Risk aversion
Risk aversion describes the behavioral preference of individuals and economic agents to favor certain outcomes over risky prospects with equivalent expected value, a tendency rooted in the concavity of utility functions under expected utility theory.[1] This preference manifests empirically in domains such as insurance purchases, where people pay premiums exceeding actuarial fair value to avoid potential losses, and portfolio choices, where investors demand higher returns for bearing volatility.[2] Formally quantified by the Arrow-Pratt measure of absolute risk aversion, defined as A(c) = -\frac{u''(c)}{u'(c)}, where u(c) is the utility of consumption or wealth c, the metric captures the intensity of aversion through the curvature of the utility function, with higher values indicating greater reluctance to accept risk.[3] The concept underpins key predictions in finance and behavioral economics, including the equity risk premium puzzle—wherein observed premiums exceed what standard models predict given estimated aversion levels—and explains why risk-averse agents diversify holdings to minimize variance.[4] Empirical studies confirm risk aversion's prevalence across populations, with genetic factors influencing sensitivity to uncertainty and developmental shifts toward greater aversion in adulthood, though anomalies like context-dependent risk-seeking in losses challenge pure expected utility frameworks.[5][6] Despite such deviations, documented in laboratory and field data, risk aversion remains a robust descriptor of human decision-making under uncertainty, informing policies from retirement savings to catastrophe insurance markets.[7]Definition and Basic Concepts
Formal Definition
In expected utility theory, risk aversion describes a decision maker's preference for a certain monetary outcome over a risky prospect offering the same expected value. Formally, given a von Neumann-Morgenstern utility function u(\cdot) defined over wealth or payoffs, an agent is risk-averse if u(\mathbb{E}[Z]) \geq \mathbb{E}[u(Z)] for any random variable Z with finite support, with strict inequality holding for non-degenerate lotteries (i.e., those with positive variance).[8] This preference implies that the certainty equivalent of the lottery—the fixed amount yielding the same utility as the expected utility of the lottery—is strictly less than the lottery's expected value.[9] The condition u(\mathbb{E}[Z]) > \mathbb{E}[u(Z)] is mathematically equivalent to u(\cdot) being strictly concave, since concavity ensures that the function lies below its tangents and satisfies Jensen's inequality in the reverse direction for expectations.[10] Strict concavity (u''(c) < 0 for all c in the domain) distinguishes risk aversion from risk neutrality (linear u) and risk-loving behavior (convex u).[11] This framework, originating from von Neumann and Morgenstern's 1944 axiomatization, assumes completeness, transitivity, continuity, and independence of preferences over lotteries.[12]Illustrative Examples
A fundamental illustration of risk aversion is the preference for a certain outcome over a gamble with equivalent expected value. For example, an individual might choose $50 with certainty rather than a 50% chance of $100 and a 50% chance of $0, both of which have an expected monetary value of $50.[13] [14] This choice reflects risk aversion because the utility function u(\cdot) is concave, satisfying Jensen's inequality: the expected utility of the gamble, \frac{1}{2}u(0) + \frac{1}{2}u(100), is less than the utility of the expected value, u(50).[14] [8] Another common example involves rejecting fair bets relative to current wealth. Most people decline a coin flip offering a gain of $1,000 on heads and a loss of $1,000 on tails, despite the expected value being zero, due to the concavity of the utility function diminishing the marginal utility of gains more than it increases aversion to symmetric losses.[13] [11] To quantify this, suppose an individual's utility is given by u(w) = \sqrt{w} where w is wealth; starting from $10,000, the expected utility of the bet is \frac{1}{2}\sqrt{11,000} + \frac{1}{2}\sqrt{9,000} \approx 99.87, while u(10,000) = 100, confirming rejection.[11] These examples demonstrate the risk premium, the amount by which the certain equivalent falls short of the expected value; for the $50 gamble, a risk-averse person might accept only $45 as certain to forgo the lottery, with the $5 difference as the premium paid to avoid risk.[8] [14] Empirical studies, such as those eliciting preferences via hypothetical choices, consistently show such behavior across populations, though the degree varies with stake size and background risk.[15]Historical Development
