Constant relative risk aversion: Meaning, Criticisms & Real-World Uses

What Is Constant Relative Risk Aversion?

Constant relative risk aversion (CRRA) is a concept within financial economics and portfolio theory that describes a specific type of risk aversion where an investor's willingness to take on risk remains constant as their wealth changes. This means that if an individual's wealth doubles, they will choose to invest the same proportion of their wealth in risky assets, rather than the same absolute dollar amount. It is a fundamental assumption in many economic and financial models, particularly those related to investment decisions and optimal portfolio allocation.

An investor exhibiting constant relative risk aversion values a percentage change in wealth consistently, regardless of their current financial standing. For instance, the disutility experienced from losing 10% of one's wealth is the same whether the initial wealth is $10,000 or $1,000,000. This characteristic makes the CRRA utility function particularly useful for modeling long-term financial behavior where an individual's wealth is expected to fluctuate significantly over time.

History and Origin

The foundational ideas underpinning constant relative risk aversion can be traced back to the broader development of expected utility theory. The concept of utility and diminishing marginal utility was first formally introduced by Daniel Bernoulli in his 1738 paper, "Exposition of a New Theory on the Measurement of Risk." Bernoulli used this framework to address the St. Petersburg Paradox, proposing that individuals do not evaluate risky outcomes based solely on their monetary expected value, but rather on the utility derived from those outcomes¹³.

While Bernoulli laid the groundwork, the modern axiomatic formulation of expected utility theory, which provides the rigorous basis for concepts like constant relative risk aversion, was developed by John von Neumann and Oskar Morgenstern in their 1944 book, Theory of Games and Economic Behavior¹². Over time, economists and financial theorists adopted utility functions, including those with constant relative risk aversion, to model individual preferences and decision-making under uncertainty, forming a cornerstone of classical finance.

Key Takeaways

Constant relative risk aversion (CRRA) implies that an investor maintains a consistent proportion of risky assets in their portfolio, irrespective of their total wealth.
The CRRA framework is widely used in financial models due to its simplicity and its ability to explain portfolio allocation behavior over time.
A key feature of CRRA is that the percentage change in utility for a given percentage change in wealth is constant.
Utility functions with CRRA properties are concave, reflecting the diminishing marginal utility of wealth.
The coefficient of relative risk aversion quantifies the degree of an investor's risk aversion within this framework.

Formula and Calculation

The utility function representing constant relative risk aversion typically takes the following form:

U(W) = \begin{cases} \frac{W^{1-\gamma}}{1-\gamma} & \text{if } \gamma \neq 1 \\ \ln(W) & \text{if } \gamma = 1 \end{cases}

Where:

(U(W)) represents the utility function derived from wealth (W).
(W) is the investor's total wealth.
(\gamma) (gamma) is the coefficient of relative risk aversion, a positive constant that measures the degree of risk aversion.

The relative risk aversion coefficient itself is defined as:

RRA(W) = -\frac{W \cdot U''(W)}{U'(W)}

For the CRRA utility function, calculating this expression yields (\gamma), demonstrating that the relative risk aversion remains constant, irrespective of the level of (W).

Interpreting the Constant Relative Risk Aversion

Interpreting the constant relative risk aversion coefficient, (\gamma), provides insight into an investor's attitude towards risk.

(\gamma = 0): This indicates risk neutrality. The investor cares only about the expected monetary value of outcomes, regardless of risk.
(\gamma = 1): This corresponds to a logarithmic utility function ((U(W) = \ln(W))). It implies moderate risk aversion where the investor's willingness to take risk scales directly with their wealth. For example, if wealth doubles, the investor will double the absolute amount invested in risky assets, keeping the proportion constant.
(\gamma > 1): This signifies higher degrees of risk aversion. As (\gamma) increases, the investor becomes more averse to risk. A higher (\gamma) suggests that the investor requires a greater risk premium to undertake a given level of risk.
(\gamma < 1) (but greater than 0): This indicates lower degrees of risk aversion than logarithmic utility.

The assumption of constant relative risk aversion implies that an investor's risk tolerance is directly proportional to their wealth. This proportionality simplifies many financial models, making the CRRA utility function a workhorse in economic analysis.

Hypothetical Example

Consider an investor, Sarah, who exhibits constant relative risk aversion with a coefficient (\gamma = 2).

Scenario 1: Sarah's initial wealth is $100,000.
Sarah considers an investment that has a 50% chance of gaining $20,000 and a 50% chance of losing $10,000.

Scenario 2: Sarah's wealth doubles to $200,000.
Sarah is now considering an investment that has a 50% chance of gaining $40,000 (20% of $200,000) and a 50% chance of losing $20,000 (10% of $200,000).

Because Sarah has constant relative risk aversion, her decision regarding the percentage of wealth to expose to risk remains consistent. If she was willing to risk losing 10% of her initial wealth in Scenario 1, she would be willing to risk losing 10% of her doubled wealth in Scenario 2. The utility impact of a 10% loss or a 20% gain is the same for her at both wealth levels, even though the absolute dollar amounts involved are different. This consistency in proportionate risk-taking is a defining characteristic of constant relative risk aversion and influences her overall financial planning.

