Skip to content

Auto Loan Payment and Vehicle Price Formulas

Review auto loan formulas for amount financed, monthly payment, trade-in equity, taxes, fees, total interest, amortization, and reverse vehicle price.

How do the Auto Loan Calculator formulas work?

This formula page defines the variables used by the Auto Loan Calculator, shows how those values are combined, and explains the order of operations behind the result. Use the worked steps to verify an estimate, compare scenarios, and understand which assumptions or rounding rules can affect the displayed answer.

Formula Overview

The calculation has two layers. The purchase layer determines trade-in equity, sales tax, upfront cash, and amount financed. The loan layer converts principal, rate, and term into a fixed payment and amortization schedule.

In reverse mode, the loan layer is solved first from the target payment. The purchase adjustments are then reversed to estimate a vehicle price.

Variable Definitions

SymbolMeaningUnit or convention
VVehicle priceCurrency
CCash incentiveCurrency
DDown paymentCurrency
TTrade-in valueCurrency
OAmount owed on trade-inCurrency
eTrade-in equity, T - OCurrency
sEntered sales-tax rateDecimal rate
FEntered feesCurrency
PAmount financedCurrency
iMonthly interest rateAnnual rate / 1,200
nNumber of monthly paymentsMonths
MMonthly principal-and-interest paymentCurrency

Amount Financed

Trade-in equity and estimated sales tax are:

e=T−Oe=T-O
Tax=VsTax=Vs

When taxes and fees are paid upfront:

P=max⁡(0,V−C−D−e)P=\max(0,V-C-D-e)

When taxes and fees are financed:

P=max⁡(0,V−C−D−e+Tax+F)P=\max(0,V-C-D-e+Tax+F)

Negative equity means e is negative. Subtracting a negative value increases principal.

Fixed Monthly Payment

M=Pi1−(1+i)−nM=P\frac{i}{1-(1+i)^{-n}}

For a zero interest rate, the calculator uses M = P / n. The current engine carries full numeric precision through the schedule and formats displayed currency to cents.

Monthly Amortization

It=Bt−1iI_t=B_{t-1}i
Qt=min⁡(Bt−1,M−It)Q_t=\min(B_{t-1},M-I_t)
Bt=max⁡(0,Bt−1−Qt)B_t=\max(0,B_{t-1}-Q_t)

It is monthly interest, Qt is principal, and Bt is the ending balance. The final principal payment is capped at the remaining balance.

Reverse Vehicle Price

The principal supported by a target payment is:

P=M1−(1+i)−niP=M\frac{1-(1+i)^{-n}}{i}

When tax and fees are financed, the price is solved from the same purchase equation:

V=max⁡(0,P+C+D+e−F1+s)V=\max\left(0,\frac{P+C+D+e-F}{1+s}\right)

When tax and fees are paid upfront, V = max(0, P + C + D + e). The reverse result estimates a price; it does not guarantee lender approval or show whether the overall transportation budget is affordable.

Total Interest, Upfront Cash, and Total Cost

Payments=MnPayments=Mn
Interest=Payments−PInterest=Payments-P
Total cost=max⁡(0,V−C)+Interest+Tax+FTotal\ cost=\max(0,V-C)+Interest+Tax+F

When tax and fees are paid upfront, upfront cash equals the down payment plus positive trade equity plus tax and fees. When they are financed, the displayed upfront amount contains the down payment plus positive trade equity. Negative equity is handled in principal rather than as positive upfront value.

Worked Calculation

For a $50,000 vehicle, $10,000 down payment, 60 months, 5% rate, 7.2% tax, and $2,800 of fees paid upfront:

  1. Trade-in equity and incentive are zero.
  2. Amount financed: $50,000 - $10,000 = $40,000.00.
  3. Monthly rate: 5 / 1,200 = 0.0041666667.
  4. The payment formula produces $754.85.
  5. Total scheduled payments are $45,290.96.
  6. Total interest is $5,290.96.
  7. Estimated sales tax is $3,600.00.
  8. Upfront cash is $16,400.00, including the down payment, tax, and entered fees.
  9. Total modeled cost is $61,690.96.

Assumptions and Exclusions

  • The loan uses a fixed annual rate and monthly payments.
  • The entered rate is used directly in the payment formula.
  • The entered sales-tax rate is applied to the modeled vehicle price.
  • The state selection is descriptive and does not override the entered tax rate.
  • The model does not automatically apply trade-in tax credits, rebate rules, exemptions, or jurisdiction-specific fees.
  • Insurance, fuel, maintenance, repairs, depreciation, optional products not entered, late fees, and lender-specific daily-interest methods are excluded.
  • The reverse calculation is not an approval or affordability decision.

Formula FAQ

Why does negative equity increase the amount financed?

The existing payoff exceeds the trade value, so the shortfall must be covered by cash or incorporated into the new transaction. For a useful Auto Loan Calculator formula result, use realistic inputs, compare relevant scenarios, and review the stated assumptions before making a decision.

Why do financed taxes and fees cost more?

They increase principal, so interest is charged on them over the loan term. For a useful Auto Loan Calculator formula result, enter realistic values, compare relevant scenarios, check the calculation details, and review the definitions, assumptions, and limitations before using the estimate for an important decision.

Why can the reverse price differ from a dealer quote?

The calculator uses only the entered adjustments and tax rate. Dealer quotes can include different taxable amounts, fees, add-ons, and credit terms. Use realistic inputs and review the Auto Loan Calculator formula assumptions before relying on this result for decisions.

Does the formula calculate APR?

No. It uses the annual rate entered in the payment field. Use the lender's disclosure for APR and finance charge. For a useful Auto Loan Calculator formula result, use realistic inputs, compare relevant scenarios, and review the stated assumptions before making a decision.

References

Use the worked auto loan examples to reproduce the formulas with complete scenarios.

Use the Auto Loan Calculator