Understanding the Kelly Criterion

Author: Edward O. Thorp Type: Essay — Wilmott Magazine columns (May & September 2008), revised reprint Raw: raw/2008_Understanding_Kelly_New (ingested).pdf

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Thorp's cautionary supplement to popular Kelly coverage (including Fortune's Formula). The criterion is simple to state; the details are mathematically subtle and commonly misapplied.

The Rule (Stated Simply)

Bet or invest to maximize expected logarithmic growth of capital after each decision. Equivalent to maximizing the expected growth rate of wealth.

The Misleading Shortcut

Many sources (Morningstar, Motley Fool, etc.) repeat:

$$f^* = \frac{\text{edge}}{\text{odds}}$$

This applies only to the special case of a two-valued payoff. General portfolios require solving for asset weights that maximize (\sum_i p_i \log(1 + R_i f)) — a nonlinear program over joint scenarios.

Stewart Enterprises Example (Pabrai)

Mohnish Pabrai's 2000 analysis of Stewart Enterprises (24-month scenarios):

Conservative lower-bound estimate gives (f^* \approx 0.975) — nearly full Kelly on a single name. Pabrai bet 10% without knowing Kelly. Would full Kelly have been right? Not necessarily.

Six Reasons Not to Bet Full Kelly (Even When (f^*) Is High)

1. Opportunity costs / portfolio Kelly

Kelly fraction for one bet depends on all other holdings and candidates. With (n) statistically identical independent opportunities, per-position Kelly is (< 1/n). Computing (f^) for Stewart alone overestimates size if the rest of the portfolio matters. Most common oversight Thorp sees — **dangerous because it inflates (f^)**.

Buffett's concentrated bets (25–40% of net worth) imply he sizes like a Kelly investor relative to opportunity set — but never from a single-name formula in isolation.

2. Risk tolerance / fractional Kelly

Full Kelly drawdowns exceed most investors' comfort. Use (f = c \cdot f^*) with (0 < c < 1). See the-kelly-criterion-thorp-2006 for half-Kelly arithmetic.

3. Scenario optimism

If true outcomes are worse than the "conservative" model, you inadvertently bet more than (f^*) — more risk, less return. Fractional Kelly hedges estimation error.

4. Black swans

Nassim Taleb's point: rare high-impact events are under-modeled. Include extreme-loss scenarios in the optimization; (f^*) drops.

5. The long run is asymptotic

Kelly dominance properties appear as time → ∞. You may not get enough independent bets for asymptotics to rescue you. Leib's paradox: non-Kelly sequences can beat Kelly with probability arbitrarily close to 1 for finite horizons — even differing at every trial.

"Essentially different" has technical meaning for constant-fraction myopic strategies; path-dependent gambling systems are trickier (see Ethier 2010).

6. Large wealth goals take time

Even Buffett-scale multiples require decades. Kelly minimizes expected time to large goals asymptotically — not next quarter.

Who Should Use Kelly?

Appropriate when:

• many repeated opportunities
• edge estimable with scenario discipline
• investor tolerates fractional sizing
• portfolio-level optimization is feasible

Less appropriate when:

• one-shot bets dominate
• correlation hidden across "different" trades
• edge is story-driven not distribution-driven

Connections

• kelly-criterion
• position-sizing
• alternative-histories
• decision-quality-vs-outcome
• the-kelly-criterion-thorp-2006
• fortunes-formula
• fooled-by-randomness

Sources

• raw/2008_Understanding_Kelly_New (ingested).pdf