Absolute risk aversion

What Is Absolute Risk Aversion?

Absolute risk aversion is a concept within the economic theory of choice that measures an investor's willingness to accept or reject risky propositions, independent of their current level of wealth. It quantifies how much an individual would pay to avoid a specific risk, where the payment amount is the same regardless of whether the individual is rich or poor. This measure is a cornerstone of utility theory, which seeks to explain how individuals make investment decisions when faced with uncertainty.

The concept of absolute risk aversion is distinct because it focuses on the monetary value of risk tolerance rather than a proportional share of wealth. A higher absolute risk aversion indicates that an individual is more averse to a given financial risk and would require a larger risk premium to undertake it. Conversely, a lower absolute risk aversion suggests a greater willingness to take on fixed-size risks, irrespective of their total financial holdings.

History and Origin

The foundational ideas leading to the concept of absolute risk aversion trace back to the 18th century with Daniel Bernoulli's work on the St. Petersburg Paradox in 1738, which posited that individuals value gambles based on their expected utility rather than their expected monetary value. This marked a significant shift from the prevailing view that the value of a lottery should be its mathematical expectation, independent of an individual's psychology. Bernoulli suggested a non-linear relationship between wealth and its utility, implying diminishing marginal utility of wealth, which forms the basis for risk-averse behavior.¹⁶

In the mid-20th century, economists Kenneth Arrow and John W. Pratt independently formalized the measurement of risk aversion. In 1964 and 1965, respectively, Pratt and Arrow introduced the measures of absolute and relative risk aversion, which are now widely known as the Arrow-Pratt measures.¹⁵,¹⁴ Their work provided a robust mathematical framework to quantify the degree of an individual's risk aversion based on their utility function, allowing economists and financial theorists to compare risk attitudes across different individuals and levels of wealth.¹³

Key Takeaways

Absolute risk aversion quantifies how much an individual would pay to avoid a fixed amount of risk, regardless of their total wealth.
It is derived from an individual's utility function, specifically the ratio of the second derivative to the first derivative, with a negative sign.
A key hypothesis, known as the Arrow hypothesis, suggests that absolute risk aversion generally decreases as wealth increases.
This measure is crucial in economic models and portfolio theory for understanding and predicting investor behavior.
It is often contrasted with relative risk aversion, which considers risk as a proportion of wealth.

Formula and Calculation

The most common measure of absolute risk aversion is the Arrow-Pratt measure of absolute risk aversion, denoted (A(w)) or (r_A(w)). It is calculated using an individual's utility function, (u(w)), where (w) represents their wealth.

The formula for absolute risk aversion is:

A(w) = - \frac{u''(w)}{u'(w)}

Where:

(u(w)) is the utility function, which represents the satisfaction or utility an individual derives from a given level of wealth.
(u'(w)) is the first derivative of the utility function with respect to wealth, representing the marginal utility of wealth. For risk-averse individuals, (u'(w) > 0), reflecting that more wealth is preferred to less.
(u''(w)) is the second derivative of the utility function with respect to wealth, representing the rate at which marginal utility changes. For risk-averse individuals, (u''(w) < 0), indicating diminishing marginal utility of wealth (i.e., each additional unit of wealth provides less additional satisfaction than the previous one).
The negative sign ensures that the measure (A(w)) is positive for risk-averse individuals.¹²,¹¹

Interpreting Absolute Risk Aversion

Interpreting the value of absolute risk aversion provides insight into an individual's risk tolerance. A higher positive value of (A(w)) indicates a greater degree of absolute risk aversion. This means the individual is more unwilling to bear a fixed-size risk, such as a gamble with a specific monetary loss potential, irrespective of their overall financial standing. Conversely, a lower positive value suggests less absolute risk aversion. A value of zero indicates risk neutrality, where the individual is indifferent to a fair gamble, valuing it solely on its expected monetary outcome.

