The volume just closed priced time; this one prices risk. Where The Rate asked what an hour of waiting is worth, The Premium asks what it is worth to bear the uncertain — the extra return demanded by those who hold what might rise or fall against them. And as that volume began with the plain fact that a sum today outweighs the same sum tomorrow, this one begins with a fact no less plain: that a certain sum is preferred to a gamble of the very same average. Offered a sure fifty, or the toss of a coin between nothing and a hundred, most will take the fifty, though the two are worth, on average, exactly alike. This preference for the certain over the merely equal-on-average is risk aversion, and it is the foundation of the entire price of risk. This plate will set out the fact and its cause — the diminishing worth of each added dollar, by which a loss looms larger than an equal gain; will show that the worth of a gamble is therefore not its average outcome but its average worth; will trace the two faces of the same dislike, the price the fearful pay to be rid of risk and the reward they must be paid to bear it; and will plant the volume's question at its root: that because risk is feared, the bearing of it is a real service, earned — but whether the price the world pays for that service is just, the volume must spend eleven further plates to weigh.
Begin with the fact itself, in its starkest form. A man is offered a choice: he may take fifty dollars, certain, or he may toss a fair coin, taking a hundred if it falls heads and nothing if it falls tails. The two are identical in their mathematical average — fifty dollars either way, for the gamble's expected value is half of a hundred plus half of nothing. Yet put the choice to a room of people and the great majority will take the certain fifty, and many will take a certain forty, or thirty, rather than face the coin. They are not being irrational; they are revealing something true about themselves — that they dislike the risk itself, the spread of the possible outcomes, and will give up real expected value to be rid of it. The most a risk-averse man will accept with certainty in place of a gamble — his "certainty equivalent" — lies below the gamble's average; and the gap between them is the measure of his aversion, the toll he will pay for calm. This is risk aversion, and almost everyone, in matters that touch them nearly, has it.
Why should this be? The answer, given first by Daniel Bernoulli three centuries ago, is that a dollar is not worth the same to a man at every level of his wealth: the more he has, the less an added dollar means to him. The hundredth dollar matters more than the thousandth; the thousandth more than the ten-thousandth. Wealth, in the economist's phrase, has a diminishing marginal utility, and the curve that plots its worth against its amount bends over, concave, rising ever more gently as it climbs. Now see what this does to the coin-toss. The hundred dollars won adds a certain quantity of worth; but the fifty foregone — the fall from the certain fifty to nothing — subtracts a larger quantity, because the dollars one loses are worth more, lying lower on the bending curve, than the dollars one might gain. The downside, dollar for dollar, outweighs the upside. And so the gamble's expected worth falls short of the worth of its average, and the certain fifty, worth its full fifty, is preferred. The worth of a gamble, it turns out, is not its expected value but its expected utility; and on a concave curve the two part ways, always to the gamble's disadvantage. This is why the volume must rest, at its base, on the idea of utility that another primitive will take up under its own name.
The dollars one stands to lose are worth more than the dollars one might gain; and so, on the bending curve of utility, the downside always outweighs the equal upside.
It is worth pausing on what this overturns. One might suppose, before reflection, that a gamble is simply worth its expected value — that a one-in-a-thousand chance at a thousand dollars is worth a dollar, flat, and a fair bet worth its stake. This is the natural first thought, and it is wrong, as risk aversion shows and as a famous paradox, which the next plate takes up, made impossible to ignore: a gamble can be constructed whose expected value is infinite — whose average payout, properly summed, exceeds any sum you care to name — and yet for which no sane person would pay more than a few coins. If expected value were the measure of worth, one ought to stake one's whole fortune on such a gamble; that no one will is the proof that expected value is not the measure. The worth of the uncertain is governed not by its average outcome but by the worth of its outcomes to the one who bears them — by expected utility, weighted by a fear that grows as the stakes grow. Risk, the volume's first lesson runs, is always more than its average; and the difference is the space in which the price of risk lives.
