Hehe, you posted an instant after I mentioned sharpe ratio, so you have my question to your answer already. How would that be linear? My goal is to give a specific "score" to each system. Do you think i should use an abritrary "risk aversion" curve (for the multiplier to the shapre which multiplies the average daily profits) to apply the sharpe ratio as i think best fits my goals?
When I ask "how would that be linear" i mean that if i get a slightly higher AVG on System2 than System1 and also System2 has a slightly higher sharpe ratio, with standard deviation in play, i'm not sure it would be linear. Or maybe it would, and i'm just bad at math? Meaning, my worry is that the slight increase in sharpe ratio in system1 (together with profits) would destroy the system1 score too much, because it doesn't change in a linear fashion. Ugh, maybe i just need to plot these scenarios on some curves and see what comes out.
There is "luck" involved in both, and based on what you've shown, there is no way to determine which was "more lucky".
I'm not really sure what you mean by 'linear' here. Sharpe ratio is by definition non-linear to scalar multiplications to returns - which I think you see - but I don't think that's a bad thing? Okay, so it now looks like you are trying to find an 'efficient' system amongst a bunch of systems. That changes the discussion again, a little. Are you sure you don't want to use sharpe ratios? I mean, you can always (all else being... simplified to the extreme) pick the highest sharpe ratio system and scale it up to the return or risk level you want. All these are really really tough problems (and frankly, I wouldn't tell you more than what you can find on wikipedia - because I think these are pretty important secret sauces). Maybe you'll do better if you can try to phrase more directly what you are trying to accomplish?
+ Best answer. Based on what he has shown if we assume that the mean of the samples equals the mean of the distribution of the mean in the sense of the central limit theorem then the probability for each system for > +10 P{P/L > +10} = 50%
Yes, you're right, i'm trying to find the most 'efficient' system. If you say you don't think that's a bad thing, do you mean that it would be a good idea to give a score to each system simply by doing: Score = AverageDailyProfits * SharpeRatio ? By most 'efficient' i mean that i expect my sample data to be insufficient and i am trying to avoid the probabilities of future major failure (drawdowns) to the greatest extent possible with what little information (samples) i have. I am trying to find a system that will have relatively stable profits with drawdowns happening over a longer period of time, rather than suddenly (i know, small stops and trailing stops and top/down trailing limits do wonders when mixed with signals, never mind that). This means I am after a slow system where drawdowns happen over longer time, never mind that they can be just as big in total as if the system had sudden drawdowns. This is mostly psychological so i don't stress out if it starts going bad. So for prolonged drawdowns, multiplying with sharpe ratio seems a good idea on the face of it, but i'm not sure due to the non-linearity of it. Regarding secret sauces: Fine! If you don't want to share your secret sauces, i'll come up with my own! ;P I didn't think something like this would be getting to the "secrets" area, but i see what you mean...
Why not just rank using the sharpe ratio? The efficient frontier is non-linear in risk/return space afterall - why do you need to linearize it? Avg * Sharpe Ratio = Avg*Avg / Std -> Not sure how you would even interpret that. I see what you are trying to do now - I should at least applaud you for doing a lot more thinking than the top 99% of ETers (not much of a compliment, I know). I'll give you a hint/question: Is the performance of system1 + system2 always (a) >= max(system1,system2), (b) >= min(system1,system2), (c) don't know.
First it does not depend on distributions of the random variables. I suppose you mean the requirement of large samples. Do you?
You sure about that? CLT requires the observations are independent and identically distributed. To clarify: independent means there can't be any mean reversion/momentum; identically means the variance can't shift There's also the small sample problem - but I figured you knew that one.