Early Conceptualizations
The St. Petersburg paradox, formulated by Nicolas Bernoulli around 1713 and popularized through correspondence among mathematicians, illustrated a fundamental tension in early probability theory: a gamble with infinite expected monetary value yet finite willingness to pay among rational individuals.[16] In this game, a fair coin is flipped until tails appears, with payoff doubling each heads (e.g., 1 ducat for first tails, 2 for heads then tails, 4 thereafter), yielding an expected value of \sum_{k=1}^\infty 2^{k-1} \cdot (1/2)^k = \infty.[16] Empirical observation showed participants typically offering only 2 to 4 ducats to play, prompting queries into why expected value maximization failed to predict behavior.[17] Daniel Bernoulli, in his 1738 exposition to the St. Petersburg Academy, resolved this by distinguishing monetary value from moral expectation, proposing that decision-makers maximize the expected value of a utility function rather than wealth itself.[18] He posited a concave utility function, such as u(w) = \ln(w) for wealth w, reflecting diminishing marginal utility: additional wealth yields progressively less subjective value.[16] For the paradox, this caps the gamble's expected utility at a finite amount (e.g., approximately 1.0986 for logarithmic utility starting from zero wealth), aligning theory with observed risk-avoiding choices.[18] Bernoulli drew analogies to annuities and inheritance, where similar discrepancies between arithmetic expectations and preferences suggested inherent aversion to variability in outcomes.[19] This framework implicitly defined risk aversion as the preference for a sure amount over a gamble with equal expected value, rooted in the concavity of utility—equivalent to Jensen's inequality, where u(E) > E[u(x)] for random x.[18] Earlier hints appeared in Gabriel Cramér's 1728 letter suggesting a square-root utility to bound the paradox, but Bernoulli's development provided the first systematic rationale, influencing later economic thought on prudence in uncertain environments.[16] These ideas predated formal probability axioms and axiomatic utility, emphasizing causal drivers like wealth-dependent valuation over mere probabilistic averaging.[19]Formalization in Expected Utility Theory
Expected utility theory, formalized by John von Neumann and Oskar Morgenstern in their 1944 book Theory of Games and Economic Behavior, represents preferences over lotteries via the expected value of a utility function u, where u satisfies axioms of completeness, transitivity, continuity, and independence.[20] Risk aversion emerges as a behavioral property under this framework: an agent is risk-averse if, for any non-degenerate random wealth \tilde{w} with mean \mu = E[\tilde{w}], the expected utility E[u(\tilde{w})] is strictly less than the utility of the expected wealth u(\mu).[21] This condition holds if and only if u is strictly concave, as guaranteed by Jensen's inequality for concave functions, which states that E[u(\tilde{w})] \leq u(E[\tilde{w}]) with equality only for degenerate distributions.[22] Concavity reflects diminishing marginal utility of wealth, leading the agent to value certain outcomes over risky ones with identical means; the certainty equivalent c satisfies u(c) = E[u(\tilde{w})], and the risk premium is \pi = \mu - c > 0.[23] To quantify the intensity of risk aversion, Kenneth Arrow and John W. Pratt independently developed local measures in the early 1960s. The Arrow-Pratt coefficient of absolute risk aversion at wealth w is A(w) = -\frac{u''(w)}{u'(w)}, where u''(w) < 0 and u'(w) > 0 ensure concavity and increasing utility.[3] [24] For small risks \tilde{\epsilon} with zero mean and variance \sigma^2, the approximate risk premium is \pi \approx \frac{1}{2} A(w) \sigma^2, providing a second-order Taylor expansion-based link between curvature and aversion.[25] Higher A(w) indicates stronger aversion, enabling ordinal comparisons of attitudes; for example, one agent is more risk-averse than another if their utility satisfies u_1 = g(u_2) for an increasing concave g.[3] The relative risk aversion coefficient R(w) = w A(w) extends this to scale wealth multiplicatively.[24]Theoretical Measures and Properties
Absolute and Relative Risk Aversion
