Two plates have laid the ground: that the fearful will pay to be rid of risk, and that the worth of a risk lives in the one who bears it, where an average swollen by rare extremes can badly mislead. This plate raises upon that ground the oldest and most benign of all the structures built to bear risk — insurance — and lays bare its engine, which is one of the deep facts of the world. The fact is this: that although no man can know whether his own house will burn this year, the fraction of a great many houses that will burn is known almost to a certainty, and known the more exactly the more houses are counted. This is the law of large numbers, and it is the quiet miracle on which the whole institution rests. For it means that an insurer who gathers a thousand separate and unconnected dreads of fire takes on, in the aggregate, almost no uncertainty at all: the individual fears are wild, but their sum is calm. This plate will set out that law and the alchemy it permits — the melting of many private storms into one public calm; will show how the insurer sells each of the fearful the certainty he craves while himself bearing little net risk, and what he charges for it; will draw the lesson that matters most for the volume, that the insurer is paid less for bearing risk than for the service of reducing it; and will mark the two places the calm fails — where the risks move together and the law of large numbers deserts the pool, and where the insurer cannot see whom or what he insures, the shadow the next volume must weigh.
Begin with the law itself, for everything follows from it. Toss a single fair coin and the outcome is beyond prediction — heads or tails, an even mystery. Toss ten thousand and the fraction that fall heads will lie so close to one half that one might stake a fortune on it; toss a million and closer still. The individual event is wild; the average of many is tame, and grows tamer without limit as the number climbs. This is the law of large numbers, proved by Jacob Bernoulli — uncle to the Daniel of the previous plate — and set down in his Ars Conjectandi of 1713. Its bearing on risk is total. No one can say whether a particular man of fifty will live to see sixty, or whether a particular ship will founder, or a particular house burn; each is a coin-toss whose fall is hidden. But the proportion of ten thousand such men who will reach sixty, of ten thousand such ships that will sink, of ten thousand such houses that will burn, is a quantity the actuary can name in advance and be scarcely wrong. The misfortune of the one is unknowable; the misfortune of the many is all but fixed. And in that gap — between the wild particular and the calm aggregate — the whole business of insurance lives.
See now what the law permits a clever party to do. A thousand householders each face the same small chance of a ruinous fire, and each, being averse to risk, would gladly pay something to be free of the dread. An insurer steps among them and makes each the same offer: pay me a modest sum each year, certain and small, and I will pay your loss, large and uncertain, should the fire come. From each he collects a premium; to each he extends a promise. Singly, the promises are terrifying — any one house might burn, and the insurer owe a fortune. But taken together, by the law of large numbers, the thousand promises resolve into a near-certainty: very close to the expected number of houses will burn, no more and no fewer, and the insurer's total payout is a figure he can foresee and has priced for. He has performed an alchemy. He has taken a thousand wild and separate uncertainties and melted them into one steady aggregate; he has sold each householder the calm he craved, converting an uncertain calamity into a certain small fee; and he himself, having pooled them, bears almost no net risk at all, for what he must pay out scarcely varies from what he expects. The risk has not been abolished — someone's house still burns — but it has been dissolved, spread so thin across so many independent shoulders that in the aggregate it scarcely trembles.
The insurer who carries a thousand separate dreads of fire need fear no fire at all, for the aggregate is steady where each part is wild.
What does the insurer charge for this service, and why will anyone pay it? The premium he sets is not the bare expected loss — not merely the average payout divided among the policyholders — but that figure plus a margin, which the trade calls the loading: an addition to cover his costs of doing business, the capital he must hold against bad years, and his profit. The premium, then, is dearer than the risk it covers is worth on average; the householder pays more, in expectation, than he is likely to lose. And yet he pays it gladly, and is right to — for here the first plate's lesson returns. Being averse to risk, he values the certainty he buys at more than its actuarial cost; the dread he sheds weighs on him more heavily than the small sure premium, and so the exchange leaves him better off though he pays above the odds. Both parties gain. The householder gains the calm he prizes above its price; the insurer gains the loading, the difference between what fear will pay for certainty and what certainty costs him to supply. The premium is the price of risk, paid — and it is a price both sides are content to strike, because risk aversion makes the shedding of risk worth more to the one than the bearing of it costs the other.
