We are all at risk!

We are all, theoretically, at risk of total financial ruin. Imagine you find a system with a 99.9% success rate, an average per-trade gain of 50% equity, and an average loss of only 0.01% of equity. You would use it, right? So would I! After all, the expectancy would be: .999 * 50 - (0.001 * 0.01) = 49.95% average equity gain on every trade!

BUT, it is not impossible for you to pick this system, and empty your account without making a penny. Think about it… you don’t expect to flip a coin and get tails 100 times in a row, but you know that it’s theoretically possible, right? Consider, too, that the average loss of 0.01% of equity is the AVERAGE loss. In theory, you could use the above system and blow your whole account on one trade. How likely this is depends on the variance associated with that average.

What can we learn from this line of thinking? Well, let’s start by getting it straight in our heads that, almost all the common statistical measures of trading success are only accurate under two conditions:

• You make an Infinite number of trades
• You can trade with a negative account balance

Obviously, neither of these conditions match reality. So, they can easily mislead you. The above example’s amazing expectancy is only guaranteed to materialize as the number of trades I make approaches infinity, and if I could somehow still trade with a $10 account.

Does this mean expectancy, and other such measures, are useless? No, but it does mean we need to take extra steps help ensure that our trading account doesn’t vanish. A small account doesn’t have the padding to weather long strings of losses. So early on, it can even make sense to choose a worse-performing system that has better-bounded worst case behavior. Right now, I’m thinking of three specific steps:

• Always risk a % of current equity
• Prefer systems with higher win rates and lower avg losses
• Prefer systems with less variance in the average loss data

1. Always risk a % of current equity

Clearly, if you want to stay in this game, you have to make money in the long run. But, you also need to never go broke in the short run. I’m going to define broke as $25k, since at that point your pattern day-trading account will be frozen. The conclusions are the same if you define broke to be $0.

Let’s say you start with $100k, and you risk a fixed 2%, or $2k, per trade. Your account will be frozen if you lose 38R. In the worst case (assuming away slippage), this means losing 1R 38 times in a row. (100k - 38*2k) = 24k = busted

Now assume you start with $100k, and you risk 2% of current equity per trade. This way, you would have to lose 69 times in a row before your account is frozen. (100K * (0.98)^69) = 24.8k = busted

Obviously, it is more likely you will go bust with a fixed risk. In the coin toss analogy, 38 consecutive tails is much more likely than 69 consecutive tails. (Again, never forget that 100 million consecutive tails is still possible, no matter how unlikely).

2. Prefer systems with higher win rates and lower avg losses

When talking about expectancy, it’s common for people to point out that you can win less than half the time, and still make money. They should really add the qualifiers “in the long run, if you don’t go broke first.” We’ll start with an absurd example, and then get more realistic:

• winRate = 3%
• avgGain = 200%
• avgLoss = 4%

… expectancy of 2.12 % equity gain per trade. Better than many systems! But, not usable in practice. At 4% avg loss, it would take 34 consecutive losing trades to bust a $100k stake. With a 3% win rate, there is (1 - 0.03)^34 = 35% chance that this will occur. WAY too risky. In general, the probability of busting an account without any wins is: (1 - winRate)^( (ln (25000/initialStake)) / (ln (1 - avgLoss)) ). Incidentally, that exponent will be the number of consecutive losing trades needed to bring down the account.

So, when comparing any two systems (or versions of the same system, if you are building one), it can be helpful to compare their probability of immediate ruin as well as their expectancy. In general (due to the equation above) higher win rates and lower average losses will perform better on the immediate ruin test.

Certainly, if two systems have the same expectancy, then you should prefer the one with the smaller probability of immediate ruin. That goes without saying. But, what about these two systems:

• winRate = 40%
• avgGain = 2%
• avgLoss = 1%

… expectancy of 0.20% equity gain per trade.

• winRate = 60%
• avgGain = 0.6%
• avgLoss = 0.5%

… expectancy of 0.16% equity gain per trade.

After 500 trades with a $100k stake, the first system is expected to make about $50k more than the second. But, it is also 2.8 x 10^79 times more likely to burn an account straight down to the $25k freeze point without winning once.

Now, at this point, you may do the math and point out that the probability of immediate ruin on the first system is only 2.5 x 10^-29 % in the first place. Isn’t that good enough that you wouldn’t want to give up $50k expected gains for more protection? I actually don’t know yet. It seems to me like the only way to answer would be to work out all 2^500 strings of trades and get a probability distribution of the outcomes. I can only say that minimizing that straight drop path’s probability should also reduce the probability of the other account busting paths, since all losing paths will have at least as many losses as the straight drop.

3. Prefer systems with less variance in the avg loss data

With the analysis we just did, on the probability of the account going straight down, we assumed we always experienced the average loss. This is another one of those statistical generalities that can mess you up. If the variance in the loss samples is large, then the average loss across a small number of trades could be way out of line with the overall average loss. Not only would that somewhat invalidate our supposed “worst case” analysis, but it could decimate our trading account!

For example, an average loss of 3% equity could have come about from either of these two strings of losses, for instance:

• string1 = 10% 5% 1% 1% 0.7% 0.25%
• string2 = 3% 4% 2% 3% 4% 2%

Across the first 2 losses, though, string1 averaged 7.5%, while string2 was still pretty much in line with 3.5%. And that’s just the difference between using 2 samples and 6! If 100’s of losses went into the expectancy calculation, then any string of 5 losses could could hurt your account far worse than the average would indicate.

As it happens, one way to remove some of the variation from the losses is to prefer systems that incorporate stops (lots of systems don’t!). This puts a cap on the loss amount, which has a side effect of making losses much more predictable. Further, in any worst-case analysis, you can assume all losing trades hit the stop, rather than assuming the average loss. That’s a very safe and conservative path to follow.

Why don’t all systems have stops, if they have so many desirable properties? Well, it would appear that lots of winning systems require you to ride out substantial paper losses on some of their trades, prior to exiting at a profit. Note that this is also why you can’t just add stops to a system that doesn’t have them… they wouldn’t be nearly as successful if they were stopped out in those cases.

But, for the small ($100k or less) account, I’m starting to think that systems without stops should be deemed too risky to use, no matter what the expectancy is. Small accounts just can’t take enough oversized losses to make it. My backtesting seems to bear this assumption out.