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I have the following formula to find the final value of compound interest with recurring contribution:

enter image description here

Where:
S: is final amount
T: my recurring value
i: interest rate
n: recurrence period

what I want is to isolate i in this formula

Marcius Leandro
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    Isolate $i$ is the most difficult thing to do, since it gives you an algebraic equation of degree $n$. – Crostul Jan 07 '21 at 00:38
  • so I came across this when I tried to isolate, and then I wondered if there was a mathematical trick that would help, using log or something more complex – Marcius Leandro Jan 07 '21 at 00:40
  • @MarciusLeandro You cannot isolate i for higher degrees than $4$. So you have to use an approximation method, i.e. Newton-Raphson method. – callculus42 Jan 07 '21 at 03:03
  • There is no closed-form solution for $i$. It's better to use a financial calculator in this case – 5201314 Jan 07 '21 at 17:06

1 Answers1

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Comment: If you mean i in term of S, T and n then let $(1+i)^n=A$, you may write :

$1+i=A^{\frac 1n}$

$$S=T\cdot \frac {A-1}{A^{\frac 1n}-1}$$

$$S\cdot A^{\frac 1n}-T\cdot A=S-T$$

That is if interest rate is 0 then $A^{\frac 1n}=A=1$

You may rewrite this as:

$$S (1+i)-T(1+i)^n=S-T$$

sirous
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