Crypto fractional Kelly sizing and loss limits

Crypto fractional Kelly position sizing multiplies a model's growth-optimal risk fraction by a smaller factor, such as one-half or one-quarter. The calculation depends on credible net outcome probabilities and payoffs; a fraction of an unreliable estimate remains unreliable. Treat the result as a model input, then apply separate capital, exposure and execution limits before translating it into an order.

The difficult part is defining the outcome distribution that deserves to be optimized. A historical win rate, an average reward-to-risk ratio and a chosen Kelly multiplier can produce a precise number while concealing losses the model cannot represent.

What the Kelly fraction measures

Kelly sizing targets expected logarithmic growth of wealth. Kelly's original paper distinguishes that objective from maximizing expected ending capital, which can favor betting everything in a favorable gamble. Neither objective promises a favorable outcome over a particular trading horizon.

For a deliberately simple two-outcome model, define:

• p: probability of a winning trade, with 0 < p < 1.
• b: net profit on a win divided by the amount lost on a loss, with b > 0.
• f: fraction of current equity lost in the modeled losing outcome.

A win multiplies equity by 1 + bf; a loss multiplies it by 1 − f. Expected log growth per trade is therefore:

g(f) = p × ln(1 + bf) + (1 − p) × ln(1 − f)

For this model, maximizing the expression over nonnegative fractions below one gives:

f_K = max(0, p − (1 − p) / b)

The two-outcome solution also appears in Busseti, Ryu and Boyd's Kelly research. Their repeated-gamble framework assumes independent, identically distributed returns. Those assumptions matter when interpreting long-run growth.

Here, f measures the modeled loss fraction. It is not automatically the fraction spent on spot assets, the margin posted, or the leveraged notional. Once a loss budget exists, the stop-distance sizing worksheet explains the separate conversion to quantity, including costs and execution allowances.

A worked example: full, half and quarter Kelly

All numbers below are hypothetical teaching inputs, not a historical strategy or a suggested allocation. Suppose a trade independently wins with probability 55%. A win earns exactly 1.5 times the amount a loss costs. These are net outcomes after every modeled fee, funding charge and execution cost.

The calculation is:

f_K = 0.55 − 0.45 / 1.5 = 0.25

Full Kelly risks 25% of current equity in the model's losing outcome. Half Kelly multiplies that fraction by 0.5; quarter Kelly multiplies it by 0.25. Quarter Kelly therefore means a 6.25% modeled loss, rather than a 25% position or a 0.25% loss.

Start with $10,000 of equity. Add an adverse scenario in which execution produces a loss twice the modeled loss amount, called a 2R loss here. This scenario lies outside the original two-outcome distribution.

The fraction scales the original solution; it does not introduce a capital floor. If the floor is $9,000, half Kelly fails even the ordinary losing outcome. Quarter Kelly passes that outcome but fails the 2R scenario.

This comparison assumes changing size preserves the net win probability and payoff ratio. Fixed charges, market impact and different fills can break that assumption, so the same inputs cannot automatically be reused at every size.

Under the original 55% and 1.5R model, expected log growth is approximately 0.04569 per trade at full Kelly and 0.03443 at half Kelly. Half Kelly retains about 75.3% of the modeled log-growth objective. These are mathematical expectations in natural-log units, not realized percentage returns, annualized forecasts or universal fractional-Kelly performance ratios.

Change the inputs before trusting the multiplier

The base example assumes the true future win probability is known. An observed 55 wins from 100 trades does not establish that probability.

Under an independent binomial sampling model, a two-sided 95% Wilson interval for 55 successes in 100 observations is approximately 45.24% to 64.39%. NIST's proportion-interval guidance supplies the method. The interval addresses sampling uncertainty under that model; it does not validate independence, account for strategy selection or establish that future trades share the same probability.

Holding the 1.5 payoff constant, the interval's lower endpoint would produce a full-Kelly fraction of about 8.74%, far below 25%. Using a lower confidence endpoint is one sensitivity check, not a complete method for selecting a fraction: the payoff itself is uncertain too.