Absolute risk aversion quantifies the degree of risk aversion at a specific wealth level through the Arrow-Pratt measure, defined as A(w) = -\frac{u''(w)}{u'(w)}, where u(w) is the von Neumann-Morgenstern utility function over wealth w.[3] This coefficient arises from a second-order Taylor approximation of the utility function around expected wealth, capturing the concavity that reflects aversion to small gambles; a higher A(w) indicates greater willingness to pay to avoid risk via the risk premium.[19] The measure is local, applying primarily to infinitesimal risks, and assumes twice-differentiable, increasing, and concave utility functions consistent with risk aversion.[11] Relative risk aversion extends this by scaling absolute risk aversion to wealth, given by R(w) = w \cdot A(w) = -w \frac{u''(w)}{u'(w)}, which assesses risk attitudes toward proportional gambles, such as percentage changes in wealth.[19] This formulation proves useful in models involving multiplicative risks, like investment returns, where decisions scale with portfolio size.[26] Unlike absolute risk aversion, relative risk aversion often exhibits constancy in empirical and theoretical applications, facilitating tractable solutions in dynamic stochastic general equilibrium models. Key properties distinguish these measures: absolute risk aversion typically decreases with wealth (decreasing absolute risk aversion, or DARA), implying that higher-wealth individuals accept larger absolute dollar risks, as observed in utility functions like quadratic or power forms.[27] For instance, the exponential utility u(w) = -\exp(-\alpha w) yields constant absolute risk aversion A(w) = \alpha, independent of wealth, while logarithmic utility u(w) = \ln(w) produces decreasing A(w) = 1/w and constant relative risk aversion R(w) = 1.[19] Power utility u(w) = \frac{w^{1-\gamma}}{1-\gamma} for \gamma > 0, \gamma \neq 1 features constant relative risk aversion R(w) = \gamma and decreasing absolute risk aversion A(w) = \gamma / w, aligning with assumptions in consumption-based asset pricing where \gamma parameters are estimated around 1-10 based on equity premium puzzles.[28] Relative risk aversion may increase (IRRA), decrease (DRRA), or remain constant (CRRA), with CRRA widely used for its homogeneity properties in growth models, though empirical meta-analyses report mean values of approximately 1 in economic contexts and 2-7 in finance, varying by methodology and sample.[28] These measures enable comparative statics: one decision-maker is more absolutely risk-averse than another if their utility satisfies u_1''(w)/u_1'(w) \leq u_2''(w)/u_2'(w) for all w, implying acceptance of smaller risks at every wealth level.[3]Implications for Decision-Making
Risk aversion, characterized by a concave von Neumann-Morgenstern utility function, implies that decision-makers reject gambles with zero expected net payoff, preferring certainty equivalents below the gamble's expected value to compensate for variance.[8] The Arrow-Pratt absolute risk aversion measure, A(w) = -\frac{u''(w)}{u'(w)}, quantifies this aversion locally: for a small risk with variance \sigma^2, the required risk premium \pi approximates \frac{1}{2} \sigma^2 A(w), determining the minimum compensation needed to accept the gamble.[29] Higher A(w) thus elevates the threshold for risk-taking, steering choices toward lower-variance outcomes across domains like consumption and asset selection.[11] In insurance decisions, risk-averse agents demand coverage exceeding actuarially fair premiums, as the utility loss from potential large losses outweighs the certain premium cost due to diminishing marginal utility; empirical calibrations confirm more averse individuals purchase greater protection against specified perils.[30] [31] This extends to investment choices, where elevated risk aversion reduces allocation to volatile assets: in mean-variance optimization, the optimal risky asset share inversely scales with A(w), favoring bonds or cash over equities when aversion intensifies, as variance penalizes expected returns more heavily.[11] Comparative statics from risk aversion properties further shape decisions: decreasing absolute risk aversion (DARA), where A'(w) < 0, implies wealthier agents accept larger absolute risks, such as insuring high-value assets while gambling small stakes; constant relative risk aversion (CRRA), with w A(w) invariant, yields wealth-independent portfolio weights, stabilizing allocation fractions amid growth.[22] These dynamics underpin precautionary motives, elevating savings buffers against income shocks, as concavity amplifies downside protection over upside pursuit.[32]Applications in Economics and Finance
Portfolio Choice and Investment
