 60 months: 59 payments of $581.67, then a last payment of $581.77. The first three months and the last two:

Interest is the sum of the schedule’s months. Here that is $4,740.55, which is also exactly the total of payments ($34,900.30) less the amount financed. The shortcut of payment × months less the loan gives $4,740.45 here, because it treats the last payment as equal to the others. The sheet never uses it.

The first payment date, and daily interest

With no first payment date, the first payment is a month after signing. If you type the date from the contract, the time to it is counted in whole months and odd days, each odd day a thirtieth of a month, and that first stretch is charged simple interest on the amount financed.

You then choose how the extra or missing days are handled. The default works the payment out again so the loan still clears in even payments. You can keep the regular payment and let the last payment take the difference, or keep it and add the extra days’ interest to the first payment.

With a first payment date you can also switch a deal to daily interest. Interest then builds up each day on a 365-day year, from signing to each due date, on what is still owed. The balance is never rounded along the way: the payment is rounded as above, and the last payment, which clears what is left, to the nearest cent. Unpaid interest is set aside the same way as above.

The APR

The APR is worked out separately from the rate you type, from the actual payments and dates, so a long first period or a rounded payment shows up in it. It uses the method that charges interest the way the schedule does (Regulation Z calls it the U.S. Rule). In the example, with a regular first month, it is 5.900%, the rate typed.

The engine also works the APR by Appendix J’s actuarial method, which treats unpaid interest as if it earned interest too. The loan details show it beside the APR, and a note on the loan points it out when the two differ by more than an eighth of a point, because a contract may print either.

An APR is cut, not rounded, to the millionth of a percent, then shown to two or three decimals. Cutting first means the figure you see is always rounded correctly from the true one.

The out-the-door price and the amount financed

The out-the-door price is the price, add-ons, doc fee, taxes and government fees. It comes before rebates and the trade-in, the way a buyer’s order shows it, because those are money toward the deal, like cash down.

The amount financed is the out-the-door price less rebates applied at signing, the trade-in’s equity (its allowance less what is still owed on it), cash down, and any charge paid separately at a state office. Negative equity, owing more than the trade-in is worth, adds to it. A rebate mailed to you after signing doesn’t lower it.

Total paid is cash down plus every payment, plus anything paid separately at an office, less any rebate paid to you later. Paying cash, it is the out-the-door price less rebates and the trade-in’s equity. Deals on the sheet are compared on total paid, so different down payments, terms and trade-ins compare fairly.

When something a figure needs is missing, such as the county where the car will be registered, the total says what it leaves out. It is never filled with a guess.

Every line of the example

Cash down pays the tax and fees first

The sheet counts your cash down against the taxes, government fees and doc fee first, and only what is left against the price. Whatever cash doesn’t cover is shown as added to the loan. This is how the sheet explains the loan; it doesn’t change the amount financed.

In the example, $5,000.00 down covers $2,459.75 of the $2,459.75 in tax, fees and the doc fee ($0.00 of them are left in the loan), and the other $2,540.25 lowers the price.

Each state’s rules

Each state’s tax is worked from that state’s own rules: what is taxed, at what rate, and what lowers the amount taxed. The rules are data, each part citing the research rows it rests on, and the engine picks the version in force on the deal’s date. These are the switches that change the answer most, read from the rules in force today:

“In some deals” means the rule depends on something about the deal, such as whether the car is new or how the doc fee is written on the order. The engine applies it from the deal’s details, and shows any assumption it makes.

The payment check

“Does your payment add up?” works backwards from a quoted payment. From the loan you agreed, the rate and the term, it works the payment those numbers need by the standard monthly method. Then it works how much loan the quoted payment can repay at that rate and term. The difference between the two loans, in dollars, is the headline. It also works the rate the quote implies over your term, and the term it implies at your rate.

