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Jim on 'fuzzy math'

I think there has been some misleading math being shown in this forum. I'm not saying anyone is purposefully misleading anyone, only that perhaps it's a case of having just enough mathematical knowledge to be dangerous. Here's an example. A common amount of an account to risk on a given trade is 0.5%. I saw a debate in here about risking 50% per trade. I know there may be some merit, for example purposes, to look at that example. But it can't be used in any way, shape or form to debate trading reality. If you actually bet 50% of your account per trade you're crazy (and I'm sure we all agree on that), end of story. 0.5% is a conservative amount. Now, if I traded three times per day, and I lost every single trade in a row, how many days would it take to draw my account down 99%, that is, to 1% of the original value? Well, it would take 918 losing trades in a row to do this, or about one and a quarter years of straight losses. First off, let me tell you that I am not too worried about that happening. Over a year and I haven't had a single winner (wow, quite a trading edge I must have had, huh?), and only now my account is down to 1%? Yep. (This assumes I can still open trades as the account value gets very low, which may not be possible in the real world, but it is an acceptable assumption for this example). But here's where it gets wild, as far as the math and the erroneous assumptions.

So, what does this imply as far as odds? If I trade for one and a quarter years I must blow out my account? No, that's silly. This is how many losers in a row I'd need to blow the account out. Let's say I'm no better than a coin flip in my winning percentages, 50/50. If I don't have an edge there would be no point to trading, so we must assume I have an edge as far as reward/risk goes, since I'm 50/50 on the winning percentages (but still, this is all irrelevant to the upcoming calculation, which is only going to look at percentage risked per trade and the case of the 918 losing trades in a row, and the odds of that happening). The argument I have seen isn't that it is impossible to have an edge, it is more that even with an edge, the random streaks that would naturally occur will necessarily blow your account out, every time. Money management decreases those odds to the point they can be much less than being struck by lightning. I still go outside even though in a given year my odds of being hit by lightning is about 1 in 600,000-700,000. What are the odds of being killed in a car accident, with perhaps almost 40,000 fatalities per year in the U.S.? But we still drive, and the small odds keep very few from driving. I contend the odds of blowing out an account, using the above parameters, based solely on a random streak of losing trades, is less than being struck by lightning or getting killed in a car accident.

See, here's where the wildly faulty and erroneous assumptions come in. If 918 losers in a row would draw the account down to 1% (I chose 1% for the example, but until you get down to a penny, you can keep going, and that would be a lot more than 1 in 918, but you need some minimum to make another trade, so let's say 1% and you are done), are my odds 1 in 918 that I'll go bust from this random drawdown, then? Nope, not even close. That's the erroneous assumption based on a misapplication of the mathematics. The talk is that if you take 918 traders who are risking 0.5% per trade, one will go bust in that one and a quarter years (I can tell right now all the traders in here are saying that's plenty good enough for me, you can stop right now). That's simply not correct. In fact, it's so far from correct that you won't even believe the 'real' mathematics here.

You see, in this 50/50 scenario, you have to figure out, as I showed Joe in the other thread, what .5^918 (one-half raised to the 918th power) is, and to calculate the odds, divide that into 1. The problem is, my calculator can't handle this. I know this will get a little boring, the math details, but I want to show them, in brief, anyway, so bear with me. I'll first do some rearranging, substituting 1/2 for 0.5, and ignoring that 1 to any power is still 1 (this is just some very basic algebra here), and we get this: 1/(2^918). Since we don't have any supercomputers handy, I'll do a little math shuffling to get an estimate. Since 2^3.32 is approximately equal to 10, we can say that 1/(2^918) is approximately equal to 1/(10^276), again using some very basic mathematical principals (2^918 = (2^3.32)^276 = 10^276). That is, the odds of actually blowing out the account with approximately three trades per day, in 918 trades total, with a 50/50 win loss ratio and a 0.5% risk per trade, based solely on a random drawdown, is approximately one in 1 x 10^276, or a 1 with 276 zeroes after it. Let's see what that looks like, assuming I counted my zeroes correctly:

1 in 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

Hmmm, looks like a lot less than my worries from lightning or a car accident, as I suspected (for perspective on 10^276, one estimate I saw said there are 10^85 atoms in the entire universe! ). So, that many traders would have to ply this approach before one would blow the account out from 'random effects' in the one and a quarter years? Yep, looks like it, if my math is correct. Well, I can say one thing, those odds are the least of my worries. You see the difference between betting 50% with winning percentage of 50% and betting 0.5% with the winning percentage 50%? Ah, the beauty of money management in trading. Most or all the examples given use figures for money management that are insane and would never, ever be used by any consistent trader who plans to stay in trading. Let's apply some real-world money management to our examples, then we can look at the results and argue if those are odds we realistically want to accept, or not. But don't tell me when the odds are like the above that I'd be insane to trade. I don't feel insane when I drive my car, and I'm not going to feel insane when I accept odds like the above in a trading plan. I'd like people to stop looking at unrealistic examples, and to use the math not to deceive but to enlighten.

