01 / Kelly Challenge · Start here
Kelly Challenge
Heads wins 60% of the time. Winning more often does not protect you from sizing too large. Try different stakes and see what changes when your risk rises and falls with your balance.
Cash: $25.00
Choose heads or tails to place your first bet.
Reach $250.00. Keep playing after the milestone.
Compare the risk you take today with a modeled risk budget.
Autobet resizes the next wager to your chosen fraction of current cash. A dollar edit changes the next wager only. Each click makes one independent flip with a 1:1 payout. Practice money only.
Inspired by Paul Butler's Kelly criterion game.
Why 20%? With these known 60/40 odds and equal payouts, full Kelly risks 20% of your current balance. Start with $25 and a $5 bet: a win raises your next bet to $6; a loss lowers it to $4.
20% is the result for this fee-free coin game, not a suggested risk level for your trades. Real trading involves uncertain odds, different payoffs and costs; even full Kelly can have deep drawdowns.
02 / Apply it to your trading
How can I apply this to my trading?
“How can I apply this to my trading so that I stop blowing up my accounts?”
That was my first thought when I saw this game. My win rate had been close to 60%, but sizing up after wins could undo my progress when losses came together. That question is why I started building these tools.
The goal: eliminate unnecessary risk from the decisions you control. Spot oversized positions, account for trading costs, and replace guesswork with a repeatable risk budget. You can improve how you prepare for a trade, even when its outcome is uncertain.
Test your risk before you take it. Enter your starting balance in Portfolio Simulator. Compare potential growth with the drawdowns you could face, then carry your chosen risk budget into TradeSizer for your next idea.
Read the full math & statistics ↓Original walkthrough by Paul Butler ↗
The full explanation: Kelly math, statistics & trading costs
Start with the intuition, then work through expected value, the outcome distribution, the growth-maximizing derivation, Kelly fractions, and trading costs. The game stays simple; the reasoning is all here when you want it.
What ℓ = 0.2 actually means
ℓ (a lowercase “ell”) is the fraction of your current balance wagered. At 0.2, that is 20%. “Autobet” updates the dollar amount after each result; you still click to place each flip.
- Start with $25.20% is a $5 wager.
- Win: balance becomes $30.The next wager grows to $6.
- Then lose: balance becomes $24.The next wager shrinks to $4.80.
For this particular 60% heads, 1:1 payout game, 20% maximizes long-run expected log growth among constant-fraction strategies. It is not a guarantee of winning a session, the safest size, or a universal 20% rule for trading. The $250 milestone does not change that long-run calculation; an objective such as reaching a target before a deadline is a different problem.
Game inspiration: Paul Butler’s interactive walkthrough. The game runs here on OnlySnipes Tools.
Start by choosing what “best” means
A bigger average balance, a better median result, a smaller drawdown, and a greater chance of reaching a target are different objectives. A strategy can lead on one and lose on another.
Butler’s walkthrough also considers whether one strategy beats another more often. Such pairwise rankings can form a cycle, like rock–paper–scissors: they do not generally identify a single best choice. Kelly instead gives us a precise objective for repeated opportunities: maximize expected logarithmic growth under a specified outcome model.
Here is the calculation from the balance update itself. Let B₀ be starting equity, f the fraction staked each round, p the probability of winning, and n the number of rounds. For now, wins and losses pay +1 and −1 times the stake, with independent flips and no costs.
1. The average can grow while most accounts fail
E[B next | B] = B × [p(1 + f) + (1 − p)(1 − f)]E[Bₙ] = B₀ × [1 + f(2p − 1)]ⁿFor p = 0.6, increasing f raises that arithmetic average. At f = 1, however, a single loss ends the account. After ten rounds, the only surviving path is ten straight wins.
| Quantity | Calculation | Result |
|---|---|---|
| Chance of surviving | 0.6¹⁰ | 0.605% |
| Chance of reaching zero | 1 − 0.6¹⁰ | 99.395% |
| Surviving balance | $25 × 2¹⁰ | $25,600 |
| Average across all paths | $25 × 1.2¹⁰ | $154.79 |
| Median balance | More than half the paths end at zero | $0 |
The rare $25,600 outcome lifts the average. It does not make $154.79 a likely individual outcome. That is why the simulator shows a distribution and downside measures alongside its median.
