Kelly Criterion

The Kelly criterion is a sizing rule for repeated bets: given a genuine edge, what fraction of capital maximizes long-run geometric growth — wealth compounded multiplicatively across many independent rounds?

Originated by John Kelly (1956, information theory). Made operational by Ed Thorp in blackjack and markets. Popularized historically in Fortune's Formula.

It is not a trade-entry rule. It answers: how much, once expected value is positive.

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The Core Idea

Kelly optimization maximizes (\mathbb[\log W]) — log utility. Each marginal dollar matters less than the last because survival and compounding through time dominate single-shot arithmetic averages.

the-jackpot-age and Bernoulli's geometric-mean insight (via fortunes-formula) say the same thing in different language: positive arithmetic EV can still destroy most paths; the geometric mean / median path governs survival.

how-to-get-rich: even 51/49, betting the entire bankroll each round eventually wipes you out. Kelly formalizes "bet a fraction of your edge."

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Formulas

Even-money binary bet

Win probability (p), lose (q = 1 - p), even payoff:

$$f^* = p - q$$ $$g(f) = p \log(1 + f) + q \log(1 - f)$$

51/49 example: (f^* = 0.02) (the-kelly-criterion-thorp-2006, how-to-get-rich).

General two-outcome (edge/odds shortcut)

$$f^* = \frac{bp - q}{b}$$

where (b) is payoff per unit risked. understanding-the-kelly-criterion warns: edge/odds applies only to this two-valued case — not general portfolios.

Continuous portfolio (excess return)

$$f^* = \frac{m - r}{s^2}$$

Expected excess return (m - r), variance (s^2), risk-free rate (r) (the-kelly-criterion-thorp-2006).

Multi-scenario

$$g(f) = \sum_i p_i \log(1 + R_i f)$$

Optimize (f) (or the weight vector) — requires portfolio-level scenario work, not isolated tickers.

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Bold vs Timid vs Kelly

Proportional betting: never stake 100% of current bankroll, so you cannot hit literal zero in one step — but wealth can still approach arbitrarily small values (fortunes-formula).

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Full Kelly vs Fractional Kelly

Full Kelly maximizes asymptotic growth but produces drawdowns most investors cannot stomach (the-kelly-criterion-thorp-2006, understanding-the-kelly-criterion).

Use (f = c \cdot f^*) with (0 < c < 1):

Reasons to fractionalize (understanding-the-kelly-criterion):

1. Opportunity costs — per-bet Kelly ignores rest of portfolio → overestimates (f^*)
2. Risk tolerance
3. Edge overestimate — true edge below model → full Kelly can yield (g \leq 0)
4. Black swans — fat tails not in scenario table
5. Finite horizon — asymptotic dominance may not arrive in time

Overbetting is always worse than underbetting (the-kelly-criterion-thorp-2006, fortunes-formula LTCM lesson).

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Kelly vs Stop-Based Sizing

Wiki default from position-sizing:

$$\text{position size} = \frac{\text{allowed loss}}{\text{distance to invalidation}}$$

Stop-based sizing does not need win-rate estimates. For discretionary trading with fuzzy edge, it is often more robust than plugging guessed (p) into Kelly.

Kelly still sets a ceiling intuition: thin-edge coin-flip quality estimates should not justify 8% account risk when (f^* \approx 2%).

Buffett-style concentration can be Kelly-consistent only when sized against the full opportunity set, not a single-name formula (understanding-the-kelly-criterion).

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When Kelly Misleads

• Unknown or unstable edge — overestimating (p) inflates (f^*) nonlinearly
• Correlated books — five trades on one macro thesis = one bet
• Path-dependent systems vs constant-fraction Kelly assumptions
• Reputational/legal ruin — how-to-get-rich; Kelly math is financial only
• LTCM-style overbetting with Nobel-approved models (fortunes-formula)

the-jackpot-age: build more edge rather than risk more size.

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Connections

• ergodicity
• position-sizing
• alternative-histories
• decision-quality-vs-outcome
• trading-edge
• risk-reward-ratio
• Edward-Thorp
• Claude-Shannon
• John-Kelly-Jr
• naval-ravikant

Sources

• the-kelly-criterion-thorp-2006 — derivation, fractional Kelly math, blackjack, sports, continuous portfolio
• understanding-the-kelly-criterion — misapplication warnings, portfolio Kelly, Pabrai example, Leib paradox
• fortunes-formula — Kelly–Shannon–Thorp history, geometric mean, LTCM overbetting
• the-jackpot-age — log-wealth / geometric mean survival
• how-to-get-rich — ruin avoidance intuition
• fooled-by-randomness — path-dependent survival, tail risk