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Car Finance Interest Rate Calculator

Back out the rate hidden in a car payment with AI-powered step-by-step solutions
What APR turns $24,000 into a $470 payment over 60 months?
Total interest on $24,000 at 6.54% over 60 months
Convert a 3.5% add-on quote on $24,000 over 5 years to an APR
Payment difference between 5% and 9% on $24,000 over 60 months

The Rate Is Locked Inside the Payment Formula

Every car loan ties four numbers together:

M=A⋅r(1+r)n(1+r)n−1M = A\cdot\frac{r(1+r)^n}{(1+r)^n - 1}

  • AA — amount financed: price plus tax and fees, minus down payment and net trade equity
  • MM — monthly payment
  • nn — number of months
  • rr — the monthly rate; the quoted APR is 12r12r

Three of the four determine the fourth — except for rr, which cannot be isolated algebraically. The equation is a polynomial of degree nn in rr, so the rate is found numerically: guess, compute the payment, and adjust.

That matters because rr is exactly the number a dealer's worksheet can leave out. A payment, a term and an amount financed are usually all visible, and those three pin the rate down completely. If a quoted payment implies a higher rate than the one on the paperwork, some cost has been rolled into AA without being named.

Interest is charged on the outstanding balance, so the total cost is simply

total interest=nM−A\text{total interest} = nM - A

Solving for the Rate, and Reading Add-On Quotes

Bisection, by hand

  1. Compute MM at a low guess and a high guess that bracket the real payment.
  2. Interpolate: r≈r1+(r2−r1)M−M1M2−M1r \approx r_1 + (r_2 - r_1)\dfrac{M - M_1}{M_2 - M_1}.
  3. Recompute at that rate and repeat. Two rounds gets within a basis point or two.

The payment rises smoothly with the rate, so the interpolation always converges from either side.

Add-on interest is not an APR

Some quotes compute interest as

I=A⋅radd-on⋅t,M=A+InI = A \cdot r_{\text{add-on}} \cdot t, \qquad M = \frac{A + I}{n}

charging the full rate on the original amount for the whole term, even though the balance is falling. The equivalent APR is close to double the add-on rate on a level-payment loan.

Rate sensitivity

∂M∂r>0 always\frac{\partial M}{\partial r} > 0 \text{ always}

but the effect on total interest grows with the term: on a 60-month loan each extra percentage point costs roughly 2.5%2.5\% of the amount financed in total interest.

Advertised rates depend on the lender, the term, and credit assessment, and they change constantly. This page works only from the numbers you enter — it does not quote or recommend financing.

Common Mistakes to Avoid

  • Dividing total interest by the amount to get a rate: 4,200/24,000=17.5%4{,}200/24{,}000 = 17.5\% over five years is 3.5%3.5\% a year add-on, not the 6.54%6.54\% APR. The two are not comparable.
  • Using the annual rate as rr: the formula wants the monthly rate, APR/12\text{APR}/12.
  • Comparing payments across different terms: a lower payment over 72 months can carry a higher rate and far more interest than a higher payment over 48.
  • Forgetting what is inside AA: rolled-in fees, negative equity and add-on products all enlarge AA, so the payment rises even at an unchanged rate.
  • Assuming a longer term is free: at a fixed rate, nM−AnM - A grows with nn because the balance is outstanding longer.
  • Confusing APR with the interest rate on a loan with fees: APR includes finance charges; the note rate does not. Ask which figure a quote refers to.

示例题目

Step 1: Try 6.0%6.0\%: r=0.005r = 0.005, (1.005)60≈1.348850(1.005)^{60} \approx 1.348850, M \approx \463.99$ — too low
Step 2: Try 7.0%7.0\%: r≈0.0058333r \approx 0.0058333, M \approx \475.23$ — too high
Step 3: Interpolate: 6+1×470−463.99475.23−463.99=6+6.0111.24≈6.535%6 + 1 \times \dfrac{470 - 463.99}{475.23 - 463.99} = 6 + \dfrac{6.01}{11.24} \approx 6.535\%
Step 4: Check at 6.5367%6.5367\%: r≈0.00544723r \approx 0.00544723, M \approx \470.00$
Step 5: Verify the present value: 470 \times \dfrac{1 - (1.00544723)^{-60}}{0.00544723} \approx \24{,}000$
Answer: About 6.54%6.54\% APR (r≈0.0054472r \approx 0.0054472 per month)

Step 1: Total paid: 60 \times 470 = \28{,}200$
Step 2: Total interest: 28{,}200 - 24{,}000 = \4{,}200$
Step 3: Add-on form: I=A⋅radd-on⋅t=24,000×r×5I = A \cdot r_{\text{add-on}} \cdot t = 24{,}000 \times r \times 5
Step 4: Set 120,000 r=4,200120{,}000\,r = 4{,}200, so radd-on=0.035=3.5%r_{\text{add-on}} = 0.035 = 3.5\%
Step 5: Payment check: (24{,}000 + 4{,}200)/60 = 28{,}200/60 = \470$ — identical
Answer: \4{,}200ofinterest;a′ of interest; a '3.5%add−on′quoteisthesameloanasadd-on' quote is the same loan as6.54%$ APR

Step 1: At 5%5\%: r≈0.0041667r \approx 0.0041667, (1+r)60≈1.283359(1+r)^{60} \approx 1.283359, M \approx \452.91$
Step 2: Total interest: 60 \times 452.91 - 24{,}000 \approx \3{,}174.58$
Step 3: At 9%9\%: r=0.0075r = 0.0075, (1.0075)60≈1.565681(1.0075)^{60} \approx 1.565681, M \approx \498.20$
Step 4: Total interest: 60 \times 498.20 - 24{,}000 \approx \5{,}892.03$
Step 5: Difference: \45.29amonthanda month and5{,}892.03 - 3{,}174.58 = $2{,}717.45$ over the term
Answer: About \45.29morepermonthandmore per month and$2{,}717moreinterest—roughlymore interest — roughly$679$ per percentage point

常见问题

Solve M = A·r(1+r)^n/((1+r)^n − 1) for r. It cannot be rearranged algebraically, so bracket it: compute the payment at two trial rates, interpolate, and repeat. Two iterations land within a basis point. The APR is 12r.

Add-on interest charges the rate on the original amount for the whole term (I = A·r·t) and ignores that the balance falls. On a level-payment loan the equivalent APR is close to double: a 3.5% add-on over five years is about 6.54% APR.

Total interest = nM − A, the sum of every payment minus the amount financed. On $24,000 at $470 for 60 months that is 28,200 − 24,000 = $4,200. It assumes every scheduled payment is made; paying early reduces it.

Not usually — longer terms are typically priced higher, and even at an identical rate the total interest rises because the balance is outstanding longer. Compare deals on amount financed, rate and total interest, not on the monthly payment alone.

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