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
- Risk a fixed percentage of capital per trade (e.g., 1-2%)
- Position size = (risk_per_trade * capital) / (entry - stop_loss)
- Example: $1M capital, 1% risk, entry $50, stop $48
- Risk amount = $10,000
- Position size = $10,000 / $2 = 5,000 shares
- Automatically sizes down after losses and up after gains (anti-martingale)
- Prevents ruin: no single trade can lose more than the fixed fraction
- Simple and widely used in systematic and discretionary trading
Volatility-Based Position Sizing
Volatility Targeting
- Set target portfolio volatility (e.g., 10% annualized)
- For each position: weight_i = target_vol / (N * sigma_i) where sigma_i is asset volatility
- More volatile assets get smaller positions; less volatile get larger
- Rebalance when realized vol deviates significantly from target
- Equalizes risk contribution from each position (if uncorrelated)
ATR-Based Sizing (Turtle Traders)
- Compute N = 20-day ATR (Average True Range) for each instrument
- Dollar volatility = N * point_value
- Unit size = (1% of equity) / dollar_volatility
- Each "unit" represents approximately 1% equity risk per day
- Maximum units per market: 4 (maximum 4% of equity at risk per position)
- Maximum units per correlated group: 6-8
Risk Parity Position Sizing
- Target equal risk contribution from each asset/strategy
- In the simple (uncorrelated) case: w_i proportional to 1/sigma_i
- With correlations: solve for w such that w_i * (Sigma * w)_i = RC_target for all i
- Risk contribution of asset i: RC_i = w_i * (Sigma * w)_i / (w' * Sigma * w)
- Equal risk contribution: RC_i = 1/N for all i
- Requires leverage for low-vol assets (bonds) to equalize with high-vol assets (equities)
- Bridgewater All Weather is the canonical risk parity portfolio
Risk Budgeting
- Assign risk budgets to each strategy or asset class (need not be equal)
- Risk budget = percentage of total portfolio risk attributed to each component
- Example: equities 40%, fixed income 30%, alternatives 30% of total risk
- Solve for weights that achieve the target risk allocation
- Component risk = w_i * marginal_risk_i = w_i * (Sigma * w)_i / sigma_portfolio
- Iterate or use optimization to find weights matching target risk budgets
Kelly with Parameter Uncertainty
Bayesian Kelly
- Instead of point estimates of mu and sigma, use posterior distributions
- Optimal bet size accounts for parameter uncertainty
- Results in smaller positions than plug-in Kelly — natural shrinkage
- With diffuse priors, Bayesian Kelly approximately equals 0.5 * plug-in Kelly
Shrinkage Approaches
- Shrink expected return estimates toward zero (or grand mean)
- Shrink covariance toward structured estimator (Ledoit-Wolf)
- Effective position size decreases as uncertainty increases
- 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