2. Count the wins, then compound the balance
Bₙ = B₀ × (1 + f)ʷ × (1 − f)ⁿ⁻ʷP(w wins) = C(n, w) × pʷ × (1 − p)ⁿ⁻ʷC(n, w) = n! / [w! × (n − w)!]The second equation assigns a probability to each possible win count. Together, the equations give the exact terminal-balance distribution for this simple model. The median is a middle outcome of that distribution; the 5th and 95th percentiles describe a wider range.
For ten flips with p = 0.6, six wins is a median win count. With 20% stakes, six wins and four losses turn $25 into $30.58. For other trade counts, n × p need not be an integer or the exact median: use the binomial probabilities rather than treating the expected win count as a guaranteed sequence.
With a constant fraction and fixed payoffs, rearranging the same wins and losses leaves the final product unchanged. It can still change peak-to-trough drawdown. Costs that vary, changing stakes, overlapping trades, and stopping rules also change the problem.
3. Derive the fraction instead of guessing it
(1/n) ln(Bₙ / B₀) = (w/n) ln(1 + f) + (1 − w/n) ln(1 − f)g(f) = p ln(1 + f) + (1 − p) ln(1 − f)g′(f) = p/(1 + f) − (1 − p)/(1 − f)g′(f) = 0 ⇒ p(1 − f) = (1 − p)(1 + f) ⇒ f* = 2p − 1g″(f) = −p/(1 + f)² − (1 − p)/(1 − f)² < 0For independent repeated flips, the observed win fraction w/n approaches p. The first line therefore approaches g(f). Logarithms turn the multiplied balance changes into an average, and the negative second derivative confirms that the interior stationary point is a maximum.
At p = 0.6, f* = 0.2. At p = 0.5, the answer is zero. Below 50%, there is no positive growth-optimal stake on heads; betting tails is a separate choice with probability 1 − p. This derivation uses 0 ≤ f < 1 and an uncertain outcome, so a loss cannot make the logarithm undefined.
The geometric multiplier per round is exp(g). For the 60/40 coin it is approximately 1.0203 at 20% risk, 1.0152 at 10%, and 0.9666 at 50%. Thus a positive arithmetic edge can coexist with negative long-run log growth when the stake is too large.
The long-run result does not promise that Kelly wins the next session. Comparing two rules on the same flips isolates sizing; comparing independent sessions adds different outcome luck. Neither a higher average nor a higher median universally proves that one arbitrary strategy beats another more than half the time.
4. State the limits before using the answer
The model assumes known probabilities, repeatable payoffs, sequential opportunities, reinvested gains, and divisible stakes. A finite deadline, a target balance, minimum order sizes, fees, gaps through stops, or correlated positions can change the appropriate policy.
With a fraction below 100%, finite idealized ±1 outcomes do not reach mathematical zero. They can still leave an account too small to trade or produce an unacceptable drawdown. “Not exactly zero” is not a useful definition of safety.
A historical win rate is an estimate, not the coin’s known probability. A finite simulation adds sampling error on top of uncertainty in the underlying strategy. No observed breach in a simulation does not establish a zero real-world probability. Smaller Kelly fractions reduce exposure; they cannot repair a nonexistent edge.
For the full original interactive argument, including its comparison diagrams and appendix, read Paul Butler’s walkthrough. The worked explanations here are written for this toolset; the source article remains available in full at that link.
Winning more often does not protect you from sizing too large
Take a fixed example: six wins and four losses, all paying 1:1, starting with $25. Change only the fraction wagered. This is a calculation for that exact outcome count, not a forecast.
At 20%: $30.58. At 50%: $17.80. You can win 60% of the time in this sequence and still lose money by risking too much.
A 50% loss needs a 100% gain to recover. Repeated returns multiply, so Kelly maximizes their expected logarithm rather than chasing the highest average dollar payoff from one bet. Betting the entire account is all-in; it is not what “full Kelly” means.
The long-run growth objective comes from J. L. Kelly Jr.’s original 1956 paper.
