Copyright 2002 Breakout Futures Trading the Risk Position Sizing and Exit Stops Michael R. Bryant,...

Copyright Copyright 2002 Breakout Futures2002 Breakout Futures

Trading the RiskTrading the Risk

Position Sizing and Exit StopsPosition Sizing and Exit Stops

Michael R. Bryant, Ph.D.

Breakout Futures

www.BreakoutFutures.com

Copyright Copyright 2002 Breakout Futures2002 Breakout Futures 22

Scope of TalkScope of Talk

• Short to intermediate-term tradingShort to intermediate-term trading

• Rational methods of position sizing Rational methods of position sizing and stop selection; mostly and stop selection; mostly quantitativequantitative

• Oriented towards futures but also Oriented towards futures but also applicable to stocksapplicable to stocks

• One market-system at a timeOne market-system at a time


What is Position Sizing?What is Position Sizing?

• Selecting the number of contracts or Selecting the number of contracts or shares of stock for the next tradeshares of stock for the next trade

• A way to reinvest profitsA way to reinvest profits

• The way traders compound their The way traders compound their returnsreturns


Methods of Position SizingMethods of Position Sizing

• Ad hoc: trade no larger than lets you Ad hoc: trade no larger than lets you sleep at nightsleep at night

• Margin plus drawdownMargin plus drawdown

• Fixed FractionalFixed Fractional

• Fixed RatioFixed Ratio

• Hybrid fixed fractional/fixed ratioHybrid fixed fractional/fixed ratio


Methods that Don’t WorkMethods that Don’t Work

• Martingale methods: increase position Martingale methods: increase position size after a loss; decrease it after a win.size after a loss; decrease it after a win.

• Equity curve methods: increase size Equity curve methods: increase size when your equity curve falls below its when your equity curve falls below its moving average (“reversion to mean”), moving average (“reversion to mean”), or increase size when you cross above or increase size when you cross above the moving average (“trade the trend the moving average (“trade the trend in equity curve”).in equity curve”).


Why They Don’t WorkWhy They Don’t Work

• Martingale and equity curve methods assume Martingale and equity curve methods assume dependency between trades.dependency between trades.

• In most cases, trades are independent of each In most cases, trades are independent of each other. The odds of the next trade being a win other. The odds of the next trade being a win are not related to whether the last trade was are not related to whether the last trade was a win or a loss.a win or a loss.

• If trades are independent, you can’t If trades are independent, you can’t determine the likelihood of the next trade determine the likelihood of the next trade being a win or a loss based on the previous being a win or a loss based on the previous trade.trade.


Margin Plus Drawdown Margin Plus Drawdown SizingSizing• The equity to trade one contract is the The equity to trade one contract is the

maximum historical drawdown multiplied maximum historical drawdown multiplied by 1.5 plus the margin requirement.by 1.5 plus the margin requirement.

• Add another contract only when the closed Add another contract only when the closed profits are equal to drawdown * 1.5 plus profits are equal to drawdown * 1.5 plus margin.margin.

• Attributable to Larry Williams; see Attributable to Larry Williams; see The The Definitive Guide to Futures Trading, Definitive Guide to Futures Trading, Volume IIVolume II..


Margin Plus Drawdown Margin Plus Drawdown (cont.)(cont.)• You always have enough money to handle You always have enough money to handle

the worst historical drawdown plus 50%.the worst historical drawdown plus 50%.

• Designed so you only increase the number Designed so you only increase the number of contracts, never reduce.of contracts, never reduce.

• Theoretically safe but doesn’t reduce Theoretically safe but doesn’t reduce contracts in a drawdown, so drawdowns contracts in a drawdown, so drawdowns can be large.can be large.

• Doesn’t take the risk of each trade into Doesn’t take the risk of each trade into account.account.


Margin Plus Drawdown Margin Plus Drawdown (cont.)(cont.)

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Fixed Fractional Position Fixed Fractional Position SizingSizing• Risk the same fraction (“fixed fraction”) of Risk the same fraction (“fixed fraction”) of

the account equity on each trade; e.g., 5%.the account equity on each trade; e.g., 5%.• Number of contracts:Number of contracts:

N = ff * Equity/|Trade Risk|N = ff * Equity/|Trade Risk|

where where ff = fixed fraction,ff = fixed fraction,Equity = account equity ($),Equity = account equity ($),Trade Risk = possible loss on trade ($)Trade Risk = possible loss on trade ($)


Fixed Fractional (cont.)Fixed Fractional (cont.)

