Expectancy and Edge: The Mathematical Core of Every Strategy

Many investors first look at win rate when evaluating a strategy — the percentage of trades that ended profitably. At first glance, that makes sense: a 70% success rate sounds better than 45%. But in practice, a strategy with a 70% win rate can lose money, while a strategy with a 40% win rate can be profitable.

Why? Because what matters is not only how often a strategy wins, but also how much it makes when it wins and how much it loses when it loses. This is captured by expectancy. If expectancy is positive over the long term and statistically robust, the strategy is said to have an edge — an advantage over random decision-making.

This is a more advanced lesson: it explains the formula, shows examples of risk/reward asymmetry, describes how to measure edge on your own data, and covers statistical honesty — including sample size, t-statistics, and the risk of overfitting.

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1. Win rate is not a strategy

Win rate (WR) is the share of profitable trades:

\[
WR = \frac{\text{number of winning trades}}{\text{total number of trades}}
\]

If a strategy made 100 trades and 55 were profitable, its win rate is 55%.

That is useful information, but it is not enough. Consider two strategies:

Strategy	Win rate	Average win	Average loss	Intuitive impression
A	70%	+1%	−4%	“Wins often”
B	40%	+5%	−2%	“Is often wrong”

Strategy A has a high success rate, but when it loses, it loses a lot. Strategy B has a low success rate, but its gains are larger than its losses. Which one is better? Without calculating expectancy, it is not clear.

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2. The expectancy formula

The basic formula is:

\[
\text{Expectancy} = WR \times \text{average win} - LR \times \text{average loss}
\]

Where:

• WR = win rate, the probability of a winning trade,
• LR = loss rate, the probability of a losing trade,
• LR = 1 − WR,
• average win = the average return of winning trades,
• average loss = the average size of losing trades, usually treated as a positive number.

Expectancy tells us how much the strategy historically made or lost per trade on average.

Calculation for strategies A and B

Strategy A:

\[
0.70 \times 1\% - 0.30 \times 4\% = 0.7\% - 1.2\% = -0.5\%
\]

Strategy A has an expectancy of −0.5% per trade. Even though it has a 70% win rate, it historically loses money.

Strategy B:

\[
0.40 \times 5\% - 0.60 \times 2\% = 2.0\% - 1.2\% = +0.8\%
\]

Strategy B has an expectancy of +0.8% per trade. Even though most trades end in losses, the winning trades are large enough to outweigh them.

That is the key lesson: win rate is a psychologically comforting metric, but expectancy is an economic metric.

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3. Risk/reward asymmetry: why a WR below 50% can work

A strategy with a win rate below 50% can be profitable if it has a favorable risk/reward (R/R) profile. This compares the average loss with the average win.

For example, trend-following strategies often work by accepting many small losses as the “cost of searching for a trend,” while occasionally capturing a large move. Such a strategy may have a WR of 35–45%, but still have positive expectancy.

Example using illustrative data:

A strategy tracks high-momentum stocks such as NVDA or MSFT, but trades only when they meet predefined quantitative rules. The numbers below are only a model example, not a recommendation.

Metric	Value
Number of trades	120
Win rate	42%
Loss rate	58%
Average win	+7.0%
Average loss	−3.0%
Expectancy	?

Calculation:

\[
0.42 \times 7\% - 0.58 \times 3\% = 2.94\% - 1.74\% = +1.20\%
\]

The strategy has a historical expectancy of +1.2% per trade. With 120 trades, that looks more meaningful than with 12 trades, but it is still not a guarantee of future results.

The important point is that a low WR requires psychological resilience. A strategy can lose six times in a row and still be mathematically positive. If an investor abandons the system after a normal losing streak, the real outcome may differ significantly from the backtest.

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4. Concrete example: a CZK 100,000 portfolio and expectancy

Imagine a model portfolio of CZK 100,000. The strategy risks 1% of capital per trade, or CZK 1,000. It uses the following rule:

• typical loss: −1R = −CZK 1,000,
• typical win: +2.5R = +CZK 2,500,
• win rate: 38%,
• loss rate: 62%.

Expectancy in R units:

\[
0.38 \times 2.5R - 0.62 \times 1R = 0.95R - 0.62R = +0.33R
\]

In Czech crowns:

\[
0.33 \times 1,000 = 330 \text{ CZK per trade}
\]

In this model, the strategy makes CZK 330 per trade before costs, slippage, and taxes. If 100 similar trades had occurred historically, the gross expected result would be CZK 33,000. In reality, however, results do not arrive evenly. There may be losing streaks, a few unusually large winners, higher transaction costs, or a change in market regime.

That is why expectancy is only the first layer. The second layer is the question: is the measured edge statistically reliable?

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5. How to measure edge on your own data

Measuring edge begins with every trade having a clearly defined reason for entry, exit, and position size. Without consistent rules, it is impossible to distinguish a strategy from a random mix of decisions.

A practical framework:

1. Strategy definition

For example: “Stocks from the S&P 500 that are above their 200-day moving average, have positive 6-month momentum, and enter after a short-term pullback.” Tickers such as AAPL, MSFT, or KO may be part of the tested universe, but they do not define the strategy by themselves.

1. Record every signal

For each trade, the data includes date, ticker, entry price, exit price, position size, costs, and exit reason.

1. Calculate the distribution of outcomes

It is not just about the average. Extremes, median, volatility of outcomes, worst losing streak, and maximum drawdown also matter.

