Kelly Criterion and Fractional Kelly: Position Sizing, Edge and Drawdown Risk

Kelly criterion: growth-optimal maths, estimation risk and drawdowns

The Kelly criterion estimates the capital fraction that maximises long-run expected logarithmic growth under a defined probability model. It does not discover an edge, protect against bad data or guarantee a comfortable drawdown.

বাংলায় পড়ুন: Kelly Criterion ও Fractional Kelly: Position Sizing, Edge ও Drawdown Risk

Learning objective: calculate a simple Kelly fraction, understand its assumptions, see how parameter error causes overbetting and evaluate fractional Kelly as a research choice rather than a universal rule.

1. Kelly begins after an edge is estimated

J. L. Kelly Jr.'s 1956 paper, “A New Interpretation of Information Rate”, connected information and the maximisation of long-run exponential capital growth. In a simplified repeated binary setup, sizing depends on:

• p: probability of a win;
• q: probability of a loss, equal to 1 − p;
• b: average net amount won for each unit lost.

The formula is:

Full Kelly fraction f = (b × p − q) ÷ b*

The output is the fraction of current capital placed at risk under the model. It is not automatically the position's notional size. A stop distance, instrument value and gap assumptions are still needed to convert cash risk into units.

2. A positive fraction requires positive expectancy

The numerator b × p − q is the simplified expectancy in loss units. If it is zero, the model has no edge after the assumptions. If it is negative, the formula does not support taking the trade in that direction.

A negative result is not permission to short automatically. The reverse trade may have different costs, probabilities, skew and execution rules.

Before using Kelly, estimate results after commission, spread, slippage, financing and realistic losses. Planned targets and stops are not enough; use actual or conservatively modelled fills.

Worked example

Assume a hypothetical strategy has:

• estimated win probability p = 45%;
• loss probability q = 55%;
• average net win 1.5R;
• average net loss 1R.

Its simplified expectancy is:

0.45 × 1.5R − 0.55 × 1R = +0.125R per trade

The full Kelly fraction is:

f = (1.5 × 0.45 − 0.55) ÷ 1.5 = 0.0833, or 8.33%*

Under exact, stable and independent assumptions, the formula allocates 8.33% of current capital to the one-unit loss risk on each trade. For a £10,000 account, that is about £833 of modelled loss risk—far larger than many traders can tolerate.

If the stop distance is £2 per share, the theoretical full-Kelly calculation would be £833 ÷ £2 ≈ 416 shares before gaps and costs. This is a mathematical output, not a recommendation.

3. Full Kelly can produce severe drawdowns

Five consecutive one-unit losses at an 8.33% fraction reduce capital approximately as follows:

Remaining capital = 0.9167⁵ ≈ 64.7%

That is roughly a 35.3% drawdown before any gap beyond the assumed loss. The strategy can still have positive expectancy while producing a losing streak of this size.

The formula maximises long-run log growth within its model. It does not minimise drawdown, volatility or emotional difficulty. A trader who abandons the process during a drawdown will not realise the theoretical long-run result.

4. Fractional Kelly scales the output

Fractional Kelly multiplies the full fraction by a chosen value:

• Half Kelly: 8.33% × 0.5 = 4.17%.
• Quarter Kelly: 8.33% × 0.25 = 2.08%.

After five assumed full losses:

• half Kelly leaves about 80.8%, a 19.2% drawdown;
• quarter Kelly leaves about 90.0%, a 10.0% drawdown.

These figures ignore gaps, changing volatility and correlated positions. The CFA Institute Research Foundation has discussed the trade-off between growth and security in fractional Kelly strategies. Reducing the fraction generally sacrifices some modelled growth in exchange for a smoother wealth path, but no fraction makes bad inputs safe.

5. Input uncertainty is the central problem

Live trading does not reveal the true win probability or payoff ratio. They are estimated from a finite, changing sample.

Suppose the real win rate is 40%, not the estimated 45%, while b remains 1.5:

f = (1.5 × 0.40 − 0.60) ÷ 1.5 = 0*

A five-percentage-point estimation error turns an apparent 8.33% full-Kelly allocation into no long-side edge under the simplified model.

Win rate and payoff may also move together. Losing trades can become larger during gaps, liquidity shocks or regime changes. Review Win Rate and Sample Size before treating a backtest estimate as stable.

6. Independence is often false

Multiple trades may share the same factor even when their tickers differ. Three long technology positions can lose together; gold, currencies and miners can respond to the same rates shock.

Applying single-trade Kelly independently to correlated positions can create excessive portfolio risk. A full portfolio treatment needs expected returns, covariance, constraints and scenario risk. In practical risk control, also cap:

• total open loss if all stops execute;
• gross and leveraged exposure;
• sector and factor concentration;
• overnight and event-gap risk;
• liquidity and maximum executable size.

7. Re-estimate without chasing noise

Updating after every small streak can create unstable sizing. Define in advance:

1. the minimum sample and data window;
2. whether estimates are rolling or expanding;
3. how costs and outliers are treated;
4. when a strategy is suspended;
5. the maximum allowed fraction regardless of the formula.

Use confidence ranges rather than a single optimistic input. One conservative approach is to calculate Kelly from lower-bound win-rate and payoff assumptions, then apply an additional fraction and hard portfolio cap. That is a research framework, not a universal prescription.

8. Stress sequence risk

Kelly depends on compounding across repeated opportunities. The order of wins and losses still shapes the path and operational survivability.

Shuffle historical trades, vary win rate and payoff, widen losses for gaps and include correlated clusters. Monte Carlo and Sequence Risk explains how to examine a distribution of possible paths instead of one backtest order.

For the unit calculation after setting a risk budget, use Position Sizing and Risk Budgeting.

Common mistakes

1. Using Kelly to invent an edge rather than size an independently tested one.
2. Feeding planned wins and losses into the formula instead of realised results.
3. Treating a small sample win rate as the true probability.
4. Applying full Kelly despite uncertain or changing inputs.
5. Confusing capital-at-risk fraction with notional allocation.
6. Calculating each correlated trade independently.
7. Ignoring gap losses beyond the one-unit assumption.
8. Increasing size immediately after a lucky streak.

Practical checklist

• Are win probability and payoff estimated after all costs?
• Is the sample large enough and representative?
• How sensitive is Kelly to modestly worse inputs?
• Does one-unit loss include gaps and slippage?
• Is the output cash risk or notional exposure?
• Which positions are correlated?
• What hard cap overrides the formula?
• Which Kelly fraction, if any, fits the tolerated drawdown?
• Has sequence risk been stress-tested?
• What evidence suspends or retires the strategy?

The Money Man View

Kelly is most valuable as a diagnostic: it shows how much a claimed edge depends on win probability and payoff. Full Kelly is highly sensitive to estimation error and can create deep drawdowns. Use conservative inputs, portfolio constraints and stress tests before considering any fraction of the mathematical output.

Educational-risk disclosure

This tutorial uses a hypothetical simplified binary model and was prepared with AI-assisted research and writing under The Money Man editorial workflow. It is educational information, not personalised investment advice or a recommendation to use Kelly sizing. Estimated edges can disappear, losses can exceed assumptions and leveraged positions can lose more than deposited capital.