Actually they aren't the same. Here's a example using the assumptions I used from above with 3% risked per-trade instead of 1% (to show the effect a little stronger). Here's a cycle of 10 trades using each method to show that the higher win % with the same expectancy and same % of equity risk has a superior return. It's caused by using the % of equity risk method which penalizes a lower winning % in favor of a higher win %. If the order of trades were reversed or jumbled, the outcome would be the same.
AAA the problem with draw downs is that they do not care whether you made much before or did not. 20% down is always 20% neither more nor less. And it can occur in the very first moment you are trading it or twenty months later when you already made 50%. And there is one other thing. We are talking always, at least I assume that, about normal percentage figures, not log normal figures. Thus if I have 100 now, make it 150 by end of next month, and then come back to 100, I think I made 50% in the first month while only loosing 33% in the second. You see the point. With small numbers this is neglectable, with bigger numbers like the ones we are talking about here, it is not. So whenever you make 500% and then you loose 100%, you're still bankrupt. Or, softer, making 100%, then loosing 50% brings you back to the start. Sorry, I came into talking - this just entered my mind ... peace
The initial question was about two systems that have the same return, yet differ in hit ratio and pay off ratio. Most likely we could have already seen at this point, if it were real systems, that they had different Sharpe ratios. Thus even if they have the same expectation value does not tell about the stability of that expectation. The expectation as such has a distribution below it, where the expectation value pops up as the peak. The distribution could be flat (or at least shows fat tails) as in system A or is more steep as for system B. I am not completely sure if this is the correct answer, honestly speaking, but I think the stability factor was not taken into account at initial question, which could be formed differently: can two systems with the same expectation value differ in the distribution of that expectation. I guess: yes. peace
Noooooooo! Equity curve and drawdown are essentially the same. The question is whether a smooth equity curve is superior to one less smooth, given equal returns. All this other stuff is just stuff and circumspection. It has nothing to do with expectancy, or profits, or ratios, or executions ar any of that. Each method provides the same return. Period. All things that go into the final result are factored in already. They have the same return. So corso wants to know if a smoother equity curve is superior. That's it. That is the whole question. Nothing else. No ratios, no expectancy, no drawdown, no profits, no slippage, no execution, just a simple question... is a smooth equity curve superior. All of the things you guys are saying are true in other circumstances, but not this question.
Of course, they can, that's simple arithmetics. I guess, acrary showed that too in some example. You can have an infinite number of different (win, loss) distributions that lead to the same expectation value.
In the formula for the expectation you have two distributions: the frequency of wins ands losses and the size of wins and losess. They are independent, for the same frequency you can have different sizes, because the frequency does not determine the size and vice versa. Because of that you can manipulate the distributions in such a way that you can always get the same expectation value. For instance, you can have smaller winners, but more frequent or bigger winners, but less frequent.
If you set the returns the same for two different methods, then there are two variables to achieve the result (forgetting about compounding for a minute), 1). Frequency of trades 2). Expected profit from each trade. Return = # trades * expected profit per-trade In my first post I had the expected profits equal and it showed the higher win% seemed to be superior. Lets see if we vary the expected profit: Method C 30% = 10 trades * 3% (80% win * 4% profit) - (20% loss * 1% loss) In this case the number of wins is 8, the modified sharpe ratio (Expected return / std. deviation) 1.42, and the reward to risk per-trade is 16 Method D 30% = 20 trades * 1.5% (50% win * 4% profit) - (50% loss * 1% loss) In this case the number of wins is 10, the modified sharpe ratio is .58, and the reward to risk per-trade is 4. It seems to me that the higher win% is superior to the lower win% since it has a smoother equity curve and better reward:risk per-trade. This is in spite of a lower number of winning trades. Now lets reverse the two: Method E 30% = 20 trades * 1.5% (80% win * 2.125% profit) - (20% loss * 1% loss) The number of wins is 16, the modified sharpe ratio is 1.17, and the reward to risk per-trade is 8.5. Method F 30% = 10 trades * 3%(50% win * 7% profit) - (50% loss * 1% loss) The number of wins is 5, the modified sharpe ratio is .71, and the reward to risk per-trade is 7. Again, it appears that the higher win% is superior due a smoother equity curve and a better reward to risk ratio. It seems to me the number of wins isn't relavant to a smoother equity curve, but the winning % is. I tried to find an example where the lower winning % is better in either reward:risk or equity curve but I can't find one. Can anybody think of an example?
acrary your analyses are so good. Fun to work thru too. Although, I am not clear on your conclusions. Given the same returns, is a smoother equity curve superior? As you have read no doubt, my premise is that all of the factors that control return are moot in this thread because the return is the same. Regardless of the means to the end, the end is the same. The only question is whether a smoother equity curve is a better equity curve. And I say it is only a matter of preference, given that profits are equal. A mathematical model is inappropriate for this question because there is not a number by which the answer is right or wrong, better or not, etc. Preference cannot be decided by a mathematical model. Preference is subjective, models are not.
If you are talking about 'returns' and 'expectancy' in absolute terms (not % returns) then of course the one with the higher DD/Net ratio is by FAR SUPERIOR, because you can more aggressively size your positions... In other words, the one with the higher DD/Net ratio, (regardless of whether the two produce the same amount of return on an absolute basis) will produce a much higher % ROE from my experiences (if, of course, you are sizing them with the same 'floor' in mind)... I personally loathe DD's and I always look at the downside first and then the upside. When I am sizing positions it is always with a 'floor' of 10-15% of total equity DD and then I view the upside potential... Anyone who says that DD's are irrelevant is simply an amateur, IMO. Only a fool views one side of the coin... BTW; who ever made that turtle reference -- it was 'ill' -- Dennis had some DD's over the years that would make anyone sick, although all you hear in the 'Market Lizards' book is how he turned several thousand into 200mil -- play back that same scenario 1k times and see how many times he busts. I will give up some upside to insure I never have to sit through watching 50-60% of my trading capital go out the window and most importantly insure there will almost definitely be a "tomorrow" for me! Of course this is all just my humble opinion, for I am no mathematician
You're right that the equity curve preference is personal. From a managed money standpoint few managers would argue that is moot, however. Investors have fled funds with erratic equity curves and shown a preference for a smoother equity curve. I think the orginal post was in regards to a higher win % or a larger win size being better. The premise of a smoother equity curve because of higher number of wins was proved to be a false assumption. I'm interested in this because if the higher % winner turns out to do better in terms of reward:risk and/or smoother equity curve on a universal basis, I may have to rethink how I've been allocating money to my trading ideas. At any rate it's been fun and something to pass the time while trading today. Hopefully they'll be another fun thread to do some brain exercises soon. Take care