A person reaches the end of a working life with a sum of money and a question the sum cannot answer by itself: how much may be spent each year? The wages arrived through time; the savings stand at a moment. Between them lies the conversion this volume has been approaching, from a stock held into a flow received, and back again. The annuity gives that conversion a contractual form. A payment now purchases a succession of payments later, perhaps for a fixed number of years, perhaps for as long as its purchaser lives. In the first case the calendar fixes the length of the stream; in the second a life does. Both are called annuities, but the difference between them changes what the purchaser is buying. A term of twenty years can end while the person still needs an income. A lifetime payment can continue into extreme old age, while leaving nothing to be collected after an early death. The size of the annual payment therefore carries several terms inside it: the rate of interest, the dates on which money arrives, the duration or contingency of the promise, and the claim, if any, that remains when the payments stop.
Begin with the calendar, and make the promise entirely certain. A fund of a hundred thousand earns five percent each year, and a payment is withdrawn at each year’s end. If the fund is to last exactly twenty years and then be exhausted, the level annual payment is about eight thousand and twenty-four. The calculation is the present-value reckoning already encountered in The Discount: bring each future payment back to the present at the stipulated rate, and make their sum equal the fund. In symbols, for principal P, annual payment A, rate r and term n, P = A[1 − (1 + r)−n]/r. The two sides are the same dated promise viewed from opposite ends. One asks what a stream is worth now; the other asks what stream a sum can support. At a zero rate the interest disappears and the same fund simply divides into twenty payments of five thousand. With a positive rate and an indefinitely continuing payment, the limiting annual amount is the interest alone: five thousand on this principal at five percent. A finite annuity can pay more because it has permission to use up what a perpetuity must keep.1
The larger payment contains the distinction on which the whole arrangement turns. In the first year the fund earns five thousand, yet pays out about eight thousand and twenty-four. The remaining three thousand and twenty-four comes from principal, leaving about ninety-six thousand nine hundred and seventy-six to earn the second year’s interest. Each payment combines income on the surviving balance with a withdrawal from that balance, until the final payment drains it. Calling the whole payment an eight-percent return would mistake consumption of the stock for earnings upon it. The account has earned five percent throughout. Conversely, preserving the original hundred thousand in nominal terms does not preserve its purchasing power if prices rise. The amount that can be spent while maintaining a real stock depends on the real return. The annuity thus gives the bathtub an exact schedule: the outflow can exceed the inflow because the water already there is being drawn down. Its generosity is partly an instruction about the ending balance, and the ending balance is part of the price.2
A lifetime annuity changes the ending condition. No individual knows the date at which a retirement fund will cease to be needed, so a private schedule must confront both the risk of exhausting the fund while alive and the possibility of dying with money left over. A pool can make those two possibilities bear upon one another. In a simple one-period example, suppose survival to the payment date has probability nine-tenths and assets earn five percent. A premium of a hundred can fairly purchase a payment of about a hundred and sixteen and two-thirds if the purchaser is then alive, with nothing paid at death: nine-tenths of that payment has the same discounted cost as the premium. Someone who holds the assets separately receives a hundred and five whether alive or dead; the survivor in the pool receives more because the contract directs the resources associated with early deaths toward those who survive. With survival certain, the extra payment vanishes. For a succession of years, each promised payment is weighted by the probability that the purchaser is still alive to receive it, as well as discounted for time. A life annuity prices a stream of contingent claims, and survival is the contingency.3
The economic case for pooling follows from that redirection. Money left at death cannot finance the deceased person’s later consumption; when consumption while alive is the only object, moving resources from death states to survival states can improve the provision for old age. Davidoff, Brown and Diamond show how strong the result can be under exact conditions. When comparable annuitized claims pay more in the survival states, wealth left at death has no value to the person, and complete markets permit the desired payments across dates and states, full annuitization follows under much weaker preference restrictions than a single familiar utility formula would suggest. The result identifies the useful service: insurance against the duration of one’s own life. It also identifies what must be held fixed in the comparison. A claim that can be sold tomorrow and one that cannot be surrendered are different resources in a medical emergency; a fixed nominal payment and an inflation-protected payment do not insure the same consumption. When available annuities have restricted payout patterns and the market is incomplete, the general conclusion about putting everything into them no longer follows. The comparison then concerns the actual paths of spending and the actual claims that can be bought.4
A bequest adds another person to the reckoning. The money remaining at death now has a purpose, and a payment to heirs competes for resources with the larger payment promised to survivors. A guarantee period or a return of unused capital can meet that purpose, but the guarantee must be paid for: resources reserved for death claims are unavailable to enlarge the life-contingent stream. This is a choice about the destinations of wealth. It sits beside choices about emergency access, the purchasing power of the payments, the insurer’s ability to honor them and the cost of administering the contract. None of these can be read from the advertised annual amount alone. Nor does a higher amount establish a better bargain when one payment begins earlier, lasts fewer years, bears more risk or leaves a smaller estate. The act of capitalization makes the contracts comparable only after their dates and contingencies have been named. A stock and its promised flow are joined by an entire set of terms, with the interest rate doing one part of the joining.5
The annuity answers the opening question by making its hidden conditions visible. A lump sum becomes a stated flow for a stated time, or for a life whose uncertain length can be pooled with others; some of the flow may be the stock itself returning to be spent. For the saver, this can turn an accumulated quantity into security through time. For the heir, it can determine how much of that quantity remains to cross a generation. And for the volume, it puts a useful restraint on the language of wealth: an income received from an asset need not be income earned by that asset, and the same stock can support different lives according to the promises into which it is converted. The next plate moves from the uses of one holding to the distribution of all holdings. People may receive similar annual wages while possessing very different reserves, claims and powers to wait. The Concentration asks how the inequality of that stock grows beyond the inequality visible in the flow.