Risk Aversion

Risk aversion is a preference for a certain outcome over an uncertain one with the same or slightly higher expected value. A risk-averse person (described as "risk-averse") will accept a lower but sure payment rather than gamble, and will demand extra reward, a risk premium, before taking on uncertainty. It is a basic assumption behind insurance, portfolio theory, asset pricing and much of modern finance.

In economics, risk aversion is usually modelled through expected utility theory: a person's satisfaction rises with wealth but at a falling rate, so a loss of a given size reduces well-being more than an equal gain increases it. It sits between two other attitudes, risk neutrality (caring only about the average) and risk seeking (preferring a gamble). It is related to, but distinct from, loss aversion, which describes a reaction to losses measured from a reference point rather than the overall shape of preferences over wealth.

Suppose a friend offers you a choice: take $50 now, or toss a coin and receive $100 on heads and nothing on tails. On average the coin toss is worth the same $50, but many people take the sure money. Some would take the sure money even if it were only $40. That willingness to give up some average value to avoid uncertainty is risk aversion.

The same instinct explains why people buy home insurance, keep savings in a bank instead of betting them, and prefer a steady salary to an uncertain commission. They are paying, directly or indirectly, to make their future more predictable. The more someone would give up to avoid a gamble, the more risk-averse they are.

The idea goes back to Daniel Bernoulli, who in 1738 argued that the value of money to a person depends on how much they already have. John von Neumann and Oskar Morgenstern gave expected utility theory its modern form in 1944. A person is risk-averse when their utility function $u(w)$ over wealth $w$ is concave, which is equivalent to diminishing marginal utility.

The certainty equivalent (CE) is the sure amount that gives the same expected utility as a gamble, and the risk premium is the gap between the gamble's expected value and that sure amount:

• $E[W]$ is the expected (average) payoff of the gamble.
• $\text{CE}$ is the sure amount the person regards as equally good.

Worked example: take $u(w) = \sqrt{w}$ and the coin toss paying $100 or $0. Expected utility is $0.5 \times \sqrt{100} + 0.5 \times \sqrt{0} = 5$. The certainty equivalent is the amount whose square root is 5, which is $25. The expected value is $50, so the risk premium is $50 - 25 = 25$: this person would accept anything above $25 for sure rather than toss the coin.

Kenneth Arrow and John Pratt, working separately in the 1960s, defined a standard measure of how strongly someone dislikes risk at a given level of wealth:

Here $u'(w)$ is marginal utility and $u''(w)$ is how fast it falls. A larger $A(w)$ means stronger absolute risk aversion. Many models assume that risk aversion relative to wealth stays roughly constant, so a richer investor holds about the same share of wealth in risky assets as a poorer one.

• Insurance. Because people prefer certainty, they will pay a premium above expected losses to transfer risk to an insurer, which pools many independent risks. This is the economic basis of insurance markets.
• Asset prices. Investors require higher expected returns on assets with more systematic risk. In the Capital Asset Pricing Model (CAPM), the market risk premium reflects the average investor's risk aversion, and the gap above the risk-free rate is the reward for bearing that risk.
• Portfolio choice. Harry Markowitz's mean-variance framework (1952) assumes investors trade off expected return against variance, so they diversify. Diversification lowers risk without necessarily lowering expected return, which risk-averse investors value. Measures such as the Sharpe ratio express this risk–return trade-off.
• Risk-on and risk-off. Market commentators describe periods when investors' collective risk appetite falls, and money shifts toward risk-off assets such as government bonds and gold, and back again.
• Suitability rules. Regulators require advisers to assess a client's risk tolerance before recommending products; examples include suitability and appropriateness tests under the EU's MiFID II and FINRA's suitability rule in the United States.

Risk aversion also shapes economic behavior beyond markets. Precautionary saving rises when the future is uncertain, firms delay investment when uncertainty spikes, and lenders charge more to riskier borrowers. Knightian uncertainty, where probabilities are unknown, can make people more cautious still.

The expected-utility account of risk aversion has well-known problems. Rajnish Mehra and Edward Prescott showed in 1985 that the historical gap between US equity and bond returns implies implausibly high risk aversion in standard models, the "equity premium puzzle". Matthew Rabin showed in 2000 that if people turn down small fair bets because of concave utility, they would have to turn down absurdly attractive large bets as well, which suggests small-stakes caution comes from something else, such as loss aversion or narrow framing.

Behavioral research also finds that risk attitudes depend on context. Under prospect theory, people are often risk-averse for gains but risk-seeking for losses, and the framing effect can flip choices. Attitudes vary with age, wealth, experience and recent market history. Defenders of standard theory reply that expected utility remains a useful benchmark, and that richer models, such as those separating risk aversion from the willingness to substitute consumption over time, can fit the data better.

• Stanford Encyclopedia of Philosophy: Decision Theory
• Nobel Prize in Economic Sciences 1990 (Markowitz, Miller and Sharpe)