# Calculating ROI - where is my mistake

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? • Perhaps you have confused a percentage point of 1/100th from a basis point of 1/10000th? 60,000 * (1.03)^5 is approximately 69600 but 60,000 + (1.0003)^5 is approximately 60090. – Toby Mak May 28 '17 at 4:56 • What is your calculation to get \67,884 ? – callculus May 28 '17 at 4:56 • The calculation is x(5) = x(0)*(1+\alpha)^5 = 60000*(1.03)^5 – Rick Joker May 28 '17 at 4:59 • @RickJoker it's slightly off though it equals to 69556 – Yujie Zha May 28 '17 at 4:59 • 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? – Toby Mak May 28 '17 at 4:59 ## 1 Answer 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

• Oh... I see...Yes that's correct. Thank you. Careless mistake. – Rick Joker May 28 '17 at 4:59
• You are welcome :) – Yujie Zha May 28 '17 at 5:01
• @Yujie Zha Your answer looks very similar to my first comment. Did you copy it? – Toby Mak May 28 '17 at 5:02
• @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 :) – Yujie Zha May 28 '17 at 5:04
• @Yujie Zha OK, perhaps this was just a coincidence. – Toby Mak May 28 '17 at 5:05