A key implication of diminishing absolute risk aversion (DARA), often observed in empirical studies and hypothesized by Kenneth Arrow, is that as an individual's wealth increases, their willingness to accept a fixed-size gamble increases.¹⁰ For example, someone with $10,000 might find a $100 gamble significant, but if their wealth grows to $1 million, that same $100 gamble becomes proportionally less important, and they may be more inclined to take it. This phenomenon is closely tied to the concept of diminishing marginal utility, where the subjective value of each additional dollar decreases as total wealth increases.

Hypothetical Example

Consider an investor, Sarah, who is evaluating a lottery ticket that costs $100 and offers a 50% chance to win $300 and a 50% chance to win nothing. The expected monetary value of this lottery is ((0.5 \times $300) + (0.5 \times $0) = $150). The cost is $100, so the net expected gain is $50.

Now, let's consider Sarah's absolute risk aversion.

Scenario 1: Low Wealth
Sarah's current wealth is $5,000. For her, risking $100 might represent a noticeable potential loss, and her high absolute risk aversion at this wealth level means she might be unwilling to buy the ticket despite the positive expected value. She values the utility of her current wealth significantly more than the potential small gain. Her decision making focuses on preserving her existing capital.
Scenario 2: High Wealth
Sarah's wealth increases to $500,000. The same $100 lottery ticket now represents a much smaller proportion of her total wealth. Due to diminishing absolute risk aversion, her willingness to engage in the fixed-size gamble of $100 (which offers a positive expected return) increases. She might now be inclined to buy the ticket, as the potential loss is less impactful on her overall financial well-being. This illustrates how absolute risk aversion can change with wealth, even for the same individual, influencing their approach to portfolio diversification.

Practical Applications

Absolute risk aversion plays a vital role in various areas of finance and economics, influencing theoretical models and practical investment decisions.

Portfolio Theory: In advanced portfolio optimization models beyond basic Modern Portfolio Theory or the Capital Asset Pricing Model, understanding absolute risk aversion helps construct portfolios tailored to individual preferences, especially concerning fixed-size bets or specific liabilities.
Asset Pricing: While risk premium is often discussed in relative terms, understanding absolute risk aversion helps explain why certain assets might command a higher premium if a significant portion of the investor base has high absolute risk aversion to the specific risks those assets carry.
Insurance: The concept is fundamental to the insurance industry. An individual's absolute risk aversion influences how much they are willing to pay for insurance to avoid a specific fixed monetary loss, such as property damage or medical expenses, regardless of their total wealth.
Economic Policy: Policymakers consider aggregate risk aversion when designing social safety nets, unemployment benefits, or disaster relief programs. These programs aim to reduce fixed financial burdens, catering to the population's absolute risk aversion.
Investor Behavior Analysis: Research often investigates how investor attitudes toward risk, including their absolute risk aversion, shift over time and in response to market conditions. For example, analyses of investor sentiment sometimes reveal periods of increased absolute risk aversion following significant market downturns.⁹ Financial regulators, such as the U.S. Securities and Exchange Commission (SEC), emphasize the importance of understanding investor risk profiles for appropriate financial advice and disclosures.⁸,⁷ The Federal Reserve also conducts research to understand how investors differ in their attitudes toward risk, which can influence market dynamics.⁶

Limitations and Criticisms

Despite its theoretical significance, absolute risk aversion, as part of broader utility theory, faces several limitations and criticisms:

Difficulty in Empirical Measurement: Directly measuring an individual's utility function and, consequently, their precise absolute risk aversion coefficient in real-world scenarios is challenging. Surveys and experimental methods often yield inconsistent results.⁵,⁴
Assumption of Rationality: The model assumes rational decision making under uncertainty, which behavioral economics has shown is not always true. Real-world decisions are influenced by cognitive biases, heuristics, and emotions that the mathematical framework of absolute risk aversion does not fully capture.
Independence of Wealth Changes: A core tenet of absolute risk aversion is that the willingness to pay for risk reduction for a fixed amount does not change with wealth. However, this is not always consistent with observed behavior. For instance, some individuals might exhibit constant absolute risk aversion (CARA), meaning their willingness to pay for a fixed risk remains the same regardless of wealth, which is a specific and perhaps unrealistic assumption for all individuals across all wealth levels.³
Focus on Monetary Outcomes: While absolute risk aversion is a monetary measure, it may not fully account for non-monetary utility or disutility from certain risks, such as reputational damage or emotional stress, which do not scale directly with income or wealth.
Context Dependence: An individual's apparent risk aversion might vary depending on the specific context of the gamble or investment opportunity, rather than being a single, stable value that applies universally. The theoretical framework of indifference curves might struggle to fully encapsulate this fluidity. The International Monetary Fund (IMF) has highlighted the complexities and challenges in accurately measuring risk aversion, especially across different demographics and economic contexts.²

