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Managing RiskJames A. Andrews April 15, 2007
___________________________________________________________________________ Welcome to the first issue of the WiserTrader Monthly Newsletter. This issue is about defining, estimating and dealing with risk. In spite of the image on the cover, I am not expecting a bear market. The scene was irresistible as it conjures up thoughts about dealing with the unexpected. The scene also provokes a feeling of, “Danger or not, this is a place where I want to be. I must find an effective way to deal with it”. I seriously doubt if there will ever be another cover image to fit the bill. However, to the subject at hand, we need to understand unexpected risk just to survive in the market. Once it is understood, it is possible to estimate trading performance under diverse conditions and identify steps needed to transform a marginal trading system into a profitable one. ___________________________________________________________________________
1.0 What is Risk?
In this article, risk is measured in dollars. The amount of risk associated with a given trade is defined as the maximum amount that one is willing to lose before the trade is abandoned. Yes, we are talking about stop loss settings for every trade. Thus, the focus is on the current bid price rather than the last traded price. Once the bid reaches our stop loss limit, we exit the trade.
It is better to trade larger when we are more likely to be successful and smaller when the chances of success are low. Funds can be allocated to less promising trades either by using a tighter stop loss or by simply trading a smaller amount. Based on personal experience, using too tight a stop loss increases the probability that a trade will be unsuccessful. Thus, it is preferable to use a trading position size that is proportional to the likelihood that a trade will succeed and adjust stop loss settings based on the trading position size and trading system history.
There is a need to distinguish between dollar risk and the so-called, emotionally charged, risk of failure. To emphasize the difference, we can refer to the probability of success for each trade. This is the probability that we will recover all our initial capital, including fees and expenses, plus some additional profit. It is unreasonable to expect every trade to be a success. No single trade is a failure in and of itself. It takes a series of trades to determine success or failure. The probability of success for each trade can be ranked relatively high, moderate or low. Once a means for defining these probabilities is established, this ranking can be used to adjust the trading position size.
We can refer to the adjustment of trading position size based on the probability of success and the adjustment of stop loss settings based on trading system history as money management. All of the money management concepts described below are included in a calculator located in the Private Archive. |
The Greatest Risk of All
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The greatest risk is to run out of trading capital. This is known as the risk of ruin, a condition to be avoided at all costs. ___________________________________________________________________________
When a catastrophic drawdown occurs, the game is over. Thus, preservation of capital is the highest priority. Without a risk model, a trader tends to place profits ahead of capital preservation. After a series of successful trades, the amount allocated to each trade tends to grow steadily larger. The result is often a severe drawdown that puts the trader out of the game. An understanding of the relationship between trading system performance, position size and risk is needed to minimize the probability of catastrophic loss, or the risk of ruin1. The probability of its occurrence is calculated below.
Optimum Amount to Risk
The exact amount to risk on a trade is the big question in all money management systems. Risk too little and your money will not grow. Risk too much and the drawdown will put you out of business. In between too little and too much risk is a range where capital will grow at its theoretical maximum safe rate. Leibfarth1 and Kelly2 provide estimates below for an optimal risk percentage of total capital where risk levels or stop loss settings can be set based on trading system performance and position size. The Leibfarth model is more conservative. Using the two calculations, one arrives at a range of values for appropriate stop loss settings.
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The key to profitable execution of a trading system is to choose the appropriate position size and stop loss setting. Calculation of the risk of ruin is explained in Section 3. Before that, a quantitative definition of trading system performance in terms of four variables is given in the next section. With this model, it is possible to calculate the impact of position size and stop loss settings on the risk of ruin. This information also allows one to estimate how much capital can be expected after a given number of trades1, as explained in Section 4. This is followed in Section 5 with Kelly and Leibfarth estimates of the optimal risk percentage. Section 6 contains an example of how one definition of the probability of success in a trend following options trading system might be applied to setting position size.
When these methods are used properly, risk is tightly controlled with built-in adjustments that transform a speculative activity into something resembling a well run business. A word of caution is in order. As with all statistical methods, the laws of large numbers prevail. This has both a positive and a negative side. While the trade size may vary according to the probability of success, over a large number of trades, the traded amount will approach an average percentage of total capital. The average margin of profit can be quite small a still have a successful trading system. However, the trading period must be short or the size of trades must be small enough to allow a large number of trades. The principle is the same as that used by a gambling house. Given a sufficiently large number of turns with only a slight edge, the house cannot lose. On the negative side, there is no guarantee on what order winning and losing trades will occur. Although the theoretical probability of being wiped out may be low, it is possible to string several losses in a row that could prove disastrous.
