At the end of every month I deposit a 1000 dollars into my bank account for a period of 15 years with a nominal interest rate of 14% per annum with half-yearly compounding that falls under a 15% tax. After 10 years I took an unknown amount $T$ from the bank account. What was the amount $T$ if after 15 years of saving we have 180 000 dollars on the account?
I used the formula I was taught for saving money that is – $$S = mx\left(1+ \frac{m-1}{2m}i\right)\left(\frac{(1+i)^n – 1}{i}\right)$$ where $m$ is the frequency of saving per compound period, $x$ is the amount you save, $i$ is the interest rate and $n$ is the amount of compound periods. So I plugged in $m = 6$, $i = 0.07\times0.85$, $n = 20$ and $x$ = 1000 for the first 10 years. I got $224 973.42832027$ dollars. So after that I decided to just use ($224 973.42832027 – T$) and use the formula one more time but only for 5 years and set it equal to 180 000 dollars. That gave me the result of $T = 227 747$ dollars. Because I can't really check if that's correct I wanted to ask if this is the right way to solve this problem.
Any tips or criticism is welcomed. I also have to mention that the problem isn't in English so it might be possible I used the wrong terminology.
Best Answer
The formula you are using appears correct to me. It assumes that the first monthly deposit gets $5$ months' interest at the end of the compounding period, the second gets $4$ months' interest, and so on.
An easier way to do the problem is to first see what the balance would have been if no withdrawal had been made. I get $481,852.10$, so we see that had we left $T$ on deposit, it would have grown to $301852.1$, that is, $$T(1+i)^{10}=301852.1$$ Solving for $T$, I get $$T=169349.76$$