II. ORIGINS OF JOINT LIFE ANNUITIES

Ancient instances of joint life contingency valuation are unknown. Likely inspired by the judicial need to enforce the Falcidian law, Ulpian’s table (Digesta 35.2.68) for valuing single life maintenances and usufructs survives as evidence that life-contingent valuations were done. The historical context surrounding inheritances suggests the need to value contingencies for joint lives was also a concern but incomplete and “often so one-sided” sources from ancient history provide no evidence for such valuations. In conjunction with the more traditional perpetual hereditary rente—rente héritable and erfrent later known as losrent—the life annuity contract—rente viagères and lijfrent—became a staple of municipal and, eventually, state finance in northern Europe.Footnote ³ As surviving Roman law sources provide no evidence of rente-type contracts, it is generally accepted such contracts evolved from the Carolingian census. Footnote ⁴ However, evolution from ninth-century census arrangements to early thirteenth-century issuance of life annuity contracts by towns and cities in the Langues d’öil of northern Europe lacks detailed study.Footnote ⁵

Conventionally, emergence of public rentes and renten is connected to the evolution of scholastic usury doctrine. Though there were some ecclesiastic dissenters—e.g., Munro (Reference Munro2003, pp. 520–523); Baum (Reference Baum1985, pp. 27–31)—accepted medieval scholastic doctrine was evidenced in 1250 by Pope Innocent IV exempting such contracts from usury sanctions, as a loan (mutuum) involved a return of what was borrowed, a feature absent in perpetual and life annuities. With some provisos, redemption of a perpetual rente was also acceptable. However, emphasis on the connection between evolution of the usury doctrine and emergence of public debt diminishes the key role played by licit private debt contracting that predates the thirteenth century, such as the annuities secured by real estate—“plots of land within the town walls” (Baum Reference Baum1985, p. 25)—in Hanse towns and elsewhere that roughly correspond to redeemable perpetual rentes. Similarly, rudimentary single and joint life annuities are reflected in corrody transactions of twelfth-century England and elsewhere, “provided by a religious institution such as a monastery, priory, abbey or hospital,” which involved an individual or couple purchasing “some agreed mixture of food, drink, heat, light, accommodation, clothing, laundry, health care, maybe a small monetary allowance and even stabling and grazing for their livestock” (Bell and Sutcliffe Reference Bell and Sutcliffe2010, pp. 142–143).

Impressive efforts by historians trolling municipal, notarial, and state administrative records from the thirteenth to the eighteenth centuries have provided a wealth of data about public finance in northern Europe. The bulk of these efforts are concerned with the role that issues of long-term debt—life annuities and redeemable perpetuities—and short-term bills played in the “financial revolutions” that contributed to the rise and consolidation of national governments (e.g., Tracy Reference Tracy1985; Munro Reference Munro2003; Fritschy Reference Fritschy2003; Gelderblom and Jonker Reference Gelderblom and Jonker2011). Consequently, limited attention has been given to methods of valuation for joint life annuity contracts prior to the later seventeenth-century contributions of de Witt and Hudde (Hendricks Reference Hendricks1853) and Halley (Reference Halley1693). Despite the absence of a systematic treatment, some evidence is available about issuance, investors, and contract design of such joint life annuities. This evidence reveals a decidedly more complicated interpretation for the general claim that early issues of life annuities were sold without reference to age, e.g., Poitras (Reference Poitras2000, p. 190).Footnote ⁶ In addition to differences in contractual terms and investor characteristics across jurisdictions and time, in some instances life annuities were marketed to investors beyond the confines of the issuing municipality, while, in other instances, sales were restricted to local residents.

Some caution is required to interpret claims about life annuities prior to the sixteenth century appearing in the secondary literature. For example, Lorraine Daston (Reference Daston1988, p. 121), correctly referencing SGN (1898, p. 209), observes that, starting in 1402, Amsterdam sold municipal annuities at regular intervals, typically “charging flat rates of 9 1/11 percent for annuities on two heads and 11 13/17 percent for one, regardless of age.”Footnote ⁷ Yet, the surviving document for this date only indicates life annuity issues were permitted to fund a loan to Duke Albert I at 10%. SGN (1898, p. 208) infers these life annuity rates from a 1464 document authorizing named Amsterdam city officials to refund outstanding loans using life annuities at “11 [years’s purchase] on two heads and nine and one half on one head.” The percent calculation for two heads is correct, but the 11 13/17 is an error in SGN, as 9.5 years’ purchase does not convert to the stated percent. Complications associated with determining contract terms is further reflected in a 1472 last survivor annuity document from Leiden for Clais Jacobs, priest, and Lysbeth, daughter of Lambrecht, where an annuity of seven gold Rhine guilders is granted but the initial capital is not recorded. The annuity is granted “in the name of the city,” with annuity payments to cumulate and be paid in full, when possible, in the event payments were suspended.

An early eighteenth-century source, Nicolas Bernoulli (Reference Bernoulli1709, ch. 4, esp. pp. 26–27), documents considerable variation in the recommended relationship between the prices of joint and single life annuities. Included among these instances is a case where, on scholastic grounds, “just pricing” for single life and last survivor annuities required sale at the same price for those of advanced age. Referencing Pierre du Moulin (1568–1658), Bernoulli observes: “If the rent should be purchased on the lifetime of two in its entirety, a considerably more ample premium … must be constituted.” Following a review of six opinions on the just price for one and two life annuities, du Moulin concludes that for individuals of comparable age and health, twelve years’ purchase is recommended for an annuity on two lives and ten years’ purchase for a single life. Without reference to the 1694 English government issue of £100 annuities on one life at £14/annum, two lives at £12/annum, and three lives at £10/annum (5&6 W&M), Bernoulli (Reference Bernoulli1709, p. 27) recognizes the 1703 English government annuity issue (2&3 Anne c.3) that charged nine years’ purchase for a single life, eleven years’ purchase on two lives, twelve years’ purchase on three lives, and fifteen years’ purchase on a ninety-nine-year term annuity.