P.S. For those that like to take 'big risks' of 2% per trade (something I would strongly recommend against, so I shouldn't even show it here, since it is getting away from what I think should be used for real trading), it would take 228 straight losing trades to draw the account down to 1% using the same assumptions as above, and the odds for that are about 1 in 10^68. Hmmm, maybe I have to rethink how much risk I am willing to take, as those odds don't sounds that bad :-)

P.S.S. Let me add another thought in here. If a trading plan is net negative outcome i.e. it loses money, then it will surely draw down over time until it is busted. That goes without saying, and isn't addressed here, and needn't be addressed in this thread. The sole purpose for this post is to see what it would take, what the odds were, that straight losers would happen to draw an account down to essentially zero, as was the example and basis for argument in another thread. Accounts can be drawn down with combinations of winners and losers, in all sorts of manners. None of those scenarios are addressed here, and the complexity of the calculations goes up with the complexity of the scenarios. All I wanted to show here was how small the odds of drawing an account straight down actually are using a reasonable per trade risk amount. All calculations are done for fun and educational purposes, and aren't to be taken as a recommendation of a trading style, or that any trading style must be profitable because of any calculations I have shown. The point was to show that of all the worries I may have about the viability of a trading plan, this type of random effect drawdown isn't one of my prime worries.
Ah, I just can't stand all the points being made in this forum that just bend math and logic rules all to heck. Say, for example, that that one uses the crazy strategy that calls for doubling down. A claim is made that that only 'works' with infinite capital. Mathematically that is 100% true. But the twisting of perception comes from 'leaving out' that you'd also have to look at an infinite time period. Without that being addressed, it makes it sound cut and dried it's a losing game, for sure. The useful calculation is what are the odds, with a given system and x amount of time, of being busted out. That's all that matters, and with some systems the odds are not that bad looking. I'm not saying I'd ever be for such a system, only that the presentation of one math fact, out of context, creates a view for the reader that is wholely inaccurate.
Sometimes I don't like knowing a lot of math. Way, way back when I was a math teacher I had to teach part of a course on deceptive use of statistics, and how by showing selected aspects of statistical analysis one could create quite a deceptive picture. Again, I'm not saying anyone is purposefully trying to be deceptive, only that by presenting selected facts out of context it is quite easy for less than the entire picture to be seen. I can see quite a few newbies struggling with what they are hearing. Let's look at the coin flip game.

I saw a game discussed that had 12 flips. The math was shown that there are 4096 possible outcomes, only one of which is all tails (heads you win, tails you lose, for simplicity here). To make the example easy, I'm going to make this a 'fair' game, such that you break even over time. Heads you win $1, tails you lose $1. Over time it's a break even game. Now, the point was made, if you play this game of 12 flips 4096 times, you bust out of the game, you lose your entire bankroll, assuming a perfect statistical distribution, which we will for the examples here. The gist of this is you'd have to be a hundred kinds of a chump to play this game 4096 or more times, as you will 'lose everything'. Now, understand, this is a finite game here, not a double down until you eventually lose or anything. Follow the game as I have set it up.

Now, I agree 100% that mathematically, as we have set it up, you will lose your entire roll once. Now, on to the context and how this can be deceptive. My point? Who cares if I lose my whole roll on that one series. This is a net neutral game (and we can't argue a trading plan wouldn't be net positive outcome, or we wouldn't be trading it) . Statistically, you can't lose over time. We decided that in the parameters. Yet, it's an issue if we 'lose it all' on one series. On one series we win the maximum amount, and that offsets the maximum loss. We have one with 11 tails, but we also have one with 11 heads to offset it. It's a 'symmetrical' series, if you will, as it all offsets and is breakeven.