In the framework of expected utility theory, risk-averse investors allocate their wealth between risky assets and risk-free assets to maximize the expected utility of final wealth, where the concavity of the utility function implies a preference for lower variance given expected return.[33] For a single risky asset with expected return μ, volatility σ, and risk-free rate r, the optimal weight π in the risky asset for an investor with constant relative risk aversion γ is given by π = (μ - r) / (γ σ²), such that higher γ reduces exposure to the risky asset.[34] This relationship, derived in continuous time by Merton (1969), holds under power utility and demonstrates that greater risk aversion leads to more conservative portfolios dominated by safe assets.[34] In multi-asset settings, risk aversion drives diversification to minimize variance for a target return, as formalized in mean-variance optimization, where the investor's objective incorporates a risk aversion coefficient A in the utility function U = E[R_p] - (A/2) Var(R_p), with higher A favoring lower-risk efficient frontiers.[35] Empirical portfolio choices reflect this: surveys indicate that investors with higher self-reported risk aversion allocate 20-40% less to equities, preferring bonds or cash equivalents, consistent with lifecycle models where risk tolerance declines with age due to shorter horizons and lower absolute risk tolerance.[11] The capital asset pricing model (CAPM) aggregates individual risk aversion to price risk, with the market risk premium equaling the product of the market's Sharpe ratio and the average relative risk aversion, implying economy-wide γ influences equilibrium returns.[36] However, the equity premium puzzle highlights a tension: the observed U.S. historical equity premium of approximately 6% (real returns over bills from 1889-1978) requires γ estimates of 10-40 in consumption-based models to match data, far exceeding microeconomic estimates from lotteries or insurance decisions, which typically range from 1-3.[37] This discrepancy persists in updated data through 2000, suggesting either underestimation of risk aversion for aggregate shocks or model misspecification, such as incomplete markets or rare disasters.[38][39]Insurance Markets and Bargaining
In competitive insurance markets, risk-averse individuals demand coverage to mitigate the variance in outcomes from uncertain losses, leading to full insurance purchase when premiums are actuarially fair—equal to the expected loss—since the certainty equivalent of the gamble falls below its expected value under concave utility.[40] With positive loading factors for costs or profits, optimal coverage becomes partial, but the quantity insured rises with the degree of absolute risk aversion, as measured by the Arrow-Pratt coefficient, because more risk-averse agents value variance reduction more highly and accept higher risk premiums. Insurers, often modeled as risk-neutral due to diversification, set premiums reflecting pooled expected losses plus loadings, with market equilibrium allocating risks efficiently under complete information, though real markets exhibit inefficiencies from asymmetric information.[13] In non-competitive or bilateral settings, bargaining over insurance terms incorporates risk aversion into negotiation dynamics, where the insured and insurer (typically risk-neutral) haggle over coverage levels and premiums. Under cooperative Nash bargaining solutions, a risk-averse insured's concave utility implies greater concessions on price for higher coverage compared to a risk-neutral bargainer, as the marginal utility loss from uninsured risk outweighs gains from premium savings; for instance, models show that relative risk aversion determines the split of surplus, with higher insured risk aversion yielding contracts closer to full coverage at higher prices.[41] [42] Dynamic game-theoretic frameworks further reveal that risk aversion amplifies the insured's willingness to accept suboptimal terms under uncertainty about insurer offers, potentially leading to persistent underinsurance in opaque markets.[43] Empirical studies support these theoretical predictions, with data from health and auto insurance markets showing that estimated Arrow-Pratt risk aversion coefficients—derived from deductible choices—predict bargaining outcomes, such as lower deductibles among high-risk-aversion households, which reflect effective negotiation for broader protection amid loading and selection pressures. Cohen and Einav's analysis of over 40,000 policies from a major Israeli auto insurer in 1993–1995 found that risk aversion accounts for 20–30% of demand variation, independent of adverse selection, implying that bargaining power correlates inversely with tolerance for residual risk. Recent extensions to asymmetric Nash bargaining confirm that risk-averse parties negotiate proportional reinsurance contracts optimizing shared risk exposure, with outcomes sensitive to bargaining weights tied to outside options and utility curvature.[44]Behavioral Alternatives and Limitations
