Bet Sizing

Purpose

Provides frameworks for determining how much capital to allocate to individual positions within a portfolio. Covers the Kelly criterion, fractional Kelly, risk budgeting, liquidity-based sizing, and conviction weighting. Proper bet sizing is critical — even a portfolio of good ideas can fail with poor sizing.

Layer

4 — Portfolio Construction

Direction

prospective

When to Use

Determining the appropriate size for a new position
Applying the Kelly criterion to a bet or investment with estimable odds
Allocating a risk budget across active positions
Setting maximum position sizes based on liquidity or risk limits
Sizing positions proportional to conviction and edge
Deciding the optimal number of positions in a concentrated portfolio
Scaling position sizes with volatility changes

Core Concepts

Kelly Criterion (Discrete)

For a binary bet with payoff odds b, win probability p, and loss probability q = 1-p:

f* = (b*p - q) / b

where f* is the optimal fraction of wealth to wager. The Kelly criterion maximizes the expected logarithm of wealth (geometric growth rate) over repeated bets.

Properties:

f* = 0 when edge = 0 (no bet when there is no advantage)
f* < 0 when negative edge (the formula tells you to bet the other side)
f* > 0 only when b*p > q (positive expected value)

Kelly Criterion (Continuous / Investment)

For a normally distributed investment return with expected excess return mu-r_f and variance sigma^2:

f* = (mu - r_f) / sigma^2

This gives the fraction of total wealth to allocate. For example, an asset with 8% expected excess return and 20% volatility: f* = 0.08 / 0.04 = 2.0 (200% of wealth — implying leverage).

Fractional Kelly

Full Kelly sizing is theoretically optimal but practically too aggressive because:

It assumes perfect knowledge of probabilities and payoffs
It produces large drawdowns (the expected drawdown of full Kelly is significant)
Estimation error in parameters can turn optimal into catastrophic

Practical approach: use a fraction of Kelly, commonly:

Half Kelly (f/2):* Achieves 75% of the growth rate with substantially lower variance and drawdown risk
Third Kelly (f/3):* Even more conservative; appropriate when parameter uncertainty is high
Quarter Kelly (f/4):* Suitable for highly uncertain estimates

The key insight: the growth rate curve is flat near the peak. Reducing from full Kelly to half Kelly only sacrifices 25% of growth but reduces risk dramatically.

Risk Budgeting

Allocate risk (not capital) across positions. The total risk budget is the maximum acceptable portfolio risk (e.g., 10% VaR or 5% tracking error).

VaR-based budgeting:

Total VaR budget: e.g., $1M at 95% confidence
Allocate across positions: Position VaR_i <= allocated VaR_i
Position VaR = w_i * sigma_i * z_alpha * Portfolio Value

Tracking error budgeting (for active managers):

Total active risk budget: e.g., 4% tracking error
Allocate across bets: each active bet consumes a portion of tracking error
Size active positions so that sum of risk contributions equals total risk budget

Maximum Position Sizes

Hard limits on individual positions to prevent concentration risk:

Liquidity-based limits:

Position < X% of average daily volume (ADV) — common limits: 10-25% of ADV
Ensures ability to exit within a reasonable time frame (e.g., 5-10 trading days)

Risk-based limits:

Position risk contribution < X% of portfolio volatility (e.g., max 10% of portfolio risk)
Single position < X% of portfolio value (common: 5% for diversified, 10% for concentrated)

Regulatory/mandate limits:

Mutual fund: no more than 5% in a single name (diversified fund) or 25% (non-diversified)
Index tracking: weight cannot deviate from benchmark by more than specified amount

Conviction Weighting

Size positions proportional to the strength of the investment thesis:

High conviction (largest positions): Strong edge, deep research, multiple confirming factors
Medium conviction: Solid thesis but some uncertainty or limited information
Low conviction (smallest positions): Early-stage idea, limited edge, or purely diversification-motivated

Framework: Score each position on edge strength (1-5) and certainty (1-5). Size proportional to the product: edge * certainty.

Optimal Number of Positions

Trade-off between diversification and conviction:

Concentrated (10-20 positions): High conviction, deep research. Each position is 5-10% of the portfolio. Appropriate when the manager has genuine skill and edge.
Diversified (50-100 positions): Lower conviction per position but broader risk reduction. Each position is 1-3%. Appropriate for systematic or factor-based strategies.
Very diversified (100+): Index-like. Risk comes from factor tilts, not individual positions.

Volatility Scaling

Adjust position sizes inversely with volatility to maintain consistent risk per position:

Adjusted size = Target risk / Current volatility

When volatility doubles, position size halves, keeping the dollar risk constant. This is a core principle in managed futures and risk-targeting strategies.

Anti-Martingale (Kelly-like) Sizing

Increase position sizes after gains (wealth grows, so Kelly fraction applied to larger base) and decrease after losses. This contrasts with martingale strategies (doubling down after losses) which can lead to ruin.

Kelly naturally implements anti-martingale sizing: bet a constant fraction of current wealth, so absolute bet size grows with wealth and shrinks with losses.

Key Formulas

Formula	Expression	Use Case
Kelly (Discrete)	f* = (b*p - q) / b	Binary bet sizing
Kelly (Continuous)	f* = (mu - r_f) / sigma^2	Investment position sizing
Half Kelly	f = f* / 2	Practical conservative sizing
Growth Rate at Kelly	g* = (mu - r_f)^2 / (2*sigma^2)	Maximum geometric growth
Growth Rate at f	g(f) = f(mu - r_f) - f^2sigma^2/2	Growth rate for any fraction
Volatility-Scaled Size	w = target_risk / sigma_i	Constant risk per position
Position VaR	VaR_i = w_i * sigma_i * z_alpha * V	Position-level risk

Worked Examples

Example 1: Kelly Criterion for a Discrete Bet

Given:

Win probability: p = 55%
Loss probability: q = 45%
Even-money payoff: b = 1 (win $1 for every $1 wagered)

Calculate: Optimal bet size

Solution:

f* = (b*p - q) / b = (1 * 0.55 - 0.45) / 1 = 0.10 / 1 = 10%

Interpretation: Wager 10% of current wealth on each bet. This maximizes long-run geometric growth.

Practical adjustment (half Kelly): f = 10% / 2 = 5% — achieves 75% of the maximum growth rate with much lower drawdown risk.

Full Kelly expected drawdown: the probability of losing 50% of wealth at some point is substantial. Half Kelly dramatically reduces this tail risk.

Example 2: Continuous Kelly for an Investment