Risk Attitudes & Arrow-Pratt Measures

Suppose you can take $50 for sure or flip a fair coin to get $100 or $0; both offers average $50. Strictly preferring the sure payment shows that you avoid this particular gamble, while being indifferent means the offers feel equal. Economists use Arrow-Pratt measures to describe preferences for small risks by comparing changes in the value of one more dollar with that dollar's current value. Try this: name the guaranteed amount that would make you indifferent to the coin flip—your certainty equivalent for this gamble, rather than a score for every risk.

Compare equal average payoffs

In expected utility theory, utility represents how much different amounts of wealth mean to someone. A risk-averse person weakly prefers a gamble's average payoff for certain; this allows indifference. With strictly concave utility and a nonconstant gamble, the preference is strict. Risk-neutral people are indifferent, while risk-seeking people prefer the gamble to its average payoff. Risk seeking does not mean accepting every gamble, regardless of its payoff.

Measure small-risk preferences

Marginal utility means the value of one more dollar. Arrow-Pratt absolute risk aversion compares how this value changes with wealth to its current level. The comparison stays the same if we rescale utility without changing preferences; a graph's raw bend does not. A positive coefficient indicates local risk aversion, while a negative one indicates local risk seeking. A zero coefficient makes the second-order small-risk premium zero; it does not by itself establish exact neutrality toward nearby gambles.

Connect the measure to a choice

A certainty equivalent is the guaranteed wealth someone values equally to a particular gamble. The risk premium is the gap between expected final wealth and that certainty equivalent. For a small fair gamble that adds or subtracts the same amount with equal probability, the premium is approximately one-half the absolute risk-aversion coefficient times the gamble's variance, a measure of its spread. This is a local approximation, not an exact rule for arbitrary large risks. It can help explain willingness to pay to avoid a specified risk without proving any insurance policy is worth its price.

Keep wealth comparisons separate

Decreasing absolute risk aversion means the same person's absolute coefficient falls as wealth rises, holding their utility function fixed. This is a property a model may have, not a prediction every person must satisfy. Comparing a billionaire with a student cannot establish it: they may have different preferences. Relative risk aversion multiplies the absolute coefficient by positive wealth; constant relative risk aversion means this product stays constant even as the absolute coefficient changes.

Apply the comparison carefully

Specify payoff amounts, probabilities and current wealth when comparing certainty equivalents. One observed choice does not reveal attitudes toward every gamble. Career and investment decisions also reflect expected returns and other benefits, so a single such decision cannot uniquely identify someone's risk aversion. The Deep Dive gives the formal definitions and assumptions.

With a twice differentiable increasing utility function u(w), absolute risk aversion is A(w) = -u''(w)/u'(w), where u'(w) > 0. Relative risk aversion is R(w) = w A(w) for w > 0. Concavity implies u''(w) ≤ 0 and A(w) ≥ 0; wherever u''(w) < 0, A(w) is positive. A(w) = 0 at one wealth level describes a zero second-order premium, not necessarily exact indifference to nearby gambles. DARA means A(w) decreases with wealth; CRRA means R(w) is constant. For a small symmetric fair risk, the local approximation is premium ≈ A(w) × variance / 2.

Arrow-Pratt measures connect preferences represented by utility functions with premiums for avoiding risk.

• Pratt (1964) examines local risk aversion, risk premia for arbitrary risks, and the concept of decreasing risk aversion, relating these ideas within a utility-for-money framework. [1]
• Eeckhoudt and Laeven (2015), Section 2, derive the expected-utility approximation for a small symmetric fair risk: its risk premium is approximately one-half its variance times -u''(w)/u'(w). Their original paper also develops counterparts under dual theory and rank-dependent utility. [2]
• Kahneman and Tversky (1979) report patterns of risky choice that conflict with expected utility theory and propose prospect theory as an alternative descriptive account, with outcomes evaluated as gains and losses. [3]

Arrow-Pratt coefficients describe local preferences within an expected-utility model with sufficiently smooth, increasing utility. A single coefficient at one wealth level does not summarize responses to every large gamble. The small-risk premium formula is an approximation whose scope must be respected. Risk attitude can vary with wealth within the model; constant risk aversion is an additional assumption. The basic setup uses specified probabilities and does not itself explain choices when probabilities are unknown. Kahneman and Tversky's prospect theory addresses observed departures from expected utility, including different responses to gains and losses [3]. Estimating a utility function also requires more information than one observed choice.