Kelly is a mathematical criterion for long-run logarithmic growth under a specified probability and payoff model. Real trading requires estimating those inputs and examining costs and possible losses beyond the model. This article separates the binary-model calculation from the decision to use a position size.

Who Kelly was and what he actually wrote in 1956

John L. Kelly Jr.’s A New Interpretation of Information Rate appeared in the July 1956 Bell System Technical Journal. It links repeated betting and information to logarithmic capital growth. In the particular binary model used here, a fraction f wins a fixed net payoff b times the stake with probability p or loses the stake with probability 1 − p. Maximizing p ln(1 + bf) + (1 − p) ln(1 − f) gives f = p − (1 − p)/b at an interior optimum. This is an application of expected value to log capital, with b > 0 and specified probabilities. If only staking or abstaining is allowed, a negative formula output means no positive stake in this model, not an instruction to reverse a market position.

What Kelly is really about, with a walk through the numbers

Kelly maximises logarithmic growth in its model, not win rate, expected terminal wealth or minimum drawdown. Its optimal stake can still produce severe drawdowns; the criterion alone is not a guarantee against ruin in a different model with estimation error, extra costs or larger losses.

For an ideal binary model, suppose win probability p = 0.55, win payoff +1.5 stake units and loss −1 stake unit. Full Kelly f = (0.55 × 1.5 − 0.45) / 1.5 = 0.25. With EUR 40,000, this means a EUR 10,000 stake inside that mathematical model. A sample ratio of average wins to average losses does not make real outcomes binary, and an ordinary stop need not deliver exactly the assumed loss. This large fraction illustrates why the model and its estimation error need scrutiny; it is not a sizing recommendation.

Costs do more than reduce a win

Subtracting a fee from both outcomes changes both capital-growth factors. A formula assuming a one-unit loss no longer describes this model.

Scroll sideways to see the full diagram

Costs on both sides of a binary Kelly modelA stake of 100 from capital 1000 returns net +140 or −110, each with probability one-half. Ending capital is 1140 or 890. Loss is 1.1 times the stake.COSTS CHANGE BOTH OUTCOMESCapital 1000; stake f = 0.10 → 100p = 0.50p = 0.50+1.50 − 0.10−1.00 − 0.10Net: +1.40Net: −1.101000 × 1.14 = 11401000 × 0.89 = 890Positive capital: 0 ≤ f < 1 / 1.10.Model maximum: f = 15/154 ≈ 0.0974.Maximum loss: 1.10 f, not f alone.

A purely hypothetical mathematical model: independent repetitions, known p = 0.50, gross outcomes +1.50 or −1.00 per stake unit. Cost is 0.10 stake units in either outcome, giving net +1.40 or −1.10. f is the fraction of capital staked, not the fraction lost at worst. Capital units are abstract; these are neither broker conditions nor strategy estimates. J.L.Kelly Jr.:A New Interpretation of Information Rate establishes a growth criterion based on probability-weighted logarithms of capital-change factors.

At illustrative f = 0.10, a positive result multiplies capital by 1.14 and a negative result by 0.89. This illustrates factors, not the computed optimum.

Here g(f) = 0.50 ln(1 + 1.40 f) + 0.50 ln(1 − 1.10 f), for 0 ≤ f < 1/1.10. Setting g′(f) = 0 gives f = (0.50 × 1.40 − 0.50 × 1.10) / (1.40 × 1.10) = 15/154 ≈ 0.0974. Maximum loss is 1.10 f = 3/28 ≈ 10.71% of capital. Without cost the optimum would be 1/6. The result depends on invented, exactly known parameters; it is not a recommendation for market position size.

How estimation and the payoff distribution change the calculation

Market probabilities and net payoff distributions are estimates, not inputs made exact by a trade count. Sampling, changing conditions and model selection can affect them. Even card-counting calculations depend on rules and implementation; they do not justify claiming that all inputs are known without error. A reduced stake still requires checking possible losses and the assumptions behind the estimate.

