An asset is bought at price $x$ and sold at a price $y$, increasing or decreasing in value $\frac{y-x}{x}$ percent ($C$). Buying/selling incurs a percentage fee, let's call it $F$. The asset is bought in exchange for "money", let's call it $M$.
To calculate the final profit/loss, one can use the following formula:
$$bought = \frac{M-MF}{x}$$
$$sold = (bought-boughtF)y$$
Because of fees, $\frac{sold - M}{M}$ will always be lower than $C$.
Example
Starting with $M=100$, with fees of $1$%, I can buy an asset that costs $x=50$:
$$bought = \frac{100-100*0.01}{50} = 1.98$$
Let's say that later the asset depreciates -10% in price, $y=45$, and it's sold:
$$sold = (1.98-1.98*0.01) * 45 = 88.209$$
The asset only depreciated -10% ($\frac{45-50}{50}$), but with fees, the loss was -11.79% ($\frac{88.209-100}{100}$).
Question
Is there a formula I can use to compute the net profit directly? I.e. knowing that the gross depreciation was -10% and there were 2 fees of 1%, it will give me -11.79%.
