The coin is fair; the consequences need not be balanced. A gain of twenty pounds may improve the evening, while a loss of twenty may leave an obligation unpaid. There is no defect in the calculation which gives the toss an expected gain of zero. The difficulty lies in treating the two changes in money as equal changes in the life through which they pass. The next pound enters a different household from the pound that has just been spent. A theory of aversion begins by asking what wealth does for its possessor, before it asks why that possessor refuses a wager.
Daniel Bernoulli gave this difference a mathematical form. In his 1738 essay, a person's valuation of a prospect depends on the utility of the wealth it can bring, not simply on the number of coins. His particular hypothesis makes an additional small amount of wealth less valuable as the fortune to which it is added grows. The resulting utility function is logarithmic. A poor person and a rich person can then assign different values to the same monetary prospect without either miscounting its chances. Bernoulli did not have to make fear an error in order to explain refusal. He had to let the value of money depend upon the position in which money was received.
A concave utility curve turns this intuition into a comparison between prospects. Draw final wealth along the horizontal axis and utility along the vertical. If the curve rises but becomes less steep, the utility of the average wealth exceeds the average utility of two different possible fortunes. A fair gamble leaves expected wealth unchanged while lowering expected utility relative to keeping that wealth certain. This is the sense in which a risk-averse person rejects it. Expected utility need not be negative: the position of zero on the utility scale is arbitrary. The comparison is between the uncertain prospect and an available certainty, measured by the same function. The theory identifies an ordering of alternatives, not a quantity of distress visible on the face of the loser.
An invented household account makes the gap tangible. Let initial wealth be one hundred pounds, and let a fair toss add or subtract twenty. Final wealth is eighty or one hundred and twenty, with equal probability. Under logarithmic utility the certain wealth giving the same utility is the geometric mean, the square root of eighty times one hundred and twenty: about £97.98. Expected wealth is still £100. The difference, about £2.02, is the risk premium for this prospect at this wealth. A person with these preferences would give up £2.02 to replace the gamble already faced with the certainty of £97.98. The example describes a hypothetical preference and an available replacement; it is not an estimate of what any actual household would pay.
The word premium needs care here because it names a gap between two valuations, before it names an invoice. The certainty equivalent is the sure final wealth judged as good as the uncertain final wealth. Subtracting it from expected wealth measures the cost, to this bearer, of the dispersion. The result does not by itself state what an insurer charges, what a new risky asset must yield in a market, or how much one may spend to acquire a gamble not yet owned. Those transactions change the initial position and introduce other parties. The £2.02 describes the cost of leaving this particular uncertainty in this particular account. A market price requires an account of who can remove it, on what terms, and what alternatives each party has.
For a sufficiently small, zero-mean risk, Pratt isolates the local force behind the premium. It is approximately half the variance of the monetary risk multiplied by a measure of absolute risk aversion, −u″(w)/u′(w). The numerator records how marginal utility changes; dividing by marginal utility prevents an arbitrary rescaling of utility from changing the answer. With logarithmic utility the measure is 1/w. At wealth of one hundred, the approximation for the twenty-pound toss is two pounds, close to the exact £2.02. The approximation is useful because it separates the dispersion of the prospect from the way its bearer values that dispersion. It remains an approximation around a wealth position, under smoothness and small-risk conditions; multiplying a large variance by local curvature cannot replace valuing the actual large gamble.
The same danger can meet different reserves. The price of relief records the meeting.
Increasing wealth changes that meeting in Bernoulli's logarithmic example. At one thousand pounds, the same twenty-pound gain or loss has a certainty equivalent of approximately £999.80 and a premium of about twenty pence. The danger in pounds has not diminished; the reserve against which it is set has grown. But it would be a mistake to infer from diminishing marginal utility alone that every richer person must pay less to avoid a fixed risk. Concavity establishes aversion. Declining absolute risk aversion is a further property. A concave exponential utility function can produce the same premium for the same additive gamble at every starting wealth. Pratt's distinctions matter precisely because several different claims can otherwise disappear into the easy phrase that the rich can afford to take chances.
The size of the stake must also remain in view. Twenty pounds is a fifth of a hundred and a fiftieth of a thousand. If the richer person instead faces a gain or loss of a fifth of wealth, the logarithmic premium rises with the fortune: about £20.20 at one thousand, compared with £2.02 at one hundred. The proportion surrendered is the same, about 2.02 percent. Absolute and relative risk aversion answer different questions because the first holds the monetary gamble fixed and the second holds its scale relative to wealth fixed. A change in the willingness to hold a particular position cannot be interpreted until we know which comparison has been made.
The relevant prospect is the person's combined position, not an isolated line on a ticket. Suppose a household's other resources will be eighty pounds in one state and one hundred and twenty in another, each equally likely. A claim paying forty in the first state and nothing in the second has an uncertain payment by itself. At a certain price of twenty, however, it leaves the household with one hundred in either state. Someone averse to the original variation can prefer buying this claim to keeping the twenty pounds. The apparent willingness to purchase a risky payment is consistent with aversion to risk in total wealth. The claim is valuable because of when it pays. Counting its own variance would miss the very service for which it was purchased.
Insurance makes the same reasoning visible from the other side. Let an uncertain monetary loss have an expected amount of one thousand pounds. A risk-averse owner may be willing to pay more than that amount for complete reimbursement, because reimbursement removes dispersion as well as replacing expected money. The highest acceptable full-insurance payment is the expected loss plus the risk premium attached to the uninsured final wealth, given a loss distribution unaffected by the contract and credible, costless settlement. The insurer's actual quotation can be below that ceiling, above it, or unavailable altogether. Administrative costs, the insurer's own exposure and the terms of competition intervene between the household's willingness to pay and the offered contract.
Fear may accompany any of these decisions, but the mathematics does not require it. A person can calmly prefer an assured amount to a dispersed prospect because the bad outcome would displace uses of money more valuable than the uses made possible by the good one. Conversely, a vivid fear need not be represented adequately by a smooth utility curve over final wealth. The account developed here neither measures anxiety nor diagnoses it. Nor is ordinary concavity the same mechanism as a special sensitivity to losses measured from a reference point. A smooth logarithm needs no kink at the current fortune to produce a premium. Keeping these explanations distinct lets us see what Bernoulli's argument accomplishes before asking it to explain every departure from a monetary average.
The premium is therefore a price of risk as borne, rather than a property that can be read from the danger alone. Probability and possible loss describe one side of the relation; resources, obligations and preferences supply the other. Calling the resulting valuation merely fear would make the holder's circumstances sound like a blemish on an otherwise complete calculation. Treating the valuation as an objective measure of suffering would go too far in the other direction. A willingness to pay expresses a preference through available means. Someone with fewer means may be unable to purchase protection even when the loss would disorganize more of life. The preference model supplies no rule that ability to pay measures deservingness.
That remaining difficulty prepares the way for the policy. If two people value the same exposure differently, a transfer may improve both positions; if many sufficiently separate exposures can be combined, a new form of protection becomes possible. The curve explains why certainty can be valuable. It does not by itself produce the resources that make certainty credible. For that, the person asking to be relieved must meet an institution able to accept the promise, collect its price, and pay when the state no one wanted finally arrives.