The Kelly Criterion

You’ve spent the last two lessons learning a cruel fact: wealth compounds multiplicatively, so the geometric mean — not the arithmetic one — governs your real long-run growth, and volatility is a tax that drags that geometric growth below the headline average. Lesson 2 left you staring at the volatility drag and probably thinking: so a bigger position isn’t always better. Correct. A bigger bet raises your expected return but inflates your volatility drag even faster, and somewhere there’s a sweet spot.

This lesson finds that sweet spot exactly. Given a genuine edge, there is a single, computable fraction of your bankroll to wager on each repeated bet that maximizes your long-run compound growth rate. Bet less and you grow slower than you could; bet more and the volatility drag eats you alive — past a certain point it doesn’t just slow you down, it drives your bankroll to zero with certainty. That magic fraction is the Kelly criterion, and for a simple binary bet it collapses to a formula so clean you’ll be able to do it in your head: $f^* = p - q/b$, the edge divided by the odds.

The bet-sizing problem — neither all-in nor all-out

Analogy. Imagine a slot machine that’s secretly rigged in your favor — over many pulls it pays back more than it takes. Wonderful. But each pull is still a gamble, and you control one dial: how much of your wallet to feed it each time. Crank the dial to “everything” and a single unlucky pull leaves you with nothing and no way back. Leave the dial at “nothing” and your favorable machine makes you exactly zero dollars. The whole game is the dial setting in between.

Definition. The bet-sizing problem is: given a repeated bet with a known positive edge, what fraction $f$ of your current bankroll should you stake on each play to make your wealth grow as fast as possible over the long run? Crucially you re-bet a fraction of your current bankroll every time, so wins compound and a loss never quite zeroes you out (as long as $f < 1$).

Worked example — the two disastrous extremes. Start with 1,000 dollars and a great edge. Suppose you go all-in ($f = 1$) each time. You might win, win, win, win — and on the very first loss your bankroll is multiplied by zero. Game over, permanently, no matter how huge the prior streak. With $f = 1$ and any chance of losing, ruin isn’t a risk; over enough bets it’s a certainty. Now the other extreme: $f = 0$. You never wager a cent, so your favorable machine multiplies your bankroll by exactly 1 forever. Growth: zero. The truth lives strictly between these cliffs — there is an interior $f$ that beats both, and the rest of this lesson pins it down.

Why maximize expected LOG wealth, not expected wealth

Analogy. Two roads to the same destination. Road A: maximize the arithmetic average of your final wealth across all possible futures. Road B: maximize the average of the logarithm of your final wealth. They sound like cousins, but they give wildly different advice — Road A says “bet everything,” Road B says “bet Kelly” — and Road B is the one that actually makes the typical you rich.

Definition. Because each bet multiplies your bankroll, after $n$ bets your wealth is a product of growth factors: $W_n = W_0 \cdot \prod_i (1 + f \cdot x_i)$, where $x_i$ is the $i$-th bet’s outcome per unit staked. Take logs and the product becomes a sum:

$\ln W_n = \ln W_0 + \sum_{i=1}^{n} \ln(1 + f \cdot x_i).$

By the law of large numbers, that sum of i.i.d. log-returns grows like $n$ times its expected value, so the long-run growth rate per bet is

$g(f) = E[\ln(1 + f \cdot x)].$

Maximizing your actual long-run growth rate means maximizing expected log-wealth — which (recall lessons 1–2) is exactly the geometric growth rate. This is the whole philosophical payload of Kelly.

Why not maximize expected wealth directly? Because $E[W_n]$ — the plain arithmetic expectation — is dominated by a vanishingly rare jackpot path and is maximized by betting 100%. A strategy can have the highest possible expected wealth while making almost every real player broke, because the average is hijacked by a one-in-a-billion path that wins every single bet. (Sound familiar? It’s the mean-vs-median, right-skew problem from the compounding lessons, in its purest form.) Expected log-wealth ignores that mirage: the log of zero is $-\infty$, so any strategy with a real chance of ruin is infinitely penalized. Maximizing $E[\ln W]$ is maximizing the growth the typical path experiences — the median fortune, not the fantasy one.