Closer inspection of the historical record reveals a diverse picture of contract variation and pricing across time and issuing location, providing some insight as to relative usage of joint as opposed to single life annuity contracts. The record reveals the claim that annuities were sold without reference to age is imprecise as some locales restricted sales by age, indicating usage of life annuities as a rudimentary old age “social safety net.” For example, in some south German towns, Remi van Schaïk (Reference van Schaïk, Boone, Davids and Janssens2003, p. 112) observes, in 1457, Augsberg sold life annuities only to those forty years of age and older, and Nuremberg restricted sales to sixty and over; van Schaïk also finds similar prices for single life and last survivor annuities arising in Zutphen (1400 to 1600), although “this did not happen very often.” Hans-Jörg Gilomen (Reference Gilomen, Boone, Davids and Janssens2003, p. 148) provides a fascinating sample of thirty-one Swiss rentes viagères contracts from 1470–71 and 1479–80—the majority from Basel (Bâle)—with nineteen contracts featuring several joint life annuity variants with husband and wife or father and son as nominees. In numerous instances, both last survivor and single life annuities sold at ten years’ purchase. Several last survivor contracts featured a lower payment when one of the nominees died. As some single life annuities were for widows, this provides some explanation for selection of single versus joint life contracts. Though ages are not given, with a few exceptions where last survivor annuities involved a father and child, the nominees appear to be older.

At some point, it is not clear when and where, the social safety net rationale for life annuity issues transitions from nominees and shareholders, usually older, being the same to include an investment vehicle motivation for shareholders using opportunistic selection of young, typically female, nominees with enhanced life expectancies. The rapid increase in Dutch debt following 1600—see, e.g., Oscar Gelderblom and Joost Jonker (2011, p. 11)—likely accelerated this transition, although the “Tableau of mortality” produced by Hudde from the Amsterdam register of life annuity nominees for the years from 1586 to 1590 indicate this transition was beginning earlier, (see, e.g., SGN 1898, pp. 80–81). Potential investment gains are reflected in Holland’s selling single life and last survivor annuities at six and eight years’ purchase in 1595 and eight and ten years’ purchase in 1608 (Fritschy Reference Fritschy2003, p. 64) without reference to age. By comparison, the initial 1672 Amsterdam single life annuity issue that took account of nominee age featured ten years’ purchase for nominees between one and nineteen years, decreasing to a low of three years’ purchase for nominees seventy-five years and older. According to Dirk Houtzager (Reference Houtzager1950), this pricing according to age was driven more by need to raise funds from older age groups, which were discouraged from purchasing life annuities sold without reference to age, rather than seeking actuarially sound pricing. The terms to annuitants were found to be so favorable to older age groups that a flood of applications ensued and a ban on sale to persons over fifty years of age became necessary.

The cost to government of the transition to young nominees in order to obtain enhanced investment returns was identified in a supplement attached to de Witt (Reference de Witt1671): “one finds with wonderment, that in practice, when the purchaser of several life annuities comes to divide his capital which he intends to invest upon several young lives—upon ten, twenty, or more—this annuitant may be assured, without hazard or risk of the enjoyment of an equivalent, in more than sixteen times the rent which he purchases” (Hendricks Reference Hendricks1853, p. 118). The practice of selecting young nominees was not isolated to the Dutch. Evidence of an overwhelming preponderance of young nominees appears in “A List of Names of the Several Nominees with Their Ages Subscribed” (Exchequer Reference Exchequer1749) for the 1693 English Million Act that featured fourteen years’ purchase for single life annuities—in combination with the possibility of also purchasing a tontine (e.g., Milevsky Reference Milevsky2015, ch. 4). One of the earliest examples of an investment fund active from the early 1770s to the late 1780s—Les trente demoiselles de Genève—involved the sale of shares in the fund organized by Genevan banks investing in French government life annuities, usually with single life annuities but in one case at ten years’ purchase with thirty nominees. These nominees were carefully selected: young females who had survived smallpox from well-to-do families in Geneva (Velde and Weir Reference Velde and Weir1992; Spang Reference Spang2015).

III. HUDDE, DE WITT, AND DE WAERDYE

One useful dictum obtainable from study of ancient history is the need to be aware of bias in interpretation arising from lack of sources. Because ancient sources are usually woefully inadequate to provide sufficient detail about historical events, historians must create a narrative for a particular event at issue based on sources “so incomplete, often so one-sided, often so naively disconnected with fundamental movements” (Frank Reference Frank1910, p. 99). A tendency to develop interpretations biased toward information in meager available sources is difficult to avoid. In contrast, historiographies of the medieval to early modern periods feature increased availability of sources: books and pamphlets, government reports, journal articles, private papers, letters, and the like. However, sorting details of a specific event is, again, guided by trolling of available sources. For example, the origins of life annuity contracts in northern Europe have been identified using contract records that have survived, though such contracts may have been sold earlier and in different locations, leaving no surviving records. Is it possible that the received intellectual history of joint life annuity valuation has been adversely impacted by the availability of sources?

