May 6, 2024

## Understanding the Difference Between Expected Value and Expected Utility in Finance

In finance, decision-making under uncertainty plays a central role. When faced with uncertain outcomes, investors and individuals often rely on two key concepts: expected value and expected utility. Although they may sound similar, these concepts differ significantly in their approach and application. In this article, we will examine the differences between expected value and expected utility, exploring their definitions, calculations, and implications in finance.

## Expected Value: A Measure of Central Tendency

Expected value, also known as mean or mathematical expectation, provides a measure of the central tendency of a random variable. It represents the long-term average outcome that can be expected from a particular investment or decision. Expected value is calculated by multiplying each possible outcome by its associated probability and summing the results.
In finance, expected value is a valuable tool for evaluating investment opportunities. By estimating potential gains and losses and weighting them by their respective probabilities, investors can estimate the average return they can expect over time. For example, when considering a stock investment, expected value can help assess potential profitability by taking into account the probabilities of different price movements and their corresponding returns.

While expected value is a useful measure of central tendency, it has limitations. It assumes that individuals make rational decisions based solely on maximizing their monetary outcomes, ignoring other factors such as risk aversion or personal preferences. This is where the concept of expected utility comes in.

## Expected Utility: Incorporating Risk Aversion and Preferences

Expected Utility takes into account an individual’s risk aversion and personal preferences, recognizing that people’s decisions are influenced by factors beyond monetary outcomes alone. It introduces the concept of utility, which represents the satisfaction or value an individual derives from a particular outcome. Unlike expected value, expected utility considers the entire distribution of possible outcomes and assigns subjective utility values to each outcome.

In finance, expected utility theory is the basis for understanding investor behavior in uncertain situations. Investors are assumed to have diminishing marginal utility, meaning that the perceived value or satisfaction they derive from each additional unit of wealth decreases as their wealth increases. This leads to risk aversion, where individuals prefer investments with more certain outcomes, even if they offer lower expected returns.

Calculating expected utility involves assigning utility values to each outcome and multiplying them by their respective probabilities. By incorporating individuals’ risk preferences and subjective values, expected utility provides a more complete framework for decision making under uncertainty.

## Comparing Expected Value and Expected Utility

The key difference between expected value and expected utility lies in the factors they consider. Expected value focuses solely on the average outcome, assuming that individuals make decisions based solely on maximizing monetary gain. In contrast, expected utility incorporates risk aversion and personal preferences, recognizing that individuals derive value from outcomes beyond their monetary value.

Expected value is a simpler concept to understand and calculate because it requires only multiplying outcomes by their probabilities and summing the results. Expected utility, on the other hand, involves assigning subjective utility values to outcomes, which can be more difficult and subjective.

While expected value is a useful measure for evaluating investment opportunities, it may overlook important factors such as risk tolerance and individual preferences. Expected utility provides a more nuanced approach that captures the multidimensional nature of decision making under uncertainty.

## Real-World Applications and Implications

In practical finance, expected value and expected utility are used in different contexts. Expected value is commonly used in scenarios where the decision maker is risk neutral or where the potential outcomes are of a similar nature and magnitude. For example, it can be used to evaluate the expected return on a portfolio of different but similar assets.

Expected utility, on the other hand, is particularly relevant when individuals are risk averse. It helps explain phenomena such as the equity premium puzzle, where investors demand higher returns from equities than from safer assets despite their higher volatility. Expected utility theory provides a framework for understanding this behavior, as investors weigh potential gains against their risk aversion.
In summary, while expected value and expected utility are both important concepts in finance, they differ in their underlying assumptions and applications. Expected value focuses on the average outcome and is useful for evaluating investment opportunities without considering risk aversion or personal preferences. Expected utility incorporates these factors and provides a more comprehensive framework for decision making under uncertainty. Understanding these concepts enables investors to make informed decisions that are consistent with their risk preferences and financial goals.

## FAQs

### What is the difference between expected value and expected utility?

Expected value and expected utility are two concepts used in decision theory and economics to evaluate uncertain outcomes. The main difference between them lies in how they measure the desirability of these outcomes.

### How is expected value defined?

Expected value, also known as the mean or mathematical expectation, is a measure of the average outcome of a random variable. It is calculated by multiplying each possible outcome by its respective probability and summing them up. In other words, it represents the long-term average value one would expect to obtain if the same situation is repeated multiple times.

### What is expected utility?

Expected utility is a concept that takes into account not only the probability of different outcomes but also the individual’s preferences or utility function. It incorporates the idea that people’s preferences may not solely depend on the monetary value of an outcome, but also on other factors such as risk aversion or personal values.

### How is expected utility calculated?

Expected utility is calculated by multiplying the utility or desirability of each possible outcome by its respective probability and summing them up. The utility function is a subjective measure of an individual’s preferences and can vary from person to person.

### What is the main difference in their application?

Expected value is commonly used in situations where the decision maker is risk-neutral, meaning they are only concerned about maximizing their average monetary outcome. On the other hand, expected utility is used when the decision maker exhibits risk aversion or has preferences that go beyond monetary considerations.

### Can expected value and expected utility lead to different decisions?

Yes, expected value and expected utility can lead to different decisions. This is because expected value only considers the average monetary outcome, while expected utility takes into account the individual’s preferences and subjective valuation of different outcomes. When faced with risky or uncertain situations, decision makers who are risk-averse may choose options with lower expected value but higher expected utility to minimize potential losses.