Translate qualitative risk attitudes into quantitative metrics. Use Arrow–Pratt measures, certainty equivalents, and HARA utility functions to calibrate portfolios, price insurance, and design financial products aligned with investor profiles.
Condition: U''(w) < 0
Prefers certain wealth to risky gambles with equal expected value. Demands positive risk premium for uncertain prospects.
Condition: U''(w) = 0
Indifferent between certain and risky options with equal expected value.
Condition: U''(w) > 0
Prefers riskier options when expected value is equal, often chasing upside potential.
Measures sensitivity to dollar changes in wealth. Declining A(w) indicates higher willingness to accept absolute risks with greater wealth.
Evaluates risk tolerance in proportional terms. Constant R(w) simplifies multi-period portfolio models with growth in wealth.
Represents the largest fair bet size an investor accepts. Useful for calibrating utility functions to survey data.
Absolute and relative aversion condense curvature information into interpretable statistics, helping advisors calibrate dollar versus proportional risk tolerances.
For a small gamble h with zero mean, the risk premium Θ satisfies U(w - Θ) = E[U(w + h)]. Using a second-order Taylor expansion yields Θ ≈ 0.5 A(w) Var(h). This links expected utility curvature to observable risk premiums demanded by investors.
The approximation clarifies why more curvature (higher A(w)) or higher variance command larger compensation in capital markets.
Learn the DerivationAbsolute risk aversion is constant; relative risk aversion increases with wealth. Suitable for fixed-dollar hedging problems.
Relative risk aversion equals 1. Captures long-horizon investment behavior with multiplicative shocks.
Constant relative risk aversion γ. Widely used in asset pricing and portfolio choice models.
Enables mean-variance analysis but implies increasing absolute risk aversion outside realistic ranges.
By tuning parameters (a, b, γ) we recover CARA, CRRA, logarithmic, and quadratic forms, enabling flexible calibration for insurance, asset pricing, and consumption models.
Assume a U.S. wealth management client with initial wealth $500,000, displaying CRRA utility with parameter γ = 3 (risk-averse).
Choice between a risk-free Treasury note yielding 4% and a diversified equity portfolio with expected return 8% and standard deviation 15%.
Using the CRRA certainty equivalent formula, CE = (E[w^{1-γ}])^{1/(1-γ)}, approximate the certain wealth level equating expected utility.
Risk premium equals expected wealth minus certainty equivalent. Compare to the risk-free payoff to gauge willingness to take equity exposure.
Real-world investors may exhibit loss aversion, probability weighting, and reference dependence. Arrow–Pratt measures provide a normative benchmark, while behavioral finance adjusts utility functions (e.g., prospect theory) to match observed behaviors in equity markets and retirement savings.
Explore Empirical StudiesFinancial advisors estimate risk aversion to match clients with appropriate asset allocations. Insurers use it to price policies that customers are willing to buy, and banks calibrate risk limits based on expected utility losses from tail events.
Risk premium is the additional expected return required to make a risk-averse investor indifferent between a risky asset and a risk-free alternative. It quantifies compensation for bearing variability and downside risk.
Hyperbolic Absolute Risk Aversion (HARA) functions encompass exponential, logarithmic, and power utilities, providing flexible modeling aligned with diverse investor behaviors and supporting analytical solutions in portfolio problems.