Auto Loan Amortization: Mathematical Framework

An auto loan is a fixed-rate, fully-amortizing instalment obligation secured against the vehicle title. The net financed principal $ L $ equals the vehicle's transaction price $ P $ minus the borrower's down payment $ D $:

\[ L = P - D \]

Let $ r = R / 1200 $ denote the monthly periodic rate, where $ R $ is the nominal annual rate expressed as a percentage, and $ n = 12N $ the total number of monthly payment periods over a term of $ N $ years. Setting the present value of the annuity cash-flow stream equal to $ L $ and solving for the monthly payment $ M $:

\[ L = \sum_{k=1}^{n} \frac{M}{(1+r)^k} = M \cdot \frac{1-(1+r)^{-n}}{r} \implies M = (P - D) \cdot \frac{r(1+r)^n}{(1+r)^n - 1} \]

The factor $ r(1+r)^n / [(1+r)^n - 1] $ is the mortgage constant — the fraction of principal remitted per period to fully retire the debt over $ n $ periods at periodic rate $ r $. It is a strictly increasing function of $ r $ and a strictly decreasing function of $ n $, encoding the fundamental tension between rate cost and term-driven payment reduction.

Total Interest Cost and Financing Efficiency

The total amount disbursed over the loan's contractual life is $ T = nM $. The total interest charge is:

\[ I_{\text{total}} = nM - (P - D) \]

For the reference scenario — vehicle price $35,000, down payment $5,000, $ R = 6.9\% $, $ N = 5 $ years — the net principal $ L = \$30{,}000 $, $ r \approx 0.005750 $, $ n = 60 $, yielding $ M \approx \$593.10 $ and $ I_{\text{total}} \approx \$5{,}586 $, representing 18.6% of the financed principal — a direct consequence of the compound interest structure of the fixed annuity.

Outstanding Balance and Negative Equity Risk

The outstanding loan balance after $ k $ monthly payments satisfies the closed-form recurrence solution:

\[ B_k = L(1+r)^k - M \cdot \frac{(1+r)^k - 1}{r} \]

This satisfies the boundary conditions $ B_0 = L $ and $ B_n = 0 $. Simultaneously, vehicle market value follows an exponential depreciation model: if $ d $ is the annual depreciation rate, the residual value after $ t $ years is $ V(t) = P(1-d)^t $. Negative equity occurs at period $ k $ when $ B_k > V(k/12) $ — a condition most acute in months 6–18 of a 72- or 84-month loan with a minimal down payment, as the interest-heavy early amortization retards principal paydown while the vehicle's depreciation curve is steepest during its first operational year.

Sensitivity Analysis: Rate, Term, and Down Payment

Rate sensitivity: A 100 basis-point increase in $ R $ on a $30,000 / 60-month loan raises $ M $ by approximately $14/month and $ I_{\text{total}} $ by approximately $840 over the life of the loan.
Term sensitivity: Extending from 60 to 72 months reduces $ M $ by approximately $78 but increases $ I_{\text{total}} $ by $1,200 while widening the negative-equity exposure window by 12 months.
Down payment sensitivity: Each additional $1,000 in down payment reduces $ M $ linearly by the scalar annuity factor $ r(1+r)^n / [(1+r)^n - 1] \approx \$19.77 \text{ / month} $ at the reference parameters — a linear relationship that holds exactly due to the multiplicative structure of the EMI formula.

APR vs. Nominal Rate: Regulatory Distinction

The nominal contract rate $ R $ is legally distinct from the Annual Percentage Rate (APR) mandated under Regulation Z (Truth in Lending Act, 15 U.S.C. § 1601). The APR incorporates origination fees, documentation charges, and mandatory add-on insurance products into a single effective cost figure. Substituting $ r_{\text{APR}} = R_{\text{APR}} / 1200 $ for $ r $ in the payment formula yields the true all-in monthly financing cost. The effective $ R_{\text{APR}} $ is solved numerically by finding the internal rate that equates net loan proceeds (principal minus all upfront fees) with the present value of all contractual payments, enabling standardised cross-lender comparison on a legally mandated, uniform basis.