Trouble arranging for interest in annuity to present value formula

Question

The formula is:

$$A = P\frac{i(i+1)^{n}}{(i + 1)^{n}- 1}$$

and I need to rearrange for $i$ because I don't know the interest. I can solve this if it were

Any help would be greatly appreciated.

Answer 1

Hint: $i = (1+i) - 1$, solve for $1+i$.

Rewrite the equation as

$$\frac{A}{P} = \frac{(1+i)-1}{1-(1+i)^{-n}}$$

Then multiply through the equation by the denominator of the RHS and rearrange to get

$$(1+i)^{n+1}-\left (1+\frac{A}{P} \right ) (1+i)^{n} + \frac{A}{P}=0$$

Now you can solve for $1+i$ as a (real, positive) root of this equation.

EDIT

When $n$ is large, in many cases $(1+i)^n$ will be much larger than $\frac{A}{P}$. We may then ignore that term will very little error and get as a very good approximation to the solution:

$$i \approx \frac{A}{P} $$

We may then find a first order approximation as follows: let $i=R+s$, where $R=\frac{A}{P}$. Then the equation takes the form:

$$s\left ( 1 + \frac{s}{1+R} \right )^n + R (1+R)^{-n}=0$$

To first order in $s$

$$i = R [1-(1+R)^{-n}]$$

Note that $(1+R)^{-n} \sim 10^{-64}$ with the current parameters, so there is no point in further approximation. But at least there is a general framework here.