Insurance
A Socratic walk-through of insurance — reasoned out one step at a time, not lectured.
The question we started with
THE QUESTION #Why is buying insurance sensible even though, on average, it loses money?
An insurer must take in more than it pays out. Claims are only part of its costs; there are also staff, capital, regulation and profit, and all of it comes from the gap between premiums collected and claims settled. So the average customer, by arithmetic and by design, hands over more than they get back.
That is not a scandal; it is the business model. But it does mean buying insurance is, in pure money terms, a losing proposition — and yet refusing to insure a house is obviously foolish. Something in the "on average" is doing less work than it appears to.
Reasoning it through
REASONING #First, how does the insurer manage to price the thing at all? Whether your house burns down this year is essentially unpredictable. Whether a hundred thousand houses do is not. Averaged over a large pool of roughly independent risks, the total settles close to its expected value, and the more policies there are the tighter that gets. The insurer is not clairvoyant; it manufactures predictability out of aggregation, and then sells certainty to people who cannot make it for themselves.
Now the buyer's side, which is the interesting half. We have been measuring the deal in pounds. Ask instead what a pound is for. The first thousand pounds of a household's money buys food and rent; the hundredth thousand buys something it would miss much less. Money's usefulness rises with the amount — but each additional pound adds less than the one before it.
Follow that where it leads. If the increments shrink as you get richer, losing a large sum costs you more usefulness than gaining the same sum would add. The two are equal in pounds and unequal in what pounds do. So a coin flip between plus and minus two hundred thousand is not a neutral proposition at all — it is a bad one, even though its average in money is exactly zero.
And now the reversal that answers the question. If a fair gamble is already worth less than its money value, then getting rid of a gamble is worth more than its money value. There is room between those two figures, and the premium lives in it. You pay a bit above the expected loss, and in exchange the distribution you face changes shape: instead of a small chance of something ruinous and a large chance of nothing, you face one modest, known, budgetable line item.
So the negative expected value is not a flaw in the argument. It is the argument. Insurance is not a bet you hope to win; it is a purchase, and what you are purchasing is the removal of the tail.
Which yields the test for whether a particular policy is worth buying. Not "will I probably claim?" — you hope not. The question is whether the loss, if it came, would land where the curve is steeply bent. Over a range you could absorb without changing how you live, that curve is essentially straight, there is no curvature to sell back to you, and the insurer's loading is simply a cost. That is the honest case against insuring a phone or extending a warranty on a kettle — and the case for insuring the house, the liability, and the loss of your income.
The analogy
THE ANALOGY #A climber carries a rope. It costs money, it costs weight, and on almost every climb it does absolutely nothing. Averaged over a career, the rope is a straight loss. It is carried anyway, because the outcome it addresses is not one you get to average over — you only need to lose that argument once.
a rope prevents the fall, whereas a policy prevents nothing and only pays afterwards, and money cannot restore everything a serious loss takes; a rope's cost is also fixed and honest, while a premium is priced off your own risk and can be loaded far above what the protection is worth.
Clarifying the model
THE MODEL #Some refinements, including the ones that limit the argument.
The curve above is a modelling device, not a measured object. Nobody has a utility function written down, and economists infer its shape from choices rather than observing it. But what the concavity encodes is something you can check against your own judgement: that you would not take a fair coin flip for a large fraction of your net worth. If that is true of you, the rest follows.
There is also a blunter version of the point that does not need a smooth curve at all. Some losses do not merely reduce your wealth; they put you in a state you cannot get back out of — bankruptcy, the loss of a home you cannot re-buy, a liability judgement larger than everything you own. Ruin is not a low point on a curve, it is a door that shuts, and avoiding it is worth paying for whatever your attitude to ordinary risk.
Two honest caveats about the other side of the transaction. Pooling only works on risks that are roughly independent. A flood, an earthquake or a pandemic strikes everyone in the pool at once, so the averaging that makes the insurer's numbers predictable simply fails — which is why those markets depend on reinsurance and state backstops, and why some of them do not really exist. And the loading is not fixed: because people who know they are high-risk buy more eagerly, and because insured people take a little more risk, insurers price defensively. Where the loading is heavy enough, even a genuinely ruinous risk can be insured on terms not worth taking.
A picture of it
THE PICTURE #How to readThe upper, bending line is how much use you get from holding a given amount — rising all the way, but flattening, so each extra thousand adds less than the last. The lower, straight line is the average usefulness of a coin flip between the two ends of the chart, £10,000 and £90,000. Look at the middle: at a certain £50,000 the curve reads about 3.9, while the gamble that averages £50,000 reads about 3.4. The gap between the two lines is the whole subject — it is what certainty is worth, and an insurer that charges less than that gap while covering its own costs leaves both parties better off. Where the curve is nearly straight, over losses small enough not to move you along it, the gap closes and there is nothing left to buy.
What became clearer
WHAT CLEARED #Expected value is the wrong yardstick for anything that could seriously hurt you, because pounds and what pounds do are not the same measure. Insurance is deliberately a losing bet in money and a winning trade in usefulness: the insurer builds predictability by pooling many independent risks, and sells you the difference between a ruinous tail and a fixed line in the budget. The corollary is the practical one — insure what would bend the curve, absorb what would not, and be suspicious wherever pooling quietly fails.
Where to go next
ONWARD #- Why the same reasoning recommends the highest excess you could comfortably pay.
- How insurers price risks they cannot pool, and what happens to flood cover when they cannot.
Key terms
TERMS #| Term | What it means |
|---|---|
| Expected value | the probability-weighted average outcome, measured in money; reliably negative for the buyer of any sustainable policy. |
| Law of large numbers | the result that an average over many independent draws converges on its expected value, which is what makes an insurer's total claims predictable. |
| Diminishing marginal utility | the principle that each additional unit of wealth adds less usefulness than the one before it. |
| Loading | the amount by which a premium exceeds the expected claim, covering the insurer's costs, capital and profit. |
Every term the collection defines is gathered in the glossary.