Why should risk command a price at all? Why should a venture that might pay handsomely or pay nothing have to offer more, on average, than a sure thing of the same expected value — why should anyone have to be paid simply to bear uncertainty? The answer was found, before economics was a discipline, in a gambling puzzle that broke the most natural theory of value there was.
The puzzle is the St. Petersburg paradox, posed by Nicolaus Bernoulli and resolved by his cousin Daniel in 1738. A fair coin is tossed until it first comes up heads; the prize doubles with every toss it takes to get there — two ducats, then four, then eight, and on. The chance of each further toss halves just as the prize it pays doubles, so every possible outcome contributes the same small slice to the expected value, and the sum of these equal slices, running on without end, is infinite. By the reigning doctrine — that a gamble is worth its mathematical expectation — a rational person should stake his entire fortune, any finite sum whatever, for the right to play. Yet no one would offer more than a trifle. The theory of value as expectation, set before this game, returned an answer that no sane person would accept.1
Daniel Bernoulli's resolution founded the modern theory of risk. What a person values, he argued, is not the money but the use of the money — its utility — and the utility of wealth rises ever more slowly as wealth grows: a ducat means more to a pauper than to a prince. This is the diminishing marginal utility met in the volume on the want, encountered here as the engine of risk. A rational person therefore weighs a gamble not by its expected money value but by its expected utility; and because the utility of the colossal prizes in the St. Petersburg game climbs so slowly, their expected utility is finite and modest — and so is the sum anyone will pay to play.2
In that resolution lies the whole reason risk has a price. If utility rises but at a diminishing rate — if the curve of utility against wealth is concave, bending over as it climbs — then a gain of a given size adds less utility than a loss of the same size subtracts. A fair gamble, one whose expected money value is exactly zero, therefore carries negative expected utility: the prospect of gain does not compensate, in felt utility, for the equal prospect of loss. A person whose utility is concave will thus decline a fair gamble, and will pay something to be rid of risk. This is risk aversion, and it follows directly from diminishing marginal utility — the very curvature that makes a second loaf matter less than the first makes the possible loss of a gamble loom larger than its equal possible gain.3
From risk aversion the premium follows almost as a matter of definition. Take a risky prospect and ask: what certain sum would leave a person exactly as well off, in utility, as facing the risk itself? That sum is the gamble's certainty equivalent. A risk-averse person values the gamble below its expected money value, so its certainty equivalent falls short of its expectation, and the gap between the two — the expected value minus the certainty equivalent — is the risk premium: the amount he would sacrifice, the discount he would accept, to be free of the uncertainty. Seen from the other side, it is the extra expected return a risky asset must offer over a safe one before he will hold it at all. The premium is the price of risk, and it is positive for precisely the reason risk aversion is: the curvature of utility.4
But notice already a feature the rest of the volume will press hard. The premium is not a property of the gamble alone; it depends on the curvature of the bearer's utility, and that curvature depends on his wealth and his circumstances. The same gamble — the very same distribution of money outcomes — carries a heavy premium for a poor man, in whom the possible loss bites deep, and a slight one for a rich man, who can absorb it; it might even be welcomed, at a discount, by one who happens to need exactly its pattern of payoffs. Risk is priced by the meeting of a distribution of outcomes with a particular bearer, and there is no single premium a gamble simply has, independent of who holds it. The price of risk is, at its very foundation, relational.5
The wager gives the volume its foundation: risk has a price because utility is concave, and the price — the premium — is the compensation a risk-averse bearer demands to face an uncertain outcome rather than a sure one of equal expectation. But the foundation already holds the seeds of the volume's deeper turns. If the premium depends on the bearer, then the price of risk is not one number but many. If it depends on the curvature of utility, it depends on something internal and hard to measure. And the gambles that matter most — the great ventures, the fortunes of a lifetime — are borne once, not repeated, so that what the premium compensates in expectation and what luck delivers in the event may bear little resemblance to each other. The plates that follow take up first how risk is tamed, by pooling and by spreading; then what, after all the taming, is left to be priced; and last, what becomes of the premium once the gamble is run, and only the survivors remain to tell of it.