Trading the Risk

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Trading the Risk

• Position Sizing and Exit Stops

Michael R. Bryant, Ph.D. Breakout Futures www.Br
eakoutFutures.com
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Scope of Talk

• Short to intermediate-term trading
• Rational methods of position sizing and stop
  selection mostly quantitative
  
• Oriented towards futures but also applicable to
  stocks
  
• One market-system at a time

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What is Position Sizing?

• Selecting the number of contracts or shares of
  stock for the next trade
  
• A way to reinvest profits
• The way traders compound their returns

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Methods of Position Sizing

• Ad hoc trade no larger than lets you sleep at
  night
  
• Margin plus drawdown
• Fixed Fractional
• Fixed Ratio
• Hybrid fixed fractional/fixed ratio

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Methods that Dont Work

• Martingale methods increase position size after
  a loss decrease it after a win.
  
• Equity curve methods increase size when your
  equity curve falls below its moving average
  (reversion to mean), or increase size when you
  cross above the moving average (trade the trend
  in equity curve).

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Why They Dont Work

• Martingale and equity curve methods assume
  dependency between trades.
  
• In most cases, trades are independent of each
  other. The odds of the next trade being a win are
  not related to whether the last trade was a win
  or a loss.
• If trades are independent, you cant determine
  the likelihood of the next trade being a win or a
  loss based on the previous trade.

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Margin Plus Drawdown Sizing

• The equity to trade one contract is the maximum
  historical drawdown multiplied by 1.5 plus the
  margin requirement.
  
• Add another contract only when the closed profits
  are equal to drawdown 1.5 plus margin.
  
• Attributable to Larry Williams see The
  Definitive Guide to Futures Trading, Volume II.

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Margin Plus Drawdown (cont.)

• You always have enough money to handle the worst
  historical drawdown plus 50.
  
• Designed so you only increase the number of
  contracts, never reduce.
  
• Theoretically safe but doesnt reduce contracts
  in a drawdown, so drawdowns can be large.
  
• Doesnt take the risk of each trade into account.

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Margin Plus Drawdown (cont.)
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Fixed Fractional Position Sizing

• Risk the same fraction (fixed fraction) of the
  account equity on each trade e.g., 5.
  
• Number of contracts
• N ff Equity/Trade Risk
• where ff fixed fraction,
• Equity account equity (),
• Trade Risk possible loss on trade ()

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Fixed Fractional (cont.)

• Trade risk may come from
• Estimate. Examples n standard deviations of the
  trade distribution largest historical loss.
  
• Size of money management stop.
• Using a money management (mm) stop to define the
  trade risk may produce greater risk-adjusted
  returns than using the largest loss.

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Fixed Fractional (cont.)
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Observations on Fixed Fractional

• As a percentage of account equity, the risk of
  each trade is the same, regardless of the number
  of contracts.
  
• Takes advantage of trade risk.
• Responsive to changes in equity (unlike margin
  plus drawdown method).
  
• The trick is determining the best value of the
  fixed fraction more on that later

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Fixed Fractional (cont.)
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Fixed Ratio Position Sizing

• Developed by Ryan Jones see The Trading Game,
  John Wiley, 1999.
  
• Based on a fixed parameter called the delta the
  profit per contract needed to increase the number
  of contracts by 1.
  
• Each contract contributes the same profit towards
  increasing the number of contracts, regardless of
  account equity.

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Fixed Ratio (cont.)

• Number of contracts
• N ½ 1 (1 8 Profit/delta)1/2
• where Profit total closed trade profit (),
• delta profit/contract to increase by 1
  contract ().

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Fixed Ratio (cont.)
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Fixed Ratio (cont.)
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Observations on Fixed Ratio

• Performance depends on total accumulated profits
  i.e., account size. It becomes more conservative
  as the account size increases.
  
• Doesnt directly depend on trade risk.

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A More Generalized Approach

• Consider the following equation for the number of
  contracts, N
  
• N ½ 1 (1 8 Profit/delta)m
• where Profit total closed trade profit (),
• delta fixed ratio parameter (),
• m 0.
• With m ½, we get the fixed ratio equation.

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A Generalized Approach (cont.)

• Consider m 0
• N ½ 1 (1 8 Profit/delta)0
• 1/2 1 1
• 1
• i.e., we get fixed contract trading (N 1).

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A Generalized Approach (cont.)

• Consider m 1
• N ½ 1 (1 8 Profit/delta)1
• 1 4 Profit/delta
• Let delta 4 Risk/ff and Equity0 Risk/ff.
• Then, N (Equity0 Profit) ff/Risk
• (i.e., the equation for fixed fractional trading)

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A Generalized Approach (cont.)

• Rate of Change of N with Profit
• ?N/?(Profit) 4m/delta (1 8
  Profit/delta)m-1
  
• m 1 ? ROC of N independent of profit e.g.,
  fixed fraction.
  
• m 1 ? N increases faster as equity grows.
• m e.g., fixed ratio.

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A Generalized Approach (cont.)
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A Generalized Approach (cont.)
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Conclusions From Generalized Approach

• m
• m 1 works best when biggest run-up comes
  late.
  
• For any sequence of trades, there is probably an
  optimal value of m. However, the sequence of
  trades and drawdowns/run-ups is unknown. (Monte
  Carlo analysis to find the best m?)

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Finding the Best Fixed Fraction

• Ad hoc e.g., 2 rule.
• Optimal f Ralph Vince, Portfolio Management
  Formulas, 1990.
  
• Secure f Leo Zamansky David Stendahl, TASC,
  July, 1998.
  