And here the volume's question meets, in insurance, an answer it will not meet so cleanly again. Is the insurer's gain earned, or is it a toll exacted upon fear? It is earned — but the manner of its earning is worth weighing closely, for it is not what it first appears. One might suppose the insurer is paid for bearing risk: that he takes the householder's dread upon himself, and is rewarded for the courage of shouldering it. But this is precisely what the law of large numbers shows to be false. The insurer, having pooled a thousand independent risks, bears almost no risk at all; his aggregate is calm, his payout foreseeable; he is not brave but, if anything, in a position of remarkable safety. What he is paid for is not the bearing of risk but the reducing of it — the alchemy itself, the service of gathering a thousand wild uncertainties into one tame aggregate and handing each man back his certainty. The loading is the wage of that service: of the pooling, the capital, the administration, the actuarial art. And this carries a lesson that will trouble the rest of the volume. For if the cleanest and most clearly-earned of all the rewards of risk — the insurer's — turns out to be the wage not of bearing risk but of reducing it, then we must ask, of every reward that calls itself the price of risk, whether it too is the quiet wage of some service that shrinks the risk it claims to bear, rather than the heroic compensation for shouldering the irreducible.
The insurer is paid not for his courage but for his arithmetic — for the alchemy of the pool, which reduces the very risk it seems to bear.
But the alchemy has a condition, written into the law that permits it, and where the condition fails the calm shatters. The law of large numbers tames an average only when the events composing it are independent — when one man's misfortune tells you nothing about the next man's. House fires are very nearly so: that one house burns does not make its neighbours more likely to. But many of the gravest risks are not independent at all; they are correlated, striking great numbers together at a single stroke. A flood drowns ten thousand houses at once; an earthquake levels a city; a pandemic sickens a nation; a financial crash ruins every holder of the same assets in the same week. Against such risks the pool is helpless, for there is no calm aggregate to retreat to: the claims do not arrive in a steady trickle whose sum is foreseeable but in one overwhelming wave that no premium was priced to meet. This is why floods and earthquakes and plagues sit at the very edge of insurability, why insurers must themselves reinsure and governments stand behind them as the bearer of last resort, and why the correlated catastrophe is the true home of irreducible risk. Here, where pooling cannot help, someone must bear the risk that will not be dissolved — and the reward for bearing that, unlike the insurer's tame loading, is the genuine and contested wage of courage the volume must still weigh. The calm of the pool is bought with independence, and ends where independence ends.
A second trouble haunts the pool, and it springs not from the correlation of risks but from the insurer's blindness to them. The law of large numbers assumes the insurer knows what risks he is gathering; but he cannot see into his applicants, and two maladies grow in the dark. The first is adverse selection. The insurer, unable to tell the careful from the reckless, must charge a single premium pitched at the average; but to the low-risk that premium is dear and to the high-risk it is a bargain, so the cautious decline the cover and the endangered rush to buy it. The pool fills with bad risks, the average loss climbs, the premium must rise to meet it, and the rise drives out yet more of the good — a spiral that, unchecked, can empty the pool of all but the doomed and bring the market down. The second malady is moral hazard. A man who has insured his house against fire takes, ever so slightly, less care to prevent one; the driver insured against collision drives a shade less warily; for the cost of the mishap now falls on another. Insurance, by shielding men from the consequences of risk, can quietly increase the very risk it covers. Both maladies have a single root — that the insurer cannot observe what his policyholders are and do — and both are the proper subject of another primitive entirely, the asymmetry of information between the two sides of a bargain, which the table must take up under its own name. The insurer fights them as he can, with medical examinations and risk-classes, with deductibles and co-payments and the fine print of exclusions; but he never wholly wins, for he is fighting his own blindness. The gathering the pool achieves is forever threatened by the unseen.
So the pool gives the price of risk its clearest and kindest form. The insurer gathers a thousand private dreads, and by the law of large numbers turns them into one calm average; he sells each of the fearful the certainty they prize, and earns his loading as the honest wage of the service — a service, the volume has been careful to note, not of bearing risk but of reducing it, the alchemy of the pool rather than the courage of the lone shoulder. In this insurance is the most benign of all the things this volume studies, and the most plainly earned. But its calm is bounded twice over. It breaks where the risks move together, in the flood and the plague and the crash, where no pool can form and the irreducible risk must find a bearer whose reward is far harder to judge. And it frays where the insurer cannot see, in the adverse selection that poisons the pool and the moral hazard that loosens care, the shadows the asymmetry of information casts across every contract written. The premium is earned, then, where it is the wage of pooling — of reducing the reducible. What of the risk that cannot be pooled away, the risk that someone must simply bear? That is the question the rest of the volume pursues: the reward demanded for holding the uncertain, the single risk the market will pay for, the puzzle of a reward grown too large, and at the last the failed gamblers we never see. The pool tames what it can; the untameable remainder is the price of risk in its hardest form. As the table says of every price, first to last: the argument remains open.