Expected values: Meaning, Criticisms & Real-World Uses

What Is Expected Values?

Expected value, often referred to as expectation, is a fundamental concept within Probability Theory that represents the anticipated average outcome of a random variable over many trials. It quantifies what one would expect to happen, on average, if an experiment or process were repeated an infinite number of times. In finance, the expected value helps to predict the average return or result of an uncertain event, such as an Investment Portfolio's future value or the payout of an Actuarial Science claim. It is not necessarily the most likely outcome, nor is it a guarantee, but rather a weighted average of all possible Outcomes, with each outcome's weight determined by its probability. Understanding expected value is crucial for making informed Decision Making under uncertainty.

History and Origin

The concept of expected value emerged from the study of games of chance in the 17th century. The foundation of modern probability theory, which includes the idea of expected value, is largely attributed to the correspondence between French mathematicians Blaise Pascal and Pierre de Fermat in 1654. Their discussions were sparked by a problem posed by Antoine Gombaud, also known as Chevalier de Méré, concerning the fair division of stakes in interrupted games. ¹⁶This "problem of points" led Pascal to explicitly reason about what is now known as an expected value.

Building on their work, Dutch mathematician Christiaan Huygens published the first formal treatise on probability in 1657, titled De Ratiociniis in Ludo Aleae ("On Reasoning in Games of Chance"). In this seminal work, Huygens systematized the concept of expected value, defining how to calculate it and framing it as the basis for fair games. ¹⁴, ¹⁵His principles were later adapted and expanded, becoming integral to fields such as insurance and finance for quantifying risks. Modern understanding of probability and its applications owes much to these pioneering efforts. For a deeper dive into these foundational principles, consider exploring resources like the MIT OpenCourseWare's Introduction to Probability.
¹³

Key Takeaways

Expected value is the long-run average outcome of a random process.
It is calculated by multiplying each possible outcome by its probability and summing these products.
In finance, expected value provides a statistical measure for anticipating returns or losses from investments and other uncertain financial events.
Expected value does not predict a single, guaranteed outcome but rather the average over many repetitions.
While a powerful tool, it has limitations, particularly when individual preferences and risk tolerances are considered.

Formula and Calculation

The expected value ((E[X])) of a Random Variable (X) is calculated as the sum of all possible outcomes, each multiplied by the probability of that outcome occurring.

For discrete random variables with (n) possible outcomes:

E[X] = \sum_{i=1}^{n} (x_i \cdot P(x_i))

Where:

(E[X]) = Expected value of the random variable (X)
(x_i) = The (i)-th possible outcome
(P(x_i)) = The probability of the (i)-th outcome occurring

For continuous random variables, the expected value is calculated using an integral:

E[X] = \int_{-\infty}^{\infty} x \cdot f(x) \, dx

Where:

(f(x)) = The probability density function of the continuous random variable (X)

This formula allows for the quantitative Risk Assessment of various scenarios by incorporating the likelihood of different events.

Interpreting the Expected Values

Interpreting expected value requires understanding that it represents a theoretical average over a large number of repetitions, not a guaranteed outcome for a single event. For example, if a game has an expected value of $5, it means that if you play the game many times, your average gain per game will be $5. You might win more or less on any given play, or even lose money, but the long-run average converges to the expected value.

In financial contexts, a positive expected value often suggests a favorable long-term proposition, while a negative expected value indicates an unfavorable one. However, it's crucial to consider the Risk Management implications and the potential range of outcomes, not just the average. For instance, an investment with a high expected value might also carry a substantial risk of significant losses, which would not be fully captured by the single expected value figure alone. Understanding the underlying Randomness and variability of outcomes is essential for a complete interpretation.

Hypothetical Example

Consider an investor deciding whether to invest in a new technology startup. The investment amount is $10,000.
There are three possible scenarios for the startup's performance after five years:

High Growth: 30% probability, leading to a gain of $50,000.
Moderate Growth: 50% probability, leading to a gain of $5,000.
Failure: 20% probability, leading to a loss of $10,000 (the initial investment).

To calculate the expected value of this investment:

Outcome 1: $50,000 * 0.30 = $15,000
Outcome 2: $5,000 * 0.50 = $2,500
Outcome 3: -$10,000 * 0.20 = -$2,000

Summing these values:
Expected Value = $15,000 + $2,500 - $2,000 = $15,500

The expected value of this investment is $15,500. This suggests that, on average, for every $10,000 invested in similar ventures, the investor could expect a return of $15,500. However, this doesn't mean the investor will receive exactly $15,500; they will either gain $50,000, gain $5,000, or lose $10,000. This calculation informs the investor's Decision Making.

