Kelly Criterion

TL;DR

Kelly Criterion: Bet a fraction of your bankroll proportional to your edge divided by the odds. Full Kelly maximizes long-run growth. Half-Kelly preserves most of the growth advantage with much lower variance.

What Is the Kelly Criterion?

John L. Kelly Jr. published "A New Interpretation of Information Rate" in the Bell System Technical Journal in 1956. Kelly was originally solving an information theory problem — how to optimally size bets in a game where a bettor receives imperfect inside information over a noisy channel. His solution turned out to have profound implications for gambling, investing, and any repeated decision under uncertainty.

The formula: f = (bp - q) / b*, where:

f* = the optimal fraction of the bankroll to wager
b = the net odds received on the wager (a bet at even odds pays b=1)
p = the probability of winning
q = the probability of losing (q = 1 - p)

Simplified for even-money bets: f = p - q = 2p - 1*

Example: A bet at even odds that you win 55% of the time has f* = 0.55 - 0.45 = 0.10 — bet 10% of your bankroll.

The Kelly criterion is optimal in the specific sense that it maximizes the expected growth rate of capital over many repeated bets. It is not designed to maximize expected value per bet — that would suggest betting everything on any positive-EV bet. Instead, it maximizes the geometric mean of wealth, which accounts for the compounding effect of sequential bets.

The key intuition: betting too little leaves growth potential unused; betting too much risks large drawdowns that reduce the compounding base. Kelly finds the exact point of maximum growth.

In practice, most professional users apply Fractional Kelly — betting 25–50% of the full Kelly amount. Full Kelly produces optimal growth but high variance (drawdowns of 50%+ are common even with a genuine edge). Half-Kelly produces roughly 75% of the growth rate with much lower variance.

How It Works

For a repeated bet with known edge:

Step 1: Estimate your edge
  p = your probability of winning
  q = 1 - p = your probability of losing
  b = the odds received (net payout per unit risked)

Step 2: Calculate full Kelly
  f* = (bp - q) / b
  Or simplified for even-money bets: f* = 2p - 1

Step 3: Apply Fractional Kelly
  Most practitioners use 25–50% of f*
  Half Kelly: bet f*/2
  Quarter Kelly: bet f*/4

Step 4: Be honest about p
  Kelly's optimality depends entirely on accurate
  probability estimates. Overestimating p produces
  overbetting, which is MORE dangerous than underbetting.
  Use reference class data, not optimism.

Example:
  Investment with 60% probability of 2x return,
  40% probability of losing all invested capital.
  b = 1 (even-money terms at 2x: nets 1x per dollar bet)
  f* = (1 × 0.6 - 0.4) / 1 = 0.20
  
  Full Kelly: invest 20% of portfolio.
  Half Kelly: invest 10%.

Real-World Examples

Example 1: Edward Thorp and Blackjack

Edward Thorp, mathematician and author of Beat the Dealer (1962), was the first person to apply Kelly Criterion to a practical gambling problem. Thorp had developed card counting — a system that gave blackjack players a small but genuine edge over the house in favorable deck configurations. The edge was real but small: typically 1–2%.

Thorp used Kelly to size his bets. With a 1% edge at even-money odds, Kelly suggests betting 1% of bankroll. When the deck was favorable and the edge rose to 4%, Kelly suggested 4% of bankroll. This variable sizing — large when edge is large, small when edge is small — is what Kelly's math optimizes.

Thorp later applied the same framework to options arbitrage and equity long-short investing. His fund, Princeton-Newport Partners, ran for 18 years (1969–1988) with positive annual returns in every year — one of the most consistent track records in financial history. He attributed a significant part of this to disciplined Kelly-based position sizing.

Example 2: Venture Portfolio Construction

A venture investor with a track record suggesting that her investments return 10x in 10% of cases and lose the full investment in 90% of cases is trying to size her position in a new deal.

b = 9 (10x return nets 9x per dollar invested) p = 0.10 q = 0.90

f* = (9 × 0.10 - 0.90) / 9 = (0.90 - 0.90) / 9 = 0

The Kelly formula says: bet nothing. The expected value is exactly break-even (9x return × 10% probability = 0.9x, minus 1.0x loss probability = negative EV adjusted for the gross math).