1. Lee Leibfarth, “Measuring Risk”, Technical Analysis of Stocks and Commodities, November 2006, pp21-26
2. L. J. Kelly, “A New Interpretation of Information Rate”, Bell System Technical Journal, July 1956 |
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2.0 Trading System Performance
The success of a trading system depends on choosing the right direction of price movement, by a useful amount, within a practical amount of time. The performance of a trading system can be defined in terms of its expectancy or average profit per trade FP, given by,
FP = FG FC + FL (1 - FC) (1)
Where, FP is the expectancy or average fractional profit per trade. FC is the fraction of winning trades. FG is the average fractional gain for winning trades. FL is a negative number equal to the stop loss or average fractional loss for losing trades.
A system must have a positive expectancy in order to be profitable. That means that gains need to be consistently greater than the amount risked.
FG FC > |FL| (1 - FC) (2)
It appears from equations (1) and (2) that there are unlimited combinations of values for FC, FG and FL that will produce a positive FP. However, FC and FG are determined by a trader’s history of selecting and executing winning trades and his average profit per winning trade with a given trading system. Based on these historical factors, the available values for FL become limited. |
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Example 1
A trader uses a system that enables him to select winning trades 55% of the time. The average profit for winning trades is 35%. His average stop loss setting is -20%.
The resulting expectancy or average long-term profit per trade from equation (1) is,
FP = FG FC + FL (1 - FC) = (0.35) (0.55) + (-0.2) (1 - 0.55) = 0.1025 or 10.25%
Example 2
With the system in example 1, the widest stop loss setting that can be used before long-term profits fall to zero can be calculated from equation (2).
|FL| < FG FC/(1 - FC)
|FL| < (0.35) (0.55) / (1 - 0.55)
|FL| < 0.4278 or 42.78%
With a stop loss setting of -42.78%, the system profit falls to zero. If the average stop loss is wider than -42.78%, the system steadily loses money over time.
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3.0 Risk of Ruin
Even with a good system and a narrow stop loss setting that provides a positive expectancy, it is statistically possible to encounter a long string of loses in a row that wipes out available capital. The probability of a large drawdown, which is frequently ignored, needs to be minimized. The risk of ruin is given by,
ROR = 1 / (1 + FP ) 1 / FR (3)
The exponent in the denominator 1 / FR is the ratio of the amount of initial capital IC to the amount at risk IR. Here it is assumed that as the amount of capital either accumulates or declines the ratio IC / IR remains the same. The fraction of total capital at risk FR is equal to the fraction of total capital per trade times the fractional stop loss |FL|. |
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Example 3
Using the system of example 1 with an expectancy FP of 0.1025, the trader chooses to commit 20% of his capital to each trade and risk 20% of each trade using a stop loss. The Risk of Ruin is calculated as follows. FR is the fraction of total capital per trade times the fractional stop loss, (0.20) (0.20) = 0.04 and 1/FR = 25.
ROR = 1 / (1 + 0.1025) 25 = 0.0872 or about 9%
The odds are 9% or 1 in 11 that a continuous string of losses will wipe out available capital even if the system and stop loss rules are followed exactly. If the fraction of available capital traded is increased from 20% to 30%, the probability of ruin doubles to 20% or 1 in 5.
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4.0 Predicting Profits
The amount of capital available after N trades is based on a variation of compound interest.
A = P (1 + FT FG)W (1 - |FL| FT) L (4)
A is the total amount of capital after N trades. P is the initial principle before any trades are made. W is the number of winning trades. L = N - W, the number of losing trades FT is the fraction of capital used per trade
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Example 4
Making one trade per week or 52 trades in one year, how much capital is there after 1 year using the above trading system if 20% of available capital is used per trade.
An initial capital value of unity can be used, P = 1.00 The number of winning trades, W = FC * N = (0.55) (52) = 29 The number of losing trades, L = N - W = 52 - 29 = 23 FT = 0.20 FG = 0.35 FL = -0.20
A = P (1 + FT FG)W (1 - |FL| FT)L
A = 1 [1 + (0.2)(0.35) ]29 [1 - (0.2)(0.2) ]23 = 2.70
Capital would increase by 2.70 - 1.00 = 170% after 52 trades, assuming that a catastrophic string of losses does not occur. Although the risk (probability) of ruin is only 9%, it is possible for the 29 wins and 23 losses to occur in any sequence.
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5.0 Optimal Risk Percentage
Two formulas are given for the optimum percentage of capital to place at risk, ORP. The results can be applied both as a percentage of total capital per trade (not all of which is placed at risk) and as a percentage of any given trade placed at risk by means of a stop loss.
