Plate ii in print

The Aversion

Arrow and Pratt, and the measure of risk aversion; how the dread of risk changes with wealth, and the rich man's steadier nerve.

How much a man would pay to be rid of a fair risk can be read from the curve of his utility — not from its bending alone, which the choice of scale makes arbitrary, but from the bending measured against the slope. That ratio is the local price of risk; and for most it falls as fortune grows, so that the gamble which frightens the poor man the rich one shrugs off — not, it may be, because he is braver, but because he stands where the curve has gone flat.

— the measure of this plate, after John Pratt, 1964

The Wager settled that risk has a price, and why: the utility of wealth is concave, a loss biting deeper than an equal gain, so that a fair gamble is a bad bargain and the averse must be paid to face one. But concavity is a yes-or-no fact. It tells us a man dislikes risk; it does not tell us how much. Two men may both decline a fair coin-flip, and one would give a dollar to be spared it while the other would give a hundred. To price risk at all, and to say how the price differs from one person to the next and shifts across a single life, the bare fact of a bending curve will not serve. The bending must be measured. The plate's question is how to measure it so that the number means something — and, once it is measured, whether the steadier nerve of the rich, which everyone has noticed, is the courage it looks like or something duller.

The obvious measure fails, and fails instructively. Take the curvature of the utility curve, the sharpness with which it bends. The trouble is that utility is fixed only up to a zero and a unit; one may rescale it freely, without altering a single preference, and the curvature rescales with it — so the bending alone reports the draftsman's choice of units as much as the chooser's mind, and two curves describing the very same man can show any curvature one likes. John Pratt, in 1964, and Kenneth Arrow, working the same ground independently a year later, found the measure that survives the rescaling: not the bending, but the bending divided by the slope — the rate at which marginal utility is falling, taken relative to its own height. This ratio, the coefficient of absolute risk aversion, is untouched by the arbitrary units, which cancel in the division; and it captures exactly what mere concavity was gesturing toward, namely how fast the next dollar is worth less than the last, at the wealth the man actually holds.1

1. The obvious measure — the curvature of the utility curve — fails: utility is fixed only up to a zero and a unit, so rescaling it (which changes no preference) rescales the curvature too. Pratt in 1964, and Arrow independently a year later, found the measure that survives: the bending divided by the slope — the rate marginal utility falls, relative to its height. This coefficient of absolute risk aversion is invariant to the units, which cancel, and captures how fast the next dollar is worth less than the last at present wealth.

The measure earns its keep at once, for it turns out to be, near enough, the price of risk itself. Pratt proved that the sum a man would pay to escape a small fair gamble — the premium he accepts in order to trade the gamble for its bare average — is approximately half the gamble's variance multiplied by his coefficient of absolute risk aversion. The coefficient is therefore no abstraction floating above behavior; it is the very factor that converts the spread of a risk into the price of being spared it. Double the variance, and the premium doubles; double the coefficient, and it doubles again. The bend of the private curve and the price asked in the open market are one quantity, seen twice.2

2. Pratt showed the risk premium — what a man would pay to escape a small fair gamble — is about half the gamble's variance times his coefficient of absolute risk aversion. The coefficient is thus the factor turning a risk's spread into the price of avoiding it: double the variance, or the coefficient, and the premium doubles. The bend of the curve and the price in the market are one quantity.
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Now the question the plate is named for can be put exactly. Does the dread of risk fall as a man grows rich? The millionaire bets sums that would sicken a clerk, and the measure confirms the impression: absolute risk aversion decreases with wealth, the rich facing larger dollar gambles than the poor with a steadier hand. But the measure also divides the impression in a way the unaided eye cannot. There are two questions hidden in it, not one. The first concerns gambles of a fixed size — a thousand at stake, win or lose — and here the rich are plainly the less averse, a fixed sum being a smaller thing held against a larger fortune. The second concerns gambles proportioned to wealth — a tenth of everything one has — and here, for a great many people, the aversion holds roughly constant: the clerk risking a tenth of his wages and the magnate risking a tenth of his pile feel, so far as one can tell, about the same reluctance. The first is absolute risk aversion, and it falls; the second is relative risk aversion, and very often it does not.3

3. The measure splits a merged impression. Gambles of fixed dollar size: the rich are less averse, a fixed sum being smaller against a larger fortune — absolute risk aversion, which falls with wealth. Gambles proportioned to wealth (a tenth of all one has): the aversion often holds constant — relative risk aversion. The eye sees one fact about the rich and risk; the measure sees two, and they need not agree.

That division is the answer to the question. The rich man's steadier nerve, set before a gamble of fixed size, is real enough, but it is not courage; it is the arithmetic of proportion. The thousand he risks so coolly is, weighed against his fortune, the same slight fraction that a few dollars are against the clerk's — and on the proportional measure, where like is at last compared with like, his nerve is no steadier than the clerk's at all. His apparent bravery is an illusion of vantage, close kin to the one the volume meets again in the survivor whose luck has come to wear the face of skill: what looks from outside like a bolder temperament is, looked at squarely, only a larger cushion. He is not braver; he is standing higher on the same bending curve, at a place where it has gone nearly flat, so that the dollar gamble which sends the poor man tumbling down a steep stretch of the curve scarcely moves him at all. Decreasing absolute risk aversion is not, then, a fact about the courage of the rich. It is the shadow, thrown across the world of bets, of the diminishing marginal utility of wealth.4

4. The rich man's coolness toward a fixed gamble is proportion, not courage: the thousand is the same fraction of his fortune that a few dollars are of the clerk's, so on the proportional measure his nerve is no steadier. His apparent bravery is an illusion of vantage — a larger cushion, not a bolder temper — close kin to the survivor whose luck wears skill's face. He stands where the curve has gone flat. Decreasing absolute risk aversion is the shadow of diminishing marginal utility, not a fact about courage.

So the measure does for the dread of risk what measurement always does for a quality: it makes the thing comparable, signable, and priceable. With it in hand one can say which of two men will take the larger risk, and how a single man's appetite for it will shift as he is insured, or enriched, or saddled with some hazard he cannot shed; one can price a small gamble from the curve of his wants alone; and one can hold apart the two senses in which the rich are loosely said to fear risk less — the absolute, in which they truly do, and the relative, in which they may not. And the measure hands back to the volume's first plate a sharper verdict than that plate could reach on its own. The premium is the price of risk; the Wager rooted that price in the concavity of utility; and the measure now shows the same concavity setting not a single price but a whole schedule of them — the identical risk dearer to the poor man and cheaper to the rich, not because their nerve is different, but because they stand at different heights on one curve that bends the hardest exactly where they have the least. The price of risk was never a fact about the risk. It was always a fact about the bearer, and about how far up the curve he had managed to climb before the throw.5

5. The measure makes risk aversion comparable, signable, priceable: which of two men takes more risk; how appetite shifts with insurance, wealth, or a background hazard; the price of a small gamble from the curve alone; the two senses of the rich fearing risk less. It returns to the Wager a schedule of prices, not one — the same risk dearer to the poor and cheaper to the rich, because they stand at different heights on one bending curve. The price of risk is a fact about the bearer, not the risk.