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Allocation

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All who traded through this last year have accumulated war stores or battle scars. Hopefully, for our subscribers, those war stories won't include the one about how you ran through your entire trading account. The battle scars won't be so deep that you've lost the ability to trade.

If you've survived this past year with your trading account intact, it's time to talk about how much traders should allocate to their trades. The easy answer? Not more than they can afford to lose.

The harder answer becomes a lot harder. That answer involves so many variables. Some trading experts believe that answer begins with setting up each trade so that it comprises a certain percentage of account equity. When the account is growing, the size of trades increases. When the account's value drops, so does the money invested in each trade.

This means trades comprise bigger dollar amounts when trades are going well, and account balances are swelling; smaller when they're not, and account balances are dropping. Sticking to a certain percentage of the account equity for each trade guarantees that traders are putting less at risk when trades appear less likely to succeed, either due to changing market environments or distractions in traders' lives that cause mistakes. I've heard trading professionals suggest anything from two to five percent of total trading equity being put at risk for each trade.

If only it were that easy. This article would be finished and subscribers would have fewer decisions to make.

It's not that easy, of course. By now, most subscribers will have already spotted one missing input in the decision-making process. Even if traders allocate only two percent of equity for each trade, they'll eventually blow through trading accounts if all of their trades are losing ones.

Notice that I didn't say "if a majority of their trades are losing ones"? Surprisingly, to some, traders can grow their accounts even if their trades are successful less than half the time.

How? Take the example that Christian B. Smart, PhD did in "Fixing the Flaws in Fixed-Fractional Position Sizing" (STOCKS & COMMODITIES, AUG 2007, pg. 32). He posed the instance of a trader whose system is successful only 40 percent of the time. If that trading system requires that losses be half of gains, Smart calculates that the "system expectancy," defined as the expected or average amount the trader anticipates making for each dollar at risk in a trade, to be $0.20 or 20 percent. The calculation is as follows, with the 40 percent winning percentage converted to the decimal 0.40 and 60 percent losing percentage converted to 0.60:

0.40*2 - .60*1 = 0.20

This means, Smart reminds readers, that traders would expect to get a return of 20 cents for each dollar put at risk. Another way of thinking of it is that the traders would expect a return of 0.20 times each amount put at risk.

Imagine that a trader's system met those parameters, including 40 percent of trades being profitable and limiting the losses to half the gains. Then that trader enacts a practice of putting four percent or 0.04 of the trading equity into each trade. If that same system expectancy or 0.20 multiple is used, the trader would expect on average a return of 0.008 or 0.8 percent of the account equity for each trade.

If that trader's account equity was $5,000 account, the trader would put $200 ($5000 x 0.04 or 4 percent) at risk on a first trade and expect a return of $40.00 ($200 x 0.2 system expectancy) for the first trade. The trader's account would then equal $5040, and the return on the next trade would be expected to be $40.32. The account would then equal $5080.32, and it goes on and on. Smart offers a more complicated formula, a geometric mean, to show the expected account values after N trades. Supposedly, the trader could plug in the number of trades and calculate what the account equity could be expected to be after that many trades.

It's still not that easy. In real-life trading, the system would always underperform, not producing the expected equity, Smart concludes. The reason proves complicated for all but math nerds, but it involves the difference between the arithmetic mean, which is the system expectancy, and the geometric mean, which is the average amount per trade that would be achieved after N trades with each trade being a fixed percentage of the account equity. Arithmetic means are always greater than or equal to geometric means, it turns out, so the system underperforms expectations.

Smart has a fix for that, but it's still not that easy. Imagine that Trader 1 began trading early this year when many indices chopped violently back and forth as they began setting up the widest portions of the triangles many formed on their daily charts. Perhaps the trader had back tested a system and found that it had that 40 percent ratio of winning trades. However, what Trader 1 actually encountered was a period of time in which a whole lot of those 60 percent of losing trades were experienced right away, right at the beginning of the time that trader began to place live trades.

Imagine that Trader 2 had begun trading two years earlier, when markets were trending up. That trader's system had the same 40 percent of profitable trades, but that trader entered the markets with live trades at a time when eight profitable trades were encountered right in a row, a loss was taken, and then another eight profitable trades were placed. It doesn't take much imagination to figure out which trader is likely to have more in account equity after 30 trades are placed.

Computations that determine an average expected gain don't go far enough. Remember when you spent a day back in third grade math class flipping a coin while a member of your group toted up the number of heads versus tails? If so, you know that the 50 percent heads/50 percent tails ratio is often not achieved until many hundreds of coins are tossed, not after the first hundred. You would have noticed many times when six heads or nine tails in a row were flipped. That can happen to traders, too, so figuring out all those system expectancies and ratio of winning trades just isn't enough.