• Trade risk may come from:Trade risk may come from:– Estimate. Examples: n standard deviations Estimate. Examples: n standard deviations

of the trade distribution; largest historical of the trade distribution; largest historical loss.loss.

– Size of money management stop.Size of money management stop.

• Using a money management (mm) stop Using a money management (mm) stop to define the trade risk may produce to define the trade risk may produce greater risk-adjusted returns than using greater risk-adjusted returns than using the largest loss.the largest loss.



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Observations on Fixed Observations on Fixed FractionalFractional• As a percentage of account equity, the As a percentage of account equity, the

risk of each trade is the same, risk of each trade is the same, regardless of the number of contracts. regardless of the number of contracts.

• Takes advantage of trade risk.Takes advantage of trade risk.• Responsive to changes in equity (unlike Responsive to changes in equity (unlike

margin plus drawdown method).margin plus drawdown method).• The trick is determining the best value of The trick is determining the best value of

the fixed fraction; more on that later…the fixed fraction; more on that later…



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Fixed Ratio Position SizingFixed Ratio Position Sizing

• Developed by Ryan Jones; see Developed by Ryan Jones; see The The Trading GameTrading Game, John Wiley, 1999., John Wiley, 1999.

• Based on a fixed parameter called the Based on a fixed parameter called the delta: the profit per contract needed to delta: the profit per contract needed to increase the number of contracts by 1.increase the number of contracts by 1.

• Each contract contributes the same Each contract contributes the same profit towards increasing the number of profit towards increasing the number of contracts, regardless of account equity.contracts, regardless of account equity.


Fixed Ratio (cont.)Fixed Ratio (cont.)

• Number of contracts:Number of contracts:

N = ½ *[ 1 + (1 + 8 * Profit/delta)N = ½ *[ 1 + (1 + 8 * Profit/delta)1/21/2]]

where where Profit = total closed trade Profit = total closed trade profit ($),profit ($),

delta = profit/contract to delta = profit/contract to increase by 1 increase by 1 contract ($).contract ($).



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Observations on Fixed RatioObservations on Fixed Ratio

• Performance depends on total Performance depends on total accumulated profits; i.e., account accumulated profits; i.e., account size. It becomes more conservative size. It becomes more conservative as the account size increases.as the account size increases.

• Doesn’t directly depend on trade Doesn’t directly depend on trade risk.risk.


A More Generalized A More Generalized ApproachApproach• Consider the following equation for the Consider the following equation for the

number of contracts, N:number of contracts, N:

N = ½ *[ 1 + (1 + 8 * Profit/delta)N = ½ *[ 1 + (1 + 8 * Profit/delta)mm ]]

where where Profit = total closed trade profit ($),Profit = total closed trade profit ($),

delta = fixed ratio parameter ($),delta = fixed ratio parameter ($),

m >= 0.m >= 0.

• With m = ½, we get the fixed ratio equation.With m = ½, we get the fixed ratio equation.


A Generalized Approach A Generalized Approach (cont.)(cont.)

• Consider m = 0:Consider m = 0:

N = ½ *[ 1 + (1 + 8 * Profit/delta)N = ½ *[ 1 + (1 + 8 * Profit/delta)0 0 ]]

= 1/2 * [1 + 1]= 1/2 * [1 + 1]

= 1= 1

i.e., we get fixed contract trading (N = i.e., we get fixed contract trading (N = 1).1).


A Generalized Approach A Generalized Approach (cont.)(cont.)• Consider m = 1:Consider m = 1:

N = ½ *[ 1 + (1 + 8 * Profit/delta)N = ½ *[ 1 + (1 + 8 * Profit/delta)1 1 ]]= 1 + 4 * Profit/delta= 1 + 4 * Profit/delta

Let delta = 4 * Risk/ff and EquityLet delta = 4 * Risk/ff and Equity00 = Risk/ff. = Risk/ff.

