Absolute Risk Aversion¶

Definition¶

For twice differentiable increasing utility with $u'(w)>0$ $0$ , the absolute risk aversion coefficient is $r(w)=-u''(w)/u'(w)$ $- u^{''} (w) / u^{'} (w)$ . It measures local curvature relative to marginal utility at wealth $w$ .

Interpretation¶

Curvature determines how much a small uncertain loss matters beyond its mean. The coefficient is unchanged by positive affine transformations of utility.

Graph of the absolute risk aversion r(w) against wealth: a flat line at alpha = 0.01 for exponential utility and a falling curve (1 - gamma)/w for power utility with gamma = 1/2.

The figure plots $r(w)$ for two utilities from the Key Results. Exponential utility with $\alpha=0.01$ $0.01$ , the coefficient of the Example, has the same curvature at every wealth level, so the certainty equivalent of a given small risk lies below $w$ by about the same amount, $\tfrac{1}{2}\alpha\varepsilon^2\operatorname{Var}(Z)$ , however wealthy the policyholder is. Power utility with $\gamma=\tfrac{1}{2}$ (chosen for the figure) has $r(w)=0.5/w$ $0.5/ w$ : it is more risk averse than the exponential utility below $w=50$ and less above, and its aversion to a fixed small risk fades as wealth grows.

Move the wealth $w$ : the exponential $r(w)=\alpha$ stays flat while the power utility's $r(w)$ falls with wealth. The approximate premium $\tfrac12r(w)\operatorname{Var}$ for a small risk of variance 100 is 0.5 at $\alpha=0.01$ $0.01$ .

Expected utility and risk aversion relates concave utility to aversion to uncertain wealth. The higher-order derivative $u''(w)$ measures that curvature; dividing its negative by marginal utility $u'(w)$ makes the local coefficient unchanged by positive affine changes of utility.

Key Results¶

Exponential utility $u(w)=-\mathrm{e}^{-\alpha w}$ has constant absolute risk aversion (CARA) $r(w)=\alpha>0$ $0$ .
Power utility $u(w)=w^\gamma$ $w^{γ}$ , $0<\gamma<1$ $1$ , $w>0$ $0$ , has $r(w)=(1-\gamma)/w$ $γ) / w$ .
For a small mean-zero risk $\varepsilon Z$ , the certainty equivalent (the sure wealth with the same expected utility as $w+\varepsilon Z$ $εZ$ ) lies below $w$ by approximately $\tfrac{1}{2}r(w)\varepsilon^2\operatorname{Var}(Z)$ , with smoothness and moment conditions for the Taylor approximation.
This local approximation is not an exact formula for arbitrary large losses.

Examples¶

With exponential coefficient $\alpha=0.01$ and a sufficiently small centred loss of variance 100, the second-order premium above the mean is approximately $0.5(0.01)(100)=0.5$ $0.5$ .

Prerequisites¶

Sources¶

Risk Insurance · Lecture 7

Aliases: Arrow–Pratt coefficient, Arrow–Pratt measure of absolute risk aversion, coefficient of absolute risk aversion, absolute risk aversion coefficient, ARA, Pratt–Arrow measure, Arrow–Pratt measure Keywords: CARA utility, constant absolute risk aversion, r(w) = -u''(w)/u'(w), small-risk premium, exponential utility, power utility Symbols: alpha, epsilon, gamma

Last updated: 17 October 2026 21:53