1. Separate in-sample and out-of-sample periods

A strategy built on 2015–2021 data should then be observed in a different period, such as 2022–2024. If it works only in the period on which it was designed, it may be overfitted.

1. Include costs

Spread, commissions, slippage, and tax effects can turn positive expectancy into zero or negative expectancy. For shorter-term strategies, this part is especially important.

Edge is therefore not a feeling that “this works.” Edge is a measurable difference between the strategy’s outcome and a realistic alternative after costs.

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6. Minimum sample size: how many trades are enough?

A common mistake is drawing conclusions from too small a sample. Ten trades with a beautiful result may simply be luck. For low-frequency strategies, this is uncomfortable because collecting data takes time.

A rough guide:

Number of trades	Interpretation
< 30	More anecdote than statistics
30–100	First signal, high uncertainty
100–300	More useful sample, still depends on variance
300+	Better foundation if it covers different market regimes

But it is not only about the number of trades. The sample should ideally include:

• bull and bear markets,
• periods of high and low interest rates,
• different sectors,
• different volatility regimes,
• a sufficient number of independent signals.

If a strategy tests 500 trades, but all of them come from technology stocks during one strong growth period, independence is lower than the number suggests.

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7. T-statistic: is the edge larger than noise?

The t-statistic helps estimate whether the average return of a strategy is large enough relative to its variability.

In simplified form:

\[
t = \frac{\text{average return per trade}}{\text{standard error of the mean}}
\]

The standard error decreases with the number of observations:

\[
SE = \frac{\sigma}{\sqrt{n}}
\]

Where:

• \(\sigma\) is the standard deviation of trade outcomes,
• \(n\) is the number of trades.

The higher the t-statistic, the lower the probability that the observed average came purely from chance. In practice, a t-stat around 2 is often treated as an approximate threshold of interest, but it is not a magic number. Financial returns are often skewed, have fat tails, and trades may not be independent.

Example:

Strategy	Avg/trade	Std. deviation	n	Approx. t-stat
X	+0.8%	5.0%	25	0.80
Y	+0.8%	5.0%	400	3.20

Both strategies have the same average expectancy, but Strategy Y has much stronger statistical support because of the larger sample. Strategy X may simply be a lucky streak.

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8. Overfitting: when a strategy explains the past perfectly

Overfitting happens when a strategy fits historical data too precisely but loses meaning outside it. It is like memorizing answers for a test instead of understanding the subject.

Typical warning signs:

• too many parameters,
• optimization of exact values such as “the 17-day average is best, not 16 or 18,”
• testing hundreds of variants and selecting the single best one,
• ignoring failed attempts,
• excellent backtest but weak out-of-sample result,
• edge disappears after costs.

More robust strategies tend to have a simple economic logic. Momentum may come from behavioral delays in the market; the value factor from overpricing pessimism; quality from resilience of earnings. Logic alone is not enough, but it helps distinguish a real hypothesis from a random curve in the data.

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How to do this in QMA

In QMA, expectancy and edge can be observed descriptively in several parts of the platform:

• /strategy

The strategies page shows metrics for individual models such as win rate (WR), average wins and losses, historical performance, alpha versus benchmark, number of observations n, and t-statistic. QMA distinguishes strategies with stronger historical validation as ✅ verified and strategies with smaller samples, shorter histories, or higher parameter sensitivity as ⚠️ experimental. The label describes the quality of evidence, not a trading instruction.

• /journal

The trading journal allows recording personal trades, entry and exit reasons, position size, and outcome. From this data, one can observe personal WR, average win, average loss, and expectancy. It is especially useful for comparing planned rules with actual investor behavior.

• /screener

The screener helps define a testable stock universe — for example by country, sector, valuation, quality, momentum, or liquidity. This reduces the risk that a strategy is created by retrospectively selecting only a few successful tickers.

• /stocks/[symbol]

A stock detail page, such as /stocks/AAPL, /stocks/MSFT, /stocks/NVDA, or /stocks/CEZ, shows fundamental, technical, and risk characteristics of a specific stock. These data points can provide context for strategy rules, not standalone proof of edge.

• /portfolio

The portfolio view shows exposures, concentration, sector allocation, and historical drawdown. Even a strategy with positive expectancy can be problematic if it is too concentrated or has losing streaks larger than the investor can psychologically and financially tolerate.

The purpose of QMA is not to tell an investor what to do. Its purpose is to show whether a rule has measurable history, how strong the evidence is, and where the weak points may be.

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9. Summary: edge is a combination of mathematics and discipline

A profitable strategy is not based on “being right often.” It is based on the fact that its wins and losses together have positive expected value after costs.

Key points:

• Win rate without average win and average loss is not enough.
• Expectancy = WR × average win − LR × average loss.
• A strategy with WR below 50% can be profitable if its wins are significantly larger than its losses.
• Edge must be measured on a sufficient sample, ideally across different market regimes.
• The t-statistic helps distinguish signal from noise, but it is not all-powerful.
• Overfitting is one of the biggest risks in backtesting.
• Real-world results also depend on costs, slippage, taxes, position sizing, and discipline.

Expectancy is like the average slope of a road. A single segment may go downhill, but if the long route statistically climbs, there is a positive direction. The investor’s task is not to find a strategy that never loses. The task is to understand whether the losses fit the rules and whether the historical advantage has survived honest measurement.

Disclaimer

Past performance is not a guarantee of future results. All investment decisions are your own responsibility.