Education
Kelly, Half-Kelly and Bankroll Ruin
The Kelly criterion answers a precise question about stake sizing under a known edge — and the lab shows exactly why most people should stake a fraction of it.
Key takeaways
- Kelly maximises the long-run growth rate of a bankroll, not profit, not certainty, not comfort.
- The growth curve is flat near its peak and steep past it — overbetting costs far more than underbetting.
- Half-Kelly keeps about 75% of the growth rate for roughly half the volatility.
- Kelly assumes p is known; every real edge is estimated, and overestimating p means overbetting.
- Ruin is a rate across many seeded paths, not a yes/no answer from one run.
What Kelly actually optimises
The Kelly fraction is (b·p − q) / b, where p is your win probability, q is 1 − p, and b is the net decimal payout. It maximises the long-run growth rate of a bankroll under a known edge. Novus Odds computes it per event and averages it over positive-EV selections only — staking a negative-EV bet is never Kelly.
The formula is exact, but its inputs are not. In the real world p is unknown, so any Kelly stake rests on an estimate, and errors in p punish you asymmetrically.
Note what Kelly does not optimise. It does not maximise expected bankroll — that would tell you to stake everything on any positive-EV bet, which busts almost surely. It maximises the expected logarithm of bankroll, which is the same as maximising the compound growth rate. The log is doing the work: it prices a halving of your bankroll as exactly as bad as a doubling is good.
A worked fraction
Take a price of +120, so the net decimal payout b is 1.2, and suppose your model puts the true probability at 50%. The implied break-even rate is 1 / 2.2 = 45.5%, so there is a 4.5-point edge. Kelly says stake (1.2 × 0.5 − 0.5) / 1.2 = 0.1 / 1.2 = 8.3% of the bankroll.
Eight percent on a single event strikes most people as enormous, and it should. That number is the mathematically optimal stake only if the 50% is exactly right. Suppose the truth is 47%: the correct Kelly fraction was 2.8%, and you have staked three times it. Suppose the truth is 45.5%: the correct fraction was zero.
The asymmetry is the point. Underbetting a good edge costs you a little growth. Overbetting turns a positive-EV process into a negative-growth one — the bankroll's expected logarithm goes negative while the expected value of each individual bet stays positive. You can be right about every bet and still go broke by sizing them wrong.
Why fractional Kelly is the sane default
Full Kelly maximises growth but tolerates gut-wrenching drawdowns. Because real edges are estimated rather than known, most practitioners use a fraction — quarter- or half-Kelly — trading a little growth for a lot less variance and a much smaller chance of ruin.
The trade is far better than it sounds, because the growth curve is a downward parabola with a flat top. Staking a fraction f of full Kelly retains f(2 − f) of the maximum growth rate. At half-Kelly that is 0.5 × 1.5 = 75% of the growth for half the volatility. At quarter-Kelly it is 44% of the growth for a quarter of the volatility.
The Model Comparison lab makes the trade-off concrete: hold the tape fixed and compare staking fractions. Full Kelly's terminal-bankroll distribution has a fatter left tail than its median suggests.
Estimation error is the real argument for fractions
The usual case for half-Kelly is psychological: full Kelly is too wild to live with. The stronger case is statistical. If your probability estimates carry error, the fraction you compute is itself a random variable, and it is biased toward overbetting — because the bets that look most attractive are disproportionately the ones where you overestimated p.
Halving the stake is a crude but effective correction for that bias. It costs a quarter of the theoretical growth rate in the case where your estimates were perfect, and it rescues the case where they were not. Given that no one's estimates are perfect, the expected trade strongly favours the fraction.
This is the same logic that shows up in shrinkage estimators throughout statistics: when your input is noisy, pull it toward the conservative value. Kelly with estimated probabilities is exactly that situation.
Ruin is a probability, not a flag
A single simulated path either busts or it doesn't — that tells you almost nothing. Our multi-path bankroll engine runs many seeded paths and reports ruin as a rate across them, plus the distribution of drawdowns. That is the honest way to talk about risk: not 'did it bust once?' but 'how often does this stake plan bust, and how deep are the holes along the way?'
Read three numbers together: the ruin rate across paths, the median maximum drawdown, and the worst-decile drawdown. The median tells you what a normal bad patch looks like. The worst decile tells you what you are actually signing up for, because one run in ten will be at least that bad and you have no way to know in advance which one you are living through.
Strictly, full Kelly never reaches zero with continuous betting and infinite divisibility — it only approaches it. Real bankrolls have minimum stakes, table limits and a human attached, all of which convert 'asymptotically small' into 'over'. The lab models the discrete case for exactly that reason.
Frequently asked
What does the Kelly criterion actually maximise?
The expected logarithm of bankroll, which is equivalent to the long-run compound growth rate. It does not maximise expected profit — that objective would tell you to stake everything and go broke almost surely.
Why is half-Kelly so popular?
Because the growth curve is flat near its peak. Staking a fraction f of full Kelly keeps f(2 − f) of the maximum growth rate, so half-Kelly retains about 75% of the growth for half the volatility — and it protects you when your probability estimate was optimistic.
What happens if I overbet Kelly?
Growth rate falls away steeply and can go negative past roughly twice the Kelly fraction, even though every individual bet still has positive expected value. Overbetting is the one way to lose money while being right about the odds.
Can a Kelly staking plan go broke?
In theory, continuous fractional staking only approaches zero. In practice, minimum stakes, limits and finite patience make deep drawdowns terminal, which is why the lab reports ruin as a rate across many seeded paths rather than a single yes or no.
Try it yourself
Everything in this article is something you can run and tweak in the lab — with your own settings and a reproducible seed.
Open the lab →