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I'm trying to calculate the interest rate in compound interest, but I find that the online formula can't get the answer I want. I don't know why. Does anyone know what the specific formula is? I know the calculator can done that, but I need to use formula for programming.

The formula I use is r = ( FV / PV )1/n - 1, but the result is 7.56 something, but not as what the picture shows.

please look at this question as a example

python learner
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  • Every month and I have already know PV, FV, how many year it takes.. it is all in the picutures. – python learner Aug 11 '22 at 15:03
  • Getting the rate involves logarithms. – Sean Roberson Aug 11 '22 at 15:04
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    here is the basic expression. You can solve for $r$ by taking logs of both sides, ending up with $\frac r{12}=e^{\ln\big(\frac {6000}{5000}\big)/30}-1$ – lulu Aug 11 '22 at 15:05
  • I don't know how to calculte radix/base number – python learner Aug 11 '22 at 15:05
  • The base is whatever you want. In my comment, I used $e$ as in the natural log, but you could use $10$ or anything else you like so long as you are consistent. – lulu Aug 11 '22 at 15:05
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    Note: always good to do the continuously compounded computation as a quick approximation. Here we'd get $e^{2.5r}=\frac 65\implies r\approx .0729$ which should be (and is) quite close to the actual answer. – lulu Aug 11 '22 at 15:07
  • very appreciate, I use your way to solve the question and it is right, now I understand how you solve this problem, very smart , thankyou very much. – python learner Aug 11 '22 at 15:13
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    Basically the formula for the future value is

    $$C_n=C_0\cdot \left(1+\frac{i}{m} \right)^n$$

    $m=12$ is the number of compoundings per year. $i$ is the nominal interest year. $C_0$ is the initial deposit. And $C_n$ is the deposit after $n$ months.

    Thus your equation is $6000=5000\cdot \left(1+\frac{i}{12} \right)^{30}$. Solving for i.

    $$i=12\cdot \left(\left(\frac{6}{5}\right)^{\frac1{30}}-1\right)=0.07315068...=7.32%$$

    (rounded to two decimal places)

     – callculus42 Aug 11 '22 at 18:16

1 Answers1

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We have the following formula for calculating the future value ($FV$) of a deposit ($PV$) with an annual interest rate of $r\%$ compounded $m$ times per year for a total of $n$ times: $$FV = PV\left(1 + \frac{r}{m}\right)^n.$$ In the example you linked we are given the following information: $PV = 5000$, $FV = 6000$, $m = 12$ and $n = 2.5 \times 12 = 30$. All that we need to find is $r$, the annual interest rate.

Plugging the known values into the formula: $$6000 = 5000 \left( 1 + \frac{r}{12} \right)^{30}.$$ We need to solve for $r$ by rearranging this equation: \begin{align} r &= 12 \left( \sqrt[30]{\frac{6000}{5000}} - 1 \right) \\ &= 0.07315\ldots. \end{align} So the annual interest rate is approximately $7.32\%$.

The general formula for annual interest $r$ (which you can get by rearranging the original compound interest formula) is $$r = m \left( \sqrt[n]{\frac{FV}{PV}} - 1\right).$$ Note that taking the $n$-th root of $FV/PV$ is the same as raising $FV/PV$ to the power of $1/n$.

mcmk
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