Trade Risk Talk

Trading the Risk Position Sizing and Exit Stops Michael R. Bryant, Ph.D. Breakout Futures www.BreakoutFutures.com Copyright 2002 Breakout Futures

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 Copyright 2002 Breakout Futures

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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 Don’t 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 Don’t 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 can’t 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. Copyright 2002 Breakout Futures

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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 doesn’t reduce contracts in a drawdown, so drawdowns can be large. • Doesn’t take the risk of each trade into account. Copyright 2002 Breakout Futures

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Margin Plus Drawdown (cont.) 140000 120000

Equity

100000 80000

1-Con Marg+DD

60000 40000 20000 0 12/31/97

12/31/98

12/31/99

12/30/00

12/30/01

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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. Copyright 2002 Breakout Futures

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Fixed Fractional (cont.) 300000

Equity

250000 200000

MM Stop Max Loss

150000 100000 50000 1/1/98

1/1/99

1/1/00

12/31/00

12/31/01

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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.) 140000 120000

Equity

100000 1-Con Marg+DD Fix Frac

80000 60000 40000 20000 0 12/31/97

12/31/98

12/31/99

12/30/00

12/30/01

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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: 1/2

N = ½ *[ 1 + (1 + 8 * Profit/delta) ] where Profit = total closed trade profit ($), delta = profit/contract to increase by 1 contract ($). Copyright 2002 Breakout Futures

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Fixed Ratio (cont.) 25

No. Contracts

20

15 Fix Frac Fix Ratio 10

5

0 0

5

10

15

20

25

30

Trade

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Fixed Ratio (cont.) 25

No. Contracts

20

15 Fixed Frac Fixed Ratio 10

5

0 0

30,000

60,000

90,000

120,000

Profit Copyright 2002 Breakout Futures

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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. • Doesn’t directly depend on trade risk.

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A More Generalized Approach • Consider the following equation for the number of contracts, N:

m

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. Copyright 2002 Breakout Futures

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A Generalized Approach (cont.) • Consider m = 0: 0

N = ½ *[ 1 + (1 + 8 * Profit/delta) ] = 1/2 * [1 + 1] =1 i.e., we get fixed contract trading (N = 1). Copyright 2002 Breakout Futures

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A Generalized Approach (cont.) • Consider m = 1: 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: m-1 ∂N/∂ N/ (Profit) = 4*m/delta * (1 + 8 * Profit/delta)

m = 1  ROC of N independent of profit; e.g., fixed fraction. m > 1  N increases faster as equity grows. m < 1  N increases more slowly as equity grows; e.g., fixed ratio. Copyright 2002 Breakout Futures

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A Generalized Approach (cont.) 450000 400000 350000

Equity

300000 m=0.5 m=1.0 m=1.5

250000 200000 150000 100000 50000 0 12/31/97

12/31/98

12/31/99

12/30/00

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12/30/01

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A Generalized Approach (cont.) 500000 425000

Equity

350000

m=0.5 m=1.0

275000

m=1.5

200000 125000 50000 12/31/98

12/31/99

12/30/00

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12/30/01

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Conclusions From Generalized Approach • m < 1 works best when worst drawdowns

come late. • 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?) Copyright 2002 Breakout Futures

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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. Copyright 2002 Breakout Futures

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Best Fixed Fraction (cont.) Optimal f: • f value that mathematically maximizes the compounded rate of return. • Doesn’t 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. Copyright 2002 Breakout Futures

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Best Fixed Fraction (cont.) 85000 75000

Equity

65000 55000 DD=9.3% 45000 35000 25000 15000 12/31/97

12/31/98

12/31/99

12/30/00

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Best Fixed Fraction (cont.) 85000 75000

Equity

65000 55000 DD=16.7% 45000 35000 25000 15000 12/31/97

12/31/98

12/31/99

12/30/00

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Best Fixed Fraction (cont.) 85000 75000

Equity

65000 55000 DD=25.6% 45000 35000 25000 15000 12/31/97

12/31/98

12/31/99

12/30/00

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Best Fixed Fraction (cont.) 85000 75000

Equity

65000 55000 DD=37.6% 45000 35000 25000 15000 12/31/97

12/31/98

12/31/99

12/30/00

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Best Fixed Fraction (cont.) 85000 75000

Equity

65000 55000 DD=46.2% 45000 35000 25000 15000 12/31/97

12/31/98

12/31/99

12/30/00

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Best Fixed Fraction (cont.) 85000 75000

Equity

65000 DD=9.3% DD=16.7% DD=25.6%

55000 45000

DD=37.6% DD=46.2%

35000 25000 15000 12/31/97

12/31/98

12/31/99

12/30/00

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

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Best Fixed Fraction (cont.) 800000 700000

Equity

600000 500000

Original

400000

Optimized Randomized

300000 200000 100000 0 12/31/97 12/31/98 12/31/99 12/30/00 12/30/01 Copyright 2002 Breakout Futures

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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. Copyright 2002 Breakout Futures

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Best Fixed Fraction (cont.) Distribution of Profit/Loss 25 20 15 10 5

60 00

50 00

40 00

30 00

20 00

10 00

0

-1 00 0

-2 00 0

-3 00 0

0

Trade P/L Copyright 2002 Breakout Futures

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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. Copyright 2002 Breakout Futures

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Best Fixed Fraction (cont.) 120

1600 1400

100 80

1000

60

800 600

40

Ave RoR (%)

P (40% DD)

1200

400 20

200

0

0 0

0.02

0.04

0.06

0.08

0.1

0.12

Fixed Fraction

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Best Fixed Fraction (cont.) 4000

120

3500

RoR at P=95%

2500

80

2000

60

1500 1000

40

500

DD at P=95%

100

3000

20

0 -500

0 0

0.1

0.2

0.3

0.4

Fixed Fraction Copyright 2002 Breakout Futures

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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. Copyright 2002 Breakout Futures

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Money Management Stops (cont.) ATR Volatility - E-mini S&P 500 60

10-day ATR

50 40 30 20 10 0 9/1/97

9/1/98

9/1/99

8/31/00

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Money Management Stops (cont.) Distribution of ATR, E-mini S&P 200 180 160 140 120 100 80 60 40 20 0 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 10-day ATR Copyright 2002 Breakout Futures

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Money Management Stops (cont.) Cumulative ATR Distr - ES 100 90 80 60 50 40 30 20 10

52

48

44

40

36

32

28

24

20

16

0 12

% of Total

70

10-day ATR Copyright 2002 Breakout Futures

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Money Management Stops (cont.) ATR Volatility - E-mini Nasdaq 400 350 10-day ATR

300 250 200 150 100 50 0 6/30/99 12/30/99 6/30/00 12/30/00

7/1/01

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Money Management Stops (cont.) Distribution of ATR, E-mini Nasdaq

60 50 40 30 20 10

0 32

0 28

0 24

5 21

5 19

5 17

5 15

5 13

5 11

95

75

55

35

0

Average True Range Copyright 2002 Breakout Futures

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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. Copyright 2002 Breakout Futures

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Performance Measures • Problem: If you simulate trading with position sizing, how does this affect performance measurements? • Short answer: Don’t 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

• Win/Loss ratio

• Largest Win

• Max Drawdown

• Largest Loss Copyright 2002 Breakout Futures

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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… Copyright 2002 Breakout Futures

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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 Copyright 2002 Breakout Futures

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Performance Measures (cont.) Number of Trades: 103 Number Winning Trades: 51 Number Losing Trades: 52 Number Skipped Trades (# contracts=0): 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 Copyright 2002 Breakout Futures

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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%

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