Expectancy

What Is Expectancy?

Expectancy, in finance, is a statistical measure that quantifies the average outcome one can expect from a series of events, typically financial transactions or investment decisions. It is a core concept in probability theory and financial mathematics, falling under the broader category of quantitative finance. Expectancy helps investors and traders determine the long-term profitability of a trading strategy or an investment approach by considering both the probability of different outcomes and the magnitude of their associated gains or losses. A positive expectancy indicates an expected profit over time, while a negative expectancy suggests an expected loss.

History and Origin

The foundational ideas behind expectancy emerged from the study of probability in the 17th century. French mathematicians Blaise Pascal and Pierre de Fermat are credited with laying the groundwork for probability theory through their correspondence in 1654, which was prompted by a gambling problem. They introduced the principle that the value of a future gain is directly proportional to the probability of obtaining it²⁵.

Building on this, Dutch mathematician Christiaan Huygens published the first comprehensive treatise on probability theory in 1657, titled "De ratiociniis in ludo aleae" (Calculations in Games of Chance). In this work, Huygens explicitly presented the idea of mathematical expectation, which is synonymous with expectancy²³, ²⁴. Daniel Bernoulli further refined these concepts in the 18th century with his work on expected utility theory, distinguishing between expected value and expected utility, and introducing the concept of declining marginal utility²⁰, ²¹, ²². His 1738 paper, "Exposition of a New Theory on the Measurement of Risk," was particularly influential¹⁹.

Key Takeaways

• Expectancy is a statistical measure of the average outcome of a series of events.
• It is crucial for evaluating the long-term profitability of investment or trading strategies.
• A positive expectancy suggests a strategy is profitable over time, while a negative expectancy indicates potential losses.
• Calculating expectancy involves considering both the likelihood and magnitude of wins and losses.
• It is a fundamental concept in risk management and decision-making under uncertainty.

Formula and Calculation

The formula for expectancy is commonly used in trading and investing to calculate the average profit or loss per trade. It considers the win rate, average profit from winning trades, loss rate, and average loss from losing trades.

The expectancy formula can be expressed as:

Where:

• Win Rate: The probability of a winning trade, calculated as (Number of Winning Trades / Total Number of Trades).
• Average Win: The average profit from all winning trades.
• Loss Rate: The probability of a losing trade, calculated as (Number of Losing Trades / Total Number of Trades).
• Average Loss: The average loss from all losing trades.

Another way to conceptualize expectancy, particularly in a broader probabilistic context, is as the sum of the products of each possible outcome's value and its probability of occurrence¹⁷, ¹⁸.

Where:

• (E[X]) = Expectancy (or Expected Value)
• (x_i) = Value of the (i)-th outcome
• (p_i) = Probability of the (i)-th outcome
• (n) = Total number of possible outcomes

Interpreting the Expectancy

Interpreting expectancy is straightforward:

• Positive Expectancy: A positive value indicates that, on average, a strategy or system is expected to generate a profit per unit of risk over a large number of trials. The larger the positive number, the more profitable the strategy is expected to be¹⁶.
• Negative Expectancy: A negative value suggests that, on average, the strategy is expected to result in a loss per unit of risk over time.
• Zero Expectancy: A zero expectancy implies that, on average, the strategy is expected to break even, with neither a profit nor a loss.

For example, if a trading strategy has an expectancy of $50, it means that for every trade executed with that strategy, an investor can expect to make an average profit of $50 over a sufficiently large number of trades¹⁵. This metric is critical for performance analysis and setting realistic return expectations.

Hypothetical Example

Consider a simplified trading strategy for a stock market investor over 100 trades:

• Total Trades: 100
• Winning Trades: 40
• Losing Trades: 60
• Total Profit from Winning Trades: $12,000
• Total Loss from Losing Trades: $8,000

First, calculate the necessary components:

1. Win Rate: (40 \div 100 = 0.40) or 40%
2. Loss Rate: (60 \div 100 = 0.60) or 60%
3. Average Win: ($12,000 \div 40 = $300)
4. Average Loss: ($8,000 \div 60 = $133.33)

Now, apply the expectancy formula:

Expectancy = ((0.40 \times $300) - (0.60 \times $133.33))
Expectancy = ($120 - $79.998)
Expectancy = ($40.002)

In this hypothetical example, the expectancy is approximately $40.00. This means that, on average, this investor can expect to make $40.00 per trade using this strategy over the long run. This positive expectancy suggests the strategy is profitable over many trades, demonstrating its utility in investment analysis.