Challenges to Expected Utility Theory
Expected utility theory (EUT), which posits that individuals choose options maximizing the expected value of a concave utility function to exhibit risk aversion, encounters significant empirical challenges from decision paradoxes and experimental data revealing axiom violations. These anomalies, replicated across studies since the mid-20th century, indicate that human preferences under risk often prioritize certainty or overweight low-probability events in ways incompatible with EUT's independence and continuity axioms.[45][46] The Allais paradox, formulated by Maurice Allais in 1953, exemplifies a core violation through inconsistent rankings of lotteries with known probabilities. In one scenario, most subjects select a certain $1 million over a prospect offering an 89% chance of $1 million, a 10% chance of $5 million, and a 1% chance of $0; however, when both options are adjusted by removing a common 89% chance of $1 million (replacing it with $0), subjects switch to preferring the riskier 11% chance of $5 million (with 89% chance of $0) over an 11% chance of $1 million (with 89% chance of $0). This reversal breaches EUT's independence axiom, which requires preferences to remain invariant under common consequences, as the expected utility calculations yield contradictions unless utility is non-concave in implausible ways. Empirical replications, including surveys of economists, confirm the paradox's robustness, with violation rates exceeding 60% in diverse populations.[45][47][45] Another critique arises from Matthew Rabin's 2000 analysis, showing that EUT's concave utility, calibrated to explain observed rejections of modest small-stakes gambles (e.g., declining a 50% chance to lose $100 for a 50% chance to gain $110 at any wealth level), implies irrational extreme risk aversion for larger stakes, such as rejecting a 50% chance to lose $1,000 for a 50% chance to gain $1,100 even at high wealth. This calibration failure persists across wealth levels, as diminishing marginal utility amplifies aversion unrealistically for high-variance outcomes, contradicting real-world behaviors like insurance purchases or gambling participation. Rabin's theorem demonstrates that no finite concave utility function reconciles small-gamble rejections with moderate large-gamble acceptance, underscoring EUT's empirical inadequacy for modeling risk aversion without ad hoc adjustments.[48][48] Meta-analyses of experimental data further quantify EUT violations, with over 70% of studies since 1980 reporting systematic deviations in risky choices, including the common ratio effect (a variant of Allais where scaling probabilities alters preferences nonlinearly). These findings, drawn from controlled lab settings with monetary incentives, suggest that EUT underestimates probability weighting distortions, where individuals overweight small probabilities and underweight moderate ones, eroding its descriptive power for risk-averse decisions.[46][46][49]Prospect Theory and Related Models
Prospect theory, formulated by Daniel Kahneman and Amos Tversky in their 1979 Econometrica paper, critiques expected utility theory as a descriptive model of decision-making under risk and proposes an alternative framework where choices are evaluated relative to a reference point, framing outcomes as gains or losses.[50] The theory's value function v(x) is S-shaped: concave for gains above the reference point, reflecting risk aversion over prospective gains, and convex for losses below it, indicating risk-seeking behavior over losses.[51] Diminishing sensitivity applies, with marginal value decreasing as gains increase or losses deepen, parameterized empirically as v(x) = x^\alpha for x \geq 0 and v(x) = -\lambda (-x)^\beta for x < 0, where \alpha \approx \beta \approx 0.88 and \lambda \approx 2.25, quantifying loss aversion as losses impacting utility roughly twice as much as equivalent gains.[51] A probability weighting function \pi(p) further modifies expected utility by overweighting small probabilities (e.g., \pi(0.01) > 0.01) and underweighting moderate to high ones (e.g., \pi(0.99) < 0.99), leading to phenomena like the certainty effect, where certain outcomes are overvalued relative to near-certain ones, contributing to observed risk aversion in gain domains and risk-seeking in loss domains.[50] This setup explains empirical deviations from expected utility, such as the reflection effect—where preferences reverse when gains become losses—and the common ratio effect violations seen in Allais paradoxes, without assuming global risk aversion.