Practical Applications

Expected value is widely used across various domains in finance and beyond:

Investment Analysis: Investors use expected value to estimate the average return of different investments, helping them compare and choose assets like stocks, bonds, or real estate. It's a key component in Financial Modeling and evaluating potential returns on capital.
Insurance: Insurance companies heavily rely on expected value to set premiums. They calculate the expected value of claims they will have to pay out for a given policy and add a profit margin to determine the premium. ¹¹, ¹²For instance, if the expected payout for a specific type of car insurance policy is $1,000 per year, the company will charge a premium higher than that to cover costs and generate profit. This application highlights how insurers manage risk by pooling many policies, relying on the Statistical Inference of average outcomes over a large number of policies.
¹⁰* Gambling and Games of Chance: Casinos and gaming operators use expected value to design games that, on average, yield a profit for the house. The expected value for a player in casino games is almost always negative, ensuring the profitability of the establishment.
Business Decisions: Companies apply expected value to assess the potential profitability of new projects, product launches, or marketing campaigns, weighing potential gains against the costs and probabilities of success or failure.
Capital Allocation: Firms use expected value in conjunction with risk assessments for Capital Allocation decisions, prioritizing projects that offer the highest expected return relative to their risk profile.

Limitations and Criticisms

Despite its widespread utility, expected value has several notable limitations, particularly when applied to individual human decision-making and real-world financial choices. One significant criticism is that it assumes individuals are "risk-neutral," meaning they only care about the average outcome and are indifferent to the variability or risk associated with that outcome. ⁹However, behavioral economics research consistently shows that individuals are often risk-averse, preferring a sure gain over a risky gain of equal or even slightly higher expected value. ⁸Conversely, they may be risk-seeking when facing potential losses.

This disconnect led to the development of Expected Utility Theory and, more prominently, Prospect Theory, proposed by psychologists Daniel Kahneman and Amos Tversky in 1979. ⁷Prospect theory highlights that people evaluate outcomes not in terms of absolute wealth, but as gains or losses relative to a reference point, and that losses loom larger than equivalent gains—a phenomenon known as Loss Aversion. Th⁵, ⁶is means a $100 loss is felt more intensely than a $100 gain. Therefore, relying solely on expected value can lead to predictions that do not align with observed human behavior in situations involving risk and uncertainty.

A⁴nother limitation is that expected value does not account for the emotional or psychological impact of extreme outcomes. A project with a high expected value but also a small chance of catastrophic loss might be avoided by a risk-averse investor, even if the expected value calculation suggests otherwise. Behavioral Economics specifically examines these deviations from purely rational economic models.

Expected Values vs. Expected Utility Theory

While closely related and often discussed together, expected value and Expected Utility Theory represent distinct approaches to decision-making under uncertainty.

Expected Value focuses purely on the statistical average of numerical outcomes. It is a mathematical calculation that tells you what you can expect to gain or lose on average, assuming a risk-neutral stance. It treats all gains and losses equally in terms of their dollar value.

Expected Utility Theory, on the other hand, recognizes that the subjective value, or "utility," of money can vary for individuals. It posits that people make decisions to maximize their expected utility, not necessarily their expected monetary value. Th², ³is theory introduces the concept of a utility function, which maps monetary outcomes to a subjective level of satisfaction or happiness. For most people, this utility function exhibits diminishing marginal utility of wealth, meaning that each additional dollar gained provides less additional utility than the previous one. Th¹is explains why a risk-averse person might prefer a guaranteed $500 over a 50% chance of $1,000, even though both have the same expected monetary value of $500. The certainty of $500 provides more utility than the uncertain, higher potential gain for a risk-averse individual.

The primary point of confusion often arises because expected utility theory uses the framework of expected value but applies it to subjective utility rather than objective monetary amounts. While expected value is a direct mathematical calculation, expected utility theory attempts to model the more nuanced, psychological reality of how individuals perceive and respond to risk.

FAQs

Q1: Is expected value always the best way to make financial decisions?

A1: No, expected value is a powerful quantitative tool, but it's not the sole determinant for all financial decisions. It assumes a risk-neutral stance and doesn't account for individual risk tolerance, Loss Aversion, or the emotional impact of extreme outcomes. For individual investors, incorporating personal preferences and risk capacity is crucial.

Q2: Can expected value be negative?

A2: Yes, expected value can be negative. A negative expected value indicates that, on average, you would expect to lose money over many repetitions of a particular event or investment. For example, in most casino games, the expected value for the player is negative, which ensures a profit for the house.

Q3: How is probability used in calculating expected value?

A3: Probability is the cornerstone of expected value calculation. Each possible outcome of an event is weighted by its probability of occurring. The expected value is then the sum of these weighted outcomes. Accurate assessment of Probability is essential for a meaningful expected value.

Q4: Does expected value predict what will happen in a single instance?

A4: No, expected value does not predict the outcome of a single instance. It represents the long-term average if the event were to be repeated many times. In any single trial, the actual outcome will be one of the discrete possibilities, not necessarily the calculated expected value.

Q5: What's the difference between expected value and mean?

A5: In the context of a random variable, "expected value" and "mean" are often used interchangeably. The expected value is the theoretical mean of a probability distribution, representing the long-run average of a random variable. The term "mean" can also refer to the average of a specific dataset (a sample mean), while expected value typically refers to the theoretical average of the underlying probability distribution.