Relative risk aversion: Meaning, Criticisms & Real-World Uses

What Is Relative Risk Aversion?

Relative risk aversion (RRA) is a concept within behavioral finance and utility theory that measures how an individual's willingness to take on risk changes as their wealth or income changes. It quantifies the degree to which an investor adjusts their risky asset allocation in response to proportional changes in their wealth. A high relative risk aversion indicates that an individual becomes significantly more risk-averse as their wealth increases, choosing to reduce their exposure to risk more drastically. Conversely, a low relative risk aversion suggests that their risk-taking behavior remains relatively stable regardless of changes in their wealth.

History and Origin

The foundational ideas behind relative risk aversion trace back to the 18th century, particularly with the work of Daniel Bernoulli. In 1738, Bernoulli published his solution to the St. Petersburg Paradox, proposing that individuals evaluate gambles based on the "moral expectation" (utility) of the outcomes, rather than their mathematical expected value. He argued for the concept of diminishing marginal utility of wealth, meaning that each additional unit of wealth provides less satisfaction than the previous one.⁹, ¹⁰ This groundbreaking idea laid the groundwork for modern utility theory, which forms the basis for understanding risk aversion.⁸

Later, economists like Kenneth Arrow and John W. Pratt further developed the formal measures of risk aversion in the 20th century, distinguishing between absolute and relative measures. The concept of relative risk aversion specifically gained prominence in economic models, particularly those dealing with consumption and saving decisions over time, as it captures how an individual's risk-taking scales with their overall financial position.

Key Takeaways

Relative risk aversion (RRA) measures how an investor's willingness to take on risk changes with their wealth.
It is a key concept in utility theory and portfolio theory.
A higher RRA means an investor becomes more cautious as wealth increases, while a lower RRA suggests more consistent risk-taking.
RRA influences investment decisions, portfolio allocation, and savings behavior.
It is distinct from absolute risk aversion, which measures risk aversion independent of wealth.

Formula and Calculation

Relative risk aversion is often characterized by the coefficient of relative risk aversion, denoted as $R(w)$ or $\gamma$. For a utility function $U(w)$, where $w$ represents wealth, the coefficient of relative risk aversion is given by:

R(w) = - \frac{w \cdot U''(w)}{U'(w)}

Where:

$w$ = current wealth
$U'(w)$ = first derivative of the utility function with respect to wealth (marginal utility of wealth)
$U''(w)$ = second derivative of the utility function with respect to wealth (rate of change of marginal utility)

A common functional form for utility that exhibits constant relative risk aversion (CRRA) is the power utility function:

U(w) = \frac{w^{1-\gamma}}{1-\gamma} \quad \text{for } \gamma \neq 1

and for $\gamma = 1$:

U(w) = \ln(w)

For these utility functions, the coefficient of relative risk aversion is simply $\gamma$, a constant value across all wealth levels. This characteristic makes the CRRA utility function particularly useful in economic and financial modeling, as it simplifies analysis and provides a consistent measure of risk preference.

Interpreting the Relative Risk Aversion

The interpretation of relative risk aversion centers on how an investor's allocation to risky assets responds to changes in their total wealth. If the coefficient of relative risk aversion is greater than 1, an investor will reduce the proportion of their portfolio allocated to risky assets as their wealth increases. If the coefficient is less than 1, they will increase their allocation to risky assets. If the coefficient is exactly 1, the proportion of wealth invested in risky assets remains constant regardless of wealth changes; this is characteristic of a logarithmic utility function.

Understanding an individual's relative risk aversion is crucial for financial advisors in tailoring appropriate investment strategies. For instance, an individual with high relative risk aversion might prefer a more conservative portfolio with a larger allocation to fixed-income securities as their wealth grows, prioritizing capital preservation over higher potential returns. Conversely, someone with lower relative risk aversion might maintain a more aggressive portfolio with a significant allocation to equities even as their wealth accumulates.

Hypothetical Example

Consider two investors, Alice and Bob, both with an initial wealth of $100,000. Alice has a high relative risk aversion coefficient of 3, while Bob has a constant relative risk aversion coefficient of 1 (logarithmic utility).

Suppose both investors' wealth increases to $200,000.

Alice (High RRA = 3): With her wealth doubling, Alice, due to her high relative risk aversion, would likely rebalance her portfolio to significantly reduce her exposure to risky assets. If she initially had 60% in stocks and 40% in bonds, she might adjust to 30% in stocks and 70% in bonds. Her increased wealth makes her even more sensitive to potential losses, leading her to seek greater security. This shift reflects her desire to preserve her accumulated wealth.
Bob (Constant RRA = 1): Bob, with a constant relative risk aversion of 1, would maintain the same proportion of his wealth in risky assets. If he started with 60% in stocks and 40% in bonds, he would aim to keep that 60/40 allocation even with $200,000. While the dollar amount invested in stocks would double (from $60,000 to $120,000), the percentage of his portfolio in stocks remains consistent, demonstrating that his risk preference is proportional to his wealth.

