Kelly and fractional Kelly
Derive the Kelly stake from expected log growth, see why staking about twice Kelly erases long-run growth even on a positive-EV bet, and weigh what fractional Kelly gives up and gains.
Learning objectives
- Compute the Kelly fraction for a binary bet from a probability and decimal odds, and confirm it by setting the derivative of expected log growth to zero.
- Evaluate expected log growth at several stake sizes and locate the zero-growth stake.
- Explain, with numbers, why a positive-EV bet can shrink a bankroll when it is over-staked.
- Quantify what half Kelly gives up in growth and gains in drawdown control using a seeded simulation.
Why maximise the logarithm of the bankroll?
A bankroll that stakes a share of itself on every pick compounds: each result multiplies it by a factor. After many picks, what matters is not the average factor but the product of all of them. Taking logarithms turns that product into a sum, and by the law of large numbers the average log return per pick settles down to its expectation. Call that expectation G(f), the expected log growth when you stake a fraction f of the current bankroll.
W_n = W_0 × (1 + f·b)^wins × (1 − f)^losses, b = o − 1G(f) = p·ln(1 + f·b) + (1 − p)·ln(1 − f)The stake that maximises G(f) makes the bankroll grow fastest in the long run: over a long enough sequence it ends up ahead of any other fixed-fraction rule with probability approaching one. Kelly derived this result in the language of information theory Kelly (1956), and it is why the rule carries his name.
Deriving the Kelly fraction
G(f) is concave in f, so it has a single peak. Differentiate and set the slope to zero:
G′(f) = p·b / (1 + f·b) − (1 − p) / (1 − f) = 0f_K = (p·b − (1 − p)) / b = (p × o − 1) / (o − 1)Read the formula as edge over odds. The numerator, p × o − 1, is the expected value per unit staked. The denominator, o − 1, is how much a winning unit pays. Two picks with the same EV get different stakes: the longer price gets the smaller share, because its outcome is more volatile.
The over-betting cliff
Take a hypothetical quote of 2.30 and an estimate of 48%. The EV is 0.48 × 2.30 − 1 = +10.4% and the Kelly fraction is 0.104 / 1.30 = 8.00%. The table shows G(f) at several stakes.
| Stake (share of bankroll) | G(f) per pick | Relative to Kelly |
|---|---|---|
| 2% | 0.00182 | 44% |
| 4% (half Kelly) | 0.00311 | 75% |
| 8% (Kelly) | 0.00413 | 100% |
| 12% (1.5× Kelly) | 0.00311 | 75% |
| 16% (2× Kelly) | 0.00004 | ≈ 0% |
| 20% (2.5× Kelly) | −0.00510 | negative |
Near the peak the curve is almost symmetric: staking 4% or 12% earns the same growth, but 12% carries three times the stake volatility. Solving for the stake at which growth returns to zero gives 16.04%, which is 2.00× Kelly.
A second case makes the trap sharper. At even money (2.00) with p = 0.52, the EV is +4% and Kelly is 4.00%. zero-growth stake (0.52, 2) returns 7.99%, again about twice Kelly. At a 10% stake, G = −0.00101 per pick: every pick still has +4% expected value, yet the typical bankroll shrinks. Over 500 picks, exp(500 × G) ≈ 0.60, so a simulated 100 Credits typically ends near 60.
G(f) ≈ f·μ − f²·σ²/2 → peak near μ/σ², zero near 2μ/σ²Fractional Kelly: trading growth for a smoother ride
Staking a fraction c of Kelly keeps roughly c·(2 − c) of the maximum growth rate for small edges. Half Kelly therefore keeps about 75% of the growth (the exact ratio at 2.30 and 48% is 0.00311 / 0.00413 = 75.1%) while halving the stake, which roughly halves the volatility of log returns.
| Stake rule | Stake | Median final | Median max drawdown | P(drawdown ≥ 50%) |
|---|---|---|---|---|
| Half Kelly | 2% | 135.0 | 32.9% | 10.0% |
| Full Kelly | 4% | 149.2 | 57.7% | 70.3% |
| 2× Kelly | 8% | 99.8 | 86.5% | 100% |
| 2.5× Kelly | 10% | 60.3 | 93.5% | 100% |
Going from half to full Kelly lifted the median ending bankroll from 135.0 to 149.2 but raised the chance of a 50% drawdown from 10.0% to 70.3%. At 2× Kelly the median ending bankroll was 99.8 — no growth — while the mean was 409.3, inflated by a handful of lucky paths. Averages hide what most paths experience.
