Arrow–Pratt Measures of Risk Aversion

The Arrow–Pratt measures of risk aversion are two local indices of the curvature of a von Neumann–Morgenstern utility function that quantify how strongly a decision maker dislikes uncertainty in wealth. They were developed independently by John W. Pratt in the 1964 Econometrica paper "Risk Aversion in the Small and in the Large" and by Kenneth J. Arrow in his 1965 Yrjö Jahnsson lectures on the theory of risk-bearing. The coefficient of absolute risk aversion, A(w) = −u″(w) / u′(w), and the coefficient of relative risk aversion, R(w) = w A(w), allow economists to compare risk attitudes across individuals and wealth levels, and to derive testable predictions in finance, insurance, and macroeconomics.

Under expected utility theory, an agent with utility function u over wealth w is risk averse if and only if u is concave, so that the agent prefers receiving the expected value of a gamble for certain to facing the gamble itself. Concavity, equivalent to u″ < 0 for twice-differentiable functions, delivers a qualitative classification but not a quantitative one. The second derivative alone is not a satisfactory measure, because expected utility representations are unique only up to positive affine transformations a + b u with b > 0, and u″ changes with the scale b. A meaningful index must be invariant to such transformations.

The ratio −u″ / u′ satisfies this requirement, since both numerator and denominator scale by b. It is also the natural quantity: it measures the rate at which marginal utility falls in proportion to its level, and it connects to the willingness to pay for insurance, as the next section shows. Related to marginal utility, the measure captures how quickly the value of an additional unit of wealth declines.

Pratt defined the risk premium π for a gamble with random payoff z̃ of mean zero as the amount satisfying E[u(w + z̃)] = u(w − π), so π is the maximum sure payment that the agent would give up to avoid the gamble. A Taylor expansion of both sides around w gives, for small risks,

π ≈ ½ σ_z² A(w),

where σ_z² is the variance of the gamble. The risk premium is thus proportional to the variance and to the local absolute risk aversion. Pratt also proved a comparison theorem: for two agents with utility functions u₁ and u₂, the following are equivalent: A₁(w) ≥ A₂(w) for all w; agent 1 has a larger risk premium for every gamble at every wealth; and u₁ is a concave transformation of u₂. The theorem converts a statement about preferences over all gambles into one about a single function of wealth.

The relative measure applies when gambles are proportional to wealth. For a multiplicative risk of small variance, the premium as a fraction of wealth is approximately ½ σ² R(w), which makes R a dimensionless number comparable across individuals with different wealth levels and across currencies.

Different assumptions about how A and R vary with wealth generate standard utility classes.

• Constant absolute risk aversion (CARA): u(w) = −e^−aw/a, with A(w) = a. The risk premium on a given gamble is independent of wealth, so wealth effects on risky holdings vanish. Together with normally distributed returns, this utility yields the mean–variance criterion in closed form.
• Constant relative risk aversion (CRRA): u(w) = w^1−γ/(1 − γ) for γ ≠ 1 and ln w for γ = 1, with R(w) = γ. Here the fraction of wealth invested in risky assets is independent of wealth, consistent with the long-run stability of portfolio shares and with balanced growth in macroeconomic models such as the Ramsey–Cass–Koopmans model.
• Hyperbolic absolute risk aversion (HARA): A(w) = 1/(c + dw), whose risk tolerance 1/A is linear in wealth. HARA nests CARA (d = 0), CRRA (c = 0), and quadratic utility (d = −1), and it is the broadest class that yields linear sharing rules in Pareto-efficient risk-sharing arrangements, a result of interest in Pareto efficiency analysis of insurance markets.

Quadratic utility has increasing absolute risk aversion and a bliss point, which most economists regard as unrealistic, though it underlies the classic mean–variance framework of modern portfolio theory.

Arrow's work focused on portfolio choice with one riskless and one risky asset. He showed that if absolute risk aversion is decreasing in wealth, the amount invested in the risky asset rises with wealth, so that risky assets are normal goods. If relative risk aversion is increasing in wealth, the fraction of wealth invested in the risky asset falls as wealth rises, and if constant, the fraction is constant. Arrow offered decreasing absolute and approximately constant relative risk aversion as plausible empirical hypotheses, a pairing that CRRA satisfies.

Other results follow. Higher risk aversion leads to greater demand for insurance and a larger precautionary savings motive when the third derivative of utility is positive, a property called prudence. The coefficients also govern asset prices: in consumption-based models, the equity premium depends on the product of relative risk aversion and the covariance of returns with consumption growth, which led to the equity premium puzzle when empirical covariances implied implausibly large values of R. In contrast, the Kelly criterion corresponds to CRRA with γ = 1, logarithmic utility, and its growth-optimal bets are larger than those of agents with γ > 1.

The Arrow–Pratt framework describes behavior under expected utility, where curvature is global and attitudes are determined by final wealth. Prospect theory, proposed by Daniel Kahneman and Amos Tversky in 1979, departs from it in several ways: utility is defined over gains and losses relative to a reference point, the value function is concave for gains and convex for losses, and losses loom larger than gains. In that case the sign of u″ switches at the reference point, so the Arrow–Pratt coefficient is positive for gains (risk aversion) and negative for losses (risk seeking). The framework thus shows which observed choices, such as the simultaneous purchase of insurance and lottery tickets, expected utility with a single concave function cannot reproduce, and it supplies the language in which prospect theory's departures are stated.

Experimental estimates of R vary widely across subjects and tasks, typically in a range from about 0.5 to 3 in laboratory and field studies, and responses depend on framing, stake size, and whether choices are made in the small or the large. Such instability is part of the motivation for non-expected-utility models, though the Arrow–Pratt coefficients remain the default way to parametrize risk attitudes in applied theory.

1. Pratt, J. W. (1964). Risk aversion in the small and in the large. Econometrica, 32(1–2), 122–136. https://doi.org/10.2307/1913738
2. Kahneman, D., & Tversky, A. (1979). Prospect theory: an analysis of decision under risk. Econometrica, 47(2), 263–291. https://doi.org/10.2307/1914185
3. Arrow, K. J. (1965). Aspects of the Theory of Risk-Bearing. Helsinki: Yrjö Jahnssonin Säätiö. https://www.worldcat.org/title/aspects-of-the-theory-of-risk-bearing/oclc/2188033
4. Markowitz, H. (1952). Portfolio selection. Journal of Finance, 7(1), 77–91. https://doi.org/10.1111/j.1540-6261.1952.tb01525.x