Absolute Risk Aversion vs. Relative Risk Aversion

Absolute risk aversion and relative risk aversion are both measures derived from an investor's utility function, but they capture different aspects of risk attitude.

Feature	Absolute Risk Aversion	Relative Risk Aversion
Focus	Willingness to forgo a fixed monetary amount to avoid a fixed monetary risk.	Willingness to forgo a fixed percentage of wealth to avoid a fixed percentage risk.
Impact of Wealth	Typically decreases with increasing wealth (Diminishing Absolute Risk Aversion, DARA). This means wealthier individuals are willing to take on larger fixed monetary gambles.¹	Often assumed to be constant or increasing with wealth (Constant/Increasing Relative Risk Aversion, CRRA/IRRA). This implies that the proportion of wealth allocated to risky assets remains stable or increases as wealth grows.
Formula	(A(w) = - \frac{u''(w)}{u'(w)})	(R(w) = - \frac{w \cdot u''(w)}{u'(w)})
Application	Useful for analyzing risks that are constant in dollar terms (e.g., insurance premiums, small gambles).	More relevant for portfolio allocation decisions where risk is scaled by total wealth (e.g., percentage of assets in stocks).
Interpretation	How much wealth an investor is willing to sacrifice to avoid a fixed dollar amount of risk.	What proportion of wealth an investor is willing to risk for a given percentage return.

The confusion between the two often arises because both describe an individual's attitude towards financial risk using their utility function. However, the critical distinction lies in whether the risk is considered in absolute dollar terms or as a proportion of total wealth. An individual could exhibit decreasing absolute risk aversion while maintaining constant or increasing relative risk aversion.

FAQs

What is the primary difference between absolute and relative risk aversion?

The main difference is in the unit of risk being considered. Absolute risk aversion measures the amount of money an individual would pay to avoid a fixed dollar amount of risk, regardless of their total wealth. Relative risk aversion, on the other hand, measures the amount an individual would pay to avoid a risk that is a fixed percentage of their wealth.

Why does absolute risk aversion typically decrease with wealth?

Absolute risk aversion typically decreases with wealth due to the principle of diminishing marginal utility. As an individual's wealth increases, each additional dollar of wealth provides less additional satisfaction. Therefore, a fixed monetary loss becomes less impactful on their overall utility as they become wealthier, making them more willing to accept that fixed risk.

Is absolute risk aversion used in real-world financial planning?

While the precise mathematical calculation of absolute risk aversion is more common in academic research and advanced economic modeling, its underlying concept influences real-world financial planning. Financial advisors implicitly consider a client's absolute willingness to bear specific dollar-amount losses (e.g., in terms of emergency funds or insurance deductibles) alongside their proportional risk tolerance for portfolio allocation.

Can absolute risk aversion be constant?

Yes, absolute risk aversion can theoretically be constant, which is known as Constant Absolute Risk Aversion (CARA). This implies that an individual's willingness to pay to avoid a fixed monetary risk remains the same regardless of their wealth level. While it simplifies mathematical models, it is generally considered a less realistic assumption for human behavior across significant changes in wealth compared to diminishing absolute risk aversion.

How does absolute risk aversion relate to expected utility theory?

Absolute risk aversion is a direct measure derived from an individual's utility function within the framework of expected utility theory. It quantifies the curvature (concavity) of the utility function, which is what determines an individual's risk-averse preferences. The more concave the utility function, the higher the absolute risk aversion.