Positive expectation — also called positive expected value — is the mathematical property of a trading system or investment strategy where the probability-weighted average outcome is positive. A strategy has positive expectation when, over many repetitions, the sum of gains multiplied by their probabilities exceeds the sum of losses multiplied by their probabilities. Positive expectation is the minimum mathematical requirement for any strategy to be sustainably profitable.

“At the heart of all trading is the simplest of all concepts — that the bottom line results must show a positive mathematical expectation in order for the trading method to be profitable.” — Curtis Faith, Way of the Turtle

Calculating Expected Value

Expected value = (probability of win × average win size) − (probability of loss × average loss size). A strategy that wins 40% of the time with an average win of $300 and loses 60% of the time with an average loss of $100 has an expected value of (0.4 × $300) − (0.6 × $100) = $120 − $60 = +$60. This positive expectation means the strategy is sustainably profitable over many repetitions, even with a minority win rate.

Why Positive Expectation Is Necessary but Not Sufficient

Positive expectation is the mathematical foundation of profitability, but it must be combined with sufficient diversification and controlled position sizing to be reliably realized. A strategy with positive expected value but extremely high variance — where a single adverse outcome can devastate the portfolio — may never produce its theoretical expected value in practice. Kelly sizing and other position sizing frameworks attempt to balance positive expectation with variance management, ensuring that the compounding of positive-expectation strategies over time produces the expected long-term wealth accumulation.

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