How to calculate total profit in a set amount of days with a 2% daily gain?

Question

I want to know if there is a formula that would give me the total amount at the end of a 365 day period with a 2% daily gain. Adding 365 days up would take for ever.

Answer 1

Yeah, it's called compound interest. Note that if you have some amount of money $x$, and it grows by 2%, that's the same as multiplying: $x \cdot 1.02$. If it does that for 365 days in a row, then you multiply $1.02$ by itself 365 times = $1.02^{365}$ and multiply the end result by $x$.

In short: if you start with $x$ amount of money and your pile of money grows by $2\%$ a day, then at the end of the year, you'll have $x \cdot 1.02^{365} = x \cdot 1377.$

EDIT: If you want to know the pure profit, you will have to subtract from that result the amount of money you started with: so you'll have made a profit of $(x \cdot 1377) -x$.

Answer 2

There is: $P = ((1+\frac{2}{100})^{365}-1) V_0$, where $V_0$ is the initial amount, and $P$ is the profit after 365 days.

You start with $V_0$. After a day you have $V_1 = (1+\frac{2}{100}) V_0$. After another day you have $V_2 = (1+\frac{2}{100}) V_1 = (1+\frac{2}{100})^2 V_0$, etc, etc. So, after 365 days, $V_0$ has turned into $(1+\frac{2}{100})^{365} V_0$. Hence the profit is $P= (1+\frac{2}{100})^{365} V_0 - V_0 = ((1+\frac{2}{100})^{365}-1) V_0$.

Roughly, $P \approx 1336 V_0$.