If looking at a system with year of history, would you rather rather trade a system making 3% a month, or one making 10% a month with same drawdown because Sharpe Ratio is better for the 3% system?
In this case, I would not be using the SR to compare these two systems at all, as they are so dissimilar. SR has its limitations and drawbacks. The sampling period for both the systems, short-term and the longer-term system would be ideally from weekly if a years worth of data is available.
If both the above mentioned systems are producing the same or nearly the same returns, then the SR can be used to determine which is better. If they are producing almost identical returns, with the StDev different, then we have the basis for comapring these two using the SR.
hehehehehe… This is your proof? This disproves the Central Limit Theorem. You deserve the nobel prize.
No, this is not proof. This is a counter example, disproving your statement you presented as fact. You asked for a counter example. I gave you one.
You really have no idea what The Central Limit Theorem means, do you? The Central Limit Theorem is about the distribution of the MEAN of the sample and not about the distribution of the sample itself. Saying that returns follow a normal distribution has absolutely nothing to do with the Central Limit Theorem and just display your limited understanding about the topic.
As usual, you are posting something which sound smart without understanding what you really are saying.
- Fanus
Sorry, Fanus, you are giving a bogus example as a real-world fact and expect me to accept that as a counter-example that disproves what follows logically from the Central Limit Theorem.
The Central Limit Theorem says that for a variable with finite variance, the distribution of means of independent samples converges to a normal distribution if the number of samples goes to infinity. I dont see any counter-examples from you that disproves it.
Fanus and Pal,
With respect to the Central Limit Theorem I may clarify a few things. This theorem says: the distribution of means that are based on samples that each consist of independent draws of the same size N from a distribution with finite variance, converges to a normal distribution when N goes to infinity.
The proof of this can be found in every basic textbook on mathematical statistics, but it requires several pages (using a lot of earlier results in the book), so it impossible to repeat that here.
Applied to the present situation, consider a system that is somehow stable in that the profits on each day can be seen as a random draw from some fixed distribution. A simple example of this would be if the system has profit $100 with probability .75 and a loss of $100 with probability .25 on each day. As Pal said, there can be some debate about which N is long enough, but let’s assume for simplicity that N = 30 is large enough (this is often suggested in textbooks as a rule of thumb). N = 30 would correspond to 30 days = 1 month. Therefore consider the monthly profits from a infinite series of months. According to the Central Limit Theorem together with the N = 30 rule-of-thumb, this set of monthly profits would be approximately normally distributed.
So I agree with Pal that an approximate normal distribution is more likely if monthly periods are used than if shorter periods are used. I agree with Fanus that for a small number of months one can easily get a deviation from normality. However, under the above assumptions, that deviation would vanish when the number of months increases. (You can of course easily write down a set of numbers that are not normally distributed, but such sets are increasingly improbable as outcomes of mean profits under the above conditions).
The essential point is that one cannot be sure that the model of drawing from the same distribution is valid. I can easily imagine that the probabilities of a winning trade and the size of the profits change with market conditions (e.g. bear versus bull). If the distribution changes during the sampling process, then the Central Limit Theorem is not applicable anymore. There are different versions of the Central Limit Theorem that may still apply then, but all versions require that some assumptions are met. Therefore I would prefer to avoid the assumption of a normal distribution whenever possible, even if it is theoretically plausible in some cases.
Jules
Pal
A counter example to disprove something does not has to be likely. It has to be possible. You cannot say something is fact because the example which disprove it is unlikely to happen. This is like saying because you are very unlikely to encounter a black swan, this is a fact that all swans are white.
What you said above about the Central Limit Theorem is correct. Please read it again. You even post it yourself and just repeat what I said. It is about the MEAN of the samples. But this has nothing to do with what you said about that returns follow a normal distribution.
You did not say the that the mean of returns follow normal distribution. You said returns follow normal distribution. Two very different things, but if you don’t understand the topic, this is easy to confuse the two. This is like you saying 1 + 1 = 3 and I tell you it is 2 and then you come back and say, but 3 + 3 is 6 and ask me to disprove it.
- Fanus
PS: This was my last post to you on this topic. I have better things to do with my time.
Fanus,
If looking at a system with year of history, would you rather rather trade a system making 3% a month, or one making 10% a month with same drawdown because Sharpe Ratio is better for the 3% system?
Well, since I look to my account balance every day, I feel much more comfortable with a little profit every day than with an earthquake system. I guess that this is the reason that most people prefer a savings account above any trading system. In other words, annual return is one quality, and stability is another quality. I want both, but not necessarily one at the expense of the other.
If both systems have no draw downs, then there is a different issue, that has been debated previously in other threads, namely that the SR is not so good at all (regardless the period) because it will also decrease as a result of large profits. I think that this problem is better solved by using another measure instead of the SR.
I don’t think that daytrade systems will necessarily have an advantage with a daily period. For example, not if they have large draw downs. My experience with non-daytrading systems is that my account value is vulnerable to all kind of daily market fluctuations. In my opinion this should be reflected in the SR.
Jules
A counter example to disprove something does not has to be likely. It has to be possible.
