Kelly Criterion

The Kelly criterion answers one of the most fundamental questions in trading and gambling: when you have an edge, how much should you bet?

Bet too little, and you leave money on the table — your wealth grows slower than it could. Bet too much, and a streak of losses can devastate your bankroll. The Kelly criterion finds the mathematically optimal balance point.

Developed by John L. Kelly Jr. at Bell Labs in 1956 (originally in the context of information theory and telephone signal noise), the criterion was quickly adopted by gamblers like Ed Thorp — who used it to beat the casinos at blackjack and then applied it to Wall Street. Today, Kelly-style position sizing is a core concept in quantitative finance and is discussed in interviews at many top trading firms.

The key insight is that the Kelly criterion maximizes the expected logarithm of wealth — equivalent to maximizing the long-run geometric growth rate. This is subtly different from maximizing expected value (which would suggest betting everything on any positive-EV bet) because it accounts for the compounding effect of sequential bets.

For a simple binary bet (you either win or lose), the Kelly criterion gives the optimal fraction of your bankroll to wager:

f* = (bp - q) / b

Where:

• f* = optimal fraction of bankroll to bet
• b = net odds received (e.g., if you bet $1 and win $2, b = 2)
• p = probability of winning
• q = probability of losing = 1 - p

This can also be written as:

f* = p - q/b = p - (1-p)/b

For the general case with continuous returns (relevant to trading), the formula becomes:

f* = μ / σ²

Where μ is the expected excess return and σ² is the variance of returns. This is the version most relevant to portfolio optimization and is closely related to the Sharpe ratio.

Important properties:

• If f* ≤ 0, don't bet (you have no edge or negative edge).
• If f* > 0, the bet has positive expected value and you should allocate a fraction f* of your capital.
• Betting more than 2× Kelly is worse than not betting at all in terms of long-run growth.
• Kelly betting is optimal only for maximizing geometric growth — it doesn't minimize variance or drawdown.

Let's walk through a concrete example to build intuition.

Scenario: You've identified a trading strategy with the following characteristics:

• When the trade wins, you make 50% on the capital deployed (b = 0.5, but in fractional form... actually let's use the decimal odds more carefully).

Let's use a cleaner example. Suppose you have a biased coin that lands heads 60% of the time. A casino offers you even-money bets on heads (bet $1, win $1). You start with $1,000. How much should you bet each round?

Parameters:

• p = 0.60 (probability of heads)
• q = 0.40 (probability of tails)
• b = 1 (even money — you win $1 for every $1 wagered)

Kelly fraction:

f* = (bp - q) / b = (1 × 0.60 - 0.40) / 1 = 0.20 / 1 = 0.20 (20%)

So you should bet 20% of your current bankroll on each flip. On the first flip, that's $200. If you win, your bankroll is $1,200 and you bet $240 next. If you lose, your bankroll is $800 and you bet $160 next.

Why not bet more? If you bet 50% of your bankroll each round, your expected geometric growth rate would actually be lower than at 20%. And if you bet 100%, a single loss wipes you out. The Kelly criterion accounts for the asymmetric effect of compounding: losing 50% requires a 100% gain to recover.

Growth rate comparison (expected log growth per bet):

• Kelly (f=0.20): g = 0.60 × ln(1.20) + 0.40 × ln(0.80) = 0.0201 (2.01% per bet)
• Half Kelly (f=0.10): g = 0.60 × ln(1.10) + 0.40 × ln(0.90) = 0.0151 (1.51% per bet)
• Double Kelly (f=0.40): g = 0.60 × ln(1.40) + 0.40 × ln(0.60) = 0.0003 (0.03% per bet)
• Triple Kelly (f=0.60): g = 0.60 × ln(1.60) + 0.40 × ln(0.40) = -0.0147 (negative!)

Notice how dramatically performance degrades above Kelly. At triple Kelly, you actually lose money on average despite having a positive edge on each bet. This is the most counterintuitive and important lesson of the Kelly criterion.

The Kelly criterion has direct applications in quantitative trading:

• Position Sizing: The most direct application. Given a strategy's expected return and variance, Kelly tells you how much capital to allocate. Most systematic trading firms use some variant of Kelly for this purpose.
• Strategy Allocation: When running multiple strategies simultaneously, the multivariate Kelly criterion (which accounts for correlations between strategies) determines the optimal capital allocation across strategies.
• Risk Budgeting: Kelly provides a principled framework for setting risk limits. If a trader is betting more than Kelly, they're taking on too much risk relative to their edge. If less, they're being too conservative.
• Leverage Decisions: Kelly naturally determines optimal leverage. If the optimal Kelly fraction exceeds 1, it suggests using leverage — but practitioners must be cautious about leverage costs, margin calls, and model uncertainty.

Fractional Kelly in Practice:

Almost no professional trader uses full Kelly. The reason is that full Kelly assumes you know the true probabilities exactly — and you never do. Since the penalty for overbetting is severe (as we saw above), practitioners typically use "fractional Kelly" — betting 25-50% of the full Kelly fraction.

This approach sacrifices some expected growth rate but dramatically reduces variance and maximum drawdown. The Sharpe ratio of a fractional Kelly strategy is actually similar to full Kelly (since you're reducing both return and risk proportionally), but the drawdown profile is much more manageable.

Ed Thorp, who pioneered applying Kelly to finance, famously used half Kelly. Many top quant firms use even less — typically quarter Kelly — because the cost of blowing up far exceeds the benefit of slightly faster growth.

The Kelly criterion is powerful but has important limitations that every quant trader should understand:

• Parameter uncertainty: Kelly assumes you know the true probability and payoff. In reality, these are estimated — often with significant error. Overestimating your edge leads to catastrophic overbetting. This is the strongest argument for fractional Kelly.
• Non-ergodicity of extreme outcomes: Kelly maximizes the expected log of wealth, but individual trajectories can still experience devastating drawdowns. A 50% drawdown (which is plausible even under full Kelly) takes a 100% gain to recover from — and may trigger forced liquidation, career consequences, or psychological damage long before the long run kicks in.
• Utility assumptions: Kelly implicitly assumes logarithmic utility — that a person's satisfaction from wealth grows logarithmically. This is a reasonable approximation but not universally correct. A trader with a $10M bonus target might rationally deviate from Kelly to maximize the probability of hitting that target.
• Correlation and fat tails: The basic Kelly formula assumes independent bets with known distributions. Real trading involves correlated positions, regime changes, and fat-tailed distributions that the simple formula doesn't capture.
• Transaction costs and market impact: Kelly doesn't account for the cost of rebalancing to the optimal fraction after every bet. In practice, frequent rebalancing incurs transaction costs and market impact.

Despite these limitations, Kelly-style thinking is indispensable. Even if you don't use the exact formula, the principle — size your bets in proportion to your edge, not your enthusiasm — is one of the most important ideas in quantitative risk management.