Position Sizing Methodologies

Position Sizing Methodologies

Kelly criterion, optimal-f, fixed fractional, volatility targeting, risk parity position sizing, and practical implementation.

When to Activate

• User designing position sizing rules for a trading strategy
• Applying Kelly criterion or fractional Kelly to determine bet size
• Volatility-based position sizing for systematic strategies
• Risk budgeting and risk parity allocation across strategies or assets
• Evaluating the tradeoff between position size, expected return, and drawdown risk
• Comparing position sizing methods for a given strategy profile

Core Concepts

Kelly Criterion

Original Kelly (Binary Outcomes)

• f* = (bp - q) / b, where b = odds, p = probability of win, q = 1-p
• f* = fraction of capital to wager
• Maximizes long-run geometric growth rate (log wealth)
• Example: p = 0.55, b = 1 (even money) -> f* = (1*0.55 - 0.45)/1 = 0.10 (bet 10% of capital)

Continuous Kelly (Normal Returns)

• f* = (mu - r_f) / sigma^2, where mu = expected return, sigma = volatility
• For a strategy with 15% expected excess return and 20% vol: f* = 0.15/0.04 = 3.75x leverage
• This is the leverage that maximizes geometric growth
• Equivalent to maximizing E[ln(1 + f*r)] over the return distribution

Multi-Asset Kelly

• f* = Sigma^{-1} * mu (vector of optimal fractions = inverse covariance times expected excess returns)
• Equivalent to mean-variance optimization with risk aversion = 1 (maximizing log utility)
• Requires estimation of expected returns and covariance matrix — both noisy

Practical Problems with Full Kelly

• Full Kelly produces extreme drawdowns: probability of 50% drawdown is approximately 50%
• Parameter estimation error: overestimated mu or underestimated sigma leads to overleveraging
• Non-normal returns: fat tails make full Kelly even more dangerous
• Strategy capacity: full Kelly may imply positions that exceed market liquidity
• Result: essentially no professional trader uses full Kelly

Fractional Kelly

• Use f = c * f_Kelly where c is typically 0.20 to 0.50 (quarter to half Kelly)
• Half Kelly (c = 0.5): 75% of the growth rate but dramatically lower drawdowns
• Quarter Kelly (c = 0.25): 44% of the growth rate but very stable equity curve
• Fractional Kelly is equivalent to full Kelly with a blended estimate (mixing with zero-return prior)
• Fractional Kelly is more robust to parameter estimation error

Optimal-f (Ralph Vince)

• Generalization of Kelly for non-binary outcomes
• f* = the fraction that maximizes terminal wealth factor (TWR) over historical trades
• TWR = product of (1 + f * return_i / |worst_loss|) over all trades
• Find f* numerically by testing f from 0 to 1 in small increments
• Related to Kelly but does not assume a distribution — uses empirical trade history
• Danger: optimal-f is computed on in-sample data and may overfit
• Like full Kelly, produces severe drawdowns; use fractional optimal-f in practice

Methodology

Fixed Fractional Position Sizing

1. Risk a fixed percentage of capital per trade (e.g., 1-2%)
2. Position size = (risk_per_trade * capital) / (entry - stop_loss)
3. Example: $1M capital, 1% risk, entry $50, stop $48
   • Risk amount = $10,000
   • Position size = $10,000 / $2 = 5,000 shares
4. Automatically sizes down after losses and up after gains (anti-martingale)
5. Prevents ruin: no single trade can lose more than the fixed fraction
6. Simple and widely used in systematic and discretionary trading

Volatility-Based Position Sizing

Volatility Targeting

1. Set target portfolio volatility (e.g., 10% annualized)
2. For each position: weight_i = target_vol / (N * sigma_i) where sigma_i is asset volatility
3. More volatile assets get smaller positions; less volatile get larger
4. Rebalance when realized vol deviates significantly from target
5. Equalizes risk contribution from each position (if uncorrelated)

ATR-Based Sizing (Turtle Traders)

1. Compute N = 20-day ATR (Average True Range) for each instrument
2. Dollar volatility = N * point_value
3. Unit size = (1% of equity) / dollar_volatility
4. Each "unit" represents approximately 1% equity risk per day
5. Maximum units per market: 4 (maximum 4% of equity at risk per position)
6. Maximum units per correlated group: 6-8

