Kelly criterion, for quant interviews

The question under the question

A “how much should you bet?” prompt is never asking for a plug-in. It is asking you to name your objective function. Fix the setup: a repeatable bet with win probability $p$ and net odds $b$ (you risk 1 to win $b$ on top of your stake), and you choose what fraction $f$ of your current bankroll to stake, over and over.

State the tension up front. Expected profit per bet is $f(pb-q)$, which is maximized by betting everything the moment the edge $pb-q>0$ — and that guarantees ruin. Kelly resolves the tension by optimizing a different quantity. On an even-money bet with $p=0.6$ and $b=1$, the EV-maximizer says $f=1$ and Kelly says $f=0.2$. That distinction is the entire point of the question.

Edge, odds & the formula

Define the pieces so you can map any prompt onto them. Edge is expected profit per unit staked, $pb-q$. Odds is $b$, the net payout (profit-to-stake), not the decimal/total payout — flag this explicitly, it is the number-one unit error. The result:

Even-money special case ($b=1$) collapses to $f^*=p-q=2p-1$. Mind the gate: if $pb\le q$ the edge is non-positive, $f^*\le 0$, and Kelly says bet zero (or take the other side). Kelly never puts money on a negative-edge game.

Worked example, actual numbers

Case A (even money). $p=0.6$, $q=0.4$, $b=1$, so $f^*=0.6-0.4/1=0.2$ — stake 20% of bankroll. The long-run growth rate is $g(0.2)=0.6\ln 1.2+0.4\ln 0.8=0.1094-0.0893=0.0201$ per bet, so the bankroll compounds at about $e^{0.0201}\approx 1.0203\times$ per bet, geometrically.

Case B ($b\ne1$). A 2-to-1 payout with $p=0.4$. The edge is $pb-q=0.4(2)-0.6=0.2>0$, and $f^*=0.4-0.6/2=0.4-0.3=0.1$ — stake 10% (cross-check: $(pb-q)/b=0.2/2=0.1$). Its growth is $g=0.4\ln 1.2+0.6\ln 0.9=0.0729-0.0632=0.0097$. The general formula is not a black box; it is just the edge divided by the odds.

Deriving it: maximize log-wealth, not wealth

Why the logarithm? Over $n$ independent bets the bankroll multiplies, $W_n=W_0\prod_i(1+bf\cdot\mathbb{1}_{\text{win}}-f\cdot\mathbb{1}_{\text{lose}})$. Take the log and average:

By the law of large numbers $g(f)$ is the almost-sure long-run growth rate — maximizing it is not a utility preference, it is the a.s. fastest-growing strategy. The one-line derivation:

The satisfying step is the $bf(p+q)$ term collapsing to $bf$ because $p+q=1$. And since $g''(f)=-\frac{pb^2}{(1+bf)^2}-\frac{q}{(1-f)^2}<0$ everywhere, $g$ is strictly concave, so this critical point is the unique maximum. That concavity is exactly what separates the two averages below.

The curve is not symmetric: overbetting

This is the single most testable intuition. $g(f)$ rises to its peak at $f^*=0.2$, then falls, crossing back through zero at $f\approx 0.389$ — just below $2f^*=0.4$ (the quadratic/Gaussian approximation puts the zero exactly at $2f^*$). Past that point you hold a positive-edge bet with negative growth: you compound yourself broke.

Underbetting merely leaves growth on the table; overbetting destroys it. At $f=1$ one loss zeroes the bankroll and $g(1)=0.6\ln 2+0.4\ln 0=-\infty$. That asymmetry — steep cliff on the right, gentle slope on the left — is why nobody serious bets full Kelly.

Fractional Kelly & variants that change the answer

Everything above assumed you knew the game. The interview usually pays off on the assumptions that break:

• You don’t actually know $p$. Kelly assumes exact $p$ and $b$. Because the curve is steep on the overbet side, an overestimated edge pushes you past $f^*$ into the punished region. Shading down to half-Kelly buys robustness: it keeps 74.7% of the maximum growth for roughly half the volatility, while $1.5\times$ Kelly gives a nearly identical 73.2% with far larger swings, and $2\times$ Kelly is −12.2% — negative.
• Continuous, one-shot, or correlated bets. For a log-normal continuous bet the analogue is $f^*=\mu/\sigma^2$ (the Merton fraction); simultaneous correlated bets share risk, so their combined Kelly stake is below the sum of the individual stakes.
• Finite horizon or genuine risk aversion. Kelly is optimal for the long-run growth rate, not for a risk-averse player with a fixed horizon or a drawdown limit. If the prompt adds “you have 10 bets” or “you can’t lose more than 20%,” Kelly is no longer the answer — that is often exactly what is being probed.
• Leverage cap. A huge edge can yield $f^*>1$; without borrowing you cap at your bankroll and the formula’s advice is truncated.

Common mistakes

• Maximizing EV instead of log-wealth. “Positive edge, so bet big” leads to bet-everything and near-certain ruin. Arithmetic EV rises with stake forever; only the geometric growth rate has an interior peak.
• Odds in the wrong units. $b$ is net odds (profit per unit staked). Feeding in decimal/total odds ($1+b$) silently mis-sizes every bet.
• Skipping the edge check. Plug in a negative-edge game and $f^*$ comes out negative; read that as “don’t bet,” not as a number to stake.
• Full Kelly on an estimated $p$. Treating a noisy point estimate as the true probability puts you on the wrong side of the peak — the one side that is actually costly.
• Assuming the curve is symmetric. Betting $2\times$ Kelly is not “twice as aggressive but still fine”; it wipes out the entire growth advantage and goes negative.