Constant absolute risk aversion

What Is Constant Absolute Risk Aversion?

Constant absolute risk aversion (CARA) is a concept within expected utility theory, a branch of behavioral finance and economics, that describes a specific characteristic of an individual's risk aversion. It implies that the amount of wealth an individual is willing to put at risk for a given gamble remains constant, regardless of their total wealth. In essence, an individual exhibiting constant absolute risk aversion would be willing to pay the same fixed amount to avoid a specific fair gamble, whether they are poor or rich. This contrasts with more commonly observed behaviors where risk aversion often decreases as wealth increases.

History and Origin

The foundational concepts underpinning constant absolute risk aversion emerged from the broader development of utility function theory in the mid-20th century. Economists Kenneth Arrow and John W. Pratt independently formalized measures of risk aversion in the early 1960s, which became known as the Arrow-Pratt measures. Their work provided a rigorous mathematical framework for quantifying how individuals value uncertain outcomes, based on the curvature of their utility functions. Constant absolute risk aversion represents a specific type of utility function where this measure remains unchanged across different levels of wealth. Their contributions, such as those detailed in academic works like the "Arrow-Pratt Measures of Risk Aversion: The Multivariate Case," provided the analytical tools to explore such specific risk preference characteristics.⁵

Key Takeaways

Constant absolute risk aversion (CARA) implies that an individual's willingness to accept or avoid a fixed-size gamble does not change with their total wealth.
It is a characteristic derived from a specific form of utility function, often represented by an exponential function.
While mathematically tractable, constant absolute risk aversion is often considered less realistic for most individuals compared to other risk aversion profiles, as it suggests the same financial sacrifice for a gamble regardless of one's economic standing.
CARA preferences simplify some economic models, making them useful for theoretical analysis in areas like portfolio optimization and investment decisions.

Formula and Calculation

The most common utility function that exhibits constant absolute risk aversion is the exponential utility function. The Arrow-Pratt measure of absolute risk aversion, denoted as (A(w)), is defined as:

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

Where:

(u(w)) is the utility function of wealth (w).
(u'(w)) is the first derivative of the utility function with respect to wealth, representing marginal utility.
(u''(w)) is the second derivative of the utility function with respect to wealth, representing the rate of change of marginal utility.

For a utility function to exhibit constant absolute risk aversion, this value (A(w)) must be a positive constant, typically denoted as (\alpha). A common form of a CARA utility function is:

u(w) = -e^{-\alpha w} \quad \text{or} \quad u(w) = 1 - e^{-\alpha w}

Where:

(e) is the base of the natural logarithm.
(\alpha) (alpha) is a positive constant that represents the coefficient of absolute risk aversion. A higher (\alpha) indicates greater risk aversion.

Let's verify this using the formula:
If (u(w) = -e^{-\alpha w}):
(u'(w) = \alpha e^{-\alpha w})
(u''(w) = -\alpha^{2 e}{-\alpha w})

Then, (A(w) = - \frac{-\alpha^{2 e}{-\alpha w}}{\alpha e^{{-\alpha w}} = \frac{\alpha}2}{\alpha} = \alpha).

Since (A(w) = \alpha) (a constant), this utility function indeed exhibits constant absolute risk aversion.

Interpreting Constant Absolute Risk Aversion

Interpreting constant absolute risk aversion involves understanding how an economic agent with such preferences would behave in the face of uncertainty. An individual with CARA preferences would view a gamble involving a fixed monetary amount (e.g., winning or losing $1,000) with the same level of apprehension, regardless of whether their starting wealth is $10,000 or $1,000,000. This implies that the maximum amount they would be willing to pay to avoid that gamble, or the minimum amount they would accept to undertake it, would remain constant.

In practical terms, this suggests that as a person's wealth increases, the proportion of their wealth they are willing to expose to risk would decrease. For example, if they are willing to risk $1,000 when they have $10,000, that's 10% of their wealth. If their wealth grows to $100,000, risking $1,000 is only 1% of their wealth, implying a diminishing proportion of risky investment decisions as wealth increases.

