Risk aversion coefficient – meaning and formula

Risk aversion coefficient – meaning and formula

When we discussed investors’ risk preferences, we distinguished between risk-averse, risk-neutral, and risk-seeking behavior. We also explained that risk-averse investors expect compensation for bearing risk, which is called a risk premium. But, how do measure risk aversion? The answer is the risk aversion coefficient. It quantifies the degree to which an individual dislikes risk.

In this post, we begin by demonstrating the link between risk aversion and utility function. Then, we introduce the formulas of Arrow-Pratt coefficients of absolute risk aversion and relative risk aversion. Finally, we discuss how economists estimate risk aversion coefficients and what they find as the typical range of estimates.

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Risk aversion and utility function

An individual’s utility function of wealth u(w) measures the utility (or happiness) the individual receives for different levels of wealth w. Of course, we expect the utility to increase with wealth, such that the wealthier the individual gets the happier she becomes.

There is an inherent link between the shape of an individual’s utility function and her risk preferences. This is illustrated in Figure 1 below. The shape of the utility function is concave for a risk-averse person, linear for a risk-neutral one, and convex for a risk-seeking one.

risk aversion utility function
Figure 1: Risk aversion and the shape of the utility function.
A risk-averse person has a concave utility function as in panel (a), a risk-neutral one has a linear utility function as in panel (b), and a risk-seeking one has a convex utility function as in panel (c).

Intuitively, this is because a risk-averse individual prefers the expected value of a gamble to the gamble itself, and that preference implies a concave utility function mathematically.

Absolute risk aversion coefficient

The Arrow-Pratt absolute risk aversion coefficient (ARA) tells us an individual’s degree of risk aversion at a particular level of wealth, hence the adjective “absolute”. Intuitively, a $10,000 gamble would mean different things to the “rich” versus the “poor”. In particular, a wealthy CEO can be expected to be much less averse to such a gamble than, say, a blue-collar worker. In such a scenario, absolute risk aversion decreases with the level of wealth.

The Arrow-Pratt absolute risk aversion coefficient is calculated by dividing the second derivative of the utility function u”(w) by its first derivative u'(w) as below.

arrow pratt absolute risk aversion coefficient
Absolute risk aversion coefficient

Note that the negative sign is used to obtain a positive value for ARA as the second derivative is negative (u”(w)<0).


Example

Let’s assume that an individual has a power utility function as follows:

u(w) = −w−1

In this case, the first and second derivatives of u(w) are as follows: u'(w) = w−2 and u”(w) = 2w−3. Then, this individual’s absolute risk aversion coefficient is:

ARA = − (−2w−3) / (w−2) = 2 / w

Notice that the individual has decreasing absolute risk aversion: ARA decreases as w gets larger.

Relative risk aversion coefficient

The Arrow-Pratt relative risk aversion coefficient (RRA) aims to capture the degree of risk aversion given a proportional change in wealth (e.g., 5% increase in wealth). As such, it is closely related to the absolute risk aversion coefficient and can be calculated by multiplying ARA by the level of wealth w as follows:

arrow-pratt relative risk aversion coefficient
Relative risk aversion coefficient

Constant relative risk aversion would imply that an individual has the same degres of aversion to, say, a 10% change in wealth, regardless of how rich or poor that person is.


Example

When discussing the absolute risk aversion coefficient, we showed that for an individual who has the utility function u(w) = −w−1, we have ARA = 2 / w. Then, the relative risk aversion coefficient for this individual is:

RRA = w ARA = w (2 / w) = 2

This means that the individual exhibits constant relative risk aversion as her RRA is always equal to 2 regardless of her wealth.

How to measure risk aversion?

Economists use various methods to estimate risk aversion coefficients. For example, an early study by Friend and Blume (1975) (see the full reference at the end) relies on cross-sectional survey data on household asset holdings. They find that the relative risk aversion coefficient is “well in excess of one and probably in excess of two” (p. 900).

Another common approach to measuring risk aversion is based on a consumption-based CAPM. For example, adopting such an approach Hansen and Singleton (1983) find that the coefficient of relative risk aversion lies between 0 and 2.

More recently, Chetty (2006) uses data on labor supply behavior to measure risk aversion. He argues that based on existing evidence of the effects of wage changes on labor supply, the relative risk aversion coefficient should be less than 2.

Risk aversion coefficient range

There are a number of studies that estimate the relative risk aversion coefficient, and results vary across these studies.

A review paper by Gándelman and Hernández-Murillo (2015) (full reference at the end) notes a wide range from 0.2 to 10 or higher. However, they add that “the most widely accepted measures lie between 1 and 3” (p. 53). According to their analysis, the coefficient of relative risk aversion varies closely around the value of 1 across the 75 countries in their sample.


RECAP

In this post, we have explained the meaning of “risk aversion coefficient”. We have also elaborated on the link between risk aversion and the utility of wealth. In particular, we have shown that a risk-averse individual has a concave utility function. Then, we moved on to providing formulas for Arrow-Pratt measures of absolute risk aversion and relative risk aversion. After that, we discussed how economists measure risk aversion in practice. We wrapped our discussion by pointing out that relative risk aversion coefficient estimates range from 0.2 to 10 or higher across studies with estimates between 1 and 3 generally accepted as the most typical.

Further reading on risk aversion coefficients

Chetty (2006) “A New Method of Estimating Risk AversionThe American Economic Review, Vol. 96 (5), pp. 1821-1834.

Friend and Blume (1975) “The demand for risky assetsThe American Economic Review, Vol. 65 (5), pp. 900-922.

Hansen and Singleton (1983) “Stochastic Consumption, Risk Aversion, and the Temporal Behavior of Asset ReturnsJournal of Political Economy, Vol. 91 (2), pp. 249-265.

Gándelman and Hernández-Murillo (2015) “Risk aversion at the country levelFederal Reserve Bank of St. Louis Review, First Quarter 2015, 97(1), pp. 53-66.

WHAT IS NEXT?

This post is part of our free course on investments. The previous post in this course focuses on investors’ risk preferences, explaining the differences between risk-averse, risk-neutral, and risk-seeking behavior. In the next post, we elaborate on an important implication of risk aversion: risk premium.

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