Intellectual history is replete with examples where seminal contributions attributed to specific individuals are later found to be inappropriate based on more detailed examination of sources, e.g., Stigler’s law of eponymy. Where definitive sources are unavailable, insights, if any, must be gleaned from context and inferences. Such is the case with Hudde, where what survives of numerous contributions to mathematics, engineering, physics, astronomy, and actuarial science are largely available from other sources, especially from appendices in books by Frans van Schooten, together with letters and correspondence of Christian Huygens, de Witt, and, to lesser extent, Gottfried Liebnitz and Baruch Spinoza. As observed in Frederick Hendricks (Reference Hendricks1853, p. 97): Hudde “seems not to have taken sufficient care in the preservation of his manuscripts.” There are no direct primary sources for Hudde for the period following his 1672 appointment by Stadtholder Wilhelm III as one four burgomasters of Amsterdam. Prior to that date, there is published work on mathematics, mostly provided by van Schooten, over the period 1654 to 1663, as well as letters about mathematics between Huygens and Hudde in 1663. There are also letters to Huygens on comets and correspondence with Spinoza on telescopes in 1665. Efforts related to life annuities appear in 1670–71 correspondence involving Hudde, Huygens, and de Witt.Footnote ⁸

Careful consideration of historical context for the correspondence between Hudde, de Witt, and Huygens raises substantive questions about relative contributions to the seminal, if not widely distributed, publication on the valuation of single life annuities, Waerdye van Lyf-renten naer proportie van Los-renten (July 1671).Footnote ⁹ Following Hendricks’s (Reference Hendricks1852) rediscovery and publication of the English translation (Value of Life Annuities in Proportion to Redeemable Annuities), this contribution has invariably been attributed to the Grand Pensionary (raadpensionaris), Jan de Witt, author of the government report.Footnote ¹⁰ Only passing mention, if any, is given by modern scholars to the contribution of Hudde. Despite the record of correspondence between de Witt to Hudde on the valuation of single and last survivor annuities being incomplete with only selected letters from de Witt to Hudde having survived, the available evidence does indicate this attribution is, at least partially, misplaced.Footnote ¹¹ From 1652, the year prior to his election as the Grand Pensionary, until the year of his tragic death in 1672, de Witt was actively engaged in various reform efforts aimed at addressing Dutch government borrowing costs (see, e.g., Gelderblom and Jonker Reference Gelderblom and Jonker2011, pp. 15–16). The work on life annuities with Hudde was aimed at critiquing a government proposal to issue single and last survivor annuities at fourteen and seventeen years’ purchase, continuing the practice of issuing such annuities without accurately accounting for nominee age.

While the timeline for Hudde and de Witt developing actuarially sound solutions for the value of last survivor annuities roughly parallels that for single lives, details do differ. Starting with the August 2, 1671, correspondence from de Witt to Hudde, it appears a solution for the last survivor annuity was actively discussed and attempts at appropriate calculations executed.Footnote ¹² Unfortunately, “the table of life annuities calculated upon two lives, in the selected class of 96 lives all aged 6 years, and also the fuller demonstration of the provisional hypothesis” provided by Hudde to de Witt in correspondence have not survived.Footnote ¹³ As the last survivor solution was decidedly more complex than for a single life, after a number of false starts de Witt proposed an ingenious, if not fully satisfactory, solution method involving the following theoretical (nearly) uniform death rate life table (Hendricks Reference Hendricks1853, p. 109):

Using an equally weighted average of the term annuity values for each lifespan and an interest rate of 4%, de Witt determines a single life annuity value (for one florin annual payment) of 17.22 florins:

$$ \frac{\;{A}_7+{A}_{15}+{A}_{24}+{A}_{33}+{A}_{41}+{A}_{50}+{A}_{59}+{A}_{68}}{8}=17.22 $$

where:

$$ {A}_N=\frac{1}{r}-\frac{1}{\left(r{\left(1+r\right)}^N\right)}=\frac{1}{r}\left(1-\frac{1}{{\left(1+r\right)}^N}\right) $$

and r is the annual interest rate. This approach is then adapted to produce solutions for last survivor annuities with young nominees of equal ages for two, three, four, up to eight lives to be determined by using a weighted average with weights calculated using binomial coefficients.

In a “Memoir” to an October 27, 1671, letter to Hudde, de Witt provides the following table for the binomial coefficients in the weighted average:

Years from Purchase to Death of Nominee

To apply these coefficients, de Witt gives worked examples for two, three, and four lives. For the two life last survivor annuity value calculation, de Witt gives the weighted average as: “the value of a life annuity upon 2 lives … is rightly and precisely equal to the value of a life annuity upon one life of a class of 28 lives, of which one life has lived 15 complete years; 2 each 24 years; 3 each 33 years; 4 each 41 years; 5 each 50 years; 6 each 59 years; and 7 each 68 years.”