This example highlights how different relative risk aversion levels lead to distinct approaches in managing a growing investment portfolio.

Practical Applications

Relative risk aversion is a fundamental concept with widespread applications in finance and economics. In asset allocation, understanding an investor's relative risk aversion helps determine the optimal mix of risky and risk-free assets in their portfolio. For instance, models for life-cycle investing often incorporate assumptions about how relative risk aversion changes with age and accumulating wealth.

Furthermore, it plays a critical role in academic research related to consumption and savings decisions. For example, some studies investigate how precautionary savings are influenced by an individual's relative risk aversion in the face of income uncertainty.⁷ Central banks and other financial institutions also monitor investor sentiment and risk appetite, which can be indirectly linked to the collective relative risk aversion of market participants. For instance, a "risk-off" mood, as reported by outlets like Reuters, suggests a general increase in risk aversion among investors, leading them to seek safer assets.⁵, ⁶

Limitations and Criticisms

While relative risk aversion is a powerful theoretical construct, it faces certain limitations and criticisms in practical application. One significant challenge lies in its empirical estimation. Accurately measuring an individual's utility function and, consequently, their relative risk aversion, can be difficult due to the subjective nature of preferences. Real-world behavior can sometimes deviate from what theoretical models of relative risk aversion predict, influenced by psychological biases or incomplete information.

Additionally, the assumption of constant relative risk aversion (CRRA), while mathematically convenient, may not always hold true across all wealth levels or life stages for every individual. Some studies suggest that risk aversion might not be constant but could vary depending on extreme gains or losses, or even be influenced by framing effects. For instance, the Bogleheads community often discusses how investors may overestimate their risk tolerance during bull markets and become more risk-averse during downturns, which might indicate a non-constant relative risk aversion in practice.³, ⁴ Furthermore, certain financial phenomena, such as the equity premium puzzle, have prompted researchers to explore alternative utility specifications or behavioral explanations, as standard models with reasonable levels of relative risk aversion often struggle to fully explain observed market returns.

Relative Risk Aversion vs. Absolute Risk Aversion

Relative risk aversion and absolute risk aversion are both measures of an individual's attitude toward financial risk, but they differ in how they relate to the level of wealth.

Feature	Relative Risk Aversion (RRA)	Absolute Risk Aversion (ARA)
Definition	Measures how an investor's willingness to take on risk changes as their wealth or income changes proportionally.	Measures how an investor's willingness to take on risk changes in response to an absolute change in wealth.
Focus	The proportion of wealth allocated to risky assets.	The absolute dollar amount of wealth allocated to risky assets.
Wealth Dependence	Varies with wealth, influencing the proportion of risky assets.	Can vary with wealth, influencing the dollar amount of risky assets.
Formula (Utility)	$R(w) = - \frac{w \cdot U''(w)}{U'(w)}$	$A(w) = - \frac{U''(w)}{U'(w)}$
Implication	For CRRA, the percentage of wealth in risky assets is constant.	For CARA, the dollar amount in risky assets is constant.

While relative risk aversion considers how risk-taking scales with overall wealth, absolute risk aversion examines how an investor's risk preference changes for a fixed amount of money, irrespective of their total wealth. Both concepts are crucial for understanding an individual's risk profile, but relative risk aversion is often more relevant for long-term investment planning, as wealth levels tend to fluctuate over time.²

FAQs

What does a high relative risk aversion mean?

A high relative risk aversion means that as an individual's wealth increases, they become significantly more cautious and will reduce the proportion of their portfolio invested in risky assets. They prioritize the preservation of their existing wealth over seeking higher returns from riskier ventures.

How does relative risk aversion influence investment decisions?

Relative risk aversion directly influences investment decisions by guiding asset allocation. Investors with high RRA tend to shift towards more conservative investments as their wealth grows, while those with low RRA might maintain or even increase their proportional exposure to riskier assets. This helps determine the balance between growth stocks and value stocks, or bonds and equities, in a portfolio.

Is relative risk aversion constant for everyone?

No, relative risk aversion is generally not constant for everyone and can vary significantly across individuals. While some theoretical models assume constant relative risk aversion (CRRA) for simplicity, an individual's actual relative risk aversion can be influenced by factors such as age, financial goals, investment experience, and current economic conditions.

Can relative risk aversion be negative?

While theoretically possible, a negative relative risk aversion implies that an individual is "risk-loving," meaning they would prefer a gamble with a lower expected return if it offered higher risk. In finance, most individuals are assumed to be risk-averse, meaning their relative risk aversion coefficient is positive.

How does relative risk aversion differ from risk capacity?

Relative risk aversion is a measure of an individual's willingness to take on risk based on their preferences and how that willingness changes with wealth. Risk capacity, on the other hand, is an objective measure of an individual's financial ability to absorb potential losses without jeopardizing their financial goals. Both are important for constructing a suitable financial plan.¹