Annuities upon Lives: or, The Valuation of Annuities upon any Number of Lives; as also, of Reversions. To which is added, An Appendix concerning the Expectations of Life, and Probabilities of Survivorship.

London: Printed by W.P. and sold by F. Fayram; Benj. Motte; and W. Pearson, 1725.

First edition of de Moivre’s influential study of annuities based upon the mortality statistics gathered by Edmund Halley in the 1690s. “De Moivre’s contribution to annuities lies not in his evaluation of the demographic facts then known but in his derivation of formulas for annuities based on a postulated law of mortality and constant rates of interest on money. Here one finds the treatment of joint annuities on several lives, the inheritance of annuities, problems about the fair division of the costs of a tontine, and other contracts in which both age and interest on capital are relevant. This mathematics became a standard part of all subsequent commercial applications in England” (DSB IX: 454). “De Moivre’s preoccupation with matters concerning the conduct of a capitalist society, such as interest, loans, mortgages, pensions, reversions or annuities, dated back at least to the 1690s, from which time a short note survives in Berlin containing his answers to a client's questions. In 1693, using the lists of births and deaths in Breslau for each of the years 1687 to 1691, Edmond Halley had published in the Philosophical Transactions a life table together with applications to annuities on lives, but the amount of calculation involved in extending this to two or more lives turned out to be immense. De Moivre replaced Halley’s life table by a (piecewise) linear function, which allowed him to derive formulas for annuities of single lives and approximations for annuities of joint lives as a function of the corresponding annuities on single lives. He published these formulas, together with the solution of problems of reversionary annuities, annuities on successive lives, tontines, and other contracts that depend on interest and the ‘probability of the duration of life’, in his book Annuities upon Lives” (ODNB). ABPC/RBH list six copies, three of which lacked the errata slip, since Norman (Sotheby’s, June 15, 1998, lot 680, $6,325).

“When writing the preface to his uncle’s Ars Conjectandi in 1713, Nicolaus Bernoulli encouraged Abraham De Moivre to work on some economic and political applications of probability. De Moivre was interested, but declined due to his teaching workload. Five years later, in his preface to his Doctrine of Chances, De Moivre suggested that Nicolaus himself was better suited to work on these kinds of problems in view of his own work in the area. He also suggested that Johann Bernoulli was well qualified to do the work … With words such as this flowing from De Moivre’s pen, it seems rather odd that seven years after the publication of Doctrine of Chances De Moivre published a short book on an economic application of probability. It turned out to be highly influential. This was the 1725 Annuities upon Lives and no reference to any Bernoulli can be found in it. The two names that are mentioned in the book are Edmond Halley and Thomas Parker, 1st Earl of Macclesfield. Halley’s name appears throughout the book; De Moivre used a model for the probability of survival that was inspired by a life table that Halley had constructed in 1693 and published in Philosophical Transactions. It is to Macclesfield that the epistle dedicatory is addressed and only there does Macclesfield’s name appear …

“There were two contracts involving annual payments that were contingent upon the survival of one or more individuals. The first and oldest was tied to the land, and is known generally as ‘leases for lives.’ A tenant could take out a lease on a piece of land where the term of the lease extended until the death of the lessee or the death of the last survivor of the lessee and up to two others named in the lease. The second contract is what we would normally consider today as a life annuity. A purchaser paid a lump sum of money—to the government or to a private company or to an individual—in order to receive an annual payment until the death of a named individual. The named individual did not have to be the purchaser of the annuity. Like leases for lives, there was also the possibility of joint-life annuities that made annual payments until the first death or until the death of the last survivor. Although these two financial ideas are quite different, they are mathematically equivalent …

“The English government needed money, a lot of it, to finance the military campaigns of King William III and his allies against France. Campaigning began in 1689 and continued yearly during the summer fighting season. The conflict, known as the Nine Years’ War, ended with the signing of the Treaty of Ryswick in 1697. The war consumed about 80% of British public revenues. The use of the life annuity for British public finance was instituted in an act of 1692 that empowered the government to raise £1 million through the sale of these annuities. Subsequently, the act was called the ‘Million Act’ and the annuities were often called Exchequer annuities since they were sold through the Office of the Exchequer. In a sharp break from the past, Exchequer annuities were not secured by land. Rather, they were secured through an excise tax on beer and liquor. Further, it was an opportunity, completely new, for people to obtain a life annuity without having to own land.