Hypothetical Example

Consider an investor, Alice, who exhibits constant absolute risk aversion with a coefficient (\alpha = 0.0005). Her utility function is (u(w) = 1 - e^{-0.0005w}).

Scenario 1: Alice's initial wealth is $50,000.
Alice is offered a gamble: she can win $1,000 with 50% probability or lose $1,000 with 50% probability.
Her current utility: (u(50,000) = 1 - e^{-0.0005 \times 50,000} = 1 - e^{-25} \approx 1 - 1.39 \times 10^{-11}) (very close to 1).

If she takes the gamble:

Utility if she wins: (u(51,000) = 1 - e^{{-0.0005 \times 51,000} = 1 - e}{-25.5})
Utility if she loses: (u(49,000) = 1 - e^{{-0.0005 \times 49,000} = 1 - e}{-24.5})

Expected utility of the gamble: (0.5 \times (1 - e^{{-25.5}) + 0.5 \times (1 - e}{-24.5}))

Scenario 2: Alice's initial wealth increases to $500,000.
She is offered the same gamble: win or lose $1,000 with 50% probability.
Her constant absolute risk aversion implies that she would still be willing to pay the same absolute dollar amount to avoid this gamble, or accept the same absolute dollar amount as a risk premium to undertake it, even though her wealth has grown tenfold. The amount of risk, in absolute terms (the $1,000 swing), feels the same to her despite her increased financial cushion. This investor's approach to risk, rather than their specific asset allocation percentages, remains fixed in terms of absolute dollar amounts.

Practical Applications

While constant absolute risk aversion may not perfectly describe all individuals, it serves as a valuable assumption in certain theoretical and practical financial models. It simplifies the analysis of investment decisions and financial planning because the optimal absolute amount invested in a risky asset remains constant, regardless of changes in wealth. This property can be useful in:

Deriving Closed-Form Solutions: In complex economic models, assuming CARA preferences allows for more straightforward mathematical solutions, providing clear insights into the qualitative behavior of economic agents under uncertainty.
Optimal Saving and Consumption: Models of intertemporal choice sometimes use CARA utility to analyze how individuals choose between current consumption and saving for future uncertain outcomes.
Insurance Theory: In insurance markets, constant absolute risk aversion can help model an individual's demand for insurance, as it implies a constant willingness to pay for a fixed reduction in risk.
Human Capital Investment: In contexts involving human capital and lifetime earnings, CARA can be used to model how individuals make decisions about education, career paths, and retirement planning, particularly in relation to their willingness to take career-related risks.
Market Behavior Analysis: While individuals may not perfectly adhere to CARA, aggregate market behavior can sometimes be approximated or studied under such assumptions to understand broad trends in risk-taking. For instance, observations in financial markets, such as those in the UK, suggest that a widespread "safety-first instinct" among investors, reflecting a low emotional capacity for risk, can lead to under-investment in riskier assets, aligning with certain aspects of absolute risk aversion.⁴

Limitations and Criticisms

Despite its analytical tractability, constant absolute risk aversion faces significant limitations and criticisms regarding its realism in describing actual human behavior.