The solution of 20.76 florins (for one florin annual payment) is determined by using a weighted average with 28 total chances:

$$ {P}_{LS}=\frac{A_{15}+\left(2{A}_{24}\right)+\left(3{A}_{33}\right)+\left(4{A}_{41}\right)+\left(5{A}_{50}\right)+\left(6{A}_{59}\right)+\left(7{A}_{68}\right)}{28}=20.766 $$

From the table, the three life case would have 56 total chances and values of 1 for 24 years, 3 for 33 years, 6 for 41 years, 10 for 50 years, 15 for 59 years, and 21 for 68 years (= 21.98; 56 chances). Solutions for a last survivor annuity on four (= 22.54; 70 chances), five (= 22.85; 56 chances), six (=23.04; 28 chances), and seven (= 23.17; 8 chances) lives follow appropriately with the value for eight lives being equal to the annuity certain value for sixty-eight years (23.26 florins). While not directly stated in the correspondence, based on the maximum eighty-year lifespan (ω) used in de Witt (Reference de Witt1671), the underlying assumption is that the lives involved are initially all twelve years old. For simplicity, de Witt uses only eight lives in calculating the solutions; de Witt refers to Hudde using calculations with eighty lives to solve the last survivor annuity, possibly also using uniformly distributed death rates. This would make for more complicated calculations but would avoid the degenerate solution given by de Witt for more than eight lives. However, details for this contribution are lost and connection to the binomial coefficient model of de Witt cannot be accurately determined.Footnote ¹⁴

How did de Witt arrive at the binomial coefficient table for estimating the last survivor annuity? While recognizing that a significant number of potential primary sources essential to definitively addressing this question have either not survived or, one hopes, not yet surfaced from the archives, substantial inferences can be drawn from five letters from de Witt to Hudde with August and October 1671 dates. That the solution to the joint life annuity price was problematic is apparent from the first letter, dated August 2: “I have perfectly understood the estimation of the value of life annuities upon one life … you will oblige me by informing me whether the computation of the value … of life annuities upon two lives, has also been made according to certain cases of several life annuities granted upon two lives.”

The statement of the binomial coefficient table appears in a “Memoir” to the last letter dated October 27. A most telling statement by de Witt appears in the October 27 letter: “I will with pleasure respond to your wish upon the subject, and in that case would establish thereon an argument a priori, although I have found it a posteriori, like in almost all inventions” (Hendricks Reference Hendricks1853, p. 108). The beginning of this letter reveals the source of empirical evidence and importance of the correspondence from Hudde to de Witt that has been lost: “Since the departure of my last letter of the 22nd, your two letters of the 21st and 22nd instant have respectively come to hand, with the table of life annuities calculated upon two lives, in the selected class of 96 lives all aged 6 years, and also the fuller demonstration of the provisional hypothesis; namely, that out of 80 young lives, about 1 dies.” As such, the binomial coefficient table provided by de Witt represents the first substantive effort to provide an analytical approximation to the price of a joint life annuity.

IV. THE METHOD OF HALLEY

The narrative for early analytical approximations to joint life annuity values is part of much broader social and economic processes taking place in the seventeenth and eighteenth centuries associated with application of numerical calculation to political reasoning (see, e.g., Deringer Reference Deringer2013; Brewer Reference Brewer1989, pt. II, ch. 4). The appearance of compound interest tables in the late sixteenth century gradually facilitated the spread of present value calculations that enabled de Witt and Hudde to estimate life annuity prices. More generally, “calculated values” were increasingly used for assessing soundness of government fiscal affairs (Deringer Reference Deringer2018b). Yet, the practical problem of pricing joint life annuities issued to fund government activities challenged available computational abilities and provided impetus for producing accurate life contingency data. Both themes appear in the seminal contribution to demography by Halley (Reference Halley1693) that accompanied emergence of the Financial Revolution in English government funding following the Glorious Revolution of 1688 that featured transplanting of Dutch fiscal methods (e.g., Dickson Reference Dickson1967; Milevsky Reference Milevsky2015).

Though the correspondence between de Witt and Hudde featured uniformly distributed death rates, it is not difficult to see the distances between yearly nodes or coefficients in the weighted average could be adjusted to reflect a theoretical life table with non-uniform death rates, as used in de Witt (Reference de Witt1671) to solve the single life annuity value as just over sixteen florins. However, being available only in correspondence not widely recognized until Hendricks (Reference Hendricks1853) in the middle of the nineteenth century, the ingenious method of solving the last survivor annuity using binomial coefficients faded into obscurity, leaving the more cumbersome and computationally intensive approach proposed by Halley to influence development of solutions for joint life annuity values. While recommending the usefulness of logarithms for “facilitating the Computation of the Value of two, three, or more Lives,” Halley does not provide completely worked solutions for any joint life annuity values. What Halley does provide is an overview of the actuarially sound compound probability method for solving joint life annuities where the ages of the nominees differ, motivated by a novel geometric analysis for individual terms in the sum that determine the joint life annuity value for two and three lives. That Halley spent considerable time and effort on the solution to the joint life annuity value is apparent. Halley (Reference Halley1693) dedicates less than two pages to value single life annuities with about four and one-half pages to joint life annuities, concentrated almost exclusively on the valuation for two and three lives.

While the conceptual approach to joint life annuity valuation proposed by Halley does have the desirable feature of allowing for unequal nominee lives, Halley recognizes that solving for the years’ purchase of a specific combination of ages is computationally demanding. Halley (Reference Halley1693, p. 604) verbally describes the brute-force method for solving the joint life annuity on two lives:

This verbal explanation is followed by a numerical illustration of the relevant calculations using the Breslaw life table for one term in the sum after eight years have elapsed and the nominees are initially eighteen and thirty-five years of age. Recognizing the difficulty involved in understanding the various calculations, Halley (Reference Halley1693, fig. 7) then provides a geometric motivation for these calculations (see Figure 1):

