Expected value (Erwartungswert)

Level 2 · Foundations

German termErwartungswert

Read firstProbability basics

Used bySpeculation vs. investing · Risk pooling · Risk of ruin / catastrophic risk · Self-insurance

Expected value (Erwartungswert, mathematical expectation) is the probability-weighted average of every possible outcome of an uncertain event: multiply each outcome by its probability and add the results. That number is what a repeated bet, insurance policy, or lottery pays per round on average, not what any single round pays.

Why it matters

Insurers price premiums above the expected loss; the difference covers claims handling, capital reserves, and profit. Lotteries price tickets below the expected payout for the same reason in reverse — the operator keeps the gap. Comparing a price to the expected value tells you which side of that gap you are standing on.

But expected value has a limit: a positive expected value can still be a bad bet for a household that cannot survive the worst-case outcome. A gamble with excellent average returns is irrelevant if one unlucky draw wipes out savings, housing, or income needed to keep functioning. This is the core reason catastrophic risks — fire, liability, serious illness — get insured even though the insurer's expected value calculation guarantees the buyer loses on average.

Worked examples

1. A lottery ticket. A ticket costs €10 and offers a 1-in-1,000,000 chance of winning €1,000,000, with no payout otherwise. Expected value: €1,000,000 x (1 / 1,000,000) = €1. The ticket's expected value is €1; the price is €10. Buying it is a bet with negative expected value by €9 — the ticket transfers €9 on average from buyer to operator, in exchange for a small chance at a large sum.

2. An insurance framing. A household faces a 1-in-200 chance per year of €20,000 in storm damage to a home, and no damage otherwise. Expected loss: €20,000 x (1 / 200) = €100 per year. An insurer offers coverage for a €150 annual premium — €50 above the expected loss. Judged purely by expected value, the premium is a bad bet: paying €150 to avoid an average loss of €100 loses €50 a year on average.

Judged by ruin risk, the calculation looks different. The uninsured household absorbs the full €20,000 in the unlucky year — a sum that may force selling assets, missing other payments, or taking on debt at a bad time. The insured household absorbs a predictable €150 every year instead. Paying above expected value converts a rare, large, disruptive loss into a small, certain, budgetable one. Whether that trade is worth it for a given household depends on how much a €20,000 hit in a single year would actually cost beyond the €20,000 itself — in missed rent, forced sales, or debt at a bad rate — not on the expected-value comparison alone.

Why insurance still makes sense: expected utility

Expected value counts euros as if losing the tenth thousand hurts exactly as much as losing the first. Households don't experience money that way — the first €20,000 of savings, the part that covers rent and food, matters far more than the twentieth. Economists capture this with expected utility: instead of averaging the euro outcomes, average the value each outcome has to the household, using a curve that rises fast at low wealth and flattens as wealth grows — a concave curve, such as log utility.

Under that curve, a rare €20,000 loss destroys far more value than its €100 expected loss suggests, because it lands when each euro is scarce. Paying a €150 premium costs only a little value, given up from a comfortable position where euros are worth less at the margin. So a bet that loses money on average — negative expected value — can still raise expected utility. That is the formal version of the ruin-risk point: insurance is rational not despite the negative expected value, but because averaging in utility, not euros, is the right measure when one bad outcome would be catastrophic.

Check yourself

A raffle ticket costs €5. It has a 1-in-500,000 chance of winning €2,000,000, and pays nothing otherwise. What is the expected value of the ticket, in euros?

A gamble has a positive expected value but a 1-in-20 chance of losing an amount the household cannot replace — for example, its entire emergency fund and a month's rent. Does the positive expected value make this an acceptable bet?

Yes — a positive expected value means the bet is worth taking regardless of the size of the possible lossNot necessarily — a loss the household cannot recover from can make an otherwise positive-expected-value bet unacceptableNo — any bet with a possible loss should be avoided

A household faces a 1-in-200 annual chance of €20,000 in storm damage (expected loss: €100/year) and an insurer charges €150/year for coverage — €50 above the expected loss. Does the premium being above the expected loss automatically mean the policyholder is being ripped off?

Yes — a fair price should equal the expected loss, so any premium above it is a rip-offNo — paying above the expected loss can still be reasonable, because it converts a rare, large, disruptive loss into a small, predictable one

A household pays a €150 annual premium to insure against a rare loss whose expected value is only €100 a year. By expected value alone the policy loses €50 a year — why can buying it still be rational?

Because expected utility weighs a catastrophic loss far more heavily than its euro-average, so insuring can raise expected utility even when expected value is negative.Because the insurer's expected-value calculation must be mistaken.Because premiums are always returned in full over enough years.