4. Expected value: the rigorous foundation of expectancy

Level 1: Probability: the language of uncertainty

Two strategies on the table. One wins 40% of the time. The other wins 80%.

The first one makes money. The second one loses it.

This is not a paradox or a contrived case. It is arithmetic, and this module makes it explicit.

A probability-weighted average

The expected value, or mathematical expectation, of a random variable is computed in three steps:

• Take every value the variable can take.
• Multiply it by its associated probability.
• Add up all those products.

It is a probability-weighted average, not a simple average.

Nor is it a forecast of the next individual outcome. It is the average you would obtain if the same uncertain situation were repeated a very large number of times, under stable conditions.

Trading expectancy is exactly this

This definition puts precise mathematical form around the concept of expectancy already used informally in the trading world.

The expected value of a trade is computed like this: the probability of winning multiplied by the average size of the wins, minus the probability of losing multiplied by the average size of the losses.

It is nothing more than applying the general expectation formula to the random variable 'outcome of a trade'. In its simplest version, that variable takes two possible values: an average gain with a certain probability, or an average loss with the complementary probability.

Winning 40% of the time and making money

A first numerical example makes this very clear. A strategy wins 40% of its trades, with an average gain of 300 dollars when it wins and an average loss of 120 dollars when it fails.

Its expected value is 0.40 × 300 minus 0.60 × 120. That is, 120 minus 72, which equals +48 dollars per trade.

The strategy makes money on average despite losing on most of its trades. The reason is that the asymmetry between the size of the wins and the size of the losses more than compensates for the low hit rate.

This is exactly the mathematical logic that underpins trend or momentum strategies: many small losses, offset by a few large gains.

Winning 80% of the time and losing money

A second example, running the other way, is just as instructive. A strategy wins 80% of its trades, with an average gain of just 50 dollars and an average loss of 220 dollars when it fails.

Its expected value is 0.80 × 50 minus 0.20 × 220. That is, 40 minus 44, which equals -4 dollars per trade.

Despite winning the vast majority of the time, the strategy loses money on average. The few losses are disproportionately large next to the routine gains.

Here is the precise mathematical basis for an uncomfortable idea: a high win rate guarantees nothing at all about profitability. The market theory course could only describe it qualitatively, when discussing the risk of poorly managed mean reversion strategies.

A per-unit figure, not a loose number

Expected value is a per-unit figure: per trade, per dollar risked. It must always be read alongside two further elements:

• How many times you can reasonably repeat that bet.
• How much size you put at risk on each repetition.

A correctly computed positive expectancy can turn out to be useless, or even ruinous. That happens if position sizes are so large that a normal losing streak — well within what is statistically expected — wipes out the capital before the favorable average has a chance to show up.

This link between expectancy and the dispersion of outcomes is precisely what variance formalizes in the next module.

Expectation adds up; variance does not

An important mathematical property of expectation is that it is additive.

The expected value of a portfolio's total result is simply the sum of the expected values of its individual parts, weighted by the size of each position. And this holds whether or not those parts are related to one another.

This point is worth underlining because it contrasts sharply with what happens to variance. Variance does not behave nearly as simply when dependence between the parts exists, as the module on independence will develop.

What expectancy does not tell you

A frequent mistake is to confuse expected value per trade with the probability that the next particular trade will be a winner.

A strategy with a clearly positive expectancy can perfectly well lose on any individual trade without contradicting anything. Expectancy is a statement about the long-run average, not about the outcome of a single attempt.

And that long-run average, it is worth remembering, is computed from probabilities and average sizes that are themselves estimates drawn from historical data. They may stop being valid if the underlying market conditions change.

Expected value is, ultimately, the number that separates a rigorously designed trading approach from a merely hopeful one. It turns the vague claim that something works into a concrete numerical statement, verifiable against data: it can be tested, compared across strategies, and monitored over time to detect when it stops holding.