Deriving the Kelly fraction for a binary bet

Setup. Consider the cleanest possible bet. With probability $p$ you win and gain $b$ times your stake (net odds $b$); with probability $q = 1 - p$ you lose your stake. You wager a fraction $f$ of your bankroll. So a win multiplies your bankroll by $(1 + b f)$ and a loss multiplies it by $(1 - f)$. The expected log-growth per bet is

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

The calculus. To find the growth-maximizing $f$, differentiate with respect to $f$ and set the derivative to zero:

$g'(f) = \frac{p\,b}{1 + b f} - \frac{q}{1 - f} = 0.$

Move the second term across and cross-multiply:

$p\,b\,(1 - f) = q\,(1 + b f).$

Expand both sides:

$p b - p b f = q + q b f.$

Gather the $f$ terms on one side:

$p b - q = p b f + q b f = b f\,(p + q).$

Since $p + q = 1$, the bracket is just 1, so $p b - q = b f$, and therefore

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

The reading that makes it unforgettable. The numerator $p b - q$ is the bet’s edge — your expected profit per unit staked (you gain $b$ with probability $p$, lose $1$ with probability $q$). The denominator $b$ is the odds. So Kelly is simply:

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

Bet a fraction of your bankroll equal to your edge divided by the payoff odds. If the edge is zero or negative, $f^* \le 0$ — meaning don’t bet. Clean, interpretable, and exactly the curve the interactive island below draws.

Worked example — the even-money coin

Setup. The most common case: $b = 1$ (win and you get even money — double your stake). Then $q/b = q$, so

$f^* = p - q = p - (1 - p) = 2p - 1.$

For an even-money bet, the Kelly fraction is just twice your edge over fifty-fifty.

Run the numbers.

At $p = 0.5$ there’s no edge and Kelly correctly says wager nothing; at $p = 0.4$ the edge is negative and Kelly goes negative, telling you to flip sides or stay out.

The growth you actually earn. Take the $p = 0.6$ case, $f^* = 0.2$. Plug back into the growth rate:

$g(0.2) = 0.6\,\ln(1.2) + 0.4\,\ln(0.8) = 0.6\,(0.1823) + 0.4\,(-0.2231).$

That’s $0.1094 - 0.0892 = 0.0202$ per bet — about 2% compound growth on every single flip. Over 100 flips that compounds to roughly $e^{2.02} \approx 7.5\times$ your bankroll. A 60/40 coin, sized correctly, is a money machine; sized at 100% it’s a guillotine.

Worked example — uneven odds, a low win-rate edge

Setup. Edges don’t require winning most of the time. Consider a bet you only win 40% of the time ($p = 0.4$, $q = 0.6$) but which pays 3-to-1 when it hits ($b = 3$) — think a long-shot lottery ticket that’s mispriced in your favor, or an out-of-the-money option you’ve correctly judged cheap.

Compute it.

$f^* = \frac{p b - q}{b} = \frac{(0.4)(3) - 0.6}{3} = \frac{1.2 - 0.6}{3} = \frac{0.6}{3} = 0.20.$

You should wager 20% of your bankroll on a bet you lose three times out of five. The edge here is $p b - q = 0.6$ per unit — a huge 60% expected profit — delivered through infrequent, fat payoffs. Notice the same $f^* = 0.2$ as the 60/40 even-money coin, arrived at by a completely different route: there, frequent small wins; here, rare big ones. Kelly cares only about edge-over-odds, not about how the edge is packaged.

The shape of growth — the inverted hump

The formula gives you the peak, but the shape of the growth-vs-bet-size curve is where the intuition lives. Drag the sliders below: at $f = 0$ growth is zero (you’re not playing), it climbs to a maximum exactly at the Kelly fraction $f^*$, then falls back to zero around twice Kelly, and dives negative beyond that. The peak is gentle; the cliff on the right is not.

The curve is the whole lesson in one picture: a hill with a sheer drop on its right flank. You want to stand on top of the hill. Standing a little short of the peak (under-betting) costs you only a sliver of growth. Standing a little past the peak (over-betting) costs you more — and walking off the right edge (past $2f^*$) doesn’t cost you growth, it reverses it.

The over-betting cliff — the one warning that matters

Analogy. Driving toward a guardrail. Easing off the accelerator before the rail costs you a few seconds. Flooring it past the rail costs you the car. The downside isn’t symmetric, and neither is over-betting.