Then, N = (EquityThen, N = (Equity00 + Profit) * ff/Risk + Profit) * ff/Risk(i.e., the equation for fixed fractional (i.e., the equation for fixed fractional

trading)trading)


A Generalized Approach A Generalized Approach (cont.)(cont.)• Rate of Change of N with Profit:Rate of Change of N with Profit:

∂N/N/∂(Profit) = 4*m/delta * (1 + 8 * Profit/delta)(Profit) = 4*m/delta * (1 + 8 * Profit/delta)m-1m-1

m = 1m = 1 ROC of N independent of profit; e.g., ROC of N independent of profit; e.g., fixed fixed fraction.fraction.

m > 1m > 1 N increases faster as equity grows. N increases faster as equity grows.

m < 1m < 1 N increases more slowly as equity N increases more slowly as equity grows; e.g., grows; e.g., fixed ratio.fixed ratio.



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m=1.5



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m=1.5


Conclusions From Generalized Conclusions From Generalized ApproachApproach

• m < 1 works best when worst drawdowns m < 1 works best when worst drawdowns come late.come late.

• m >= 1 works best when biggest run-up m >= 1 works best when biggest run-up comes late.comes late.

• For any sequence of trades, there is For any sequence of trades, there is probably an optimal value of m. However, probably an optimal value of m. However, the sequence of trades and the sequence of trades and drawdowns/run-ups is unknown. (Monte drawdowns/run-ups is unknown. (Monte Carlo analysis to find the best m?)Carlo analysis to find the best m?)


Finding the Best Fixed Finding the Best Fixed FractionFraction

• Ad hoc; e.g., 2% rule.Ad hoc; e.g., 2% rule.

• ““Optimal f”: Ralph Vince, Optimal f”: Ralph Vince, Portfolio Portfolio Management FormulasManagement Formulas, 1990., 1990.

• ““Secure f”: Leo Zamansky & David Secure f”: Leo Zamansky & David Stendahl, TASC, July, 1998.Stendahl, TASC, July, 1998.

• Monte Carlo simulation: Bryant, Monte Carlo simulation: Bryant, TASC, February, 2001.TASC, February, 2001.


Best Fixed Fraction (cont.)Best Fixed Fraction (cont.)

Optimal f:Optimal f:• f value that mathematically maximizes f value that mathematically maximizes

the compounded rate of return.the compounded rate of return.• Doesn’t take the drawdown into Doesn’t take the drawdown into

account.account.• Typically results in very large – and Typically results in very large – and

dangerous – f values.dangerous – f values.• Theoretically sound but not practical to Theoretically sound but not practical to

trade.trade.



Secure f:Secure f:• f value that maximizes the compounded f value that maximizes the compounded

rate of return subject to a limit on the rate of return subject to a limit on the maximum drawdown; e.g., “what f value maximum drawdown; e.g., “what f value gives the greatest rate of return without gives the greatest rate of return without exceeding 30% drawdown?”exceeding 30% drawdown?”

• Improvement on optimal f.Improvement on optimal f.• Only problem:Only problem: the drawdown calculated the drawdown calculated

from the historical sequence of trades is from the historical sequence of trades is not very reliable.not very reliable.



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DD=9.3%



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DD=16.7%



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DD=25.6%



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DD=46.2%



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DD=9.3%

DD=16.7%

DD=25.6%

DD=37.6%

DD=46.2%



• Historical sequence: 14% max drawdown Historical sequence: 14% max drawdown on 2 contracts, starting with $50k.on 2 contracts, starting with $50k.

• Find the fixed fraction that maximizes Find the fixed fraction that maximizes the RoR of the historical sequence with the RoR of the historical sequence with no more than 30% drawdown no more than 30% drawdown f = 8.2% f = 8.2%

• Try f=8.2% on some randomized Try f=8.2% on some randomized sequences of the original trades. One sequences of the original trades. One result: max drawdown = 76%!result: max drawdown = 76%!



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Randomized



Monte Carlo Simulation:Monte Carlo Simulation:

• Replaces random variables in a simulation Replaces random variables in a simulation with their probability distributions.with their probability distributions.

• Distributions are randomly sampled many Distributions are randomly sampled many times.times.

• Output of simulation is a distribution.Output of simulation is a distribution.

• Can be used to find the “best” fixed fraction Can be used to find the “best” fixed fraction by replacing the by replacing the tradetrade with the distribution with the distribution of trades.of trades.



