The Kelly criterion

The bet size that wins the long game.

Five bankrolls start at 1× and race over thousands of bets — each staking a different multiple of the Kelly fraction f*. Full Kelly grows fastest. Over-bet, and the same edge drives you to ruin.

Edge (b·p − q)

—

Kelly f*

—

g(f*) log-growth

—

Ruin @ 2·f*

—

0.25 · f* cautious

0.50 · f* half

1.00 · f* Kelly

1.50 · f* overbet

2.00 · f* ruin path

No edge (f* ≤ 0).
The model's verdict: don't bet.

wealth × starting bankroll · log scale · median bold, paths faint bet 0 → 1,000

The two numbers that set the edge

p — win probability0.55

b — payoff odds (win = b × stake)1.0

Simulation depth

N — bets per path1,000

M — paths per fraction50

Median terminal wealth · ruin rate

g(f) — log-growth vs fraction

How to read it

Drag p (win probability) and b (payoff multiple). The Kelly fraction f* updates live, and five wealth trajectories race from quarter-Kelly to 2× Kelly. Watch the over-bet lines (yellow, red) shoot ahead early, then collapse — the same positive edge that builds the green Kelly line destroys them through compounding losses. The g(f) pane shows why: growth peaks exactly at f* and turns negative past it.

The Kelly formula

For a binary bet where you win b × stake with probability p and lose stake with probability q = 1 − p, the fraction of wealth that maximises long-run growth is

f* = (b·p − q) / b.

Equivalent form: f* = (p(b + 1) − 1) / b. The numerator is the "edge" (expected return on stake) and the denominator is the payoff odds.

The expected log-growth at fraction f is

g(f) = p · ln(1 + f·b) + q · ln(1 − f)

maximised at f = f*, zero at f = 0 and at a second "ruin-equivalent" point 2·f*·b/(b+1)·(approximately) where the geometric mean drops back to 1.

If you've seen Martingale

The Martingale trap is the opposite of Kelly. Martingale increases the bet after every loss, on the theory that one win recovers everything. The math destroys it: bet size grows exponentially, table limits and finite bankrolls cap the recovery, and a long enough losing streak (which arrives with certainty over time) wipes the player out.

Kelly does the opposite. It scales the bet to the edge that exists, not to the losses already taken. When you have no edge, Kelly tells you to bet zero. When the edge is small, Kelly tells you to bet small. This is why the same math that destroys gamblers builds positions for Edward Thorp and Warren Buffett.

Why most professionals run fractional Kelly

Full Kelly is technically growth-optimal, but its drawdowns are brutal. The expected time spent below half-peak wealth under full Kelly is roughly 1/3 of all time. Most fund managers run quarter to half Kelly — gives up some growth, gets dramatically smoother equity curves.

The quarter-Kelly trick also gives huge robustness to mis-estimation of p and b. Real edges are uncertain; betting half of your Kelly fraction protects against overestimating your own skill.

Try setting p = 0.51 and b = 1.0. The 2×-Kelly path collapses even though the edge is positive — that's the leveraged-ETF decay problem, same mechanism.

Citations

Kelly, J.L. (1956). "A new interpretation of information rate." Bell System Technical Journal, 35, 917–926. (The original derivation, framed in information theory.)

Thorp, E.O. (2006). "The Kelly criterion in blackjack, sports betting, and the stock market." Reprinted in Handbook of Asset and Liability Management, vol. 1.

MacLean, L.C., Thorp, E.O. & Ziemba, W.T. (2010). The Kelly Capital Growth Investment Criterion: Theory and Practice. World Scientific.

Poundstone, W. (2005). Fortune's Formula. (Popular history of Kelly and Thorp.)

Built by Joshua Chua · Fable rebuild · vanilla JS + Canvas