The Kelly formula with the actual numbers worked out: what f* really maximizes, why half Kelly gives up only a quarter of the growth rate, why doubling your stake can turn a real edge into a slow bleed, and why crypto trading breaks the formula's clean assumptions.
Ready to size a real position?Opere en Bybit: gran liquidez, tarifas bajas y un bono de bienvenida de hasta $30 000.Abrir cuenta Bybit →Kelly isn't a risk rule — it's a growth-maximizing rule. For a bet with win probability p, loss probability q = 1 − p, and a reward-to-risk ratio b (you win b units per unit risked), the Kelly fraction is:
f* = (b·p − q) ÷ b
That fraction of your bankroll, staked on every repeated bet, maximizes the expected tasa de crecimiento logarítmico of your capital over many trades — not your expected dollar profit on any single trade, which is a different (and much riskier) thing to maximize. John Kelly derived this in 1956 at Bell Labs for noisy telephone lines; Ed Thorp carried it into blackjack and then Wall Street in the 1960s. The formula hasn't changed since — only the bets have.
Say your honest, fee-adjusted stats are a 55% win rate with an average 1.5:1 reward-to-risk (win $1.50 per $1 risked). Plug into the formula:
f* = (1.5 × 0.55 − 0.45) ÷ 1.5 = (0.825 − 0.45) ÷ 1.5 = 0.375 ÷ 1.5 = 0.25
Full Kelly says risk 25% of your bankroll on this trade. That number alone should make you uncomfortable — and it should, which is exactly why the next section matters more than the formula itself.
Run your own numbers →Check your break-even win rate →Using that same 55% / 1.5:1 edge, here is the exact expected logarithmic growth per trade — E[ln(bankroll multiplier)] — at different stake sizes, computed directly from p·ln(1+b·f) + q·ln(1−f):
| Stake (fraction of bankroll) | Per-trade growth (log) | Per-trade multiplier | % of full-Kelly growth |
|---|---|---|---|
| Tenth Kelly — 2.5% | 0.00886 | ×1.0089 | 19.4% |
| Quarter Kelly — 6.25% | 0.02024 | ×1.0205 | 44.3% |
| Half Kelly — 12.5% | 0.03443 | ×1.0350 | 75.4% |
| Full Kelly — 25% | 0.04569 | ×1.0468 | 100% |
| 1.5× Kelly — 37.5% | 0.03396 | ×1.0345 | 74.3% |
| 2× Kelly — 50% | −0.00413 | ×0.9959 | negativo |
Two things jump out. First, the curve is flat-topped and asymmetric: cutting your stake in half from full Kelly costs you only a quarter of the growth rate, which is the mathematical reason half- and quarter-Kelly are the practical standard. Second, the curve is not symmetric around the peak — overbetting is punished far harder than underbetting by the same distance. At exactly double full Kelly (50% staked here), the expected growth rate turns negativo: the bankroll shrinks by about 0.41% per trade on average, despite a genuinely positive edge. Overbetting Kelly doesn't just grow you slower — past 2× it guarantees you lose, edge and all.
Every number above assumed p and b were known exactly. In live trading they're estimates from a finite, noisy sample — and backtests are notorious for overstating them (survivorship bias, curve-fitting, and look-ahead leaks all inflate the apparent win rate). Here's why that matters more than the overbetting curve above: if you size at 25% believing p = 55%, but your true out-of-sample win rate is actually 35% at the same 1.5:1 payoff, your real per-trade growth isn't positive at all — it's −0.0755, a sharp, compounding bleed, because you're staking a quarter of your account on a trade that no longer has an edge (breakeven win rate at 1.5:1 is 40%; 35% is already on the losing side).
This is the actual danger in Kelly, not the formula: it has zero margin for being wrong about your own edge. A rule that is mathematically optimal for a known edge becomes a bankroll shredder the moment the edge was never as good as the backtest said. Verify the edge itself — out of sample, on a real forward sample — with the calculadora de expectativa comercial before trusting any Kelly output.
None of this means don't use Kelly — it means use a fraction of it. Quarter Kelly is the common conservative default for discretionary crypto trading specifically because it keeps 44% of the theoretical growth rate while needing your edge estimate to be roughly 4× more wrong before it turns the real trajectory negative.
For a win probability p, loss probability q = 1 - p, and a reward-to-risk ratio b (how many units you win per unit risked), the Kelly fraction is f = (b·p − q) ÷ b. It is the fraction of your bankroll to risk on the next bet that maximizes the long-run logarithmic (compounded) growth rate of your capital.
With a 55% win rate and a 1.5:1 reward:risk, full Kelly stakes 25% of the bankroll and produces an expected per-trade log-growth of about 0.0457. Half Kelly (12.5%) produces about 0.0344, which is 75.4% of the full-Kelly growth rate, while the standard deviation of outcomes is roughly half. The relationship is not linear: cutting the stake in half costs only a quarter of the growth.
With the same 55%/1.5:1 edge, full Kelly is 25% of the bankroll. Betting double that, 50%, produces a NEGATIVE expected log-growth of about -0.0041 per trade — the position slowly bleeds the account even though the underlying edge is genuinely positive. Overbetting Kelly does not just reduce growth, it can flip a winning edge into a losing bankroll trajectory.
The formula itself is correct, but it assumes you know your true win rate and payoff exactly and that trades are independent and identically distributed. Crypto breaks all three: win rates are noisy estimates from small, overfit samples, payoffs have fat tails and slippage through stops, and funding payments and correlated positions link trades together. That is why fractional Kelly (half or quarter) combined with a hard per-trade risk cap is the practical standard, not full Kelly.
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