Compound Interest Formula Deducting Relative Fees Annually From Investment Portfolio

Question

I'd like to know the compound interest formula for the following scenario:

I invest $\$10,000$ in an investment portfolio. The fund produces an annual return of $6 \% $ over $20$ years. Annual fees of $2 \% $ are subtracted from the investment portfolio value at the end of each year, that is, after the return is applied to the previous amount. No withdrawals are made.

Is there a single formula (preferred with variables) that calculates the future value of my investment portfolio after $X$ years, considering that fees are deducted each year?

Any help is very appreciated! Thank you.

Answer 1

You can calculate the net rate of return as follows. To apply the $6\%$ return, you multiply by $1.06$, and to subtract the $2\%$ fee you multiply by $0.98$. Since $1.06\cdot 0.98=1.0388$, the net rate of return is $3.88\%$ annually. Your balance at the end of $20$ years is then $10000\cdot 1.0388^{20}$.

In variables, if the principle is $P$, the annual rate of return is $r$ (as a decimal, e.g., $0.06$ for $6\%$), the annual fee percentage is $f$ (again, as a decimal), and the number of years is $X$, the ending balance is $P\cdot ((1+r)(1-f))^X$.

Answer 2

This is simply an arithmetic progression. The ratio of the geometric progression is: $q=\frac{106}{100}-\frac{2}{100}\cdot\frac{106}{100}=\frac{2597}{2500}$. The general term of a geometric progression is $I_n=10^4\cdot q^X$ with $n\in N$ and $n\geq0$. So, I have: $$I_n=10^4\cdot \left( \frac{2597}{2500}\right)^X$$