I disagree. In the world of statistics and probability, a likely event has to be probable which is what leads to certainty.
ps: The first range of evidential continuum is covered by the concept “possible.” A conclusion is “possible” if there is some, but not much, evidence in favor of it, and nothing known contradicts it. This last condition is obviously required-a conclusion that contradicts known facts is false-but it is not sufficient to support a verdict of “possible.” There are countless gratuitous claims in regard to which one cannot cite any contradictory fact, because they are inherently detached from facts; this does not confer on such claims any cognitive status. For an idea to qualify as “possible,” there must be a certain amount of evidence that actually supports it. If there is no such evidence, the idea falls under a different concept: not “possible,” but “arbitrary.”
Like all cognitive claims, possibilities are asserted within a context. Should it change, the verdict must change accordingly: the initial possibility may be weakened (even erased), or it may be strengthened. If favorable evidence continues to be discovered, at a certain point the claim stops being merely “possible.” It becomes probable.
“Probable” indicates a higher range of the evidential continuum. A conclusion is “probable” if the burden of a substantial body of evidence, although still inconclusive, supports it. In this case, there are not merely “some” supporting data, but a relatively extensive amount, although these data have not yet reached the standard of proof. Because they have not, there are still objective grounds to remain in doubt about the final verdict.
Like possibilities, probabilities are asserted within a context and may be weakened or strengthened as it changes. If favorable evidence continues to be discovered, at some point the cognitive climax will be reached, The conclusion ceases to be a hypothesis and becomes knowledge. Such a conclusion in CERTAIN.
The concept of “certainty” designates knowledge from a particular perspective: it designates some complex form of knowledge considered in contrast to the transitional evidential states that precede them, (By extention, the term may be applied to all knowledge, perceptual and conceptual, to indicate that it is free of doubt.) A conclusion is "certain; when the evidence in its favor is conclusive; i.e., when it has been logically validated. At this stage, one has gone beyond “substantial” evidence. Rather, the total of the available evidence points in a single direction, and this evidence fulfills the standard of proof. In such a context, there is nothing to suggest even the possibility of another interpretation. There are, therefore, no longer any grounds for doubt.
Certainty, like possibility and probability, is contextual. It is a verdict reached within a definite framework of evidence, and it stands or falls with the evidence.
Is man capable of certainty? Since man has a faculty of knowledge and nonomniscience is no obstacle to its use, there is only one rational answer: certainly.
Hi Jules
You might be more comfortable with small daily gains. Someone else might be more comfortable with “earthquake” systems.
For me this, is not really the issue. The problem for me is that the Sharpe Ratio say the one is better than the other one and this value is used to rank systems and for all practical puproses only highlight systems which appeal to some people. I think this would be better to not use Sharpe Ratio in the rankings, but keep it available in the statistics page so that people who are interested in it can look at it after finding systems based on things like annualized returns, drawdowns, average trade, realism factor, W:L, etc.
All in all, I think monthly periods is a good compromise and by eyeballing the systems on the best systems list, I think it is doing a decent job.
And remember, since daytrade systems positions are closed at the end of each day, their intratrade drawdowns are not displayed and have no effect on the Sharpe Ratio. Only their closed trade drawdowns are used. But for longer term systems, their intra-trade drawdowns are used for the calculation. This is why I prefer monthly periods as this somewhat even the playing field between daytrade and longer term systems.
Regards
- Fanus
Jules: This is directed specifically to you, as I respect your analysis of this more than, ahem, some others…
I think I’m having difficulty comparing, in concept regarding Sharpe, systems that trade frequently, .vs. longer term position systems. Assuming that the individual trades from each system are reasonably consistent over time, I’d think that the short term system would compare more favorably over time than the long term system. Consider trades within each portfolio that are basically the same: Up 5%, down 10%, up 3%, down 2%, up 12%, closed out. System A experiences all of this in one day, System B over a month. If the portfolio is measured EOD for the daytrading system, you’d see more consistent results for the EOD system, as it’s fluctuations are hidden from the sampling interval. The daily sampling interval doesn’t see much of the drastic equity swings. But for a postion trading system those swings are measured and evaluated daily, and all the ups and downs are exposed.
OK, another example, one that should warm the heart of a mathemetician. Let’s say we are flipping coins. If you measure return over a very short period, you’ll see fluctuations in equity, as lucky and unlucky streaks play out. After all, even if it’s a 50/50 chance, over the short haul you will be either ahead or behind of the game. If, OTOH, you measure return over the long haul, there will be very little fluctuation, as - over many trials - the results will be quite consistent. Std Dev taken over each 3 flips should be greater than Std Dev taken over 100 flips. Std Dev measured over each 100 flip interval should be negligible. The means will even out, but the variation short term .vs. long term should be greater.
Jules - does this make any sense?
Hans. (who studied advanced statistics in college 35 years ago, and has forgotten most of it…)
Fanus,
You address this post to Pal, but I’m a little afraid that it was in response to my post. So I feel obligated to answer - although I don’t want to encourage the dispute between you and Pal.