Full Kelly, half Kelly, quarter Kelly—and your plan
Full Kelly means 100% of the formula’s answer, not 100% of your account. Half and quarter Kelly scale that answer down. All three can use the same trading strategy and win probability.
| Rule | Risk per trade | After 5 consecutive losses |
|---|---|---|
| Full Kelly | 20% of current equity | 67.2% drawdown |
| Half Kelly | 10% of current equity | 41.0% drawdown |
| Quarter Kelly | 5% of current equity | 22.6% drawdown |
| Your plan | Your selected dollars or percentage | Depends on your size and compounding rule |
Why not always use full Kelly? Its growth optimum assumes the probabilities and payoff distribution are right. In trading, they are estimates. Five losses can occur even with a positive edge, and several correlated positions can lose together. Smaller fractions trade some modeled growth for lower exposure to losses and estimation error.
“Your plan” is the plan the tool is testing. A fixed $100 remains $100 even after your balance falls. A fixed 1% becomes $90 when a $10,000 account falls to $9,000. The model-sizing mode searches candidate percentages under your simulated drawdown tolerance; its answer does not have to equal a named Kelly fraction.
Full Kelly can have the highest theoretical long-run growth and still finish behind a smaller fraction over a particular finite run. Half Kelly does not always outperform full Kelly, and it is not automatically safe.
For research on explicitly trading off growth and drawdown, see Busseti, Ryu & Boyd, Risk-Constrained Kelly Gambling. Our finite simulation is a scenario test, not that paper’s formal risk bound.
Bring the idea into trading
Win rate alone is not enough. A strategy winning 40% at +2R and losing 60% at −1R has +0.2R expected profit before costs. At 25% with the same payouts, it has −0.25R expectancy. Raising a target on paper does not prove that the win rate will survive that change.
A win rate near 60% can be a useful starting input for your research, including my calls. It is not a known coin probability: check the sample, actual fills, realized rewards and losses, fees, slippage, and how the strategy behaves in different markets.
Try the assumptions
Hypothetical long setup: every win makes the same gross R and every stop loses 1R before costs. Default fees are 0.045% taker on each fill’s notional, plus a separate 0.03% slippage allowance per side. These are editable modeling assumptions, not a claim about actual fills or OnlySnipes performance.
- Expectancy after costs
- +0.125R
- Full Kelly
- 12.6%
- Half / quarter Kelly
- 6.3% / 3.1%
These assumptions have a positive modeled edge. The percentages are formula benchmarks, not recommended live risk limits.
Modeled costs: 0.076R on a win and 0.074R on a stop. Risk percentages refer to the planned price loss before costs, not position value or margin. Tighter stops mean more notional for each dollar risked, so fees consume more R. Fees apply to both entry and exit values; leverage does not reduce them. Funding is not modeled here.
The formula and its assumptions
Let p be win probability, W the net profit on a win in R, and L the total loss on a loss in R. If f is the fraction of current equity exposed to one nominal R, expected log growth is:
g(f) = p ln(1 + fW) + (1 − p) ln(1 − fL)Setting the derivative to zero gives f* = p/L − (1 − p)/W, with a zero allocation when expectancy is nonpositive. For L = 1 and W = b, this becomes f* = p − (1 − p)/b. For the coin, p = 0.6 and b = 1, so f* = 0.2.
For positive W and L, the feasible range is 0 ≤ f < 1/L. The actual loss fraction at the modeled stop is f × L; f alone is not the all-in loss budget when costs make L greater than 1.
The calculator uses W = gross win R − winning-exit costs in R and L = 1 + stopped-exit costs in R. For each outcome, costs = quantity × (entry price + exit price) × (fee rate + slippage allowance). It assumes independent, repeatable outcomes and no withdrawals or capital limits. Real trading has variable outcomes and simultaneous exposure; substituting a historical win rate and typical winning R is an approximation.
Find a size your strategy can sustain
Advanced risk comparison lets you compare your current plan with Kelly benchmarks using your win rate, reward:risk and trading costs. Test different trade counts and see the modeled balance and drawdown ranges. Use it to challenge a strategy before increasing risk—and to recognize when the inputs do not justify a trade.
There is no profitable sweet spot for every strategy. The value is a repeatable process: measure an edge, account for costs, size it conservatively, and revisit the assumptions as evidence changes.