I recently got into investing and I'm trying to calculate my ROI, it doesn't seem to go very well and I don't see my mistake.
I am interested in the mathematics of this question moreso than the actual result. This is about mathematics, not finance.
Ok so: Suppose I invested an initial sum $x(0)$ and I want to know $x(t)$ where $t$ stands for work days since initial investment. Each day my investment earns some profit, and that profit is then invested as well.
Suppose the investment grows at fixed percentage. Meaning $x(t) = x(t-1)+\alpha x(t-1)$ where $\alpha$ is some (hopefully positive) constant.
I managed to show that $x(t) = x(0)(1+\alpha)^t$ via simple induction.
From the definition we have $x(1) = x(0)(1+\alpha)$, now suppose $x(t-1) = x(0)(1+\alpha)^{t-1}$, which agrees with the $x(1)$ case. Then from the definition $x(t) = x(t-1)(1+\alpha)$ and applying the assumption we get $x(t) = x(0)(1+\alpha)^t$
Mathematically this makes sense and is fairly straightforward, but in real world scenarios it really doesn't seem to work out.
Suppose I invested $60000\$$ 5 days ago, and I noticed that each day I was earning roughly $0.03$ of the investment sum the previous day. For instance, $x(1) = 60020\$$ so $\alpha = 0.03$.
Following that logic, $x(5)$ which is where we are today, should be $67884\$$. Sadly for me, it is not. It is a substantially lower amount, much closer to $60000$.
Where is the mistake here?
I think you messed up with $0.03$ v.s. $0.03 \%=0.0003$
Your equation is
$$x(t)=x(0)(1+\alpha)^t$$
And now you get $60020$ by setting $\alpha = 0.03 \%$
$$60020 \approx 60000 \times 1.0003$$