Practical Applications

Constant relative risk aversion is a widely used concept across various areas of finance and economics:

Portfolio Choice and Asset Allocation: The CRRA utility function is fundamental in determining optimal asset allocation strategies for investors. It implies that the optimal proportion of wealth allocated to risky assets remains constant over time, regardless of how an investor's total wealth changes¹⁰, ¹¹. This makes it a popular choice for long-term investment models.
Intertemporal Consumption and Saving: In macroeconomic models, CRRA preferences are used to analyze how individuals make decisions about consumption and saving over their lifetime. The constant relative risk aversion assumption helps model how consumption patterns respond to changes in income and wealth, maintaining a consistent attitude towards risk in these decisions⁹.
Asset Pricing Models: Many theoretical asset pricing models, such as the Capital Asset Pricing Model (CAPM) and consumption-based asset pricing models, often incorporate CRRA preferences to explain observed phenomena like the equity premium puzzle.
Policy Analysis: Government agencies and policymakers use models based on CRRA to evaluate the welfare implications of various economic policies, particularly those related to taxation, social security, and environmental regulations. For instance, the value of a statistical life (VSL) used in cost-benefit analysis often relies on assumptions about individuals' CRRA⁸.
Behavioral Finance Research: While classical finance uses CRRA extensively, the field of behavioral finance often contrasts it with alternative preference models like Prospect Theory. However, some experimental studies have found strong support for CRRA preferences in certain contexts of investment decisions ⁷.

Limitations and Criticisms

Despite its widespread use, constant relative risk aversion has several limitations and has faced criticism:

Unrealistic for Extreme Wealth Levels: The CRRA function assumes that individuals maintain the same relative risk attitude across all wealth levels, from very low to extremely high. Critics argue that this may not hold true in reality; an individual's behavior might change drastically if they are near poverty versus being incredibly wealthy. For instance, some research suggests that CRRA utility with coefficients outside a narrow range (e.g., 0.75-1.15) can lead to paradoxical choices that most individuals would not make⁶.
Independence from Investment Horizon: In standard models with constant investment opportunities, CRRA implies that the optimal asset allocation is independent of the investment horizon. This "myopic" behavior is often challenged by empirical evidence and behavioral observations, where investors might adjust their risk exposure based on how long they plan to invest⁴, ⁵.
Inability to Explain Certain Puzzles: While used in asset pricing, CRRA models sometimes struggle to fully explain observed market phenomena like the equity premium puzzle without requiring unrealistically high coefficients of risk aversion³.
Simplification of Reality: As a theoretical construct, CRRA simplifies complex human decision-making. It does not account for psychological biases, cognitive errors, or varying states of mind that can influence an individual's actual risk aversion and subsequent choices¹, ².

Constant Relative Risk Aversion vs. Constant Absolute Risk Aversion

Constant relative risk aversion (CRRA) and constant absolute risk aversion (CARA) are both measures of risk aversion used in utility function analysis, but they differ in how an investor's risk attitude scales with their wealth.

Feature	Constant Relative Risk Aversion (CRRA)	Constant Absolute Risk Aversion (CARA)
Definition	Risk attitude is constant relative to wealth.	Risk attitude is constant in absolute terms, regardless of wealth.
Utility Function	(U(W) = \frac{W^{1-\gamma}}{1-\gamma}) or (U(W) = \ln(W))	(U(W) = -\frac{1}{\alpha}e^{-\alpha W})
Coefficient	(\gamma) (gamma) - constant	(\alpha) (alpha) - constant
Risk-Taking Behavior	Invests a constant proportion of wealth in risky assets.	Invests a constant absolute amount in risky assets.
Implication	As wealth increases, the absolute amount invested in risky assets increases proportionately.	As wealth increases, the absolute amount invested in risky assets remains the same.
Context	Often used in models where percentage returns and proportionate wealth changes are key, such as long-term portfolio theory.	More applicable when fixed monetary outcomes are considered, regardless of initial capital.

The main point of confusion often arises because both imply "constant" risk aversion. However, CRRA means constant relative to wealth (percentage), while constant absolute risk aversion (CARA) means constant in absolute dollar terms. A CARA investor with more wealth would invest a smaller proportion of their wealth in risky assets, whereas a CRRA investor would maintain the same proportion.

FAQs

What does a high constant relative risk aversion coefficient mean?

A high constant relative risk aversion coefficient ((\gamma)) indicates that an investor is highly risk-averse. They require a significantly larger expected return for taking on a given level of risk, and they will be more sensitive to percentage losses in their wealth.

Why is constant relative risk aversion important in finance?

Constant relative risk aversion is crucial in finance because it provides a tractable and consistent framework for modeling investor behavior in various theoretical and practical applications, particularly in portfolio theory, asset pricing, and intertemporal investment decisions. It simplifies analysis by assuming a stable risk preference across different wealth levels.

Does constant relative risk aversion account for behavioral biases?

Generally, traditional models using constant relative risk aversion do not directly account for behavioral finance biases like loss aversion, framing effects, or mental accounting. It assumes rational decision-making based purely on expected utility maximization. More recent research in behavioral economics attempts to integrate these biases with classical utility theories.

Can individuals have different constant relative risk aversion coefficients?

Yes, different individuals can and typically do have different constant relative risk aversion coefficients. These coefficients reflect their unique risk tolerance and preferences. Financial advisors often try to ascertain an individual's effective risk aversion level when developing a financial planning strategy.