Distribution of Profit/Loss

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Applying Monte Carlo to Fixed Fractional Trading:Applying Monte Carlo to Fixed Fractional Trading:• Randomize the sequence of trades, and, for each Randomize the sequence of trades, and, for each

sequence, calculate the return and max drawdown sequence, calculate the return and max drawdown using a given value of f.using a given value of f.

• The drawdown at 95% confidence is the drawdown The drawdown at 95% confidence is the drawdown such that 95% of sequences have drawdowns less such that 95% of sequences have drawdowns less than that.than that.

• The return at 95% confidence is the return such The return at 95% confidence is the return such that 95% of sequences return at least that much.that 95% of sequences return at least that much.

• Find the f value that maximizes the return at 95% Find the f value that maximizes the return at 95% confidence while keeping the drawdown at 95% confidence while keeping the drawdown at 95% confidence below your drawdown limit.confidence below your drawdown limit.



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Money Management StopsMoney Management Stops

• Lesson from fixed fractional trading:Lesson from fixed fractional trading: a money management stop defines a money management stop defines the trade risk, which enables more the trade risk, which enables more precise position sizing.precise position sizing.

• How do we choose the size of the How do we choose the size of the money management stop? One money management stop? One approach: approach: volatilityvolatility..


Money Management Stops Money Management Stops (cont.)(cont.)

ATR Volatility - E-mini S&P 500

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Distribution of ATR, E-mini S&P

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Cumulative ATR Distr - ES

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ATR Volatility - E-mini Nasdaq

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Distribution of ATR, E-mini Nasdaq

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Trailing StopsTrailing Stops

Some ideas for trailing stops:Some ideas for trailing stops:• Try basing the size of the stop on volatility, Try basing the size of the stop on volatility,

as suggested for money management as suggested for money management stops, but use a smaller value.stops, but use a smaller value.

• Try tightening the stop sharply after a big Try tightening the stop sharply after a big move in your favor (but not before).move in your favor (but not before).

• If the trailing stop is tighter than the mm If the trailing stop is tighter than the mm stop, wait until the market has moved in stop, wait until the market has moved in your favor by some multiple of the ATR your favor by some multiple of the ATR before applying the trailing stop.before applying the trailing stop.


Performance MeasuresPerformance Measures

• Problem:Problem: If you simulate trading with If you simulate trading with position sizing, how does this affect position sizing, how does this affect performance measurements?performance measurements?

• Short answer:Short answer: Don’t rely on the Don’t rely on the TradeStation performance summary.TradeStation performance summary.


Performance Measures Performance Measures (cont.)(cont.)

If given in dollars, some performance If given in dollars, some performance statistics could be skewed by the higher statistics could be skewed by the higher equity and larger number of contracts equity and larger number of contracts at the end of the equity curve:at the end of the equity curve:

•Average Trade

•Largest Win

•Largest Loss

•Win/Loss ratio

•Max Drawdown



• Solution:Solution: Calculate equity-dependent Calculate equity-dependent performance statistics by recording performance statistics by recording the trade profit/loss as a percentage the trade profit/loss as a percentage of the equity at the time the trade is of the equity at the time the trade is entered.entered.

• Consider my Consider my FixedRiskFixedRisk and and MonteCarloMonteCarlo EasyLanguage user EasyLanguage user functions…functions…


Performance Measures Performance Measures (cont.)(cont.)* MM ANALYSIS: PERFORMANCE OF HISTORICAL SEQUENCE ** MM ANALYSIS: PERFORMANCE OF HISTORICAL SEQUENCE * NQ_0_V0B.CSV (Daily Data), 4/19/2002NQ_0_V0B.CSV (Daily Data), 4/19/2002

TRADING PARAMETERS:TRADING PARAMETERS:Initial Account Equity: $50000.00Initial Account Equity: $50000.00Position Sizing Method: Fixed FractionalPosition Sizing Method: Fixed FractionalRisk Percentage (fixed fraction): 4.00%Risk Percentage (fixed fraction): 4.00%

PERFORMANCE RESULTS:PERFORMANCE RESULTS:Error Code: 0Error Code: 0Total Net Profit: $119572.00Total Net Profit: $119572.00 Gross Profit: $319002.00Gross Profit: $319002.00 Gross Loss: $-199430.00Gross Loss: $-199430.00 Profit Factor: 1.60Profit Factor: 1.60Final Account Equity: $169572.00Final Account Equity: $169572.00


