Expected Value and Risk Aversion

How to calculate the average outcome of a gamble, why most people prefer a sure thing anyway, and how that explains insurance and diversification.

Would you rather have 50 dollars for certain, or a coin flip that pays 100 dollars on heads and nothing on tails? Most people choose the certain 50 dollars, even though, on average, the two options are worth the same. Understanding why is central to economics.

Expected value

The expected value of an uncertain outcome is its average result if you could repeat it many times. You multiply each possible outcome by its probability and add them up. The coin flip has an expected value of 0.5 times 100 plus 0.5 times zero, which equals 50 dollars.

Risk aversion

A person who prefers the certain 50 dollars to the coin flip is risk averse. Most people are risk averse for important sums of money. Economists explain this with diminishing marginal utility: each extra dollar adds less to your wellbeing than the one before. Losing 100 dollars you need for rent hurts more than gaining an extra 100 dollars helps.

The extra amount a risk-averse person would give up to avoid risk is called a risk premium. Someone might accept 40 dollars for certain rather than take the coin flip. The 10 dollar difference is what they would pay to avoid the risk.

Why insurance makes sense

Suppose there is a 1 percent chance your house burns down, causing a loss of 200,000 dollars. The expected loss is 2,000 dollars a year. An insurer might charge 2,500 dollars. On average you pay more than you expect to lose, yet buying insurance is sensible. Losing your home uninsured would be devastating, while paying the premium is manageable. Risk aversion is why insurance exists.

Applications

Risk aversion explains many financial choices:

Insurance: people pay more than the expected loss to avoid catastrophic outcomes.
Diversification: spreading savings across many investments reduces the chance of a large loss.
Returns on investments: riskier investments, like shares, must offer higher average returns than safe ones, like government bonds, to attract risk-averse investors.
Job choices: many people accept a lower but stable salary over a higher but uncertain income.

Behavioural economists add that people are not always consistently risk averse. Many buy lottery tickets, which have a negative expected value, while also buying insurance.

Assuming the option with the highest expected value is always best

Expected value tells you the average outcome, not what will happen to you. When a bad outcome would be devastating, choosing a safer option with a lower expected value can be perfectly rational.

Key takeaways

Expected value is the probability-weighted average of possible outcomes.
Most people are risk averse, preferring a certain amount to a gamble with the same expected value.
Diminishing marginal utility explains risk aversion.
Risk aversion explains insurance, diversification and the higher returns on risky investments.