“It was at this point that Edmond Halley, by chance, walked onto the life annuity stage. Caspar Neumann, the Lutheran pastor in the city of Breslau in Silesia, had collected data on the total number of births and the number of deaths at each age from the church registers of that city. The data covering the years 1687 to 1691 eventually found their way to the Royal Society for Halley to analyze. Regarding life annuities, Halley did three things with Neumann’s data. First, he calculated a life table, or more correctly a population table, which gives estimates of the number of people alive in Breslau at each age up to 84. For some reason Halley provided no information for ages 85 and up other than the total number of people in that age group. Halley’s table is probably the first life table ever to be calculated based on population data.

“The second thing Halley did was to use the life table to calculate the present value of a life annuity at various issue ages. From reading the minutes of Royal Society meetings, one gets the impression that this was possibly an afterthought. Halley presented his life table to the Royal Society on March 8, 1693. It was not until a week later that he presented his annuity calculations. The third thing Halley did was that he indicated how joint-life annuities, for two and three lives, could be calculated from his table … Using his life table and tables of logarithms, and with paper and quill pen only, Halley calculated some values for the present value of a life annuity at [given] rate of interest of an amount 1, payable annually at the end of the year to a person aged x until death occurs. He did this at a 6% rate of interest, the legal rate at the time, and at age x = 1 and the ages x = 5 quinquennially through 70. The burden of hand calculation for these valuations is enormous …

“After the government entered the life-annuity business, the floodgates soon opened. Some enterprising individuals decided to finance their business schemes by the sale of life annuities, where the only security was in the success of the enterprise … Others decided to market annuities by alienating land to provide the security … None of these private schemes were properly funded and many went broke. Within only four or five years of the 1692 Million Act, individuals were looking to buy or sell life annuities on the open market, often, but not always, secured by land. Between 1715 and 1720, the open market for life annuities was firmly established …

“De Moivre’s interest in life annuities was almost certainly motivated by financial issues related to land. As we shall see, the motivation comes out very subtly in his book. The ability to evaluate life annuities sold on the open market was an added bonus from which De Moivre benefited in later years. De Moivre made a very simple insight into Halley’s published mortality data. The insight can be obtained immediately by looking at a plot of the number alive at each age in the city of Breslau. After age 30, the curve is approximately linear, so De Moivre made the natural assumption [that] a constant number of deaths occur at each age” (Bellhouse, pp. 155-161). With this assumption, De Moivre obtained a simple formula for the value of a life annuity, which could easily be evaluated using tables of logarithms. “Using some numerical examples, De Moivre showed that he could provide a good approximation to Halley’s annuity values … the calculation of the value of a life annuity, which for Halley was an onerous undertaking, now requires only a few calculations using logarithms. The saving in time is enormous. Using the same linear assumption, De Moivre also obtained an easily computable expression for the present value of an annuity paid for the life of the annuitant or a fixed term, whichever comes first. Further, De Moivre showed how to carry out the evaluation of [the value of annuities] when [the population vs. age curve] is piecewise linear. It took a little more effort in calculation but, in terms of Halley’s life table, it was a realistic assumption when ages under 30 are included” (ibid., p. 162). The problem for joint lives proved more difficult, and “De Moivre changed his assumption on survivorship to get an easily calculable result. He assumed that the survivor function has an exponential shape rather than a linear one and that, for convenience of finding sums, the potential length of life was limitless … Using only one example, De Moivre checked the accuracy of making the incompatible linear and exponential assumptions together … After that, he never questioned the incompatibility of the two assumptions.

