Kelly criterion and risk of ruin
Kelly gives the mathematically optimal stake for long-run growth — and full Kelly is far too aggressive for anyone whose edge estimate might be wrong. Why practitioners use a fraction, how estimation error compounds, and how to size for a drawdown you can survive.
The Kelly criterion sets stake as a fraction of bankroll proportional to your edge divided by the odds. It provably maximises the long-run growth rate of a bankroll — under the assumption that your probability estimate is correct. Because that assumption is false in betting, practitioners stake a fraction of Kelly, typically a quarter to a half.
The formula
f* = (b × p − q) / bb = decimal odds − 1; p = your win probability; q = 1 − p. f* is the fraction of bankroll to stake.Worked example: price 2.50 (so b = 1.5), your estimate 45%. f* = (1.5 × 0.45 − 0.55) / 1.5 = 8.3% of bankroll. On a £1,000 bank that is £83 on a single bet — which is the moment most people notice something is off.
Kelly is doing exactly what it promises. It maximises the expected logarithm of wealth, which is the correct objective if you will place infinitely many bets, never need to withdraw, and know p exactly.
Why does full Kelly fail in betting?
- Your edge estimate is wrong. Kelly staking is linear in the estimated edge, so a 2× overestimate produces a 2× overstake. Staking double Kelly reduces long-run growth to zero and beyond; the growth curve is a downward parabola with roots at zero and 2f*.
- Drawdowns are brutal by design. Under full Kelly, a drawdown of 50% of bankroll is an ordinary event rather than a crisis. Very few people can hold a position through it.
- Bets are not sequential and independent. Multiple bets on the same day, and correlated bets within one match, break the derivation.
- Bankroll is not perfectly divisible or infinitely patient. Real bettors withdraw money, hit limits, and stop.
The asymmetry that decides the argument: underbetting costs you growth linearly, overbetting destroys the bankroll geometrically. When your inputs are uncertain, the two errors are not equally bad — so the rational response to uncertainty is to stake less than the formula says, every time.
Fractional Kelly
| Fraction | Share of full-Kelly growth | Drawdown behaviour |
|---|---|---|
| Full (1.0) | 100% | Extreme. Assumes a perfect edge estimate |
| Half (0.5) | ≈ 75% | Roughly halves drawdown |
| Quarter (0.25) | ≈ 44% | Comfortable for most; the common professional default |
| Tenth (0.1) | ≈ 19% | Very conservative; appropriate when the edge is genuinely uncertain |
The trade is favourable. Half Kelly keeps about three quarters of the growth for roughly half the drawdown, because growth falls off quadratically while risk falls off linearly as you reduce the fraction. That curvature is the whole reason fractional Kelly is standard practice.
Risk of ruin
Risk of ruin is the probability that a bankroll reaches zero, or a floor you have chosen, before growing. It rises with stake size, falls with edge, and rises sharply with variance.
The practical version most bettors need is not a closed-form formula but a simulation: take your win rate, your average odds and your stake rule, replay several thousand sequences, and count how often the bank breaches your floor. That also produces the distribution of drawdowns, which is the number you actually need for sizing a bank.
A useful sanity check that requires no mathematics: if a single loss changes how you feel about the strategy, the stake is too large — whatever the formula says.
Flat staking and why serious records use it
Every figure CONSENSUS publishes assumes one unit flat: the same stake on every signal regardless of confidence or price. This is not because flat staking is optimal — Kelly is, given a correct edge — but because a published record must be reproducible.
A record computed with variable stakes cannot be audited by a reader, and it can be improved after the fact by adjusting historical stake sizes. Flat staking removes that degree of freedom entirely: anyone can recount our record from the log with arithmetic. Stake sizing is a decision for the reader, using their own bankroll and their own tolerance — which is why the front page also lets you replay the same history at your own unit size.
Check any of this against our record
Every signal CONSENSUS publishes carries the bookmaker odds fixed before the event starts and the settled result afterwards — including the drawdowns and the losing runs. The running total is on the front page and every entry is in the log.