Unrealistic Wealth Effects: The most prominent criticism is its implication that the absolute amount an individual is willing to put at risk is independent of their wealth. In reality, most people's risk tolerance tends to increase with wealth. A millionaire is generally more willing to risk $1,000 on a gamble than someone with only $10,000. This suggests that a constant absolute amount of risk is not realistic. Empirical evidence and behavioral observations often indicate that individuals exhibit decreasing absolute risk aversion (DARA), meaning they are willing to take on a larger absolute dollar amount of risk as their wealth increases.³
Inconsistent with Portfolio Allocation: If an individual's wealth increases significantly, a strict CARA preference would imply they invest a smaller percentage of their wealth in risky assets. For example, if someone always risks $1,000, then at $10,000 wealth, they risk 10%, but at $100,000, they risk only 1%. This contrasts with observed behavior where individuals often maintain a relatively constant percentage of their portfolio in risky assets, particularly in the context of diversification and long-term capital accumulation.
Behavioral Anomalies: CARA, like other traditional expected utility theory models, struggles to explain various behavioral anomalies observed in finance, such as the disposition effect (the tendency to sell winners too early and hold losers too long) or phenomena related to loss aversion. Academic discussions highlight issues with expected utility as a purely descriptive model of choice, urging economists to explore alternative models that better capture the nuances of risk aversion.²
Ignores Scale: The model implicitly assumes that the scale of a gamble doesn't influence the decision beyond the absolute dollar amount. However, for many investors, the relative size of a potential loss compared to their total wealth is a crucial factor. This limitation leads many financial practitioners to focus more on relative risk measures.

Constant Absolute Risk Aversion vs. Constant Relative Risk Aversion

Constant absolute risk aversion (CARA) and constant relative risk aversion (CRRA) are two distinct assumptions about an individual's risk tolerance that describe how their utility from wealth changes. The key difference lies in how risk-taking behavior scales with wealth:

Feature	Constant Absolute Risk Aversion (CARA)	Constant Relative Risk Aversion (CRRA)
Definition	The absolute amount of wealth an individual is willing to put at risk is constant, regardless of total wealth.	The proportion (percentage) of wealth an individual is willing to put at risk is constant, regardless of total wealth.
Formula	Absolute Risk Aversion coefficient (A(w) = \alpha) (a constant)	Relative Risk Aversion coefficient (R(w) = \gamma) (a constant)
Utility Function	Exponential utility: (u(w) = -e^{{-\alpha w}) or (u(w) = 1 - e}{-\alpha w})	Power utility: (u(w) = \frac{w^{1-\gamma}}{1-\gamma}) (for (\gamma \ne 1)), or Logarithmic utility: (u(w) = \ln(w)) (for (\gamma = 1))
Implication for Risky Investment	As wealth increases, the percentage of wealth invested in risky assets decreases.	As wealth increases, the percentage of wealth invested in risky assets remains constant.
Realism	Generally considered less realistic for most individuals across a wide range of wealth levels.	Often considered more realistic as it aligns with the observation that people tend to maintain a consistent proportion of their portfolio in risky assets as their wealth changes.

While CARA is mathematically convenient, CRRA is often preferred in models aiming for greater psychological realism, as it better reflects how many investors approach their portfolio optimization as their financial situation evolves. For example, a common approach in asset allocation is to maintain a fixed percentage of stocks and bonds, which aligns with CRRA behavior.¹

FAQs

Why is constant absolute risk aversion considered unrealistic?

Constant absolute risk aversion is often considered unrealistic because it suggests that an individual would be willing to risk the same fixed dollar amount on a gamble, whether they have $10,000 or $1,000,000. In reality, most people become more willing to take larger absolute risks (though perhaps not larger proportional risks) as their total wealth increases, as the impact of a fixed loss is less severe.

How does constant absolute risk aversion affect portfolio decisions?

With constant absolute risk aversion, as an investor's wealth grows, the proportion of their portfolio allocated to risky assets would tend to decrease. This is because the absolute dollar amount they are willing to risk remains constant, making that fixed amount a smaller percentage of their increasing total wealth. This is a key consideration in investment decisions.

What type of utility function exhibits constant absolute risk aversion?

The exponential utility function is the most common form that exhibits constant absolute risk aversion. It is characterized by its property that the Arrow-Pratt measure of absolute risk aversion yields a constant value, regardless of the individual's current wealth.

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

While the strict assumption of constant absolute risk aversion is less common in direct financial planning applications due to its unrealistic implications for wealth effects, the underlying mathematical framework of utility theory from which it derives is fundamental. Financial advisors typically assess an individual's risk tolerance using more flexible approaches that allow for changes in risk preference as wealth and life circumstances evolve.