Some differences have ordinary reasons, and the check allows for them. Rounding: what one cent of payment can repay, rounded up. The first payment date: interest for the days more or less than a month to the first payment. With a date, that is worked from the date, and the check also works the payment on daily interest for those dates. Without one, the check allows for the longest and shortest first periods a lender may treat as a regular month, which depend on the term:

The check then reads where the difference falls: within rounding, it matches; within the first-payment allowances, the date explains it (or, with no date, could); beyond them, the payment carries more, or less, than the agreed numbers need, and the check says by how much beyond the allowances. It reports a reading, never a verdict about the dealer.

An example

The example loan above, quoted at $601.67 a month with no first payment date: This payment carries $1,036.95 more than your agreed numbers need.

The same two readings of one gap: more in the loan at the agreed rate, or a higher rate on the agreed loan. The check shows both and doesn’t choose between them.

What you can afford

Tax comes in slices, bands and brackets, so there is no formula to run backwards from a monthly budget. Instead the engine searches whole cents of price, each try worked as the full deal with your state’s tax and fees, trade-in and cash down, for the highest price whose regular payment fits the budget. Put that price back into the deal and it pays the budget or less; one cent more pays more. The search stops at the largest price the calculator takes.

With the example deal’s own payment, $581.67, as the budget, the answer is $32,700.19. At that price the payment is $581.67; a cent more on the price makes it $581.68. The answer can sit a few cents above the deal’s own price, because a payment rounded to the cent doesn’t move for every cent of price.

Where the data comes from

Every tax rate, fee and doc-fee figure starts as a research row: the value, the government or statute page it came from, and the day someone last checked it there. The rules the engine reads cite those rows, and every default on the sheet shows its source and checked date. Local rates and fees come from each agency’s own tables, kept with the day each was pulled.

Two federal figures are read too:

• Banks’ average rate on a 60-month new-car loan (Federal Reserve, G.19), measured August 2026: shown beside the rate box as a yardstick, and the rate a loan deal is worked on, marked assumed, until you type your own: 7.54%. Next checked by Dec 7, 2026.

  federalreserve.gov · checked Nov 11, 2026

• The Truth in Lending threshold for the year: above it, the lender may not have to show an APR box, and the calculator says so: $73,400.00. Next checked by Jan 1, 2027.

  federalregister.gov · checked Oct 2, 2026

When a loan deal has a price but no rate and no term typed, the calculator works it on Federal Reserve G.19 figures (the banks’ average rate on a 60-month new-car loan, and the common term nearest the finance companies’ average maturity on new-car loans), marks each one “assumed” with its source and date, and stops using it the moment you type your own.

How often it is checked again

Each local table and each figure that changes on a known date carries the day it is due to be checked again, usually the day its agency publishes the next one: most local rate tables every quarter, fees on the dates the law moves them, the federal rate on each release. An automated check fails as soon as a re-check date passes, or a row the rules cite is more than a year old, and the fix is always to read the source again.

When a figure changes, the new value goes in as a new dated version beside the old one, so a deal is worked with the rules in force on its own date. Nothing before Oct 1, 2026 is loaded: a deal dated earlier is worked from figures you type.

States worked from what you type

A state is worked from its rules only while it is verified, and verified is a test, not a judgment: its rules reproduce the research’s worked examples, every row it cites is current, its local tables cover every county or place, and every unconfirmed row has been reviewed. A state that stops passing comes off the list until it is refreshed. Today all 51 (DC counts as one) are worked from their rules.

A state that comes off the list, and any deal dated before the rules begin, are worked from the tax rate and government fees you type, labelled as not yet verified. The typed rate is applied to the price, doc fee and add-ons, less the trade-in allowance where you say it lowers the tax (that switch starts on), never below zero. Rebates don’t lower it.

What the calculator leaves out

• Leases. A deal is paid with a loan or in cash.
• Private sales. The calculator works a purchase from a dealer, and assumes the dealer is in the state where the car will be registered.
• Vehicles other than passenger cars and light trucks, such as motorcycles, RVs and commercial vehicles.
• A tax credit for selling your old car yourself, special cases such as new residents or military buyers, and specialty plates.
• Charges billed after you buy, such as a yearly property tax, are listed under the total but never added to it. Charges a state has that the total doesn’t include are listed under it too.
• Late-payment penalties, and limits on the rate a lender may charge. The calculator works the rate you type.