In the third row, the unconstrained expression is −5%. With only the original trade direction and cash available, the model chooses zero. A negative calculation does not authorize reversing the trade; a reverse strategy has its own costs and outcome distribution.

At the last row's probabilities, keeping the original quarter-Kelly size of 6.25% gives expected log growth of approximately −0.00288 per trade. Reducing an estimated positive edge does not create an edge when the assumed trade has none.

Changing market behavior adds a separate uncertainty. ADF and KPSS stationarity tests can inform specific time-series questions, but they do not certify that a trading strategy's future win probability or payoff distribution will remain constant.

Average wins and losses do not define the distribution

The simple formula is exact for the stated two-outcome model. Real trades may have small wins, large wins, partial exits, several loss sizes and outcomes beyond a planned stop. Replacing those observations with only an average win and an average loss generally changes the log-growth problem.

For net outcomes expressed as multiples x of one risk unit, the broader objective is E[ln(1 + fx)]. Every modeled outcome must leave positive wealth for the logarithm to exist. An average payoff cannot identify the probability or severity of a loss that makes 1 + fx approach zero.

For example, treating every loser as exactly −1R removes the hypothetical −2R event used above. Multiplying the resulting estimate by one-quarter does not put that missing event back into the distribution. Reconcile the net trade ledger, include costs at the intended size, and inspect the lower tail before treating the binary approximation as useful. The expectancy and win-rate guide helps separate frequency from payoff magnitude.

Apply a capital floor as its own constraint

A floor check begins with positive current equity E above a fixed minimum equity B, and a positive assumed maximum loss of a risk units on the next trade. To retain at least the floor under that particular loss assumption:

E × (1 − af) ≥ B, so f ≤ (E − B) / (aE)

For $10,000 equity, a $9,000 floor and a 2R assumed maximum loss, the cap is 1,000 / 20,000 = 5%. That explains the last row of the first table. Requiring equity strictly above the floor would require a smaller fraction. Additional existing exposure consumes some of the same headroom.

At or below the floor, there is no positive remaining loss allowance under this rule. Reaching it calls for stopping new risk and reviewing the exposure already present.

This is a conditional, next-trade constraint. A 2R stress assumption is not proof that real losses stop at 2R. If larger losses, collateral changes or inaccessible exits remain possible, the check does not guarantee the floor. Nor does leverage turn a planned stop into a binding loss limit.

Recompute the allowance after each change in equity. In the quarter-Kelly example, one ordinary loss leaves $9,375. Repeating the same 6.25% fraction and losing again leaves $8,789.06. After the first loss, the remaining $375 above the floor permits only $187.50 of modeled risk under the 2R assumption: 2% of $9,375. A fixed fraction of current equity does not preserve a fixed dollar floor indefinitely.

The distinction also appears in risk-constrained Kelly research: scaling Kelly bets and imposing a specified drawdown-probability constraint are different procedures. That paper's bound depends on its assumed return model; it is not an exchange guarantee. For horizon and failure-event definitions, consult the risk-of-ruin assumption guide.

Validate the portfolio before accepting a size

Several simultaneous trades can lose together. Adding separately calculated Kelly fractions does not optimize their combined wealth process. Shared collateral, overlapping signals and common market exposure need portfolio-level treatment.

Start by inspecting how capital is distributed. The portfolio concentration HHI worksheet measures concentration in weights; it does not replace a joint-outcome model. Many similarly exposed positions can still consume the same capital-floor allowance during one shock.

Before using a Kelly-derived number, retain a short decision record:

1. Define the equity pool, net payoff units, observation period and treatment of open trades.
2. State why the probability and payoff estimates apply to the next decision, including evidence that could invalidate them.
3. Compare weaker-input and larger-loss scenarios, keeping missing events visible.
4. Apply current portfolio exposure, floor headroom, liquidity and venue constraints; recalculate after quantity rounding.

If a small change in plausible assumptions turns the result from substantial risk to zero, record that instability before choosing a multiplier. The useful output is a size that survives the stated validation process, or a documented decision that the evidence does not support a positive size.