[52] Cumulative prospect theory, an extension developed by Tversky and Kahneman in 1992, addresses limitations in the original model's handling of rank-dependent probabilities and stochastic dominance by replacing separable decision weights with cumulative weighting functions w^+(p) for gains and w^-(p) for losses, applied to ordered outcomes.[53] In this formulation, the prospect value is \sum \pi_i v(x_i) transformed to cumulative form, preserving loss aversion and diminishing sensitivity while resolving inconsistencies like non-monotonic weighting for intermediate ranks.[54] Risk attitudes emerge from interactions between the value function's curvature and the weighting capacities, which are subadditive for gains (promoting risk aversion) and superadditive for losses (promoting risk-seeking), with empirical parameters showing capacities crossing at probabilities around 0.3–0.4.[53] Related models, such as rank-dependent utility theory (preceding and influencing prospect theory), incorporate similar cumulative weighting but lack the reference-dependent value function, focusing instead on probability distortions alone to model risk attitudes.[52] These frameworks collectively highlight how reference dependence and nonlinear transformations deviate from constant relative or absolute risk aversion in expected utility, better capturing behavioral data where individuals reject small-probability gains but accept equivalent small-probability losses.[54]Empirical Evidence
Experimental and Survey Data
Experimental studies consistently demonstrate risk aversion in choices under uncertainty, where participants prefer a certain outcome over a gamble with equal or higher expected value. In the canonical Holt-Laury multiple price list task, subjects select between a fixed certain payment and a lottery with varying probabilities of high and low payoffs; the point at which they switch to the lottery reveals their risk aversion coefficient under constant relative risk aversion (CRRA) assumptions, with median estimates often yielding CRRA values around 0.5 to 1.0 for small stakes.[55] Larger-stakes experiments, such as those involving real financial decisions in game shows like "Deal or No Deal," confirm risk aversion but show it diminishes as stakes increase, challenging the constant absolute risk aversion implied by some expected utility models.[56] Field experiments further substantiate these patterns; for instance, in a study of Thai rice farmers facing repeated weather risks, participants exhibited decreasing risk aversion over time, with initial CRRA estimates averaging 1.2 but declining with experience.[57] However, Rabin's calibration theorem highlights limitations: even mild risk aversion over small gambles implies unrealistically extreme aversion for larger stakes under expected utility theory, as evidenced by hypothetical choice data where subjects reject modest gambles but accept substantial ones in principle.[12] These inconsistencies suggest that experimental risk aversion may partly reflect non-EU factors like probability weighting, though EU models still fit aggregate data reasonably well in high-stakes settings.[55] Survey-based measures of risk aversion, often derived from hypothetical income gambles or self-reported tolerance, yield CRRA estimates typically between 0.6 and 0.8 across diverse populations. In the U.S. Health and Retirement Study, responses to questions about accepting a 50% chance of gaining X versus losing Y imputed relative risk tolerance levels consistent with moderate aversion, correlating with actual asset allocation in retirement portfolios.[58] Korean household surveys similarly estimate CRRA at 0.6-0.8, with lower values among males, younger individuals, and higher-income respondents, though these self-reports exhibit low stability over time compared to incentivized experiments.[59] Simple survey instruments, such as asking for reservation prices on lotteries, provide Arrow-Pratt measures aligning with experimental findings but are prone to hypothetical bias, overestimating aversion relative to revealed preferences.[60]| Study Type | Key Finding | CRRA Estimate | Source |
|---|---|---|---|
| Holt-Laury MPL (lab) | Switch point indicates aversion for small stakes | 0.5-1.0 | [55] |
| Game show field (high stakes) | Aversion decreases with stake size | Variable, lower than small stakes | [56] |
| HRS survey (hypothetical gambles) | Moderate aversion linked to portfolios | Implied tolerance ~0.6-0.8 | [58] |
| Korean household survey | Demographic variations in aversion | 0.6-0.8 overall | [59] |