1. In terms of your analogy, the CLS says that black swans are unlikely, so you cannot disprove it by pointing out a black swan. (But I’m not sure that you wanted to disprove the CLS. If you just wanted to prove that black swans exist, then pointing out one is sufficient of course).
2. The confusion between mean and total profit was something I was already afraid of when I wrote my post. The mean (M) is related to the total (T) by M = T/N. If M is normally distributed, then T is normally distributed too, and conversely. So as far as the shape of the distribution is concerned, there is no need to distinguish them.
3. All together I agree with you that we better not assume a normal distribution though.
Jules
Fanus,
Oops, I see that in the mean time you replied to my other post so perhaps my previous one (assuming you confused me with Pal) was a little premature… 
Jules
Jules: I forgot, should have added, in my example just given, that the daytrading system after experiencing wide daily fluctuations, has somewhat reasonably consistent EOD returns. The long term system, by comparison, has reasonably consistent EOM returns. If the sampling period is daily, wouldn’t the daytrade system have a much better Sharpe, even if it is in reality seeing similar equity fluctuations?
Hans.
Fanus and Hans,
I think you both have an argument in common, namely the difference between open and closed trades. If the periods are days, then the SR of daytrade systems would be based on closed trades while the SR of swing systems would be based on open trades. I admit that I didn’t think of that.
Hmm. The equity curves have the same property, but there has also been a debate about that.
The basic question is then wether we want the SR to reflect the daily equity curve, or to give more long term information condensed in longer periods and (probably) closed trades.
My personal style is to look at the equity curve and EOD account value (including open positions), but I agree that this may well be entirely personal. So I cannot claim that daily is objectively better than monthly. It is just my personal preference. If you appreciate monthly periods, I’m fine with that too.
Jules
Hans,
Yes, your point makes sense. See my previous post. But note that by the same argument, the trader of an ETF system (at least one exists in C2), holding positions for several months, could argue that we should take years. But then I agree with Fanus that months would be a fair compromis.
Jules
I guess what I’m getting at is that trading system results will probably resemble a sawtooth pattern, to some extent. With a daytrading system, the individual ‘teeth’ of the sawtooth will be hidden within and between the sampling period (day), while the ‘teeth’ in the sawtooth pattern are exposed in a long term position trading system and thus will create a greater std dev. In essence, if the sampling period is considerably longer than the ‘teeth’, the system will have a good Sharpe. If the sampling period is shorter than the ‘teeth’, Sharpe will suck. Or so it seems… Ya catch my drift?
Hans.
Yes, you’re right, but I see no easy way to avoid that. One possibility would be to adapt the period to the average trade length. But if you take the average trade length as the basic period, then you will probably still have open trades at the end of a period (because some trades are longer, and because the periods do not necessarily start at the beginning of a trade). I expect that it would be difficult to program too.
Another disadvantage of adaptive periods is that SRs of different systems would be difficult to compare in some sense, since some would be based on months and others on 13.9 days and others on 5.14 hours. So at least they should be annualized. But even then I doubt whether it is wise to assign the same SR to the two systems in your example (having the same pattern in a day versus a month). For me that would be quite different systems. But I admit that that is more a feeling than a rational argument, as their risk/reward is logically the same.
I think we should distinguish two kinds of analysis here: per period and per trade. The SR is by definition per period, and what you want is per trade, if I understand your correctly. That’s OK, but I would suggest not to call it a SR.
Thinking further about this, if you want a pure per-trade measure, this is not so difficult: Make a list of the profits of all closed trades (where a loss is a negative profit); compute the mean and standard deviation of the profits; divide the mean by the standard deviation. There are two problems with this:
1. How to enter the risk-free return (but this can probably be solved)
2. What if the system has a basket of trades that are done at approximately the same time to spread the risk? (E.g. with options there will often be many losing and one winning trade). This is harder to solve since you don’t know the intention of the trades, i.e. wether they belong together or not. The only way to see that is whether they are done in the same period, but then we’re back to a per period analysis.
Well, perhaps we should discuss this further another time. I’m really very tired now (I spent the whole European night on this) and the SR of me making sense becomes smaller with the minute…
Good night,
Jules
Yes, good night, Jules. I’ve been tending to a sick freezer trying not to loose several hundred dollars of meat. But I’m throwing in the towel, and headed for a good night’s sleep.
Hans. (Perhaps I should stick to bread and toast, and not worry about freezers. (grin). )
Hans,
Why would my sharpe be bad? I have a trade record around 90% and have never had a drawdown over 20%? I have had a few down trades, but overall I’ve been pretty consistent, and that is what it shows.
Sincerely,
Craig
SR of me making sense becomes smaller with the minute…
SR has its uses, but if one is looking for a performance measure to compare accross systems, it is totally invalid even if the same sampling period is used for all systems. There are other good performance measures that can be used to compare accross systems namely, the Expectancey Score (ES) which most of us are aware and also the Sortino Ratio or UPI, though if it calculated based on closed EOD trades (day-trading) for some systems and open EOD trades for some systems (long-term trend following), it would bias the results in favor of the closed EOD trades system unlike the ES: http://www.sortino.com/htm/satchell.pdf
