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.