• Monte Carlo simulation Bryant, TASC, February,
  2001.

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Best Fixed Fraction (cont.)

• Optimal f
• f value that mathematically maximizes the
  compounded rate of return.
  
• Doesnt take the drawdown into account.
• Typically results in very large and dangerous
  f values.
  
• Theoretically sound but not practical to trade.

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Best Fixed Fraction (cont.)

• Secure f
• f value that maximizes the compounded rate of
  return subject to a limit on the maximum
  drawdown e.g., what f value gives the greatest
  rate of return without exceeding 30 drawdown?
• Improvement on optimal f.
• Only problem the drawdown calculated from the
  historical sequence of trades is not very
  reliable.

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

• Historical sequence 14 max drawdown on 2
  contracts, starting with 50k.
  
• Find the fixed fraction that maximizes the RoR of
  the historical sequence with no more than 30
  drawdown ? f 8.2
  
• Try f8.2 on some randomized sequences of the
  original trades. One result max drawdown 76!

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

• Monte Carlo Simulation
• Replaces random variables in a simulation with
  their probability distributions.
  
• Distributions are randomly sampled many times.
• Output of simulation is a distribution.
• Can be used to find the best fixed fraction by
  replacing the trade with the distribution of
  trades.

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

• Applying Monte Carlo to Fixed Fractional
  Trading
  
• Randomize the sequence of trades, and, for each
  sequence, calculate the return and max drawdown
  using a given value of f.
  
• The drawdown at 95 confidence is the drawdown
  such that 95 of sequences have drawdowns less
  than that.
  
• The return at 95 confidence is the return such
  that 95 of sequences return at least that much.
  
• Find the f value that maximizes the return at 95
  confidence while keeping the drawdown at 95
  confidence below your drawdown limit.

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Best Fixed Fraction (cont.)
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Best Fixed Fraction (cont.)
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Money Management Stops

• Lesson from fixed fractional trading a money
  management stop defines the trade risk, which
  enables more precise position sizing.
  
• How do we choose the size of the money management
  stop? One approach volatility.

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

• Some ideas for trailing stops
• Try basing the size of the stop on volatility, as
  suggested for money management stops, but use a
  smaller value.
  
• Try tightening the stop sharply after a big move
  in your favor (but not before).
  
• If the trailing stop is tighter than the mm stop,
  wait until the market has moved in your favor by
  some multiple of the ATR before applying the
  trailing stop.

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Performance Measures

• Problem If you simulate trading with position
  sizing, how does this affect performance
  measurements?
  
• Short answer Dont rely on the TradeStation
  performance summary.

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Performance Measures (cont.)

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

Average Trade Largest Win Largest Loss
Win/Loss ratio Max Drawdown
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Performance Measures (cont.)

• Solution Calculate equity-dependent performance
  statistics by recording the trade profit/loss as
  a percentage of the equity at the time the trade
  is entered.
• Consider my FixedRisk and MonteCarlo EasyLanguage
  user functions

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Performance Measures (cont.)

• MM ANALYSIS PERFORMANCE OF HISTORICAL SEQUENCE
  
  
• NQ_0_V0B.CSV (Daily Data), 4/19/2002
• TRADING PARAMETERS
• Initial Account Equity 50000.00
• Position Sizing Method Fixed Fractional
• Risk Percentage (fixed fraction) 4.00
• PERFORMANCE RESULTS
• Error Code 0
• Total Net Profit 119572.00
• Gross Profit 319002.00
• Gross Loss -199430.00
• Profit Factor 1.60
• Final Account Equity 169572.00

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Performance Measures (cont.)

• Number of Trades 103
• Number Winning Trades 51
• Number Losing Trades 52
• Number Skipped Trades ( contracts0) 0
• Percent Profitable 49.51
• Largest Winning Trade () 16.02 (9400.00)
• Largest Winning Trade () 24400.00 (14.54)
• Average Winning Trade () 5.85
• Average Winning Trade () 6254.94
• Max Consecutive Wins 5
• Largest Losing Trade () -6.77 (-12805.00)
• Largest Losing Trade () -12805.00 (-6.77)
• Average Losing Trade () -3.10
• Average Losing Trade () -3835.19
• Max Consecutive Losses 5

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Performance Measures (cont.)

• Ratio Avg Win()/Avg Loss() 1.89
• Ratio Avg Win()/Avg Loss() 1.63
• Average Trade 1.33
• Average Trade 1160.90
• Max Contracts 18
• Avg Contracts 5
• Max Closed Trade Drawdown 21.13 (43351.40)
• Date of Max Drawdown 4/1/2002
• Max Closed Trade Drawdown 43351.40 (21.13)
• Date of Max Drawdown 4/1/2002
• Return on Starting Equity 239.14

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Performance Measures (cont.)

• MM ANALYSIS MONTE CARLO ANALYSIS
• INPUT DATA
• Initial Account Equity 50000.00
• Risk Percentage (fixed fraction) 4.00
• Number of Trades 103
• Rate of Return Goal 100.00
• Drawdown Goal 30.00
• Probability Goal 95.00
• Number of Random Sequences 1000

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Performance Measures (cont.)

• OUTPUT/RESULTS
• Error Code 0
• Average Rate of Return 249.48
• Average Final Account Equity 174741.00
• Probability of Reaching Return Goal 100.00
• Probability of Reaching Drawdown Goal 85.10
• Probability of Reaching Return and Drawdown
  Together 85.10
  
• Rate of Return at 95.00 Probability 195.31
• Drawdown at 95.00 Probability 35.16