From this one dislike spring two faces, which look opposite and are the same. The first is the willingness to pay to be rid of risk. A man who owns a house he could not bear to lose will pay a yearly sum to an insurer to carry that risk for him — a sum larger, and knowingly larger, than the average loss he is insuring against, for he will pay the average loss plus something more to convert an uncertain calamity into a certain small fee. That something more is the price of risk, paid. The second face is the demand to be paid to bear risk. A man choosing between a safe government bond and a volatile share will not hold the volatile share unless it offers him a higher expected return — enough more to compensate him for the spread of outcomes he must endure. That something more is the price of risk, received. The insurance premium and the risk premium are one phenomenon seen from its two sides: because risk is disliked, shedding it costs, and bearing it pays. The whole architecture of this volume — the pooling of fears, the premium on the risky asset, the reward of the equity holder — rises from this single root. Risk aversion is the reason risk has a price at all.
And here the volume's question makes its first appearance, in the form of an answer. For if risk is genuinely disliked — if the certain is genuinely preferred to the equal-on-average — then the one who agrees to bear risk performs a real service. The insurer who takes the householder's dread of fire upon himself, the investor who holds the volatile share the cautious will not touch, the entrepreneur who stakes his capital on a venture that may come to nothing — each takes upon himself what the risk-averse will pay to be free of, and relieves another of a burden that genuinely weighs. To bear risk, on this account, is not to do nothing and collect; it is to render a service, the service of carrying fear, and the premium is its wage. This is the volume's "earned" answer at its very foundation, the counterpart to the earlier volume's finding that to wait is a real sacrifice: here, to bear uncertainty is a real service, and the reward for bearing it is, at its root, the honest wage of providing calm to those who dread the storm. The premium begins its life as something earned.
To bear risk is not to do nothing and collect; it is to render a service — the carrying of a fear that genuinely weighs — and the premium is its wage.
But it does not stay so simple, and the volume's question is whether it stays so at all. For two shadows fall across the clean picture of the premium as the earned wage of bearing fear, and the eleven plates to come are largely the chasing of them. The first is that risk aversion, though real, is a fact of human psychology, and human psychology is not always the cool reckoner the picture assumes; people fear in ways that do not match the odds, dread some losses far out of proportion to their size, and the premium the market actually demands may be larger — much larger, as a later plate will show the equity premium to be — than any sober weighing of risk could justify. If the premium is swollen by panic beyond the price of the service, the excess is not earned but exacted by fear. The second shadow is luck. Much of what looks, after the fact, like the just reward of brave risk-bearing is merely the winnings of those who gambled and happened to survive — and the failed gamblers, who bore the very same risk and lost, are not there to be counted. How much of a great fortune is the wage of courage, and how much the mere residue of survival, is the question the volume will end upon. So the premium begins as earned — the honest wage of carrying fear — but whether the price the world actually pays for risk is the just price of the service, or is swollen by panic and gilded by luck, is the contest this volume opens. As with every price in the table, the answer will not be single.
So the price of risk is begun. Its foundation is the fact set down here: that people prefer the certain to the equal-on-average, because the dollars they might lose are worth more than the dollars they might gain, and so the worth of a gamble falls below its average and risk acquires, for everyone who feels its weight, a price. From this single dislike spring the two faces — the premium paid to shed risk and the premium earned to bear it — and from them the whole volume rises: the paradox that first forced the reckoning, the pooling of many fears into one calm average, the premium on the risky asset, the strange free lunch of diversification, the single risk the market deigns to pay for, the puzzle of a premium too large, the profit that rewards the unforeseeable, the rare catastrophe the averages hide, and, at the last, the failed gamblers we never see and the question of how much of any fortune is courage and how much survival. The bearing of risk begins as a service honestly waged; whether it ends so is the argument. As the table says of every price, first to last: the argument remains open.