Figure 1. Halley’s Method for a Joint Annuity on Two Lives

The whole area of $ ABCD\;\left(={\mathrm{L}}_{\mathrm{x}}{\mathrm{L}}_{\mathrm{y}}\right) $ is the total number of chances for two lives at $ t=0 $ where $ {\mathrm{L}}_{\mathrm{x}}\left({\mathrm{L}}_{\mathrm{y}}\right) $ is the number of lives at starting age x (y).Footnote ¹⁵ The area of the inner rectangle $ HGIB\;\left(={\mathrm{L}}_{\mathrm{x}+\mathrm{m}}\;{\mathrm{L}}_{\mathrm{y}+\mathrm{m}}\right) $ is the total number of chances both are alive at $ t=m $ . The two side rectangles $ IDGE\;\left(={\mathrm{L}}_{\mathrm{x}}\left({\mathrm{L}}_{\mathrm{y}}-{\mathrm{L}}_{\mathrm{y}+\mathrm{m}}\right)\right) $ and $ HAFG\;\left(={\mathrm{L}}_{\mathrm{y}}\left({\mathrm{L}}_{\mathrm{x}}-{\mathrm{L}}_{\mathrm{x}+\mathrm{m}}\right)\right) $ are the chances one nominee is alive while the other is dead, leaving the upper rectangle EGFC to be the chances both nominees have died. Subtracting the ratio of EGFC to ABCD from one determines the weight—the survival rate—applicable to the discounted cash flow associated with $ {\left(1/\left(1+r\right)\right)}^m $ .

Formally, because no allowance is made for the portion of joint annuity payments that would be made when there are no chances the eldest nominee will be alive, this geometric argument fully covers only valuation of the last survivor annuity where the two nominees have equal ages.Footnote ¹⁶ Significantly, Halley does provide an explicit recognition of “what value ought to be paid for the Reversion of one life after another.” Stating Halley’s method algebraically gives the following compound probability specification for t = m:

$$ 1-\left[\left({\mathrm{L}}_{\mathrm{y}}-{\mathrm{L}}_{\mathrm{y}+\mathrm{m}}\right)\left({\mathrm{L}}_{\mathrm{x}}-{\mathrm{L}}_{\mathrm{x}+\mathrm{m}}\right)/\left({\mathrm{L}}_{\mathrm{x}}{\mathrm{L}}_{\mathrm{y}}\right)\right]\Big)=\left({\mathrm{L}}_{\mathrm{y}+\mathrm{m}}/{\mathrm{L}}_{\mathrm{y}}\right)+\left({\mathrm{L}}_{\mathrm{x}+\mathrm{m}}/{\mathrm{L}}_{\mathrm{x}}\right)-\left(\left({\mathrm{L}}_{\mathrm{y}+\mathrm{m}}{\mathrm{L}}_{\mathrm{x}+\mathrm{m}}\right)/{\mathrm{L}}_{\mathrm{x}}{\mathrm{L}}_{\mathrm{y}}\right) $$

Halley did not observe that multiplying this result by $ {\left(1/\left(1+r\right)\right)}^m $ and summing from $ t=1 $ to the maximum possible age gives the value of the two nominee last survivor annuity being equal to the sum of the single life annuities for each nominee minus the value of the joint survivor annuity. As the method for solving the single life annuity value is available, the valuation of the last survivor annuity requires a solution to the joint survivor valuation problem, and vice versa.

From this, Halley (Reference Halley1693, fig. 8) extends the two-dimensional analysis for a joint life annuity on two lives to a three-dimensional analysis for three lives (see Figure 2).

Figure 2. Halley’s Method for a Joint Annuity on Three Lives

For Halley (Reference Halley1693, p. 606), the discussion for the two-dimensional case “is the Key to the Case of Three Lives.” Extending the analysis to three dimensions, the fraction of total volume where all three nominees are dead is associated with the cube KLMNOPI. Unlike the two-dimensional case, Halley does not provide numerical calculations for a given payment. It is apparent the calculations involved are too numerous and tedious to warrant a complete resolution. Like the joint life annuity with two lives, Halley does not expand the geometrical discussion algebraically to three nominees with starting ages x, y, and z at t=m:

$$ {\displaystyle \begin{array}{l}\left({\mathrm{L}}_{\mathrm{y}+\mathrm{m}}/{\mathrm{L}}_{\mathrm{y}}\right)+\left({\mathrm{L}}_{\mathrm{x}+\mathrm{m}}/{\mathrm{L}}_{\mathrm{x}}\right)+\left({\mathrm{L}}_{\mathrm{z}+\mathrm{m}}/{\mathrm{L}}_{\mathrm{z}}\right)-\left(\left({\mathrm{L}}_{\mathrm{y}+\mathrm{m}}{\mathrm{L}}_{\mathrm{x}+\mathrm{m}}\right)/{\mathrm{L}}_{\mathrm{x}}{\mathrm{L}}_{\mathrm{y}}\right)-\left(\left({\mathrm{L}}_{\mathrm{y}+\mathrm{m}}{\mathrm{L}}_{\mathrm{z}+\mathrm{m}}\right)/{\mathrm{L}}_{\mathrm{z}}{\mathrm{L}}_{\mathrm{y}}\right)-\\ {}\left(\left({\mathrm{L}}_{\mathrm{z}+\mathrm{m}}{\mathrm{L}}_{\mathrm{x}+\mathrm{m}}\right)/{\mathrm{L}}_{\mathrm{x}}{\mathrm{L}}_{\mathrm{z}}\right)+\left(\left({\mathrm{L}}_{\mathrm{y}+\mathrm{m}}{\mathrm{L}}_{\mathrm{x}+\mathrm{m}}{\mathrm{L}}_{\mathrm{z}+\mathrm{m}}\right)/{\mathrm{L}}_{\mathrm{x}}{\mathrm{L}}_{\mathrm{z}}{\mathrm{L}}_{\mathrm{y}}\right)\end{array}} $$

Consequently, Halley did not discuss or observe the extension to more than three nominees. Having laid this rough foundation, Halley never returned to published efforts on this subject, leaving the next stage of development to Abraham de Moivre.

