Adjusted expected value

LINK_POOL:

• Risk Tolerance
• Expected Utility Theory
• St. Petersburg Paradox
• Utility Function
• Marginal Utility
• Decision Theory
• Prospect Theory
• Risk Aversion
• Risk-Adjusted Return on Capital (RAROC))
• Capital Allocation
• Capital Budgeting
• Risk Management
• Regulatory Compliance
• Behavioral Economics
• Economic Capital

What Is Adjusted Expected Value?

Adjusted expected value is a concept within the broader field of behavioral economics that modifies the traditional notion of expected value by incorporating an individual's subjective preferences, attitudes toward risk, and other psychological factors. While standard expected value calculations focus solely on the probability-weighted average of monetary outcomes, adjusted expected value acknowledges that people do not always make decisions based purely on monetary gain. Instead, it considers the "utility" or subjective satisfaction an individual derives from those outcomes, making it a more realistic framework for understanding human choice under uncertainty. This adjustment is crucial because the perceived value of money can diminish as wealth increases, a principle known as marginal utility.

History and Origin

The foundational concept for adjusted expected value stems from the development of Expected Utility Theory. This theory has its roots in the 18th century with the work of Daniel Bernoulli, who proposed it as a solution to the St. Petersburg Paradox. Bernoulli's insight was that individuals do not evaluate risky outcomes based purely on their monetary values, but rather on the subjective satisfaction or "utility" they derive from those outcomes. He argued that the marginal utility of money decreases as an individual's wealth increases, meaning an additional dollar provides less satisfaction to a wealthy person than to a poor one.³², ³³

While Bernoulli laid the groundwork, the modern axiomatic formulation of expected utility theory, which provides a rigorous basis for adjusted expected value, was developed by mathematician John Von Neumann and economist Oskar Morgenstern in their 1944 book, "Theory of Games and Economic Behavior."²⁸, ²⁹, ³⁰, ³¹ Their work provided a set of axioms for rational decision-making under uncertainty, demonstrating that if an individual's preferences satisfy these axioms, then their choices can be represented by maximizing the expected value of a utility function.²⁶, ²⁷

Key Takeaways

• Adjusted expected value incorporates subjective preferences and risk attitudes into the calculation of expected outcomes.
• It is rooted in [Expected Utility Theory], which posits that individuals make decisions to maximize their expected utility, not just monetary value.
• Unlike traditional expected value, adjusted expected value accounts for diminishing [marginal utility] of wealth.
• It helps explain why individuals might choose a lower monetary expected value if it offers higher perceived satisfaction or reduced risk.
• This concept is a key component of [behavioral economics], seeking to describe actual human decision-making rather than purely rational models.

Formula and Calculation

The adjusted expected value is fundamentally derived from the expected utility formula. While the exact form of the utility function can vary depending on the individual's preferences, the general concept remains the same:

Where:

• ( AEV ) = Adjusted Expected Value
• ( p_i ) = The probability of outcome ( i ) occurring
• ( x_i ) = The monetary or real-world outcome ( i )
• ( u(x_i) ) = The utility (subjective value or satisfaction) derived from outcome ( x_i )

This formula differs from a simple expected monetary value by applying a utility function, (u(x_i)), to each outcome before summing the probability-weighted values. This function transforms monetary values into units of subjective satisfaction.

Interpreting the Adjusted Expected Value

Interpreting the adjusted expected value involves understanding that it represents an individual's subjective "worth" of a particular risky prospect. A higher adjusted expected value implies a more desirable outcome for that specific individual, considering their personal risk tolerance and preferences. For instance, a person with high risk aversion might prefer a sure, smaller gain over a risky, larger potential gain, even if the latter has a higher traditional expected monetary value. The adjusted expected value reflects this preference by assigning a higher utility to the certain outcome. It provides a framework for evaluating choices not just on objective financial returns, but on the subjective satisfaction they are expected to provide. This helps explain various observed behaviors in [decision theory] that deviate from purely rational economic models.

Hypothetical Example

Consider an individual, Sarah, who is faced with two investment opportunities:

Investment A: A guaranteed return of $1,000.
Investment B: A 50% chance of gaining $2,500 and a 50% chance of gaining $0.

From a traditional expected monetary value perspective:

• Expected Value of A = 1.00 * $1,000 = $1,000
• Expected Value of B = (0.50 * $2,500) + (0.50 * $0) = $1,250

A purely rational agent maximizing monetary value would choose Investment B. However, Sarah is quite risk-averse. Her personal utility function for money is (u(x) = \sqrt{x}).

Now, let's calculate the adjusted expected value for Sarah:

Adjusted Expected Value for Investment A:

Adjusted Expected Value for Investment B:

Based on the adjusted expected value, Sarah would choose Investment A (31.62 utils) over Investment B (25 utils), despite Investment B having a higher traditional expected monetary value. This demonstrates how her [risk aversion] is incorporated into the decision, as the guaranteed gain provides her with more subjective satisfaction than the uncertain, higher potential gain.

Practical Applications

Adjusted expected value and the broader [Expected Utility Theory] framework have several practical applications across finance and economics, particularly in situations involving [risk management] and strategic [capital allocation].

