Risk aversion

[박대선]

In economics and finance, risk aversion is the tendency of people to prefer outcomes with low uncertainty to those outcomes with high uncertainty, even if the average outcome of the latter is equal to or higher in monetary value than the more certain outcome.[1]

Risk aversion explains the inclination to agree to a situation with a lower average payoff that is more predictable rather than another situation with a less predictable payoff that is higher on average. For example, a risk-averse investor might choose to put their money into a bank account with a low but guaranteed interest rate, rather than into a stock that may have high expected returns, but also involves a chance of losing value.

Example[edit]

Utility function of a risk-averse (risk-avoiding) individual

Utility function of a risk-neutral individual

Utility function of a risk-loving (risk-seeking) individual

CE – Certainty equivalent; E(U(W)) – Expected value of the utility (expected utility) of the uncertain payment W; E(W) – Expected value of the uncertain payment; U(CE) – Utility of the certainty equivalent; U(E(W)) – Utility of the expected value of the uncertain payment; U(W0) – Utility of the minimal payment; U(W1) – Utility of the maximal payment; W0 – Minimal payment; W1 – Maximal payment; RP – Risk premium

A person is given the choice between two scenarios: one with a guaranteed payoff, and one with a risky payoff with same average value. In the former scenario, the person receives $50. In the uncertain scenario, a coin is flipped to decide whether the person receives $100 or nothing. The expected payoff for both scenarios is $50, meaning that an individual who was insensitive to risk would not care whether they took the guaranteed payment or the gamble. However, individuals may have different risk attitudes.[2][3][4]

A person is said to be:

• risk averse (or risk avoiding) - if they would accept a certain payment (certainty equivalent) of less than $50 (for example, $40), rather than taking the gamble and possibly receiving nothing.
• risk neutral – if they are indifferent between the bet and a certain $50 payment.
• risk loving (or risk seeking) – if they would accept the bet even when the guaranteed payment is more than $50 (for example, $60).

The average payoff of the gamble, known as its expected value, is $50. The smallest guaranteed dollar amount that an individual would be indifferent to compared to an uncertain gain of a specific average predicted value is called the certainty equivalent, which is also used as a measure of risk aversion. An individual that is risk averse has a certainty equivalent that is smaller than the prediction of uncertain gains. The risk premium is the difference between the expected value and the certainty equivalent. For risk-averse individuals, risk premium is positive, for risk-neutral persons it is zero, and for risk-loving individuals their risk premium is negative.

Utility of money[edit]

In expected utility theory, an agent has a utility function u(c) where c represents the value that he might receive in money or goods (in the above example c could be $0 or $40 or $100).

The utility function u(c) is defined only up to positive affine transformation – in other words, a constant could be added to the value of u(c) for all c, and/or u(c) could be multiplied by a positive constant factor, without affecting the conclusions.

An agent is risk-averse if and only if the utility function is concave. For instance u(0) could be 0, u(100) might be 10, u(40) might be 5, and for comparison u(50) might be 6.

[박대선]

Mathematically, we should find a 50–50 flip of the coin for $100 to be worth as much as a sure promise of $50. After all, the expected value of the gamble over a number of trials is $50. However, nearly everyone but committed gamblers would rather have the sure thing than a risky prospect. For many people, a 50–50 flip of a coin even for $200 might not be worth as much as a sure promise of $50, even though the gamble is mathematically worth twice as much!

[박대선]

Risk preference is sometimes reversed: People prefer to take chances when trying to prevent a negative outcome. They would rather take a 50–50 gamble on losing $100 than accept the certain loss of $50. Thus they will risk losing a lot of money at trial rather than settle out of court.