V. THE APPROXIMATIONS OF DE MOIVRE

The proximate reason given by Halley (Reference Halley1693, p. 654) for “not thinking of Methods for facilitating the Computation of the Value of two, three or more lives” is that the Breslaw life table was not based on “the Experience of a very great Number of Years.” However, even though such a task would be “very worth while,” it “seems (as I am inform’d) a Work of too much Difficulty for the ordinary Arithmetician to undertake.” To this end, Halley “sought, if it were possible, to find a Theorem that might be more concise than the Rules there laid down, but in vain.” It was this task de Moivre addressed, primarily by extending series methods used to approximate the single life annuity value assuming arithmetically declining survival rates (uniformly distributed death rates). However, while the easy-to-calculate approximation for a single life annuity works well for ages in the middle of the life table, approximations for joint life annuities were either decidedly more complicated to calculate or lacked precision, leading to considerable subsequent analysis by Simpson, de Moivre, and others seeking approximation methods and formulas for computationally demanding valuation from available life tables.

Precisely when and how Halley and de Moivre became acquainted is unclear. Drawing from the Matthew Maty (Reference Maty1755) biography (Bellhouse and Genest Reference Bellhouse and Genest2007), and the Ivo Schneider (Reference Schneider1968) examination of letters and papers by de Moivre, Hald (Reference Hald1990, pp. 397–398) and Schneider (Reference Schneider2004) find Halley and de Moivre becoming acquainted, possibly friends, in 1692, “likely introduced through the London Huguenot community, some of whose members were Halley’s friends and neighbors in London” (Bellhouse and Genest Reference Bellhouse and Genest2007, p. 114). A credible inference is that de Moivre was a person that “inform’d” Halley of the “Difficulty” of calculating values for joint life annuities. It is interesting that de Moivre (Reference de Moivre1725, p. ii) recognizes:

From this starting point, de Moivre (Reference de Moivre1725) proceeds in Problem I to provide an easy-to-calculate approximation to the value of a single life annuity exploiting the assumption of arithmetically declining survival rates demonstrating solutions for various ages from one to seventy with a maximum age at death (ω) of eighty-six years. Specific comparisons to values calculated by Halley (Reference Halley1693, p. 603) are provided; e.g., with r = .06 and x = 30, the approximation is 11.61 comparable with 11.72 computed by Halley (Reference Halley1693) from the Breslaw table. In stating the solution where k = ω - x:

$$ E{\left[{A}_x\right]}_L=\frac{1}{r}\left(1-\frac{1+r}{k}{A}_k\right)=\frac{1}{r}\left(1-\frac{1+r}{k}\left[\frac{1}{r}-\frac{1}{r{\left(1+r\right)}^k}\right]\right) $$

de Moivre refers to “a general Theorem in the Doctrine of Chances, pag. 132” (de Moivre Reference de Moivre1718, p. 15) on a series solution “not so vulgarly known” but does not carry out a proof.

Following Halley, de Moivre typically focuses on survival rates, ignoring potential simplifications provided by death rates. Hald (Reference Hald1990, p. 521) provides details required to complete the complicated proof of Problem I, correctly observing that if uniform death rates are used instead of survival rates, then proving the approximation is less complicated. Observing $ Prob\left[x,t\right] $ , the probability of $ x(y) $ surviving at time $ t=n $ corresponds to the survival rate $ \left({\ell}_{\mathrm{x}+\mathrm{n}}/{\ell}_{\mathrm{x}}\right) $ , the actuarially sound calculation of the single life annuity price $ E\left[{A}_x\right] $ follows from multiplying by $ {\left(1+r\right)}^{-t} $ , and summing gives the required result, which holds for any life table. The relationship between pricing with survival rates and the death rates $ \left(\left({\ell}_{\mathrm{x}+\mathrm{n}+1}-{\ell}_{\mathrm{x}+\mathrm{n}}\right)/{\ell}_{\mathrm{x}}\right) $ used by de Witt follows:

$$ E\left[{A}_x\right]=\frac{1}{\ell_x}\sum_{n=1}^{\omega -x-1}{A}_n{d}_{x+n}=\frac{1}{\ell_x}\sum_{n=1}^{\omega -x-1}{d}_{x+n}\sum_{t=1}^n\frac{1}{{\left(1+r\right)}^t} $$

$$ \hskip7.3em =\frac{1}{\ell_x}\sum_{t=1}^n\sum_{n=1}^{\omega -x-1}{d}_{x+n}\frac{1}{{\left(1+r\right)}^t}=\frac{1}{\ell_x}\sum_{n=1}^{\omega -x-1}{\ell}_{x+n}\frac{1}{{\left(1+r\right)}^t} $$

where $ {\ell}_{\mathrm{x}+\mathrm{i}} $ is the number living i years past the starting age x, and $ {d}_{\mathrm{x}+\mathrm{i}} $ is the corresponding number dying for that year. Reconciling the arithmetically declining survival rate, the single life annuity approximation given by de Moivre with $ E{\left[{A}_x\right]}_L $ calculated using uniform death rates (k = ω − x) gives:

$$ E{\left[{A}_x\right]}_L=\frac{1}{k}\sum_{n=1}^{\varpi -x-1}\left[\frac{1}{r}-\left(\frac{1}{r{\left(1+r\right)}^n}\right)\right]=\frac{1}{rk}\left(\sum_{n=1}^{\varpi -x-1}\left[1-\left(\frac{1}{{\left(1+r\right)}^n}\right)\right]\right)=\frac{1}{r}\left[1-\frac{1+r}{k}{A}_k\right] $$

where the last equality can be verified by induction or by evaluating the sum and manipulating.

