Money management - the square root rule 

Many traders believe they should invest a fixed percentage - such as 1%, or for the more daring, 2% - of their account balance per trade. This is one of the most common reasons of account crashes and unexpected margin calls, especially with automated systems. Here's why:

The maximum drawdown of any trade system increases over time. The longer you trade, the higher is the probability of a long loss streak and the bigger the depth of the drawdown. That's why a system, tested over 10 years, has a worse maximum drawdown than the same system tested over only 5 years. When modeling drawdown depth mathematically with a diffusion model, the maximum drawdown of a break-even system is proportional to the square root of the number of trades, and therefore also to the square root of the trading time. This also means that drawdowns have no limit. A trading system will suffer a drawdown of any depth when you wait long enough.

Drawdown also increases with the invested amount: When investing twice the volume you'll get twice the drawdown. Thus, when you reinvest a fixed percentage of your balance, the maximum drawdown grows with the balance. And the balance of a profitable system also grows proportional to the trading time.

When summing up both effects, you'll get an overproportional drawdown growth with trading time: the drawdown grows proportionally to time to the power of 1.5. The 1 comes from the reinvested profit, the 0.5 from the square root of the number of trades (in fact the exponent will be slightly higher than 1.5 as reinvested profit also grows overproportionally, but that shall not bother us here). In any case your drawdowns will grow faster than your account balance. At some point, a drawdown will inevitably exceed the balance, causing a margin call. That will happen later or sooner, dependent on the system and the reinvested percentage.

Therefore better don't invest a fixed balance percentage, no matter how often it's recommended in trading books or seminars. There are several methods to overcome the drawdown growth issue. One method is to reinvest only an amount proportional to the square root of the capital growth. Thus when your capital doubles, increase the trade volume only by a factor of about 1.4 (the square root of 2), i.e. 40%. Example: You're trading with a Margin of $50. Your account doubles from an initial $1000 to $2000. You can now increase your Margin to $70 (= 1.4 * $50) for reinvesting your gain.

Another method is investing a variable percentage - for instance the OptimalF factor - that is calculated from the real equity curve and regularly updated so that it decreases when the drawdowns increase. In both cases, the drawdowns of your system will then only grow at the same rate as your account balance, so you stay away from a margin call. In the Black Book you can find examples of several methods for correctly (nor not) reinvesting profits.

Withdrawing profits

If you do not reinvest, but withdraw your profits regularly, keep a part of your profit on the account for the very same reason. As explained above, the expectancy value of the maximum drawdown depth grows with the square root of trading time. So your account balance must grow by the same factor for keeping pace with the expected drawdown. Thus, when your account doubled, you can remove only 60% of your profit and should let 40% stay on the account. This lets your account grow by the required factor 1.4 (again, the square root of 2).

Example: You start with a capital of $1000 and want to withdraw profit whenever the system won $300. Thus the first withdrawal is at $1300 account balance. Your investment grew by factor 1.3; the square root of 1.3 is 1.14. $1140 must stay on the account and you can withdraw $160. - Now your system made another $ 300. The account balance is now $1440, but the total growth (without the withdrawn amount) is $1600 / $1000 = 1.6. The square root of 1.6 is 1.265. $1265 must stay, and $175 can be withdrawn.

What if you want to withdraw not the whole amount, but reinvest the rest? Here are some simple formulae that help you calculate what you can withdraw, what you can reinvest, and what should remain on the account:

Balance on account:  C+P-W
Must stay on account: C*f
Available to withdraw:   C+P-W - C*f
Available to reinvest:    C*f - W/f

where C is your initial capital at the first start of the system, P the total profit so far, W the total withdrawal so far (including what you're just withdrawing), and f the square root growth factor sqrt(1+P/C). Example: You start with a capital of $1000 and won $300, so your balance is now $1300. How much capital do you have for reinvestment when you withdraw $50?

Balance on account:  C+P-W = $1300
Must stay on account: C*f = $1000 * sqrt(1+$300/$1000) = $1000 * 1.14 = $1140
Available to withdraw:   C+P-W - C*f = $1300 - $1140 = $160
Available to reinvest:    C*f - W/f = $1140 - $50/1.14 = $1096

So when withdrawing $50 from a $300 win, you can increase your investment from $1000 to $1096 (not to $1250 as you might have expected). Consider the difference a tax that you pay to the god of statistics. Unfortunately you'll have to pay a real tax for that, too...

See also:

Win/Loss, OptimalF, Markowitz, Performance, Tutorial


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