Risk Parity Position Sizing

1. Target equal risk contribution from each asset/strategy
2. In the simple (uncorrelated) case: w_i proportional to 1/sigma_i
3. With correlations: solve for w such that w_i * (Sigma * w)_i = RC_target for all i
4. Risk contribution of asset i: RC_i = w_i * (Sigma * w)_i / (w' * Sigma * w)
5. Equal risk contribution: RC_i = 1/N for all i
6. Requires leverage for low-vol assets (bonds) to equalize with high-vol assets (equities)
7. Bridgewater All Weather is the canonical risk parity portfolio

Risk Budgeting

1. Assign risk budgets to each strategy or asset class (need not be equal)
2. Risk budget = percentage of total portfolio risk attributed to each component
3. Example: equities 40%, fixed income 30%, alternatives 30% of total risk
4. Solve for weights that achieve the target risk allocation
5. Component risk = w_i * marginal_risk_i = w_i * (Sigma * w)_i / sigma_portfolio
6. Iterate or use optimization to find weights matching target risk budgets

Kelly with Parameter Uncertainty

Bayesian Kelly

1. Instead of point estimates of mu and sigma, use posterior distributions
2. Optimal bet size accounts for parameter uncertainty
3. Results in smaller positions than plug-in Kelly — natural shrinkage
4. With diffuse priors, Bayesian Kelly approximately equals 0.5 * plug-in Kelly

Shrinkage Approaches

1. Shrink expected return estimates toward zero (or grand mean)
2. Shrink covariance toward structured estimator (Ledoit-Wolf)
3. Effective position size decreases as uncertainty increases
4. More robust to estimation error than naive Kelly

Examples

Kelly Criterion for a Trading Strategy

Strategy backtest: annual excess return = 12%, annual volatility = 18%
Full Kelly: f* = 0.12 / 0.18^2 = 3.7x leverage
Expected growth rate at full Kelly: mu - 0.5*sigma^2*f = 12% - 0.5*3.24%*3.7 = 6%
Expected max drawdown at full Kelly: approximately 50%+

Half Kelly: f = 1.85x leverage
Expected growth rate: approximately 4.5%
Expected max drawdown: approximately 25%

Quarter Kelly: f = 0.93x leverage (no leverage needed)
Expected growth rate: approximately 3%
Expected max drawdown: approximately 12%

Recommendation: half Kelly (1.85x) if drawdown tolerance is 25%;
quarter Kelly (0.93x) if more conservative.

ATR-Based Position Sizing

Account equity: $500,000
Risk per unit: 1% = $5,000

Crude oil (CL): ATR(20) = $2.50, point value = $1,000/point
  Dollar vol = $2.50 * $1,000 = $2,500
  Unit size = $5,000 / $2,500 = 2 contracts

E-mini S&P (ES): ATR(20) = 45 points, point value = $50/point
  Dollar vol = 45 * $50 = $2,250
  Unit size = $5,000 / $2,250 = 2 contracts (round down)

Gold (GC): ATR(20) = $25, point value = $100/point
  Dollar vol = $25 * $100 = $2,500
  Unit size = $5,000 / $2,500 = 2 contracts

Each position risks approximately $5,000 (1% of equity) per ATR move.
Maximum 4 units per market = 4% equity risk per market.

Risk Parity Allocation

Three-asset portfolio: Equities (vol=16%), Bonds (vol=5%), Commodities (vol=20%)
Assuming zero correlations (simplification):

Equal risk contribution target: 33.3% each
w_equity proportional to 1/16 = 0.0625
w_bonds proportional to 1/5 = 0.2000
w_commod proportional to 1/20 = 0.0500

Normalized: w_equity = 20%, w_bonds = 63%, w_commod = 16%
Sum = 99% (approximately 1x leverage)

To target 10% portfolio vol:
Unlevered portfolio vol = sqrt(0.20^2*16^2 + 0.63^2*5^2 + 0.16^2*20^2) = 5.4%
Leverage = 10% / 5.4% = 1.85x
Levered weights: equities 37%, bonds 117%, commodities 30%

Quality Gate

• Kelly fraction must be computed with realistic (after-cost, after-slippage) return estimates
• Never use full Kelly in practice — maximum recommended is half Kelly for well-estimated strategies
• Parameter uncertainty must be accounted for: use Bayesian Kelly or fractional Kelly
• Position sizes must respect market liquidity constraints (max % of ADV)
• Volatility estimates for sizing must use appropriate lookback (20-60 day realized vol or EWMA)
• Risk parity weights must be rebalanced when volatility or correlation estimates change materially
• Total portfolio leverage must be within acceptable bounds regardless of what Kelly implies
• Sizing rules must be backtested with transaction costs and slippage included
• Maximum position size limits must exist as a hard constraint independent of the sizing formula
• Document the sizing methodology and parameters; review quarterly