The Kelly Criterion: Bet Sizing as a Solved Problem

Most people obsess over which bets to take. Kelly answers the question that determines survival: how much? It is also a favorite interview topic at trading firms — SIG famously discusses poker and bet sizing at length — because sizing errors, not signal errors, are what kill traders.

The setup and the answer

You have a bet that wins with probability $p$ (paying $b$-to-1) and loses with probability $q = 1 - p$. You'll face it repeatedly and can stake any fraction $f$ of your bankroll each time. What $f$ maximizes long-run growth?

Wealth after $n$ bets with $W$ wins: $(1 + fb)^W (1 - f)^{n-W}$. Maximizing the expected log growth rate

$$g(f) = p \ln(1 + f b) + q \ln(1 - f)$$

via $g'(f) = 0$ gives the Kelly fraction:

$$f^* = \frac{pb - q}{b} = \frac{\text{edge}}{\text{odds}}$$

For an even-money bet ($b=1$): $f^* = p - q$. A 55/45 coin → bet 10% of bankroll. For continuous return streams the analogue is $f^* \approx \mu / \sigma^2$ — expected excess return over variance — which for typical market numbers implies eye-watering leverage, a first hint to treat full Kelly with suspicion.

Why log? Because repetition forces it

The log isn't a utility-function taste choice; it emerges from compounding. Your terminal wealth is a product of random factors, so its long-run growth rate is the average of logs (by the law of large numbers on $\frac1n \sum \ln(\text{factor}_i)$). Maximizing expected wealth instead says "bet everything, every time" — which ends in certain ruin with repetition, because expectation is dominated by vanishing-probability jackpot paths. Kelly is what maximizing typical (median) long-run wealth looks like.

The asymmetry every trader must internalize

Plot $g(f)$: it rises to a peak at $f^*$, then falls, crossing zero at exactly $2f^*$. The consequences are not symmetric:

• Half Kelly ($f^*/2$): you keep 75% of the optimal growth rate at half the volatility. Cheap insurance.
• Double Kelly ($2f^*$): growth rate zero — infinite volatility, no compounding.
• Beyond $2f^*$: negative growth. You can have a genuine edge on every single bet and still go broke, guaranteed, purely from oversizing.

Hence the asymmetry: under-betting costs you a little growth; over-betting costs you everything. And since real edges are estimated — usually overestimated — betting "full Kelly on your estimated edge" is over-betting on your true edge with high probability. This is why practitioners run fractional Kelly (a quarter to a half), and why "what fraction of Kelly and why" is a better interview answer than the formula itself.

Worked micro-example

An interviewer offers: 60/40 coin, even money, $1,000 bankroll, 100 flips. Sizing?

$f^* = 0.6 - 0.4 = 0.2$. Full Kelly stakes 20% each flip; growth rate $g = 0.6\ln 1.2 + 0.4 \ln 0.8 \approx 2.0\%$ per flip. Recommended answer: bet 5–10% (quarter-to-half Kelly), citing estimation risk on $p$ and the 75%-of-growth-at-half-vol trade-off. Mentioning that betting 50% per flip — despite a 20% edge — would lose money on average per log-dollar is the detail that lands.

Beyond coin flips

Kelly generalizes: simultaneous bets shade each bet's size down with correlation (the portfolio version reduces to mean-variance with a specific risk aversion); a strategy's Kelly leverage is $\mu/\sigma^2$, so a Sharpe-1 strategy at 10% vol "wants" 10× leverage at full Kelly — and the fact that no sane person runs that is the practical case for fractional sizing in one sentence. The same $2f^*$ cliff also explains a real failure mode: funds that added leverage after good (lucky) years were increasing $f$ exactly when their estimated $\mu$ was most inflated.