Plate iii in print

The Pool

insurance and the law of large numbers: how pooling turns the unforeseeable fate of the one into the near-certain average of the many — and where the pooling breaks.

What is wildly uncertain in the single case becomes nearly sure in the aggregate; the more numerous the trials, the more the average is governed by the true probability and the less by chance.

— the principle of this plate, after Jacob Bernoulli, 1713

If risk has a price because individuals are averse to it, the first great response is not to bear it alone but to share it — and the institution of sharing, insurance, rests on a mathematical fact that turns what is wildly uncertain in a single life into something very nearly certain across many.

The fact is the law of large numbers, proved by Jacob Bernoulli. The fate of one house — whether it burns this year — is deeply uncertain; but the fraction of ten thousand similar houses that burn is, year upon year, remarkably steady and close to the true probability of fire. As the number of independent cases grows, the average outcome converges on its expectation and the relative fluctuation shrinks toward nothing. What cannot be foreseen for the one can be foreseen, with precision, for the many.1

1. The law of large numbers (Jacob Bernoulli, Ars Conjectandi, 1713): the average of many independent trials converges to its expected value, and the relative variability falls as the number of cases rises. The individually unforeseeable becomes collectively predictable.

On this rests insurance. A thousand people each face a small chance of a large loss — a fire, a death, a wreck — which any one of them, being risk-averse, would pay dearly to avoid. Let each instead pay a small, certain premium into a common fund; the unlucky few who suffer the loss are paid out of it; and because the number of losses is, by the law of large numbers, nearly certain in advance, the fund can be set to cover them. Each person has exchanged an uncertain large loss for a certain small payment — exactly the trade a risk-averse person, who values a gamble below its expectation, is glad to make. Risk has not been destroyed; it has been pooled, spread across so many that each now bears a sliver where before there loomed a catastrophe.2

2. The insurance principle: pooling independent risks lets each member trade an uncertain large loss for a certain small premium, and the law of large numbers makes the aggregate loss predictable, so the pool stays solvent. Each insured gains because, being risk-averse, he values the certain premium below the expected loss it covers.
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What, then, is the insurer paid for? In part for its administration and its capital; but in essence the premium it charges exceeds the expected loss by a margin, and that margin is the price of risk — paid now not by the insured, who has shed his risk, but to whoever stands behind the pool and guarantees it against the chance that the losses run beyond the fund. Insurance does not abolish the premium met in the first plate; it relocates it. The risk-averse individual pays a little above the actuarially fair price to be rid of his risk; the insurer, better able to bear what remains — because pooling has made the aggregate nearly certain, and because its capital can absorb the residual swing — collects that margin as the reward for bearing it. Risk flows from those who fear it to those equipped to hold it, and the premium is the toll levied along the way.3

3. The insurer's margin over the expected loss is the price of risk, relocated from the risk-averse insured to the risk-bearing pool. Pooling reduces but never wholly eliminates aggregate risk — losses can still exceed the fund — and the insurer's capital bears that residual in exchange for the margin.

But the magic works only under a condition easily forgotten: that the risks be independent. The law of large numbers tames the average only when the cases are uncorrelated — when one house burning does not make the others more likely to burn. Against correlated risks — an earthquake that levels a whole city, a panic that strikes every borrower at once, a pandemic that sickens all at the same season — pooling fails, because the losses arrive together and overwhelm the fund; there is no safety in numbers when the numbers all fall at once. This is why the deepest risks, the systemic ones, cannot be insured away in the ordinary manner, and why a later plate must distinguish the risk that pooling and spreading can destroy from the risk that no sharing can touch.4

4. Pooling requires independence; correlated or systemic risks — earthquakes, panics, pandemics — defeat it, since the losses arrive together and swamp the fund. The distinction between diversifiable and undiversifiable risk, central to the later plates, begins here. Compare The Basket and The Systematic.

Insurance has a second vulnerability, of an altogether different kind, which the volume on asymmetric information will pursue: the pool depends on knowing whom it insures, and on their conduct not changing once they are insured. If those likeliest to suffer loss are the keenest to buy — the ailing seeking health cover, the reckless seeking liability cover — the pool fills with bad risks and the premium must climb, driving the good risks out; and if being insured makes people careless, the losses themselves swell. These are not flaws in the mathematics of pooling but in the information behind it, and they mark the boundary of what insurance can do.5

5. Insurance is undone not only by correlation but by information: adverse selection (the worst risks the most eager to insure) and moral hazard (insurance dulling the care that would prevent loss). Both belong to the economics of asymmetric information — see the forthcoming volume The Asymmetry.

Insurance is the first and greatest institution built upon the price of risk: it lowers no one's aversion, but it lets the averse hand their risk to those who, through pooling and capital, can bear it more cheaply, and the premium is what changes hands. Yet pooling has a hard limit — it tames only independent risks, and shatters against correlated ones — and a second response to risk works from the opposite side: not by sharing one risk among many people, but by splitting one fortune among many risks. The next plate takes up diversification, and finds that it can do something pooling cannot — destroy a whole kind of risk outright, and for free.