Historical context is useful to interpret analytical development of approximations to single and joint life annuity values. Despite increasing availability of methods for actuarially sound pricing, French and British government issues of joint and single life annuities throughout the eighteenth century did not accurately account for age. While contributions such as Essai sur les probabilités de la durée de la vie humaine by Antoine Deparcieux (Reference Deparcieux1746) and, to a lesser extent, Calcul des Rentes viagères by Nicolaas Struyck (Reference Struyck1740) (see SGN 1912), substantively increased the quality of data on mortality, life annuity valuation was largely a concern for mathematicians until emergence of the life insurance industry in the later part of the eighteenth century. The numerous approximations relevant for single and joint life annuities in de Moivre (Reference de Moivre1725) were benchmarked to the rudimentary life table provided by Halley. Often motivated by de Moivre’s desire to achieve mathematical solutions, some of his approximations were more successful than others, and Simpson expended considerable effort showing that direct calculation making use of actual life tables was substantially better for pricing the joint life annuity (Hald Reference Hald1990, p. 532).

The key step in solving for a joint life annuity on two lives involves using the compound probability given by Halley to solve:

$$ E\left[{A}_{x\;y}\right]=E\left[{A}_x\right]+E\left[{A}_y\right]-E\left[{}_x{A}_y\right] $$

where $ E\left[{A}_{x\;y}\right] $ and $ E\left[{}_x{A}_y\right] $ are prices of two nominee last survivor and joint survivor annuities, respectively. This exact result, applicable to any life table, follows from applying the compound probability rule given by Halley:

$$ \left[1-\Big(1- Prob\right[x,n\left]\left)\right(1- Prob\right[y,n\left]\Big)\right]= Prob\left[x,n\right]+ Prob\left[y,n\right]-\left( Prob\left[x,n\right] Prob\left[y,n\right]\right) $$

to calculate the price of the last survivor annuity $ \left(E\left[{A}_{x\;y}\right]\right) $ on two lives at time $ t=0 $ . In contrast to approximation for single life annuity values, de Moivre struggled to achieve comparable results for joint life annuities. Taking pains to distinguish between “real lives” calculated from a life table and “fictitious lives” from the approximation, de Moivre (Reference de Moivre1725) used two approaches to solve for approximations to $ E\left[{}_x{A}_y\right] $ . One approach takes $ Prob\left[i,j\right] $ —“the Decrements of Life”—to be geometrically declining; the other approach uses arithmetically declining survival rates.

More precisely, if $ E{\left[{A}_x\right]}_{\mathrm{G}} $ and $ E{\left[{A}_y\right]}_{\mathrm{G}} $ are the values of single life annuities calculated assuming geometrically declining survival rates, and $ E{\left[{}_x{A}_y\right]}_{\mathrm{G}} $ is the value of the “joint continuance” using geometrically declining “life probabilities,” then de Moivre (Reference de Moivre1725, Problem IV) is able to show:

$$ E{\left[{}_x{A}_y\right]}_G=\frac{E{\left[{A}_x\right]}_G\;E{\left[{A}_y\right]}_G\;\left(1+r\right)}{\left(E{\left[{A}_x\right]}_G+1\right)\left(E{\left[{A}_y\right]}_G+1\right)-\left(E{\left[{A}_x\right]}_G\;E{\left[{A}_y\right]}_G\;\left(1+r\right)\right)} $$

This result follows from the geometrically declining probability single life solution for $ E{\left[{A}_x\right]}_{\mathrm{G}} $ (given in Problem III). This solution is obtained by observing that for x < 1, then the geometric series $ x+{x}^2+{x}^3+\dots =\left(x/\left(1-x\right)\right) $ . Where $ Prob\left[{g}_{\mathrm{x}}\right] $ is the rate of geometric decline, then the solution for the $ E{\left[{A}_x\right]}_{\mathrm{G}} $ follows:

$$ E{[{A}_x]}_G=\sum_{t=1}^{\mathrm{\infty}}{(\frac{Prob[{g}_x]}{1+r})}^t=\frac{\frac{Prob[{g}_x]}{1+r}}{1-\frac{Prob[{g}_x]}{1+r}}=\frac{\frac{Prob[{g}_x]}{1+r}}{\frac{1+r-Prob[{g}_x]}{1+r}}=\frac{Prob[{g}_x]}{1+r-Prob[{g}_x]\hskip2pt } $$

Solving:

$$ Prob\left[{g}_x\right]=\frac{\left(1+r\right)E{\left[{A}_x\right]}_G}{E{\left[{A}_x\right]}_G+1}\hskip2.72em Prob\left[{g}_y\right]=\frac{\left(1+r\right)E{\left[{A}_y\right]}_G}{E{\left[{A}_y\right]}_G+1} $$

Using the compound probability, the approximation for a two nominee joint survivor annuity value assuming geometrically declining survival rates is:

$$ E{[{}_x{A}_y]}_G=\frac{Prob[{g}_x]\hskip2pt Prob[{g}_y]}{1+r-(Prob[{g}_x]\hskip2pt Prob[{g}_y])} $$

Substituting for $ Prob\left[{g}_{\mathrm{x}}\right] $ and $ Prob\left[{g}_{\mathrm{y}}\right] $ and cancelling give the desired result for $ E{\left[{}_x{A}_y\right]}_{\mathrm{G}} $ . While this approach leads to a relatively easy-to-calculate approximation for $ E{\left[{A}_{x\;y}\right]}_{\mathrm{G}} $ , casual inspection reveals that “real” survival rates do not decline geometrically. It follows that comparison with the more complicated solution using arithmetically declining survival rates $ \left(E{\left[{A}_{x\;y}\right]}_{\mathrm{L}}\right) $ is needed.