Performance Measures Performance Measures (cont.)(cont.)Number of Trades: 103Number of Trades: 103 Number Winning Trades: 51Number Winning Trades: 51 Number Losing Trades: 52Number Losing Trades: 52 Number Skipped Trades (# contracts=0): 0Number Skipped Trades (# contracts=0): 0 Percent Profitable: 49.51%Percent Profitable: 49.51%

Largest Winning Trade (%): 16.02% ($9400.00)Largest Winning Trade (%): 16.02% ($9400.00)Largest Winning Trade ($): $24400.00 (14.54%)Largest Winning Trade ($): $24400.00 (14.54%)Average Winning Trade (%): 5.85%Average Winning Trade (%): 5.85%Average Winning Trade ($): $6254.94Average Winning Trade ($): $6254.94Max # Consecutive Wins: 5Max # Consecutive Wins: 5

Largest Losing Trade (%): -6.77% ($-12805.00)Largest Losing Trade (%): -6.77% ($-12805.00)Largest Losing Trade ($): $-12805.00 (-6.77%)Largest Losing Trade ($): $-12805.00 (-6.77%)Average Losing Trade (%): -3.10%Average Losing Trade (%): -3.10%Average Losing Trade ($): $-3835.19Average Losing Trade ($): $-3835.19Max # Consecutive Losses: 5Max # Consecutive Losses: 5


Performance Measures Performance Measures (cont.)(cont.)Ratio Avg Win(%)/Avg Loss(%): 1.89Ratio Avg Win(%)/Avg Loss(%): 1.89Ratio Avg Win($)/Avg Loss($): 1.63Ratio Avg Win($)/Avg Loss($): 1.63Average % Trade: 1.33%Average % Trade: 1.33%Average $ Trade: $1160.90Average $ Trade: $1160.90Max # Contracts: 18Max # Contracts: 18Avg # Contracts: 5Avg # Contracts: 5

Max Closed Trade % Drawdown: 21.13% ($43351.40)Max Closed Trade % Drawdown: 21.13% ($43351.40)Date of Max % Drawdown: 4/1/2002Date of Max % Drawdown: 4/1/2002Max Closed Trade $ Drawdown: $43351.40 (21.13%)Max Closed Trade $ Drawdown: $43351.40 (21.13%)Date of Max $ Drawdown: 4/1/2002Date of Max $ Drawdown: 4/1/2002Return on Starting Equity: 239.14%Return on Starting Equity: 239.14%



* MM ANALYSIS: MONTE CARLO ANALYSIS ** MM ANALYSIS: MONTE CARLO ANALYSIS *

INPUT DATA:INPUT DATA:Initial Account Equity: $50000.00Initial Account Equity: $50000.00Risk Percentage (fixed fraction): 4.00%Risk Percentage (fixed fraction): 4.00%Number of Trades: 103Number of Trades: 103Rate of Return Goal: 100.00%Rate of Return Goal: 100.00%Drawdown Goal: 30.00%Drawdown Goal: 30.00%Probability Goal: 95.00%Probability Goal: 95.00%Number of Random Sequences: 1000Number of Random Sequences: 1000


Performance Measures Performance Measures (cont.)(cont.)OUTPUT/RESULTS:OUTPUT/RESULTS:Error Code: 0Error Code: 0Average Rate of Return: 249.48%Average Rate of Return: 249.48%Average Final Account Equity: $174741.00Average Final Account Equity: $174741.00Probability of Reaching Return Goal: 100.00%Probability of Reaching Return Goal: 100.00%Probability of Reaching Drawdown Goal: 85.10%Probability of Reaching Drawdown Goal: 85.10%Probability of Reaching Return and Drawdown Together: Probability of Reaching Return and Drawdown Together:

85.10%85.10%Rate of Return at 95.00% Probability: 195.31%Rate of Return at 95.00% Probability: 195.31%Drawdown at 95.00% Probability: 35.16%Drawdown at 95.00% Probability: 35.16%

Copyright 2002 Breakout Futures Trading the Risk Position Sizing and Exit Stops Michael R. Bryant,...

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Transcript of Copyright 2002 Breakout Futures Trading the Risk Position Sizing and Exit Stops Michael R. Bryant,...