Risk Aversion - Example

Example

Utility function of a risk-neutral individual. Utility function of a risk-affine (risk-seeking) individual. CE - Certainty equivalent; E(U(W)) - Expected value of the utility (expected utility) of the uncertain payment; 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 without. In the guaranteed 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. A person is said to be:

• risk-averse (or risk-avoiding) - if he or she 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 he or she is indifferent between the bet and a certain $50 payment.
• risk-loving (or risk-seeking) - if the guaranteed payment must be more than $50 (for example, $60) to induce him or her to take the guaranteed option, rather than taking the gamble and possibly winning $100.

The average payoff of the gamble, known as its expected value, is $50. The dollar amount that the individual would accept instead of the bet is called the certainty equivalent, and the difference between the expected value and the certainty equivalent is called the risk premium. For risk-averse individuals, it becomes positive, for risk-neutral persons it is zero, and for risk-loving individuals their risk premium becomes negative.