Let say I want to earn 2000€ with an investment done each day in a year (i.e. a percentage every day, for 365 days). I have a initial budge of X amount.
Which percentage (per day) I have to gain to get +2000€ in a year?
For example, starting with a budget of 850€, which percentage of gain I need to use every day reaching that exact sum? Of course, what I gain every day I'll re-invest the next day.
Its seems a sort of exponential percentage...
On
You are correct that it is not linear, but you can use the formula $$FV = PV\left(1+\frac {r}{n}\right)^{nt}$$ where $FV = \text {future value} \\ PV = \text {present value} \\ r = \text {interest rate} \\ n = \text {number of times interest is compounded} \\ t = \text {length of time}$
and solve for $r$.
ETA: Per your request, $$r=\bigg(\bigg(\frac {FV}{PV}\bigg)^{\frac {1}{nt}}-1\bigg)$$ so $$r=\bigg(\bigg(\frac {2000+850}{850}\bigg)^{\frac {1}{365}}-1\bigg) \approx 0.00332 \ (0.332\%)$$
where $FV = 2850 \ (2000 + 850) \\ PV = 850 \\ n = 365 \ (\text {number of times interest is compounded}) \\ t = 1 \ (\text {one year})$
for example, let the percentage earned be 5%, the total amount be A, and the initial amount be X. After one day, the amount is $A = X + X*(0.05)$, or after factoring, $A = X(1.05)$. To get the amount after day 2, we can repeat this procedure to get $A = X(1.05)(1.05)$. Repeating, we get the formula $A = X(1.05)^{days}$
instead of dealing with 1.05, let p = the percentage + 1, so we have $A = X(p)^{days}$. Now let's solve the problem
You want an earned amount of 2000 after 365 days, so A-X = 2000 and A = X*p^365
Subtract X from both sides of the latter equation, to get $$ A-X = Xp^{365} - X $$ but we know A-X is 2000, so $$ 2000 = Xp^{365} - X $$ rearranging for p, we get $$ \sqrt[365]{\frac{2000+X}{X}} = p $$ or just $$ \sqrt[365]{\frac{2000+X}{X}} - 1 = percentage $$