There's a little something called "risk of ruin," a term borrowed from gamblers, that measures how likely traders are to experience ruin, or the loss of their entire trading equity, due to those periods of drawdowns. Risk of ruin calculations for traders vary from source to source, and can be complicated or simple. One of the simplest but perhaps not most accurate, for a reason I'll explain later, can be found on page 66 of Perry Kaufman's SMARTER TRADING. Kaufman introduces the calculator by warning that the greatest risk of ruin exists at the beginning of a period of trading. Why that's true is clear from the hypothetical examples of Trader 1 and Trader 2, mentioned previously. Kaufman notes that "once profits accumulate, the chance of ruin decreases."

His formula doesn't include everything some others do, but it does illustrate a couple of important points and doesn't involve calculated mathematics. This formula calculates risk of ruin as a probability.

Risk_of_Ruin=((1-Edge)/(1 + Edge))^Units_Capital

Kaufman's description of edge differs from his use in sample calculations, at least as far as I could tell. In those calculations, he employs the probability of win as the "Edge," and that's what we'll do. He defines units of capital by providing an example. Perhaps a unit of capital might be $10,000, he suggests, allowing calculations for risk of ruin to be made for investments of 1 unit or $10,000 or 2 units or $20,000. This can be done to see how increasing the size of trades increases or decreases the risk of ruin.

Let's make some sample calculations to see how this works. Imagine that Trader 3 has a trading account of $5,000. Trader 3 wants each trade to put at risk 4 percent of the trading account or $200. That amount will now constitute one unit of capital. Trader 3's system has a 40 percent or 0.40 probability of winning or being profitable.

Risk of Ruin by this calculation for Trader 3 is ((1 - 0.40)/(1 + 0.40))^1 = 0.4286 or 42.9 percent.

That's a fairly hefty risk of ruin. This calculation has some shortcomings since it doesn't factor in whether Trader 3 keeps losses small in proportion to profits. That's why I noted earlier that it's perhaps not the best calculation to use, at least when traders are attempting to calculate their own risk of ruin. Still, we're going to stick with it because following several sample computations still shows some important points.

Now imagine that Trader 3 tinkers around with the trading system and achieves a 50 percent or 0.50 probability of winning or being profitable.

Risk of Ruin for Trader 3, with the new, improved trading system is ((1 - 0.50)/(1 + 0.50)^1 = 0.333 or 33.3 percent.

Imagine that Trader 4 also has a 50 percent or 0.5 probability of winning or being profitable and invests the same 4 percent of the trading account. Trader 4, however, has a $10,000 trading account, so Trader 4's four percent equals $400. Since our unit of capital is $200, Trader 4 would have 2 ($400/$200) units of capital.

Risk of Ruin for Trader 4 is ((1-0.50)/(1 + 0.50))^2 = 0.111 or 11.1 percent.

This tells us that the trader with the larger trading account has a smaller risk of ruin. Of course, the mathematicians among you have already noted that if Trader 3 doubles the amount put at risk with each trade but keeps the arbitrary definition of a unit of capital the same, that trader would seemingly lessen the risk of ruin to the same as Trader 4's risk, and that doesn't make sense intuitively.

Logic tells us that risk of ruin won't work that way, that this computation is too simplistic, not taking into account all the parameters and employing an arbitrary definition of a unit of capital. If Trader 3 regularly risks 80 percent of the $5,000 trading equity, or $4000, which would be 20 units of capital, the calculation would produce an even lower risk of ruin, but we know intuitively that some one with a $5000 trading account, regularly risking $4000 for each trade, is going to come to ruin sooner rather than later.

On succeeding pages, Kaufman's formulas become more complex, incorporating more and more inputs. Other writers talk about the complexity of the risk-of-ruin formulas, too. A search of the Internet turns up some risk-of-ruin calculators, although most are geared toward gamblers, and articles in magazines such as FUTURES have covered the topic, too.

However, the necessary points have been made. A conservative approach to determining how much should be risked on each trade involves risking a certain small percentage of an account on each trade. It's a type of diversification, if you will. Furthermore, the smaller the trade or the lower the percentage of winning to losing trades, the higher the risk of ruin. Risk of ruin also rises when losses are not kept small in relationship to profits, something not addressed in Kaufman's original and most simplistic formulas, but certainly addressed in all those calculators on gambling sites.

You might be drawing an unfortunate but true conclusion by now. That unfortunate truism is that the smaller the account, the higher the risk of ruin. The calculators prove it, but we already knew it intuitively. Traders with small accounts just can't hold out through a prolonged period of draw downs. Commissions eat up a larger percentage of a small account than they do of a larger account.

What can traders with small accounts do? For one thing, they can be vigilant about setting stops so that losses don't grow too large in relationship to gains. Although much of the previous discussion has not strictly applied to those who trade credit spreads or other types of combination trades, as I do, this admonition certainly applies. How many months of profits does it take to make up for one 10-point credit spread gone bad and not exited in time?

Traders with small accounts can keep checking the percentage of winning versus losing trades. That percentage will drop when the inevitable string of losing trades occurs, but over time, it should remain near the percentage originally settled upon, if not improving.

They can determine that they will not trade unless setups are stellar. That sometimes means trading less often even in the best of times and may mean trading infrequently in the current climate.
 

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