What it means
Imagine being offered a coin toss that pays either $160,000 or $40,000. The average payout is $100,000, but most people do not value the extra dollars as much as they dislike losing the first ones.
Expected utility theory captures this by replacing money with a utility score, which measures how much a given amount of wealth is worth to the person. The key idea is diminishing marginal utility: each extra dollar adds a little less satisfaction than the one before.
When this is true, the utility curve bends, and a person is risk averse, meaning that they prefer a certain amount to a gamble with the same average. If the curve is straight, the person is risk neutral, and if it bends the other way, they are risk seeking.
To use the method, list every possible outcome, give each a probability, convert each outcome to utility, and then take the probability-weighted sum. The option with the highest expected utility is the one the person should prefer.
A common simple utility function is the square root of wealth, which is easy to calculate and has the right shape. The approach is the foundation of much of modern finance and insurance.
It explains why people buy insurance, why investors demand a higher return for riskier assets, and why portfolio theory focuses on both return and risk. The gap between the average value of a gamble and the certain amount that gives the same utility is known as the risk premium.
The nuance is that real people do not always behave in line with the theory. Behavioural research shows that people overweight small chances, dislike losses more than they like gains and respond to how choices are framed.
Expected utility is therefore best treated as a useful benchmark for rational choice, and not as a perfect description of what happens.
In practice
Real-world examples.
Example
A founder is offered $90,000 for her small business today, or a deal that pays $160,000 or $40,000 depending on future sales. Using expected utility she realises that the certain offer is just as attractive to her as the risky one, even though the risky deal has a higher average payout. She accepts the certain sum.
Example
An insurance company prices a policy for a homeowner. Because the homeowner is risk averse, he will pay more than the expected loss to avoid a rare but large bill. The difference between the premium and the expected loss is the insurer's margin.
Example
A pension fund compares two investment strategies with the same average return but different volatility. Trustees who care about the members' retirement security choose the steadier strategy, because the extra swings reduce expected utility even when the average return is identical.
Formula
Calculation
Expected utility = Sum of (Probability of each outcome x Utility of that outcome)
Suppose utility equals the square root of wealth, and a person faces a 50% chance of $160,000 and a 50% chance of $40,000. The utility of $160,000 is 400, and the utility of $40,000 is 200, so expected utility = 0.5 x 400 + 0.5 x 200 = 300. A certain amount with a utility of 300 is 300 x 300 = $90,000, so that is the certainty equivalent. The expected money value is 0.5 x 160,000 + 0.5 x 40,000 = $100,000, so the risk premium is 100,000 - 90,000 = $10,000.Case study
Seen in the real world.
Halcyon Orchards is an illustrative, fictional fruit grower whose owner must decide whether to buy frost insurance. Without insurance, a 20% chance of a severe frost would cut her income from $200,000 to $80,000.
The expected income without insurance is 0.8 x 200,000 + 0.2 x 80,000 = $176,000. The insurance costs $28,000 a year and guarantees an income of $172,000 after the premium whatever the weather. On average she is better off uninsured, because 176,000 is more than 172,000.
Using a square root utility, however, the uninsured outcome has an expected utility of 0.8 x 447.2 + 0.2 x 282.8, which is about 414.3, while the insured outcome has a utility of about 414.7. In this illustrative case the insured option scores slightly higher even though its average income is $4,000 lower, so she buys the policy, and the lesson is that a small premium can be worth paying to remove a large swing in income.
Watch out
Common mistakes.
- Treating expected utility as the same thing as expected value, when expected utility accounts for attitudes to risk and expected value does not.
- Assuming that a utility score is a measurable amount of money, when it is a way of ranking preferences.
- Believing that everyone has the same utility function, when risk attitudes differ between people and between situations.
Questions
People also ask.
Why do people buy insurance if it costs more than the expected loss?
Because a large loss hurts more than a small premium, so the certain small cost gives higher expected utility than the gamble.
What is a certainty equivalent?
It is the guaranteed amount that gives the same utility as a risky choice, and the gap to the expected value is the risk premium.
Does expected utility predict real behaviour perfectly?
No, behavioural studies show people often depart from it, so it is used as a benchmark rather than as an exact description.
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