• Investment Decisions: Investors, especially those with varying degrees of [risk aversion], often use a qualitative form of adjusted expected value. While not always explicitly calculated, the principle guides choices towards investments that align with their [risk tolerance] and subjective preferences, even if it means foregoing the highest potential monetary return.
• Insurance: The decision to purchase insurance is a classic example of maximizing adjusted expected value. Individuals are willing to pay a premium (a certain loss) to avoid the small probability of a large, financially devastating loss, because the utility gained from avoiding the large loss outweighs the disutility of paying the premium.
• Project Valuation and [Capital Budgeting]: In corporate finance, concepts like [Risk-Adjusted Return on Capital (RAROC)] are applications of risk-adjusted valuation. RAROC measures the profitability of an investment or project relative to the capital at risk, effectively adjusting the expected return for the project's specific risk profile. This helps companies decide where to allocate funds to maximize returns while considering risk.²³, ²⁴, ²⁵
• Regulatory Frameworks: Financial regulators, such as the Federal Reserve, integrate risk assessment frameworks that implicitly consider risk-adjusted outcomes when evaluating the capital adequacy and stability of financial institutions. These frameworks ensure that institutions have sufficient capital to absorb potential losses under stressful conditions, aligning with a broader goal of systemic stability rather than just maximizing raw financial returns.¹⁸, ¹⁹, ²⁰, ²¹, ²² Furthermore, the SEC requires public companies to disclose material risks to investors, reflecting the importance of understanding risk-adjusted prospects.¹³, ¹⁴, ¹⁵, ¹⁶, ¹⁷

Limitations and Criticisms

While adjusted expected value, particularly through [Expected Utility Theory], provides a powerful framework for understanding decision-making under uncertainty, it faces several limitations and criticisms, especially from the field of [behavioral economics].

One major criticism is that observed human behavior often deviates systematically from the predictions of Expected Utility Theory.¹² These deviations are highlighted by various paradoxes, such as the Allais Paradox, which demonstrate that people's choices can be inconsistent with the theory's core axioms.¹⁰, ¹¹ For instance, individuals often overweight outcomes that are certain compared to outcomes that are merely probable, a phenomenon known as the "certainty effect." This can lead to [risk aversion] for potential gains and risk-seeking behavior for potential losses.⁸, ⁹

Furthermore, Expected Utility Theory assumes that individuals are consistent in their preferences and that these preferences are stable over time. However, research in behavioral economics suggests that preferences can be fluid and highly context-dependent, influenced by factors like how a choice is "framed" or presented.⁶, ⁷

In response to these empirical inconsistencies, Daniel Kahneman and Amos Tversky developed [Prospect Theory] in 1979.², ³, ⁴, ⁵ Prospect theory proposes an alternative model where value is assigned to gains and losses relative to a reference point, rather than to final asset states, and probabilities are replaced by "decision weights."¹ This theory better accounts for phenomena like loss aversion, where the psychological impact of a loss is felt more intensely than the pleasure of an equivalent gain, and the reflection effect, which describes how individuals' attitudes toward risk can reverse depending on whether they are facing potential gains or losses.

Adjusted Expected Value vs. Expected Value

The primary distinction between adjusted expected value and traditional expected value lies in their foundational assumptions and what they aim to measure.

Traditional expected value calculates a probability-weighted average of all possible monetary outcomes, assuming that each dollar gained or lost has the same objective value, regardless of an individual's financial situation. It provides a purely quantitative measure of a gamble or decision.

In contrast, adjusted expected value modifies this by applying a utility function to each outcome. This transformation accounts for the fact that the subjective value of money can change depending on an individual's wealth or other psychological factors. For example, a gain of $100 might mean more to someone with little wealth than to a billionaire. Therefore, while expected value tells you the average monetary outcome, adjusted expected value tells you the average subjective satisfaction or utility derived from that outcome.

FAQs

Why is "adjusted" necessary for expected value?

The "adjusted" aspect is necessary because traditional expected value assumes people are purely rational and only care about maximizing monetary outcomes, which isn't always true. Adjusted expected value factors in human psychology, specifically an individual's subjective value for money and their [risk tolerance], providing a more accurate model of real-world decision-making.

How does risk tolerance affect adjusted expected value?

[Risk tolerance] significantly impacts adjusted expected value. Individuals who are [risk averse] will generally assign a lower subjective utility to highly uncertain outcomes, even if they have a high traditional expected monetary value. This means their adjusted expected value calculations will favor options with lower risk and more predictable returns, reflecting their preference for security over potentially larger but less certain gains.

Is adjusted expected value only for individuals?

While the concept of adjusted expected value originated in modeling individual decision-making, its principles are applied in various corporate and regulatory contexts. For example, financial institutions use frameworks like [Risk-Adjusted Return on Capital (RAROC)] to evaluate projects and allocate [economic capital], implicitly adjusting for risk to align with strategic objectives beyond just maximizing raw returns.

What is the main difference between adjusted expected value and prospect theory?

Adjusted expected value is based on [Expected Utility Theory], which suggests people make decisions to maximize their subjective utility, typically over absolute levels of wealth. [Prospect Theory], developed as a critique of expected utility, proposes that people evaluate outcomes as gains or losses relative to a specific reference point, and that they are more sensitive to losses than to equivalent gains (loss aversion). So, while both adjust for subjective value, Prospect Theory introduces the crucial element of a reference point and different sensitivity to gains versus losses.

Can adjusted expected value be negative?

Yes, the adjusted expected value can be negative. If the potential losses in a scenario, combined with their probabilities and the individual's subjective disutility for those losses (which can be amplified by [loss aversion]), outweigh the utility from potential gains, then the adjusted expected value would be negative. A negative adjusted expected value would suggest that, for that specific individual, the perceived cost or disutility of the risky prospect outweighs its potential benefits.