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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?

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  • $\begingroup$ Perhaps you have confused a percentage point of $1/100$th from a basis point of $1/10000$th? $60,000 * (1.03)^5$ is approximately $69600$ but $60,000 + (1.0003)^5$ is approximately $60090$. $\endgroup$ – Toby Mak May 28 '17 at 4:56
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    $\begingroup$ What is your calculation to get $\$67,884$ ? $\endgroup$ – callculus May 28 '17 at 4:56
  • $\begingroup$ The calculation is $x(5) = x(0)*(1+\alpha)^5 = 60000*(1.03)^5$ $\endgroup$ – Rick Joker May 28 '17 at 4:59
  • $\begingroup$ @RickJoker it's slightly off though it equals to 69556 $\endgroup$ – Yujie Zha May 28 '17 at 4:59
  • $\begingroup$ Your formula is correct: this is a simplified version of the compound interest (en.wikipedia.org/wiki/Compound_interest) formula where $n=1$. This version is commonly used in IGCSE and IB exams. How did you plug the variables into the formula? $\endgroup$ – Toby Mak May 28 '17 at 4:59
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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 $\alpha = 0.03 \%$

$$60020 \approx 60000 \times 1.0003$$

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  • $\begingroup$ Oh... I see...Yes that's correct. Thank you. Careless mistake. $\endgroup$ – Rick Joker May 28 '17 at 4:59
  • $\begingroup$ You are welcome :) $\endgroup$ – Yujie Zha May 28 '17 at 5:01
  • $\begingroup$ @Yujie Zha Your answer looks very similar to my first comment. Did you copy it? $\endgroup$ – Toby Mak May 28 '17 at 5:02
  • $\begingroup$ @TobyMak Of course not. It took me a while to type, that's why I was 40-ish secs slower than your comment - if I see yours and type, I would not have been that fast :) $\endgroup$ – Yujie Zha May 28 '17 at 5:04
  • $\begingroup$ @Yujie Zha OK, perhaps this was just a coincidence. $\endgroup$ – Toby Mak May 28 '17 at 5:05

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