The Average Is the Wrong Question

Expected value is the first serious idea in probability and the one most often applied where it does not belong. Take each outcome, multiply by its probability, add them up. The result is the average you would get over many repetitions.

For a decision you will make thousands of times, with stakes small enough that no single one matters, it is exactly the right guide. A casino uses it, an insurer uses it, a market maker uses it. All three are running the same play: a small edge, repeated enormously often, with no single event large enough to matter.

For a decision you will make once, it can be almost irrelevant.

Consider a bet with a ninety-nine per cent chance of doubling your money and a one per cent chance of losing everything. The expected value is excellent. Take it repeatedly and you are certain to be wiped out; the only question is when. Averages assume you survive to collect them, and ruin is the state in which you stop playing.

This is why leverage destroys funds whose underlying strategies were sound on paper. Long-Term Capital Management had two Nobel laureates and a strategy with a positive expectation, and in 1998 it lost most of its capital in four months because the leverage meant an unusual few weeks was enough to end the game. It is also the content of the Kelly criterion, which sizes bets by reference to what you can afford to lose rather than by what you expect to win, and which recommends much smaller positions than the expectation alone would.

The shape of the distribution is not a refinement on the average. It is frequently the whole story.

Human heights are what statisticians call thin-tailed. Nobody is ten times the average height, a sample of a thousand people tells you what the population looks like, and one more observation cannot move the mean appreciably.

Wealth is not like that. Nor are city sizes, insurance losses, book sales, war deaths, or pandemic mortality. In these, a single observation can exceed the sum of everything else in the sample. Add the richest person in the world to a room of a thousand and the average wealth is now a fact about him and nothing else.

In a fat-tailed world the sample average is unstable, thirty quiet years tell you very little about the bad one, and the usual statistical reflexes (take the mean, quote a standard deviation, assume the next observation resembles the last) quietly stop working. Catastrophe modellers therefore work with the tail directly rather than with averages, and the phrase 'in an average year' is a warning sign in any field whose distribution has a tail.

Which brings us to insurance, and to the fact that it is a bad bet almost everyone should take.

The premium exceeds the expected payout. It has to: that difference is the insurer's costs and profit, and without it there would be no insurer. So on expected value alone you should never insure anything.

But you are not buying an average. You are converting a small probability of a loss you could not absorb (the house, the liability claim, the decade of earnings) into a certain small cost you can. The shape of your exposure has changed, and nothing in the expected value records that.

The same reasoning tells you when not to insure. The extended warranty on a washing machine is a loss you can survive, so paying someone a margin to carry it is simply paying a margin. The rule is not 'insure' or 'don't insure'; it is 'insure what would end you, self-insure what would annoy you'.

Bernoulli got here first, in 1738, with the St Petersburg game. It pays two pounds if the first head comes on the first toss, four on the second, eight on the third, doubling indefinitely. Each term contributes a pound to the expectation, so the expected value is infinite, and nobody will pay fifty pounds to play.

His resolution was that the value of money is not linear (the millionth pound is worth less to you than the first) so the utility of the game converges even though its monetary expectation does not. That is the beginning of utility theory, and it is the formal version of the thing everyone already knows: a pound is not the same size to everyone, and an average is not a decision.