Leibfarth provides the more conservative estimate of Orp,
ORP_Leibfarth = FP FG / |FL| (5)
The Kelly formula for Orp usually provides a larger value,
ORP_Kelly = FC - (1 - FC) / (FG/|FL|) (6)
In both formulas, |FL| is the historical average percentage loss for losing trades. It may also be equal to the historical stop loss percentage. The Leibfarth and Kelly results form a bracketed range for the stop loss percentage to use for each trade.
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Example 5
Calculate the Kelly and Leibfarth estimates of the optimum risk percentage for the trading system from Examples 1 through 4.
The Leibfarth model:
FP = 0.1025 FG = 0.35 |FL| = 0.20
ORP_Leibforth = FP FG / |FL| = (0.1025) (0.35) / (0.20) = 0.179 or 17.9%
The Kelly Model:
FC = 0.55
ORP_Kelly = FC - (1 - FC) / (FG/|FL|) = 0.55 - (1 - 0.55) / (0.35/0.20) = 0.293 of 29.3%
Based on the interpretation used for individual trades, the historical 20% stop loss setting and 20% of capital per trade are close to the lower end of the range for optimal risk percentage.
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6.0 Position Size & the Probability of Success
The probability of success is estimated based on the type of trading system used. This can be most easily illustrated for a trend following trading system having a history that was detailed in the examples above. For a short term trend following trading system, a trader might be inclined to trade, not only in the direction of a stock’s trend, but also in the direction of the major averages since stocks move with significant movements of the major averages for short durations. The direction of significant shifts in the major averages over a short span of days can be quite important. The major averages can trend upward, downward or sideways. On top of this motion is a possible cyclical movement, which will be ignored for this discussion.
In order to assess how the market is trending, one needs a consistent and objective way to measure movement. One can use the ADX trend indicator that can be adjusted for the expected trading period. For trades lasting a week or less, a 10-day indicator can be used. It is best to use indicators for which the mathematics is well understood. A trend can simply be defined when the cumulative 10-day percentage gain for an index is greater than 1% for an uptrend or less than -1% for a downtrend. If the change is between 1% and -1%, the market is moving sideways.
Consider that the major averages have a historical upward bias. For example, over the first three months of 2007, the averages trended upward 53% of the time, sideways 28% and downward 19% of the time. Based on this information, an upward trending market is assigned as having the highest probability of a successful trade with a long position. The trader might commit 20% of his capital to a long trade in an up trending market because it occurs most often.
The probability of a reversal from one day to the next is greatest when the market moves sideways. Stocks have a tendency to revert to their mean in a sideways market. This most treacherous behavior gives a sideways market the lowest probability of success in either a long or a short trade. A trader might decide to commit only 10% of his capital per trade in a sideways market. The trader might decide not to trade at all in sideways markets.
Therefore, the down trending market for short trades ranks second in the probability of successful trades. For a down trending market, a trader might decide to commit 15% of his capital for any short position.
Table 1 illustrates the effect the above reasoning on position size as a percentage of total capital for the first quarter 2007 market. The hypothetical capital allocation in Table 1 would change with market conditions. |
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Table 1 Position Size versus Probability of Success |
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Trade \ Condition |
Up Trending Market |
Sideways Market |
Down Trending Market |
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Long (or Call option) |
Moderate probability Trade size: 20% |
Lowest Probability Trade Size: 10% |
Lowest Probability - no trade - |
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Short (or Put option) |
Lowest Probability - no trade - |
Lowest Probability Trade Size: 10% |
Medium probability Trade Size: 15% |
7.0 Summary
Money management is the most important part of successful trading. It evaluates the risk and reward of a trade and tells you how much money to place at risk. It is the difference between great trading performance and poor performance. It will make the difference between making money and going broke. The use of a stop loss along with careful position sizing is needed for effective money management. Neither used alone will suffice.
Two systems from gambling theory are the Antimartingale system and the Martingale system. The Antimartingale system has you increase your risk when you win and it decreases your risk when you are losing. The Martingale system increases money at risk during a losing streak. After a loss, you increase the money on the next bet. The assumption is that after a string of losses you will eventually win. Both systems work. The problem with the second system is drawdown.
The model described above belongs in the Antimartingale category. It shows how to quantify statistical trading system performance, estimate profitability and minimize the risk of a drawdown based on trading position size and stop loss settings. As with all statistical models, the law of large numbers makes the results more valid. The numbers work best over a large number of trades. For the short term, a string of losing trades is quite possible. For these reasons, trade small and often. The money management concepts described above are included in a calculator located in the Private Archive. |
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