Hold the hypothetical fixed payoff b = 1.5. Changing p from 0.55 to 0.50 changes f from 0.25 to 1/6 ≈ 0.1667; changing p to 0.60 gives 1/3 ≈ 0.3333. These are formula outputs, not safer or survivable recommendations. A different payoff distribution changes the optimum even with identical average winners and losers. Model A has +1.5 with probability 55% and −1 with 45%. Model B has +1.5 with 55%, −1/7 with 35%, and −4 with 10%. In B the conditional mean loss is (0.35/7 + 0.10 × 4)/0.45 = 1. Both means are +0.375 per stake unit, yet f = 0.25 wipes out capital in B’s −4 branch. Its correct objective is 0.55 ln(1 + 1.5f) + 0.35 ln(1 − f/7) + 0.10 ln(1 − 4f), for 0 ≤ f < 0.25. Its derivative is zero at f = 0.10 and its second derivative is negative. Model A’s optimum is 0.25. These invented, exact distributions explain why replacing outcomes with two averages can fail; neither fraction is recommended for trading.

In a hypothetical independent model with win probability 55% and exact losses of 25% of current capital, three consecutive losses leave 0.75³ = 42.1875%, a drawdown of 57.8125%. A specified three-trade window consists entirely of losses with probability 0.45³ = 9.1125%. Overlapping windows are not separate waiting times: the expected number of trades until the first three-loss run is (1 − 0.45³)/(0.55 × 0.45³) ≈ 18.13. None of these quantities guarantees a particular run within a finite sample.

Fractional Kelly — a compromise that sometimes makes sense

Fractional Kelly multiplies the model optimum by a chosen fraction: if full Kelly is 25%, half is 12.5% and quarter is 6.25%. The reduction trades modelled logarithmic growth for lower exposure, but it does not universally halve drawdowns, cost exactly a quarter of growth or make a real account’s risk bearable. Compare model assumptions and estimation error. See risk-management basics.

A historical sample supports parameter estimates and sensitivity checks, not a known safe multiplier. No minimum trade count or willingness to accept 20–30% drawdowns establishes that fractional Kelly suits an account. Check the distribution of actual net outcomes and possible losses beyond the model.

What a fixed-fraction rule does and does not assume

A fixed-fraction rule such as 1% initial planned risk does not require a Kelly probability estimate, but it also does not establish positive expectancy or guarantee survival. ESMA’s 2018 intervention cited historical retail-CFD loss figures; those figures do not isolate sizing as the cause or prove that one sizing rule systematically beats another.

Treat a 1% or 2% rule as a chosen initial risk limit with explicit execution assumptions, not as mathematically optimal or automatically conservative for every account. Professional status and a large journal do not remove estimation error. See the two-percent versus one-percent comparison.

What to do tomorrow

  1. Define the sample, net outcomes and exclusions in the journal. Estimate the input uncertainty and whether a binary payoff model describes the exercise. Neither 200 trades nor a larger count guarantees a representative distribution of future outcomes.
  2. Calculate the binary-model value only when its payoff assumptions fit the intended exercise. A fractional value near 1% does not prove safety; a higher value does not by itself prove that win probability was overestimated. Check net outcomes, input uncertainty and loss limits separately.
  3. Run a sensitivity exercise with p = 0.50, 0.55 and 0.60 at fixed b = 1.5. The corresponding interior fractions are 1/6, 1/4 and 1/3. Compare the assumptions and possible loss sizes; this exercise does not establish that any other rule is the only sensible choice.
  4. Document any daily or monthly review threshold as a chosen plan input. A threshold does not guarantee loss stops at that amount, especially with open positions, gaps, costs or a worse fill. Do not treat an example percentage as a universal limit.
  5. Record when inputs or the model are reviewed and why. A calendar interval alone does not validate the estimates, and an accumulated journal does not remove uncertainty about future outcomes.