Definition. Because the growth curve crosses back through zero at roughly $2f^*$, betting more than twice the Kelly fraction turns your long-run growth rate negative — even though the underlying bet still has a positive expected value. Negative growth rate means your bankroll trends to zero: you go broke with probability 1, slowly but inexorably, by winning bets too aggressively.

Worked example. Back to the 60/40 even-money coin, where $f^* = 0.2$. Twice Kelly is $f = 0.4$. Check the growth there:

$g(0.4) = 0.6\,\ln(1.4) + 0.4\,\ln(0.6) = 0.6\,(0.3365) + 0.4\,(-0.5108) = 0.2019 - 0.2043 \approx -0.002.$

Essentially zero — you’ve handed back all of your edge. Push to $f = 0.5$ and growth is clearly negative; the very same favorable coin now drains your account. You didn’t lose your edge — you over-bet it into a death spiral.

Pitfalls — why nobody bets full Kelly

Kelly is mathematically optimal under assumptions that almost never hold perfectly. Three pitfalls explain why the real world bets less.

Pitfall 1 — you don’t actually know $p$ and $b$. The formula assumes you know your win probability and payoff exactly. In reality you estimate them, and estimation error is asymmetric in its damage: because the cliff is on the over-betting side, overestimating your edge (betting too big) is far more dangerous than underestimating it. A confidently wrong $p$ that’s 5 points too high can quietly push your bet past the danger zone. This is the single biggest reason for fractional Kelly (bet half or a quarter of $f^*$) — the topic of the next lesson.

Pitfall 2 — Kelly maximizes growth, not comfort. Full Kelly is brutally volatile. A full-Kelly bettor routinely suffers drawdowns of 50% or more on the way to long-run dominance. Mathematically optimal growth and emotionally survivable investing are different objectives; most humans can’t psychologically hold a position that the math says is “correct” while it’s cut in half.

Pitfall 3 — it needs many independent, repeated, re-bettable plays. Kelly’s optimality is a long-run statement resting on the law of large numbers. It assumes a long series of bets and the ability to re-stake a continuous fraction of your current bankroll each time (continuous rebalancing). A single one-shot bet, or a bet where you can’t size continuously, isn’t the world Kelly was derived for — apply it literally there and the “optimality” guarantee evaporates.

When it matters

Sports and poker bankroll management. A handicapper who’s identified a mispriced line, or a poker player with a known win rate, faces a literal repeated +EV bet. Kelly is the standard tool for sizing each wager so a hot streak compounds and a cold streak can’t bust the roll — and the reason serious gamblers bet a fraction of Kelly is pitfall 1: they never know their true edge precisely.

Position sizing in trading. Replace “win probability” with the statistical edge of a strategy and “payoff odds” with its average win-to-loss ratio, and Kelly becomes a position-sizing engine. It’s why “bet big on your best ideas” has a precise mathematical form: bet proportional to edge-over-odds — bigger edge, bigger position — but never so big that a string of correlated losses (the trading version of the over-bet cliff) wipes you out.

The general lesson. Anywhere you face repeated, sized exposures to a favorable-but-risky process, the question isn’t only “is this +EV?” but “how much?” — and Kelly is the answer that maximizes how fast your wealth actually compounds, while screaming at you to never, ever bet past the cliff.

Putting it together

The Kelly criterion answers the bet-sizing question with a single number. Because wealth compounds multiplicatively, the right objective is maximizing expected log-wealth — the geometric growth rate — not arithmetic expected wealth (which absurdly recommends betting everything and ruining the typical player). For a binary bet that wins $b$-to-1 with probability $p$, maximizing $g(f) = p\ln(1 + bf) + q\ln(1 - f)$ gives the clean optimum $f^* = (pb - q)/b = p - q/b$, readable as edge over odds. The growth curve is an inverted hump: zero at $f = 0$, peaking at $f^*$, back to zero near $2f^*$, and negative beyond — the over-betting cliff where a real edge still drives you to certain ruin. Because the downside is so asymmetric and your edge is never known exactly, almost everyone bets a fraction of Kelly — next lesson’s subject.

Next up — Fractional Kelly — we take the cliff seriously. Because you never know your true edge and full-Kelly drawdowns are savage, real practitioners deliberately bet a fraction of $f^*$ — half, a quarter — trading a small, well-understood slice of growth for a large reduction in volatility and a margin of safety against the over-betting cliff. We’ll quantify exactly what that trade buys you.