# The Kelly Criterion: How to Size Bets for Maximum Long-Term Growth Cite as: Gorak, R. (2026). The Kelly Criterion: How to Size Bets for Maximum Long-Term Growth. Tradicted. https://www.tradicted.com/research/thorp-kelly-2006/ Paper: Edward O. Thorp — *The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market* Published in: Handbook of Asset and Liability Management (2006) Original: https://sites.oxy.edu/lengyel/M330/thorp/paper1.pdf Key finding: Thorp (2006) reports that a Kelly-managed convertible-hedging investment partnership compounded at approximately 20 percent annually for about 28.5 years, from November 3, 1969 through May 1998, turning $10,000 into $18 million tax-exempt. --- The Kelly criterion is a bet-sizing rule that maximizes a bettor's expected long-run growth rate by maximizing expected logarithmic wealth. A gambler facing a coin biased 53 percent in his favor should wager 6 percent of his bankroll on each flip, the paper's Example 2.1. In *The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market*, Thorp (2006) describes managing a Kelly-sized investment partnership for about 28.5 years. From November 3, 1969 through May 1998, the partnership compounded at approximately 20 percent annually, turning $10,000 into $18 million tax-exempt. ## What the Study Found For a coin biased 53 percent in the gambler's favor, the Kelly-optimal fraction is f* = .06, or 6 percent of the bankroll, per bet. Half-Kelly betting retains 75 percent of the maximum growth rate while cutting the chance of losing half the bankroll from 1/2 to 1/8. A 1994 field test of a casino sports betting system turned a $50,000 bankroll into a $123,000 profit over 101 days. A Kelly-optimal reallocation would have gained 117.6 percent with no rebalancing versus the board's actual 73.5 percent gain over the same period. Over about 28.5 years managing a Kelly-sized convertible-hedging partnership, Thorp compounded capital at approximately 20 percent annually with about 6 percent volatility. ## Methodology The securities-market case study uses monthly return data for Berkshire Hathaway, BioTime, the S&P 500, and T-bills. The case study covers 63 months of data, and the sports betting field test covers 101 days of live wagers. The case study spans 3/31/92 through 6/30/97, and the sports betting test ran during the first four and a half months of 1994. Portfolio allocations were constrained by margin requirements of 50 percent initial and 30 percent maintenance. The sports book test used a deliberately conservative expectation estimate of about 6 percent. ## Key Statistics | Metric | Finding | Context | |---|---|---| | Kelly optimal fraction (even-money bet) | f* = p − q | Example 2.1; p=.53 gives f*=.06 | | Sports betting field test profit | $123,000 on a $50,000 bankroll | 101 days, first 4.5 months of 1994 | | Half-Kelly growth rate vs. full Kelly | 75% of g(f*) | Section 7(d), fractional Kelly | | Probability of losing half the bankroll | 1/2 (full Kelly) vs. 1/8 (half Kelly) | Section 7(d), equation (7.13) | | XYZ Corp. recommended vs. actual portfolio gain | 117.6% (no rebalance) / 199.4% (one rebalance) vs. 73.5% actual | 8/17/97-3/31/98, Tables 8.2-8.4 | | Convertible-hedging partnership compounding | ~20% annually, sd ~6% | Nov 3, 1969-May 1998 (~28.5 years) | | Continuous-time Kelly fraction | f* = (m−r)/s² | Equation (7.3), securities markets | | Berkshire Hathaway price growth, 1964-1998 | 20 → 70,000 (3500x), ~27% annualized | Example 7.3 | ## Why This Matters The Kelly criterion offers a principled alternative to ad hoc position-sizing rules used by traders and portfolio managers. Because overbetting is punished far more severely than underbetting, practitioners uncertain about their true edge are better served erring toward a fractional Kelly approach. The framework generalizes from single bets to portfolios of correlated securities, which is why it underlies leverage and asset-allocation decisions beyond simple gambling contexts. Its core insight is that maximizing expected wealth differs from maximizing expected growth of wealth. That distinction is foundational to how quantitative investors think about bet sizing and capital allocation. Applying the criterion requires your own win rate and payoff ratio as inputs. A [Kelly criterion calculator](/tools/kelly-criterion-calculator/) runs the fraction once you have those two numbers. ## FAQ ### What is the Kelly criterion? f* = p − q is the Kelly-optimal fraction of a bankroll to wager on a favorable even-money bet. Here p is the win probability and q is the loss probability. Thorp (2006) shows the rule maximizes the expected logarithm of wealth rather than expected wealth itself. For p=.53, f*=.06. ### How does fractional (half) Kelly compare to full Kelly betting? 75 percent of the full Kelly growth rate is retained by betting half the Kelly fraction (f*/2), according to Thorp (2006). Half-Kelly also cuts the probability of ever losing half the starting bankroll from 1/2 down to 1/8. Thorp reports that most blackjack players and investors he has worked with prefer half-Kelly's reduced volatility despite the lower growth rate. ### What did Thorp's Kelly-based investment partnership achieve over the long run? 20 percent annual compounding with a standard deviation of about 6 percent is what Thorp's market-neutral convertible-hedging partnership achieved. The period ran from November 3, 1969 through May 1998, about 28.5 years. An initial $10,000 grew to $18 million tax-exempt, with about $80 billion in cumulative trading volume across roughly 1.25 million bets. ### How did a Kelly-sized casino sports betting system perform in real-world testing? $123,000 was the profit on a $50,000 test bankroll after Thorp's group bet on 101 days of games. The test ran during the first four and a half months of 1994. Total action was about $2,000,000, with bets sized using the Kelly criterion and a typical expectation of about 6 percent. ## Related Research Coming soon ## Source Thorp, E. O. (2006). The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market. In *Handbook of Asset and Liability Management*. [Read the full paper →](https://sites.oxy.edu/lengyel/M330/thorp/paper1.pdf)