A recent review of Hawk-Fx reminded me of a discussion that we had several months ago about the Annualized %. There are a lot of new people now, so perhaps the opinions have changed
At least my opinion has changed somewhat.
I think, as does that reviewer, that C2s formula for Annualized % is biased if the system uses its accumulated capital to trade as most systems do. If the system has a constant growth rate, then its capital will grow exponentially. The C2 formula assumes a linear growth, however. This leads to a bias: the Annualized % can be severely over-estimated both in the beginning and in later stages of the system. This is misleading for everyone who is not aware of this effect, and probably even for those who are aware of it. I think this is one of the reasons why people are often disappointed even when the system functions normally.
The C2 formula seems to be
Annualized % = (Cumu $) (365) / (#days the system is active * $100K).
For example, consider for simplicity two systems that both have a constant growth rate and that re-invest their profits continuously. Say, system A grows with 0.2% per day and system B grows with 0.4% per day (I tried to pick easy parameters that roughly mimic the present returns of Extreme-os and Hawk-fx)
After one year, system A will have 207.36% of its original capital. The Annualized%, however, will develop like this over time:
1 month 1291.82%
6 month 288.50%
1 year 207.36%
2 year 214.98%
So in the beginning there is a sever over-estimation, this drops quickly, after a year the estimate is correct and later it will gradually increase and become an over-estimate again. The minimum is 198.24% after 500 days, which is a small under-estimation.
After one year, system B will have 429.34% of its original capital. The Annualized%, however, will develop like this over time:
1 month 1371.46%
6 month 414.72%
1 year 429.34%
2 year 921.68%
Here we see the same pattern, but the bias is even stronger. The minimum is 396.08% after 250 days, which is a small under-estimation.
So the Annualized % can be an over-estimate or an under-estimate, but which of these two happens depends on the age and the growth rate. In any case it seems rather easy to overrate the performance of either system.
This is for a simple model with constant growth rate. The reality is more complex of course. But even so Id think that a correction would give more realistic expectations.
A correction is somewhat tricky, because I expect that it depends on whether you correct on basis of days, weeks, or months. Also, one might argue that an exponential model would not be fair to systems that do not re-invest their profits. My impression is that the large majority of systems re-invests and so my opinion is that a correction of the formula is preferable. Otherwise this post can perhaps make some people aware of the problem.
Jules, very nice summary, thank you.
I admit, at first glance I thought the reviews were a bit out of line, in that a review shouldn’t point out C2 flaws that have nothing to do with the system specifically being reviewed.
On second glance, I realized the systems reviewed were re-investing profits (as you pointed out, as most systems here do) and that the basic C2 annual % return calculations were incorrect.
In any case, I wouldn’t consider this an “error” or “miscalculation” (or however it was put in the reviews) on C2’s part, but if this discrepancy is not pointed out in the text description of a system, then that fact can and should be called out in a system’s review, as was done.
Not being mathematically inclined myself, it was also instructional to see the proper formula expressed for compounded gain
But I made an error in the computations. I shouldn’t do these things at the end of the night when I was actually already in bed. Wait a minute.
The error was that I didn’t subtract the initial capital in Cumu$, i.e. I interpreted it as the cumulated capital while it is only the cumulated profit. My apologies.
After this correction the figure is much easier: The Annualized % increases all the time. This is (fortunately) like I claimed a few months ago. Despite my stupid error I maintain my conclusion that the Annualized % is misleading for compounding systems…
For system A, assuming a growth of 0.2% per day, the estimate of the Annualized% develops like this:
1 month 75.16%
6 month 87.95%
1 year 107.36%
2 year 164.98%
For system B, assuming a growth of 0.4% per day, the estimate of the Annualized% develops like this:
1 month 154.79%
6 month 214.17%
1 year 329.34%
2 year 871.68%
So the conclusion should be this:
- When the system is younger than 1 year, then the Annualized% under-estimates how the system will actually perform in a one year period.
- When the system is older than 1 year, then the Annualized% over-estimates the return of a one year period.
A naive user who looks at the Annualized % when the system is 2 years old, will expect that the profits of system A will be 164.98% per year. But actually it is more reasonable to expect 107.36%, even if the user compounds continuously too. This is only 65% of what he expects. For system B he will expect 871.68% per year while 329.34% would be more realistic. This is only 38% of what he expects.
The ratio between the two systems changes too: from 2.06 after one month to 5.28 after two year. I.e. after one month it seems like system B is 2 times more profitable, but after 2 year it looks 5 times more profitable.
So it is hard to compare systems with different ages and different growth rates on basis of the Annualized%, and it doesn’t predict what you will have after one year.
We see sometimes systems that start with an extremely high Annualized % that soon drops, but this must have other explanations - which have been discussed frequently.
I think this is another reason why we should use a constant $100K portfolio size.
Hans.
