Risk aversion and utility

In short

• Offered a sure amount or a gamble with the same expected value: the risk-averse investor takes the sure amount, the risk-neutral investor is indifferent, the risk seeker takes the gamble.
• Markets price assets as if the typical investor is risk averse; we assume risk aversion from here on.
• Risk tolerance is the opposite of risk aversion: more tolerance, more willingness to take risk.
• Utility $U = E(r) - \tfrac{1}{2}A\sigma^2$: $A > 0$ risk averse, $A = 0$ risk neutral, $A < 0$ risk seeking.
• Utility only ranks choices for one investor; it cannot be compared or added across people.
• A risk-free asset ($\sigma = 0$) gives the same utility to everyone: its return.