Practical Applications

Expectancy has several practical applications across various financial domains:

• Trading Strategy Evaluation: Traders use expectancy to assess the profitability of their trading systems and algorithmic trading models. By calculating expectancy, they can compare different strategies and allocate capital more effectively to those with higher positive expectancy¹⁴.
• Position Sizing and Risk Management: Expectancy aids in determining appropriate position sizing and setting stop-loss orders and take-profit levels. Knowing the average expected return per trade allows for better management of portfolio risk¹³.
• Option Pricing: In derivatives markets, expectancy is implicitly used in option pricing models such as the Black-Scholes model, where the expected value of future payouts is discounted back to the present.
• Gambling and Games of Chance: While not directly financial markets, the concept originated here and helps illustrate the principle. Casinos, for instance, design games with a slightly negative expectancy for players, ensuring long-term profitability for the house.
• Insurance Underwriting: Insurance companies use expectancy to price policies. They calculate the expected value of claims based on historical data and probabilities to ensure premiums cover potential payouts and generate a profit.
• Capital Budgeting: Businesses may use expected values when evaluating capital budgeting projects, weighing the probabilities of different revenue and cost scenarios to determine the expected net present value of an investment.

For example, a momentum-based algorithmic trading strategy that aims to capitalize on market trends can use expectancy to validate its profitability based on historical data. If the strategy has a positive expectancy, it indicates that, on average, the algorithm generates a profit per trade¹².

Limitations and Criticisms

While expectancy is a powerful tool in finance, it has several limitations and criticisms:

• Reliance on Historical Data: Expectancy calculations are based on past performance, and there is no guarantee that historical probabilities and outcomes will repeat in the future. Market conditions can change, rendering past expectancy less relevant for future predictions.
• Assumption of Independence: The calculation often assumes that each trade or event is independent, which may not always be true in complex financial markets where consecutive trades can be influenced by preceding ones or broader market trends.
• Ignores Risk Aversion and Utility: Expectancy, particularly in its simplest form (expected value), does not account for an individual's risk aversion or the concept of utility¹⁰, ¹¹. A rational investor may choose a lower expected value if it comes with significantly lower risk, demonstrating the importance of expected utility theory in behavioral finance.
• Impact of Outliers: A few large winning or losing trades can significantly skew the expectancy calculation, especially over a small sample size, potentially providing a misleading view of a strategy's true long-term profitability.
• Does Not Account for Transaction Costs or Slippage: Simple expectancy formulas often omit transaction costs, such as commissions and slippage, which can significantly erode profitability in real-world trading⁹.
• Rational Expectations Fallacy: While related, it's important to distinguish expectancy from the concept of rational expectations in macroeconomics. Rational expectations theory assumes individuals use all available information efficiently to form unbiased expectations, but in reality, investors may have biases or imperfect information, leading to deviations from purely rational behavior⁶, ⁷, ⁸.

Expectancy vs. Expected Value

The terms "expectancy" and "expected value" are often used interchangeably, particularly in financial contexts, and represent the same mathematical concept. Both refer to the weighted average of all possible outcomes, where each outcome is weighted by its probability of occurrence.

However, a subtle distinction can be made when considering the broader academic context. "Expected value" is the more general statistical term used across mathematics, statistics, and various scientific fields to denote the long-run average of a random variable. "Expectancy," while mathematically identical, gained prominence in fields like trading and investing to specifically describe the average profit or loss per unit of risk over a series of trades or financial events⁵. This slight difference in usage emphasizes the practical, application-oriented nature of "expectancy" in the financial world, particularly within trading psychology and system optimization.

FAQs

What does positive expectancy mean in trading?

A positive expectancy in trading means that, on average, your trading strategy is expected to generate a profit over a large number of trades. For instance, an expectancy of $0.50 means you expect to make 50 cents for every dollar risked⁴.

How does expectancy relate to risk management?

Expectancy is a critical tool for risk management as it helps traders and investors understand the potential average outcome of their decisions. By knowing the expectancy of a strategy, they can make informed choices about position sizing and overall exposure, ensuring that potential losses are managed relative to expected gains³.

Is a high win rate always better for expectancy?

Not necessarily. While a high win rate is desirable, expectancy also heavily depends on the average size of your wins versus your losses. A strategy with a lower win rate but significantly larger average profits from winning trades can have a higher expectancy than a strategy with a high win rate but small average profits and disproportionately large losses².

Can expectancy guarantee future profits?

No. Expectancy is based on historical data and probabilities and does not guarantee future profits. Financial markets are dynamic, and past performance is not indicative of future results. It is a statistical tool to assess potential long-term averages, not a predictor of individual trade outcomes¹.

How often should I calculate my trading expectancy?

The frequency depends on your trading style and the volatility of the markets you operate in. For active traders, calculating expectancy periodically (e.g., monthly or quarterly) or after a significant number of trades (e.g., every 50-100 trades) can provide valuable insights into the ongoing effectiveness of their strategy and help in strategy adjustment.