“From these basic results, using the linear assumption, possibly in combination with the exponential assumption, much of the remainder of De Moivre’s results follow; only the situation changes. De Moivre considered three additional situations with respect to survivorship: (1) reversions, (2) successive lives, and (3) renewal of lives. The first two of these situations pertain both to life annuities bought and sold on the open market and to annuities related to land tenure. The third one pertains only to land—in particular, to leases for lives. In the first type of survivorship, a reversionary annuity for a given person is one that makes lifetime payments to that person beginning on the death of another person. There are variations on this annuity. It could be paid to one person after the death of the last survivor of two other people. It could be paid on the joint lives of two people beginning on the death of a third. In a succession of lives, the annuity is initially paid to a given annuitant. On that person’s death, the payment is made to a successor named by the original annuitant. When there is an option for the renewal of lives, payments could continue indefinitely. Without renewal, payments would be made until the last survivor dies. If the annuitants exercise their option of renewal, then on the death of one of the annuitants, an amount is paid (a fine) to replace that person with someone else. The renewal of lives only occurs in the case when there was a lease for lives …

“The connection of De Moivre’s Annuities upon Lives to land tenure is not explicit in the book. This is a source of confusion to many when trying to connect the book to applications. Without any explicit connection, the seemingly most obvious possibility for the modern reader is the open market in life annuities, as the book begins with the evaluation of life annuities. However, the section in the book on successive lives and reversions … points directly to the practice of copyhold leases and not to the usual annuity market. For example, De Moivre’s Problem 16 in Annuities upon Lives is a direct answer to the problem of valuing a lease on three lives when there is a fine to be paid on the replacement of any expired life. The problem reads, ‘If there be three equal Lives, and A or his Heirs are to have a Sum f paid them upon the Vacancy of any of those Lives, what is the Expectation of A worth in present Money.’ De Moivre’s method of solution is based on the exponential survivorship model …

“De Moivre’s Annuities upon Lives had impact very soon after its publication. It occurred in three ways: (1) his results resonated among his friends and connections, one of whom was highly placed politically; (2) more publications, directly inspired by De Moivre’s book, appeared on the valuation of life annuities; and (3) some mathematicians, as well as others, began to offer their services to the public as consultants in the valuation of life annuities. The writers of annuity books were typically not necessarily professional, or even accomplished, mathematicians. On the other hand, they did have an intimate knowledge of either the London financial industry or of landed estates … Newton owned a copy of Annuities upon Lives. So did Halley” (ibid., pp. 163-166).

A French Huguenot, Abraham de Moivre (1667-1754) was jailed as a Protestant upon the revocation of the Edict of Nantes in 1685. When he was released shortly thereafter, he fled to London. In London he became a close friend of Sir Isaac Newton and the astronomer Edmond Halley. De Moivre was elected to the Royal Society of London in 1697 and later to the Berlin and Paris academies. Despite his distinction as a mathematician, he never succeeded in securing a permanent position but eked out a precarious living by working as a tutor and a consultant on gambling and insurance.

“In the Philosophical Transactions for 1711 de Moivre published a long article, ‘De mensura sortis’, which was followed by the Doctrine of Chances. The second, much extended, edition of the Doctrine (1738) contained his normal approximation to the binomial distribution that he had found in 1733. This special case of the central limit theorem he understood as a generalization and a sharpening of Bernoulli's Theorema aureum, which was later named the law of large numbers by Poisson. De Moivre's central limit theorem is considered as his greatest mathematical achievement and shows that he understood intuitively the importance of what was later called the standard deviation. Crucial to this theorem was a form of the so-called Stirling formula for n!, which de Moivre and Stirling had developed in competition which ended in 1730” (ODNB).



8vo (198 x 119 mm), pp. [ii], 4, viii, 108, [2], errata slip mounted on final blank leaf. Contemporary 'royal crest' stamped full calf (rubbed and spine joints tender but holding). Rare in such good condition.

Item #3291

Price: $4,500.00