Could an argument be made that if that 'total loss' happens in any place but the end you can't continue to play? No, because we would never put ourselves into a position that if we had a 'full' losing streak we'd have to quit. This would be a crazy starting position, and we'd never do it. If we did, we would be asking to be busted almost every time. But what has that to do with common sense in real-world trading? I say nothing. Picture this game simplified with just two flips. I know in the end, every time, I'll be where I started. With two flips I can have the following scenarios:

H,H Win $2

H,T Flat

T,H Flat

T,T Lose $2

Here's what the money would look like:

1, 1 = +2
1, -1 = 0
-1, 1 = 0
-1, -1 = -2

Net overall I'm breaking even. It doesn't matter if I play this with 2 flips or 200 flips, that doesn't change. So, what's the issue? To play I have to put up $1 per flip. If I start with $2 and I happen to hit T,T first, I'm busted out and I am net down $2 in a game that was net breakeven How can this be? I busted out of a fair game? And one in four tries this will happen? Overall, I'd be a loser then, and I better not play this game. Now let's start with $3. Guess what. No bust outs, not possible, can't happen, and, statistically, no matter how long I play, I'll always be right back at $3. So, by simply having a little cushion I took what was painted to me as a 1 in 4 chance of busting out to a never can bust out, not mathematically possible, game. See how easy it is to paint something one way when it can so easily be another?

If one does stupid, assinine things that violate money management rules, they will almost for sure bust out. If you do not create a plan that is sound, you will almost surely bust out. But by looking at out of context examples that clearly violate sound rules, and then using that to show 'mathematically' how trading is a losing game, it makes no sense. I am trying to show the other side, and how easy it is to be fooled by mathematical hijinks. Read what you will, decide what you will, but don't be fooled by out of context math as you try to make informed decisions.
him smart!
Probably more just goofy in the freakin' head with too much time on my hands...
I'm just sitting here enjoying this GLOBEX action and I keep thinking about all this great discussion. I was thinking about all the various permutations and examples we have all been discussing. I realized that I mentioned above about betting 50% per trade, when, on looking back on the example that got me to thinking of all that, it was a double down example, which is different. It doesn't really make any difference to the above example I did because I was looking at a 0.5% wager, which is a realistic one, and the same coin flip 50/50 winning percentage, and seeing what it would take to draw that down.
Originally posted by jimkane

I think there has been some misleading math being shown in this forum. I'm not saying anyone is purposefully misleading anyone, only that perhaps it's a case of having just enough mathematical knowledge to be dangerous.

I guess this is a jab at me. Well, I fall outside the category of knowing "just a little math." My graduate math work was done in perturbation methods (

Anyway, arguing about credentials is pointless. The best way to get one's point across is to show examples. I'll make an effort to put together a little computer program to run the coin toss simulations. I'll post both the program and its results here.

It's much easier for people to see what's going on when you can demonstrate the actual results of running 4,000 coin tosses, using different amounts of money and different betting methods.
Okay, just make sure you use realistic betting percentages and win/loss ratios. Use 0.5%, 1.0%, 1.5% betting percentages, and throw in some 70% winners, 60% winners, and 50% winners, and some 2 to 1 reward/risk ratios, some 3 to 1 reward/risk ratios, and the like, and then I'd be thrilled to see the simulations. Those are parameters that fit with successful trading plans. Anything else is not much use to us as traders, and there is no point to posting it in here.

As far as credentials, the course I taught pointed out to me that the more knowledge a person had, the deadlier they were at being able to show only that which would make the point they wanted, since they understood what would show their point and what would harm it (they didn't show all they found, they showed only what helped their point, like it was the entire story). Who would be the most dangerous killer? Probably a doctor.
If you have $1,000 and you bet 1 penny per coin flip, then of course you would expect it too take a lot of coin flips before seeing a big drawdown on your account.

But, as you well know, on a fair coin toss, placing a constant bet is pointless because the expected outcome is break even (no profit, no loss.)

Agreed. Most people wouldn't be able to tell whether your math is correct or mine.

Originally posted by jimkane

As far as credentials, the course I taught pointed out to me that the more knowledge a person had, the deadlier they were at being able to show only that which would make the point they wanted, since they understood what would show their point and what would harm it (they didn't show all they found, they showed only what helped their point, like it was the entire story). Who would be the most dangerous killer? Probably a doctor.

So, run those simulations with the parameters I mentioned above and post those results in here and that will be worth discussing. But post all the results, not just some of them. Don't just post the maximum drawdown cases, show everything. And don't bother posting results for simulations that aren't listed above. If you want to do that, post them in your section. Let's keep this an area for what seems to be realistic trading parameters.
Sorry I wasn't clear on that PT, and it was good that you caught it, in the interest of accuracy for all the readers. I dig into this quite a bit in my Trade Management book (mentioned only for reference, not as a promotion, as I do none of that in this forum) for just the reason you mention, it is so rarely discussed anywhere. Seeing the net results as a function of the expected value, the combination of winning % and reward/risk, and seeing how they are inversely related and pretty much equal a constant (all other things being equal) is a really key concept. I love it when this forum provides great info for traders. That's why I'm here. That's why most of us are here. It's great when it works :-)