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009_position_sizing

Position Sizing: Optimal Capital Allocation Theory

Position Sizing: Optimal Capital Allocation Theory

I. Introduction

Position sizing determines how much capital to allocate to each trade. It is often considered more important than entry/exit timing, as improper sizing can lead to ruin even with a profitable strategy.

II. Basic Position Sizing Models

Definition 2.1 (Fixed Dollar Risk)

PositionSize=RiskAmountStopLossDistance×PointValuePositionSize = \frac{RiskAmount}{StopLossDistance \times PointValue}

Definition 2.2 (Fixed Percentage Risk)

RiskAmount=AccountBalance×RiskPercentRiskAmount = AccountBalance \times RiskPercent PositionSize=AccountBalance×RiskPercentStopLossDistance×PointValuePositionSize = \frac{AccountBalance \times RiskPercent}{StopLossDistance \times PointValue}

Example 2.1

  • Account: $10,000
  • Risk: 2% = $200
  • Stop Loss: 50 pips
  • Point Value: $10/pip (1 standard lot EUR/USD)

PositionSize=20050×10=0.4 lotsPositionSize = \frac{200}{50 \times 10} = 0.4 \text{ lots}

III. Kelly Criterion

Theorem 3.1 (Kelly Formula)

Given:

  • p = Probability of winning
  • b = Win/Loss ratio (R:R)

The optimal fraction of capital to risk: f∗=p(b+1)−1b=p−1−pbf^* = \frac{p(b+1) - 1}{b} = p - \frac{1-p}{b}

Proof (Maximum Growth Rate)

We maximize the expected logarithmic growth: G(f)=E[ln⁡(1+f⋅X)]G(f) = E[\ln(1 + f \cdot X)]

Where X = +b with probability p, X = -1 with probability (1-p).

G(f)=pln⁡(1+fb)+(1−p)ln⁡(1−f)G(f) = p \ln(1 + fb) + (1-p) \ln(1 - f)

Taking derivative and setting to zero: G′(f)=pb1+fb−1−p1−f=0G'(f) = \frac{pb}{1+fb} - \frac{1-p}{1-f} = 0

Solving: f∗=p(b+1)−1bf^* = \frac{p(b+1) - 1}{b} ∎

Theorem 3.2 (Kelly Properties)

  1. f* = 0 when p = 1/(b+1) (break-even edge)
  2. f* < 0 when edge is negative (don't trade)
  3. f* > 0 when p > 1/(b+1) (positive edge)

Table 3.1 (Kelly Fractions)

| Win Rate | R:R | Kelly % | |----------|-----|---------| | 50% | 2:1 | 25% | | 60% | 1:1 | 20% | | 55% | 1.5:1 | 18.3% | | 40% | 3:1 | 13.3% |

IV. Fractional Kelly

Definition 4.1 (Fractional Kelly)

To reduce volatility, use a fraction of full Kelly: ffractional=k×f∗f_{fractional} = k \times f^*

Common choices:

  • Half-Kelly: k = 0.5
  • Quarter-Kelly: k = 0.25

Theorem 4.1 (Growth Rate Trade-off)

Using fraction k of Kelly: G(kf∗)=G(f∗)×k(2−k)G(kf^*) = G(f^*) \times k(2-k)

At half-Kelly: G = 0.75 × G_max with significantly lower variance.

V. Volatility-Based Position Sizing

Definition 5.1 (ATR-Based Sizing)

PositionSize=RiskAmountk×ATR×PointValuePositionSize = \frac{RiskAmount}{k \times ATR \times PointValue}

Where k is a multiplier (typically 2-3).

Theorem 5.1 (Volatility Normalization)

ATR-based sizing ensures consistent risk across different volatility regimes: σPortfolio=const regardless of σMarket\sigma_{Portfolio} = const \text{ regardless of } \sigma_{Market}

VI. Anti-Martingale Systems

Definition 6.1 (Fixed Ratio Method)

Increase position by 1 unit after every δ dollars of profit: Units=1+⌊Profitδ⌋Units = 1 + \lfloor \frac{Profit}{\delta} \rfloor

Definition 6.2 (Percent Volatility Model)

PositionSize=AccountBalance×TargetVolatilityInstrumentVolatility×PointValuePositionSize = \frac{AccountBalance \times TargetVolatility}{InstrumentVolatility \times PointValue}

VII. Portfolio Position Sizing

7.1 Maximum Correlation Limit

When trading multiple positions: TotalRisk=∑iRiski×ρijTotalRisk = \sum_i Risk_i \times \rho_{ij}

If positions are correlated, reduce individual size.

7.2 Heat Rule

Maximum total portfolio risk: ∑i∈OpenPositionsRiski≤MaxHeat\sum_{i \in OpenPositions} Risk_i \leq MaxHeat

Typical MaxHeat = 6% of account.

VIII. Survival Probability

Theorem 8.1 (Ruin Probability)

With edge e > 0 and fraction f per trade, probability of ruin: Pruin≈(1−e1+e)B/fP_{ruin} \approx \left(\frac{1-e}{1+e}\right)^{B/f}

Where B is initial bankroll in risk units.

Corollary 8.1 (Never Risk Too Much)

Even with positive edge, risking too much leads to eventual ruin: lim⁡f→1Pruin=1\lim_{f \to 1} P_{ruin} = 1

IX. Practical Guidelines

9.1 Risk Limits

| Trader Type | Max Risk/Trade | Max Risk/Day | |-------------|----------------|--------------| | Conservative | 0.5% | 1.5% | | Moderate | 1-2% | 4-6% | | Aggressive | 2-3% | 6-10% |

9.2 Drawdown Recovery

| Drawdown | Gain Needed to Recover | |----------|------------------------| | 10% | 11.1% | | 20% | 25.0% | | 30% | 42.9% | | 50% | 100.0% |

X. Exercises

Exercise 1: Calculate Kelly fraction for p=0.55, R:R=1.5:1.

Exercise 2: Given $25,000 account, 1.5% risk, 40 pip stop, calculate position size for EUR/USD.

Exercise 3: Prove that recovery requirement R after drawdown D is: R = D/(1-D).

Exercise 4: Compare equity curves of 1%, 2%, and 5% risk over 100 trades with 55% win rate and 1.5:1 R:R.

XI. References

  • Kelly, J.L. (1956). A New Interpretation of Information Rate
  • Vince, R. (1992). The Mathematics of Money Management
  • Tharp, V.K. (2007). Trade Your Way to Financial Freedom
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