How to solve for interest rate $r$ in the amortization equation below?

$$A = \dfrac{P(r(r+1)^n)}{(r+1)^n - 1}$$


closed as off-topic by José Carlos Santos, Martín Vacas Vignolo, erfink, John B, Dando18 Apr 28 '18 at 22:04

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  • $\begingroup$ I added the phrase "interest rate" from the title to the body of the Question, which you presumably label as $r$. There is no simple expression for $r$ in terms of $n,A,P$. Note that solving for $r$ amounts to solving an $n+1$ degree polynomial equation. Despite having a simple form, the most practical approach is not an explicit expression for the root but a numerical root solver. This problem has been discussed several times previously here. I will find you one of those previous Questions. $\endgroup$ – hardmath Apr 28 '18 at 16:35

First of all you we can get rid off the denominators.

$A = \dfrac{P(r(r+1)^n)}{(r+1)^n - 1}$

Substitution: $r+1=z\Leftrightarrow r=z-1$

$A = \dfrac{P((z-1)z^n)}{z^n - 1}$

$Az^n-A=P\cdot z^{n+1}-Pz^n$

$0=P\cdot z^{n+1}-Pz^n-Az^n+A$

$0=P\cdot z^{n+1}-(P+A)z^n+A$

Here you have to find the roots of a polynomial with degree $n+1$ (the non-zero coefficients are $P, -(P+A)$ and $A$). Unfortunately it cannot be done algebraically. Thus you have to apply an approximation method like the Newton–Raphson method.


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