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A while ago, a rather sneaky car salesman tried to sell me a car financing deal, advertising an 'incredibly low' annual interest rate of 1.5%. What he later revealed that this was the 'flat rate' (meaning the interest is charged on the original balance, and doesn't decrease with the balance over time).

The standard for advertising interest is APR (annual percentage rate), where the interest charged decreases in proportion to the balance. Hence the sneaky!

I was able to calculate what the interest for the flat rate would be (merely 1.5% of the loan, fixed over the number of months), but I was unable to take that total figure of interest charged and then convert it to the appropriate APR for comparison.

I'm good with numbers but not a mathematician. To the best of my knowledge I would need to use some kind of trial and error of various percentages (a function that oscillates perhaps?) to find an APR which most closely matched the final interest figure.

What would be the most appropriate mathematical method for achieving this?

Please feel free to edit this question to add appropriate tags - I don't know enough terminology to appropriately tag the question.

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3 Answers 3

up vote 3 down vote accepted

There isn't a single correct answer for your question - in fact, the method by which financial firms calculate APRs vary too. However, if you're willing to use the following definition as an estimate (and if you ignore the time value of money - that is, you value one dollar today as the same as one dollar a year later) you can use the following idea.

  • Calculate the total amount that you will have to pay at the 1.5% interest rate. For example, for a $10,000 loan over 10 years, you will be paying a total of

$$$10,000 \times (1+0.015\times10)=$11500$$

  • Assuming equal monthly installments, calculate your monthly installment payments.

$$\frac{$11500}{120}=$95.83$$

Substitute that value as $c$ in the monthly mortgage payment formula. This formula calculates the monthly installments you would make on a loan where the interest charge depends on the balance. The equation can be found here, but I've typed it out for you:

http://en.wikipedia.org/wiki/Mortgage_calculator#Monthly_payment_formula

$$c=\frac{rP}{1-(1+r)^{-N}}$$

  • $r$ is $\frac{1}{12}$ the annual interest rate
  • $P$ is the loan principal - in this case $10,000
  • $N$ is the number of payments to be made, in this case 120 months.

There is no analytic way to solve this problem.

However, if you are just interested in the answer, this function in EXCEL will do the trick: =RATE(120, -95.83, 10000, 0)*12, or more generally, =RATE(N, -C, P, 0)*12 to give you the annual rate.

For this example of a 10 year loan, the APR is only 2.86% - sounds OK to me! :)

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I just want to correct something from Vincent Tjeng's response. The example of: 10,000 x (1+.015x10) = 11500 should be as follows: 10,000 x (1+.015x120) = 18000 Payment amount would be as follows: 18000/120 = 150 The interest rate using the EXCEL RATE function: =RATE(120,-150,10000,0)*12 = 13.11672473% APR Still not a bad APR compared to credit card rates, but much greater than the 2.86% APR. –  dzdeano Mar 23 at 14:50
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First, you summarize the cash flow. We normalize the total loan to $1$, since its magnitude doesn't affect the calculation: So you pay an interest of $f=0.015/12$ per month. Let's say you pay the whole thing back in equal installments over $m$ months: Then the cash flow can be summarized as $$c(t)=\begin{cases}-1&\text{at } t=0,\\\frac1m+f&\text{at }t=\frac1{12},\frac2{12},\ldots,\frac{m}{12}.\end{cases}$$ At an effective interest rate $r$ you should have $$\sum_t c(t)(1+r)^{-t}=0,$$ which in the present case becomes (after some manipulation – you need to know the sum of a finite geometric series) $$\Bigl(\frac1m+f\Bigr)\frac{\rho-\rho^{m+1}}{1-\rho}=1\qquad\text{where }\rho=(1+r)^{-1/12}.$$ You will have to solve that by some numerical scheme (Newton's method for example).

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I was playing around with the numbers and realised that at interest of 1.5%, the APR is maximum at 50 months (2.88463%). Do you have any idea why? I thought that it would increase without limit. –  Vincent Tjeng Feb 28 '13 at 12:40
    
@VincentTjeng The longer your loan term, the less benefit you get by reducing interest on a monthly basis, so you have more payments that are closer to the flat rate. Offhand, I suspect that might be why. –  Arkamis Feb 28 '13 at 18:06
    
I think one way to understand it is that in the limit $m\to\infty$, the $1/m$ term drops out, so it's like you're just paying the interest forever and making no down payments. But the present value of a fixed monthly payment from now to infinity is finite, and after a while you are beginning to see the beneficial side of making no down payments. –  Harald Hanche-Olsen Feb 28 '13 at 18:07
    
@Arkamis and Harald: thank you, it makes more sense now. –  Vincent Tjeng Mar 1 '13 at 2:18
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Rather than getting down to this level that requires a Masters degree in Applied Mathematics, I use a simple rule of thumb that closely equates one with the other. Apart from extremely low or high interest rates, it is close enough the draw a good comparison. Put simply, divide the variable interest by 2 and then add 2 to get the eqivalent flat interset rate. Eg a 30% variable rate approximates to 30 / 2 + 2 = 17% flat. 20% approxiamtes t0 20 / 2 + 2 = 12% flat. How easy is that!

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