How to calculate time needed for relative sum reduction?

Question

Let's assume that a company has a net worth of $x$. For maximum tax effiency, shareholders decide to withdraw dividens $8\%$ of net worth every year. Here is the problem: since net worth is reduced every year, how do we calculate how many years does it take until net worth reaches $y$? We can assume that company has no revenue or other expenses during this time. I have tried to create formula but so far have been unable to do so, could I have some help?

Answer 1

The inequality is $W^N(t)=x\cdot 0.92^t \leq y$, where WN(t) is the net worth of the company and t is the number of years. To solve the inequality for $t$ take the logarithm at both sides of the inequality.

$\log\left(x\cdot 0.92^t\right)\leq \log\left(y\right)$

$\log\left(x\right)+\log\left( 0.92^t\right)\leq \log\left(y\right)$

$\log\left(x\right)+t\cdot \log\left( 0.92\right)\leq \log\left(y\right)$

$t\cdot \log\left( 0.92\right)\leq \log\left(y\right)-\log(x)$

Next we divide the inequality by $\log\left( 0.92\right)$. Since it is negative the inequality sign turns around.

$t\geq \frac{\log\left(y\right)-\log(x)}{\log\left( 0.92\right)}$