Worked example
Worked example · Kelly, half Kelly and the cliff for a single hypothetical price
A hypothetical quote of 2.30 on Harbour City to beat Northvale. Your model says 48%, and for this exercise we pretend that number is exact. Stakes are shares of a simulated bankroll of virtual Credits.
- Break-even probability
1 / 2.3043.48% - Expected value per unit
0.48 × 2.30 − 1+10.4% - Kelly fraction
(0.48 × 2.30 − 1) / (2.30 − 1) = 0.104 / 1.308.00% of the simulated bankroll - Check the derivative at 8%
0.48 × 1.30 / 1.104 − 0.52 / 0.92 = 0.5652 − 0.56520 (the peak) - Growth at Kelly and half Kelly
G(0.08) = 0.00413; G(0.04) = 0.00311Half Kelly keeps 75.1% of the growth - Typical growth over 100 picks
exp(100 × 0.00413) vs exp(100 × 0.00311)×1.51 at Kelly vs ×1.36 at half Kelly - Zero-growth stake
zero-growth stake (0.48, 2.30)16.04% (2.00× Kelly); G(0.20) = −0.00510 - Five losses in a row
Probability 0.52⁵ = 3.80%; bankroll 0.92⁵ at 8%, 0.96⁵ at 4%65.9% left at Kelly vs 81.5% at half Kelly
If 48% were exact, 8% would be the growth-maximising stake and anything above 16% would destroy growth. Half Kelly gives up a quarter of the growth for a much shallower losing run. Because 48% is really an estimate, 8% is a ceiling, not a target.
Practice · See the growth–drawdown trade-off
Open the Kelly and Drawdown lab with the preset (estimated 52%, actual 52%, lower bound 50%, odds 2.00, 500 picks). Compare full Kelly, half Kelly and flat 1% on median ending bankroll, median maximum drawdown and the chance of a 50% drawdown. Then explain why the lower-bound rule stakes nothing.
Lab 3 · Kelly and drawdown
Synthetic · deterministicChoose inputs and run the simulation. The default sets a model that believes 55% when the truth is 50%: a 5-percentage-point overestimate (10% relative) at even money.
Educational calculator. Uses only the numbers you enter or seeded synthetic data — no live prices, no staking, no real money.
Show what you should find
In our reference run (400 paths, seed 1, bankroll 100): full Kelly stakes 4%, median ending bankroll 149.2, median maximum drawdown 57.7%, 70.3% of paths fall 50% from a peak and 36.0% finish below the start. Half Kelly stakes 2% (identical to the fixed 2% rule here): median 135.0, median drawdown 32.9%, 10.0% hit a 50% drawdown, 23.5% finish below start. Flat 1% of the starting bankroll: median 120.0, median drawdown 15.8%, 0.3%. The lower bound of 50% at even money has zero EV, so Kelly on it is 0% and the bankroll stays at 100. If the lab runs a different number of paths, expect the same pattern with slightly different figures.
Common mistakes
- ✗ Treating full Kelly as the stake you ‘should’ use.Kelly maximises long-run growth only when p is known exactly. Half Kelly keeps about 75% of the growth with far shallower drawdowns (32.9% vs 57.7% median in the reference run).
- ✗ Believing a bigger stake is always better when EV is positive.Past roughly 2× Kelly, expected log growth turns negative. At p = 0.52 and odds 2.00, a 10% stake has G = −0.00101 per pick despite +4% EV.
- ✗ Judging a staking rule by the mean simulated outcome.The mean is driven by rare lucky paths. At 2× Kelly the reference run's mean was 409.3 while the median was 99.8. Use medians and tail probabilities.
- ✗ Applying the binary formula to markets that can push or void.A push outcome changes the optimum. Value 7 picks are strictly win-or-lose, so the binary formula fits there, but general bookmaker whole lines need the three-outcome version.
When it fails
Supporting research
- Kelly (1956): A New Interpretation of Information RateBell System Technical Journal · Peer-reviewed theoryMaximising expected log wealth gives the fastest long-run growth; the origin of the Kelly rule.
- MacLean, Thorp & Ziemba (2010): Long-term capital growth: the good and bad properties of the Kelly and fractional Kelly capital growth criteriaQuantitative Finance · Peer-reviewed theoryKelly is risky in the short run; fractional Kelly trades growth for lower risk.
- Thorp (2006): The Kelly Criterion in Blackjack, Sports Betting, and the Stock MarketHandbook of Asset and Liability Management, Vol. 1 (Elsevier) · Book chapterPractitioner account of Kelly's growth properties and the common use of fractional Kelly.
Check your understanding
Practice only: saved on this device, no XP and no account record.