I would appreciate an explanation or example so I can wrap my head around this problem.
Consider an asset that costs \$10,000. In 3 years it generates \$40,000, but it has to be replaced every 3 years. So, in a 3-year time horizon, the ROI would be $[40000-10000]÷10000=300\%$. From this, the annualised ROI would be $[(1+ROI)^{1/n}-1]$, so $[(1+3)^{1/3}-1]=58.74\%$.
Now, if the time horizon is now 9 years, we have to replace the asset 3 times. The ROI remains the same $[120000-30000]÷30000=300\%$, however, the annualised ROI is now $[(1+3)^{1/9}-1]=16.65\%$.
Why does the annualised ROI get smaller and smaller the longer the time horizon. Are there flaws in my logic, or flaws in my understanding?
Thank you for any help and explanation!
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3 Years, 1 Purchase: 300% ROI, 58.74%
9 Years, 3 Purchases: 300% ROI, 16.65%
Addendum added that responds to the OP's comment question.
I agree with quasi's answer, but wish to express it differently.
Your equation of
$$[(1+3)^{1/9}-1]=16.65\%$$
is wrong. The fund quadruples every three years. Therefore, after 9 years, the fund has increased from $(1)$ to $(64)$, which is an increase of $(63)$.
Therefore, your equation should be
$$[(1+63)^{1/9}-1]=58.74\%$$
Addendum
Responding to the OP's comment question.
After receiving your comment/question, I re-read your original posting and discovered that I apparently mis-interpreted your intent. As I now read your original question, you intend the following:
Invest $(10,000)$ into a fund.
After 3 years, the fund has grown to $(40,000)$.
At that time, remove all $(40,000)$. Now, the fund has a balance of $(0)$.
Now, re-invest $(10,000)$. After 6 years (i.e. an additional 3 years) the fund has again grown to $(40,000)$. Again, remove all $(40,000)$. Now the fund again has a balance of $(0)$.
Now, again re-invest $(10,000)$. After 9 years (i.e. an additional 3 years after you cleaned out the fund the 2nd time) the fund the fund has again grown to $(40,000)$. Again, remove all $(40,000)$. Now the fund again has a balance of $(0)$.
The reason that I am interpreting your original posting in this manner is that in your original posting, you use the following math expression:
$$[120,000 - 30,000] \div 30,000 = 300\%.$$
I assume from this two things:
After 9 years, you now have $120,000$, rather than (for example) $(4^3 \times 10,000 = 640,000)$.
You have made three separate investments of $(10,000)$. The 1st investment at year-$0$, the 2nd investment at year-$3$, and the 3rd investment at year-$6$.
What this means is that at year-$3$, when you removed the $(40,000)$ from the fund, that money never had the chance to continue to generate interest.
Therefore, your calculation of $\displaystyle (1 + 3)^{(1/9)} - 1 \approx 0.1665$ is inappropriate.
Assume that you invest $(1)$ for $n$ years, leaving all of the money in the fund, with no deposits or withdrawals. Further assume that after $n$ years, your fund has increased from $(1)$ to $(D + 1),$ for a profit of $(D)$.
Then, the formula for ROI is $\displaystyle \left[(1 + D)^{(1/n)} - 1\right].$
However, in your original question, you (in effect) terminated the fund after 3 years, when you withdrew all of the money. This money that you withdrew never had the chance to continue to earn interest. Then, you re-started the fund at year-$3$. This analysis of your actions is the only way to explain your total accumulation of $(120,000)$, after 9 years.
That is, you must have accumulated $(40,000)$ 3 separate times.
Therefore, you have 3 separate transactions, where the appropriate ROI formula for each transaction is
$\displaystyle \left[(1 + 3)^{(1/3)} - 1\right].$
So:
$(4^3 \times 10,000)$ is not the total return.
By terminating/restarting the fund (twice), you prevent the interest earned in the first 3 years from continuing to earn interest and you prevent the interest earned in the second 3 years from continuing to earn interest.