This is a sobering application: Kelly confirms what many venture LPs observe — most individual venture bets are not positive EV when the full probability distribution is honestly assessed. Positive returns in venture come from information advantages, portfolio diversification, and deal selection that genuinely shifts p above the base rate.

When to Use It

✅ For any repeated investment or betting decision where you have a genuine, estimable edge. Kelly provides the theoretically optimal position size for this case.

✅ As a sanity check on existing position sizes. If your Kelly calculation says 5% and you're holding 40%, either your edge estimate is wrong or you're overbetting.

✅ For portfolio-level position sizing. Kelly generalizes to multi-asset portfolios (though the math becomes more complex).

❌ When the probability estimate is highly uncertain. Kelly's optimality depends on accurate p estimates. Uncertain p produces uncertain f*, and overbetting (due to overestimated p) is extremely costly.

❌ When the position is not truly repeatable. Kelly maximizes long-run growth over many repetitions. For genuinely one-off decisions, the Kelly framework applies differently.

Model Combinations:

Combine with	Effect
Expected Value	EV tells you whether to bet; Kelly tells you how much to bet
Asymmetric Risk	Asymmetric payoffs (large b) allow profitable Kelly bets at lower p
Margin of Safety	Apply margin of safety to p estimates before calculating Kelly; fractional Kelly is itself a margin of safety on the position size

Common Misuses and Limitations

Misuse 1: Overestimating p. The most dangerous Kelly error. If you estimate p at 0.65 when it's actually 0.55, the Kelly-optimal bet at 0.65 significantly overbets your true edge, producing much worse outcomes than a more conservative position. The recommendation to use fractional Kelly is largely a hedge against p overestimation.

Misuse 2: Applying it to one-time decisions. Kelly is a long-run growth optimizer for repeated bets. For a single irreversible bet, a risk-adjusted expected value approach is more appropriate.

Limitation — assumes independent bets. Kelly assumes each bet's outcome is independent of others. In financial markets, assets are correlated; during crises, correlations rise sharply. The standard Kelly formula underestimates risk in correlated environments.

Expected Value: Kelly extends EV — EV tells you the sign of the bet; Kelly tells you the magnitude.

Asymmetric Risk: High b values (large asymmetric payoffs) enable profitable Kelly positions even at low p.

FAQ

Why do most practitioners use Half-Kelly instead of Full Kelly?

Full Kelly maximizes the long-run growth rate but produces very high short-run volatility — drawdowns of 50%+ are common and expected even with a genuine edge. Most investors and bettors cannot tolerate this psychologically, and excessive volatility can also produce forced liquidation at bad prices. Half-Kelly produces roughly 75% of Full Kelly's growth rate with approximately 50% of the variance — a much more tolerable trade-off for most practical applications.

How does Kelly handle correlated bets or multiple simultaneous positions?

The single-bet Kelly formula doesn't handle correlation. For portfolios with multiple correlated positions, the multivariate Kelly criterion applies — it requires estimating the full covariance matrix of returns, which is computationally complex and data-intensive. In practice, practitioners apply simple fractional Kelly to individual positions and maintain total portfolio leverage below Full Kelly to account for correlation risk.

What is the best resource for learning Kelly Criterion?

William Poundstone's Fortune's Formula (2005) is the most accessible popular treatment, covering Kelly's original paper, Edward Thorp's applications, and the broader history. Edward Thorp's A Man for All Markets (2017) describes his direct application of Kelly to blackjack and investing. For the mathematical treatment, Kelly's original paper 'A New Interpretation of Information Rate' (Bell System Technical Journal, 1956) is available online.

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What Is the Kelly Criterion?​

How It Works​

Real-World Examples​

Example 1: Edward Thorp and Blackjack​

Example 2: Venture Portfolio Construction​

When to Use It​

Common Misuses and Limitations​

Related Models​

FAQ​

Apply This Model with AI​

Further Reading​