Precision Money Management
This article describes the model of a natural relationship between trading system performance, trade position size, stop loss settings and profit goals. The model consists of algebraic equations that specify the trade size and stop loss settings needed to meet profit goals over a specified time period for any consistently used trading system for which historical performance data is available.
Most of us think of a trailing stop loss when the term money management is mentioned. William O'Neil in his book, "How to Make Money in Stocks", used a value from 7 to 8%. Many stock advisories, including Stansberry and Associates, Outstanding Investments and the Oxford Club, typically use a 25% trailing stop loss. Option advisories use still higher values in the 35% range, as is done by Michael Lombardi, and up to as high as 50%, as used by Dr. Stephen Cooper. Trailing stops are typically used along with a maximum percentage of capital per trade to avoid large portfolio draw-downs in the event that a given trade goes badly.
Beyond this precaution, there is little theory to explain how position size and trailing stop losses should be arrived at, leaving the impression that they can be arbitrarily chosen based on one's risk comfort level. However, this is not the case. Too narrow a stop loss setting can eat into profits by exiting volatile trades too early. Too wide a stop loss setting can eat into trading profits by consuming too much capital. A systematic way is needed to choose an optimum position size and stop loss setting to achieve a precise level of money management.
Intuitively, the higher the success rate in correctly choosing the direction of trade and the higher the average gain per trade, the looser one can afford to set his stop loss. However, when one has a specific earnings goal, this relationship needs to be more precise. Fortunately, the availability of consistent trading system performance data allows the use of an engineering approach. This approach enables us to define a very precise relationship between the average return for a series of trades, the percentage of correct choices in the direction of a trade, the size of each trade, profit goals and the appropriate stop loss settings.
The model introduced here for precision money management is based on average values of historical trading system performance and is only applicable when a trading system is consistently followed. The model should not be applied to unstructured trading across a variety of instruments requiring varying trading techniques. Each trading system or technique generates a unique set of statistics to which this methodology can be applied on an individual basis.
The model is derived based on fractional averages from information readily available to anyone that uses a trading system consistently. A pair of concise algebraic relationships evolves in the process. Finally, examples are provided to show the roles of position size and stop loss settings in meeting profit goals.
FP is defined as the average fractional profit for all historical trades being taken into consideration. FP is equal to the sum of the fractional gains and losses for all trades divided by the total number of trades N,
FP = (sum of fractional gains + sum of fractional loses) / N
In order for this to be valid, each trade must involve very close to the same amount of capital that we will assign an average value C. For example, if there were 3 historical trades resulting in +25%, -15% and +30% gains, the average fractional profit would be (0.25 - 0.15 + 0.30)/3 = 0.133. Of course, a much larger statistically significant number of trades would be used in practice.
Since the sum of fractional gains is equal to the number of gains NG times the average fractional gain FG, and the sum of fractional loses is equal to the number of loses NL times the average fractional loss FL, the definition can be expressed as,
FP = (NG FG + NL FL)/ N
It is understood that NG + NL = N. The value of NG divided by N equals FC, the fraction of trades chosen in the correct direction. NL divided by N equals (1 - FC), the fraction of trades chosen in the wrong direction. So N divided into NG and NL leaves the following form.
FP = FC FG + (1 - FC) FL . . . . . . . . . .(1)
FP is the average fractional profit for N trades that each uses an average amount of capital C
FC is the fraction of trades chosen in the correct direction
FG is the average fractional gain for NG winning trades
FL is the average fractional loss for NL losing trades
The fractional quantities can each be expressed individually as percentages but they should be expressed as decimal fractions in the equation.
In order to use equation (1), a profit goal must be established over a definite period of time. The profit per trade needed to meet a specific profit goal in a given amount of time depends on the number of promising trades likely to be identified by the trading system over that time period. The number of promising trades that become available within a given time period must be estimated judiciously because the last thing we want to do is force a trade under less than ideal conditions. In other words, we need to remain true to whatever system we are using.
For N trades each valued at an average capital amount C, the average fractional profit can also be defined by the total dollar profit goal DG divided by the dollar sum of all N trades DS,
FP = DG / DS
Since DS is equal to the average capital amount C times the number of trades N, this becomes,
FP = DG / (C N) . . . . . . . . . .(2)
Let us suppose that we have done a sufficient number of trades using our system to determine that the average fractional profit is 10%, the average gain per trade has been 29% and the fraction of times we chose the correct trading direction was 70%. Further let us set a goal to earn $3,000 per month. By our estimate, we figure that we can safely enter an average of 3 trades a week and remain within trading system guidelines. This equates to 3 trades per week times 4.33 weeks per month or an average of 13 trades per month.
Variables: FP = 0.1
N = 13
DG = $3,000
FC = 0.7
FG = 0.29
Solving equation (2) for C gives us the average size of each trade,
C = DG / (FP N) = $3,000 / [(0.1) (13)] = $2307.69 for the average size of each trade
Rearranging equation (1), the average stop loss setting FL must be,
FL = (FP - FC FG) / (1 - FC)
= [0.1 - (0.7) (0.29)] / (1 - 0.7) = - 0.3433 or -34.33%
Using essentially the same situation, we can look at what the effect of certain improvements in trading would have on the profits. Say we habitually exit winning trades too early and could possibly increase the average fractional gain FG from 29% to 36%. From the same relationship used for example 1, the resulting stop loss setting FL could then be widened to,
FL = (FP - FC FG) / (1 - FC)
= [0.1 - (0.7) (0.36)] / (1 - 0.7) = - 0.5066 or -50.66%
Let's suppose that for a series of potentially high yielding trades we know that an extra wide stop loss setting of -60% is needed and we want to know what the effect will be.
First we might want to look at the effect of a wider stop loss setting on profits with everything else remaining constant. We do this by equating the right sides of equations (1) and (2) and solving for DG,
DG = (C N) [FC FG + (1 - FC) FL] . . . . . . . . . .(3)
= ($2307.69) (13) [(0.7) (0.29) + (1 - 0.7) (-0.6)] = $689.99
Clearly, our original monthly profit goal of $3,000 can not be met without some additional changes, such as an increase in the number of trades from 13 to 57 over the month period. But this is not feasible since it was already estimated that the maximum number of trades identified by the trading system would be only 13 per month.
Next, since the trades in example 3 are believed to be potentially high yielding trades, we might look at the increase in the fractional gain per trade FG needed to justify the wider stop loss setting of -60% and still meet the original profit goal. By rearranging equation (1),
FG = [FP - (1 - FC) FL] / FC
= [0.1 - (1 - 0.7) (-0.6)] / 0.7 = 0.4 or 40%
So the average fractional gain for winning trades FG would need to increase from 29% to 40% to justify a widening of the stop loss from -34.33% to -60%, keeping everything else the same while meeting the monthly profit goal.
The foregoing examples give insight into trading system characteristics that affect position size and stop loss settings. Narrow stop loss settings imply a smaller fraction of trades chosen in the correct direction or a smaller fractional gain for winning trades. Wider settings imply the opposite. Stop loss settings should not be arbitrarily set independently of position size, trading goals and trading system performance. Stop loss levels more or less define future profits for a given set of trading rules, whether the user realizes it or not. While it is laudable that traders are encouraged by their advisors to adopt money management, the recommendation of a specific stop loss value without knowing the profit goal and average position size can be misleading. When a trading system is used consistently, this model enables precise money management.