# How do I calculate APR from monthly instalments?

I have been given a quote for a car loan in Thailand. I want to calculate the APR but am not sure how. (All of the tools and explanations I’ve seen are going in the other direction, from a percentage to calculating monthly payment.) The salesman quoted 2.6% but I couldn’t ascertain exactly what that figure represents. Language was a barrier here.

For this plan, the total amount borrowed for the car is 511,200 baht over five years. The monthly payment is 9,628 baht, which means I would pay a total interest of 66,480 baht on top of the 511,200.

I’ve arrived at a percentage rate of about 5% by trial and error with an amortisation loan spreadsheet, but would like to know how it’s calculated properly.

With monthly rate `r` and

``````principal   s = 511200
payments    d = 9628
no. months  n = 5*12
``````

Equating net present values

Solving for `r`

``````s = (d - d (1 + r)^-n)/r

∴ r = 0.00409911
``````

Nominal interest compounded monthly is `12*r = 4.91893 %`

Effective annual interest is `(1 + r)^12 - 1 = 5.03136 %`

The former is APR in the US and the latter is APR in Europe.

• As gnasher729 notes `(d*n/s - 1)/5 = 2.6 %` but that's a strange way to calculate interest. – Chris Degnen Jun 1 '19 at 13:26
• I've actually been present when a car dealer offered 10% "APR" for an £8,000 loan over five years with £12,000 repayment. Obviously, the 10% was NOT APR. APR was about 20%. I still regret not calling trading standards. – gnasher729 Jun 1 '19 at 16:24

Excel and LibreOffice Calc have a function for this: RATE(). Running your numbers through it says that the rate is 4.92%.

My calculation ended up with quite exactly 5.04%, using the rules that would be applied in Europe.

This is how APR would be calculated: You start with 511,200. Every month, 9,628 is subtracted from your debt (and no interest added at this point). Then, after twelve months, the interest is calculated. You owed 511,200 for the first month, 501572 for the second month, and so on. You add the 12 numbers, divide by 12, multiply by the APR / 100, and that is your interest for the first year, which gets added. And you repeat this five times.

Note that the rules that APR is using mean that you don't pay compound interest (you don't pay interest on the interest that is accumulated through the year). Which means the number looks high.

What the dealer quoted you is a nominal interest rate. It's 2.6%, but it is 2.6% of the whole loan amount for the whole of the five years. So after four years, when you only owe about 100,000, you still pay 2.6% of 511,200 in interest instead of 2.6% of 100,000. Which is let's say naughty - that is what APR is for, to make things comparable.

As a rule of thumb, you can take total interest paid, divided by loan amount, divided by the number of years, times 2, times 100 - that would be 5.202% in your case, and that will be a tiny bit higher than the APR. That formula works reasonably well as long as the loan is not for too long, and the interest rate not too high.

• Your explanation really helped me understand what’s going on, thank you. – Kit Johnson Jun 3 '19 at 5:32