Is Fixed-Fractional Position Sizing Ill-Fated?
Fixing The Flaws In Fixed-Fractional Position Sizing
by Christian B. Smart, PhD
Fixed-fractional position sizing is a time-tested method for money management, but in the long run, it will never achieve system expectancy. Here's how you can fix this flaw.
Fixed-fractional position sizing is a popular and time-tested method for money management. In the strategy, a fixed percentage of equity is risked per trade. The formula is given here as:
where f is the fixed percentage of equity risked per trade.Amount risked per trade = Equity * f
Fixed-fractional money management is an intuitive method in which bet size increases when equity increases and bet size decreases when equity decreases. This form of money management is conservative in that it dramatically decreases risk of ruin.
A concept related to money management is system expectancy. A system's expectancy is the average, or expected, amount of money an investor expects to make per dollar risked. For example, a trading system with a winning percentage of 40%, whose average win is equal to twice the average loss, has an expectancy approximately equal to 0.40 * 2 - 0.60 = 0.80 - 0.60 = 0.20.
On average, the system returns 20 cents for every dollar risked. If an investor uses fixed-fractional position sizing and risks 2% of equity per trade, then the average expected return per trade is (2%) * 0.20 = 0.40% of equity. The expected equity for an investor with $100,000 of initial risk capital is $100,400 after the first trade, $100,801 (=$100,400 * 1.004) after the second, and $100,000 * (1.004)N after the Nth trade.
With fixed-fractional position sizing, the system does not achieve this expectancy in the long run, but an amount less than the system expectancy. Risking 2% per trade in a system where all losses are the same size, all wins are the same size, and wins are twice as large as losses, equity either increases by 4% (0.02 * 2) or decreases by 2% (0.02 * 1) on each trade.
After three trades -- a win, a loss, and a win -- the account equity is increased by (1.04) * (0.98) * (1.04) = 1.059, or approximately 6%. After N trades with M wins and N - M losses, the total return is (1.04)M(0.98)N-M. In the long run, M will be 40% of N, so for sufficiently large N, the return will approximately be (1.04)0.4N(0.98)0.6N times the original equity.
With a progressive betting system like fixed-fractional sizing in which returns are reinvested, the total return is the product of a series of numbers. The average of a product of a series of numbers is the geometric mean, which is simply the Nth root of the product of N values.
For the series (1.04)0.4N(0.98)0.6N, the geometric mean is:
which reduces to 1.040.40.980.6 = 1.003573, or a return of 0.3573% per trade. This is less than the system expectancy of 0.40% with 2% risked per trade. While this may not seem like a large difference, it makes a noticeable impact after only a few dozen trades. Figure 1 contains the fixed-fractional expected return vs. the expected equity after 1, 10, 100, and 1,000 trades. The system is losing over six percentage points after only 100 trades with this form of position sizing because of this flaw.
...Continued in the August issue of Technical Analysis of STOCKS & COMMODITIES
Excerpted from an article originally published in the
August 2007 issue of Technical Analysis of
STOCKS & COMMODITIES magazine. All rights reserved. © Copyright
2007, Technical Analysis, Inc.
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