# How do I find the rate (r) when I only have the Payment (Pmt), term (t) and financed amount (p)?

I know the simple interest formula is, $I = prt$, but I don't know the interest to be able to use that formula.

$$p = \92,000.00$$

$$t = 72 \text{ months}$$

$$\mathrm{Pmt} = \1461.00$$

$$r = ?$$

This is a simple interest loan paid monthly and the rate is a fixed annual percentage.

• You must be more precise. We're just guessing at what you might mean. What's the interest scheme? How frequent are the payments? Is this a loan with life $t$? More information, please. – MPW Nov 6 '15 at 3:06
• Thank you MPW, I updated the question with more details. – SirOracle Nov 6 '15 at 3:12
• This question didn't have anything to do with (boolean-algebra)... – Math1000 Nov 6 '15 at 3:33
• I didn't know what tag to put on it but it required one and I'm not a math whiz..... Sorry – SirOracle Nov 6 '15 at 3:35

I suppose that you need to use the classical formula $$A = P\frac{i(1 + i)^n}{(1 + i)^n - 1}$$ where $A$ is the monthly payment amount, $P$ the amount of the loan, $i$ the monthly interest and $n$ the total number of payments.

Everything is simple to compute using this formula except the interest (otherwise you would not ask the question) and, as far as I know, only numerical methods would be able to solve the problem.

Since $i$ is small compared to $1$, we can use Taylor expansion around $i=0$ and get as an approximation $$A=\frac{P}{n}+\frac{ (n+1) P}{2 n}i+\frac{ \left(n^2-1\right) P}{12 n}i^2+O\left(i^3\right)$$ which reduces to quadratic equation in $i$. Applied to your numbers $(A=1461$, $P=92000$, $n=72)$, the positive root is given by $$i=\frac{4 \sqrt{1175358765}-125925}{2980225}\approx 0.00376113$$ Using this number would lead to $A=1460.98$ which is not bad.

A rigorous calculation, based on Newton method, would lead to $i\approx 0.00376161$.

You can notice that, neglecting the second order term of the expansion, we would have obtained $$i=\frac{1649}{419750}\approx 0.00392853$$ which is too high but already quite accurate.

If you want to stay simple, you could use the simplest Pade approximant and write $$A=P \frac{\frac{i (n+2)}{3 n}+\frac{1}{n}}{\frac{1}{6} i (1-n)+1}$$ For your numbers, this would give $$i=\frac{9894}{2635579}\approx 0.00375401$$

• These formulas are for compound interest, are they not? I think the OP was asking about simple interest. – David K Nov 6 '15 at 14:35
• @DavidK. May I confess that I don't know anything in the financial area, interest and, thanks to that, I guess that the bank blesses me everyday ! – Claude Leibovici Nov 7 '15 at 3:11
• For what it's worth, I don't think I've ever seen simple interest except maybe in a textbook or a homework problem. The formulas you gave are the ones that I think would apply in real life. – David K Nov 7 '15 at 4:41