The approach in de Moivre (Reference de Moivre1725) to $ E{\left[{A}_{x\;y}\right]}_{\mathrm{L}} $ —the last survivor annuity “whose Decrements are in Arithmetic Progression”— is unusual. The solution is not developed in the same fashion as other results, only stated without a derivation for “two lives” as a “Remark” added to Problem VII for the last survivor annuity for three lives, accompanied by some calculations to compare the arithmetic and geometric approximations. The solution requires the maximum age $ \omega\;\left(=86\right) $ with starting ages x and y where $ k=\unicode{x03C9} -x $ and $ j=\unicode{x03C9} -y $ . Assuming $ x>y $ the solution provided is:Footnote ¹⁷

$$ E{\left[{A}_{xy}\right]}_L=\frac{1}{r}-\frac{1+r}{rk}\left({A}_k+{A}_j\right)+\frac{1+r}{r^2 jk}\left(2j-\left(2+r\right){A}_j\right) $$

A numerical example for two lives aged forty and fifty is provided and, with r = 5%, the values of $ E{\left[{A}_{xy}\right]}_{\mathrm{L}} $ for arithmetically and $ E{\left[{A}_{xy}\right]}_{\mathrm{G}} $ for geometrically declining cases are solved as 14.53 and 14.55 years’ purchase, respectively. No further comparisons are provided, though Simpson (Reference Simpson1742, pp. 36, 61) did later show the linear approximation has a better fit to values for $ E\left[{A}_{x\;x}\right] $ using a “life table” determined from “ten Years Observations on the Bill of Mortality of the City of London.” No indication is given in de Moivre (Reference de Moivre1725) about a solution for three lives. The $ E{\left[{A}_{xy}\right]}_{\mathrm{L}} $ formula was dropped from later editions (de Moivre 1743, 1750, 1752).

Driven by the desire to provide simple-to-calculate joint life annuity formulas, de Moivre (Reference de Moivre1725) provides an extension to three lives for the geometrically declining case. The connection between joint survivor and last survivor annuities follows appropriately:

$$ E\left[{A}_{x\;y\;z}\right]=E\left[{A}_x\right]+E\left[{A}_y\right]+E\left[{A}_z\right]-E\left[{}_x{A}_y\right]-E\left[{}_y{A}_z\right]-E\left[{}_x{A}_z\right]+E\left[{}_{x\;y}{A}_z\right] $$

where $ E\left[{A}_{x\;y\;z}\right] $ is the value of the last survivor annuity for three nominees and $ E\left[{}_{x\;y}{A}_z\right] $ is the value of the joint survivor annuity with three nominees. As for the two nominee case, this result is general and applies to any life table. Extending the solution for the two nominee case, de Moivre (Reference de Moivre1725, Problem VI) gives the geometrically declining survival rate result:

$$ E{\left[{}_{xy}A\right]}_G=\frac{\left(E{\left[{A}_x\right]}_G+E{\left[{A}_y\right]}_G+E{\left[{A}_z\right]}_G\right){\left(1+r\right)}^2}{\left(E{\left[{A}_x\right]}_G+1\right)\left(E{\left[{A}_y\right]}_G+1\right)\left(E{\left[{A}_z\right]}_G+1\right)-\left(\left(E{\left[{A}_x\right]}_G+E{\left[{A}_y\right]}_G+E{\left[{A}_z\right]}_G\right){\left(1+r\right)}^2\right)} $$

While no results or calculations are given for the three life joint survivor annuity using arithmetically declining life probabilities, de Moivre does indicate, with demonstration, the solution methodology for “finding the Values of as many joint Lives as may be assigned” (de Moivre Reference de Moivre1725, pp. 44–45, Corollary I). The simplification provided by assuming the nominees’ lives are equal is also recognized. Life tables from Halley, Willem Kersseboom, Antoine de Parcieux, and John Smart are included only in the Appendix to de Moivre (Reference de Moivre1756b).

VI. SIMPSON’S IMPROVEMENTS

Detailed examination of de Moivre’s contributions to joint life annuity valuation would be incomplete without considering Simpson (Reference Simpson1742, 2nd ed. 1775). As evidenced in the preface of de Moivre (Reference de Moivre1725; with corrections 1743), the contents of Simpson (Reference Simpson1742) incensed de Moivre. Without identifying Simpson by name, de Moivre makes the following observations clearly directed at Simpson: “he mutilates my Propositions, obscures what is clear, makes a Shew of new Rules, and works by mine, in short confounds in his usual way, every thing with a croud of useless Symbols” (p. xii). This attack by de Moivre compelled Simpson to include in future printings an additional “Appendix containing some remarks on Mr. Demoivre’s book … with answers to some personal and malignant misrepresentations, in the preface thereof.” In this appendix, Simpson claims:

Simpson makes a strong case for the credibility of his contributions that is difficult to deny. In addition, it is likely that results and comments in the appendix led de Moivre to make some substantive changes in the 1743 corrected edition of 1725 and later versions.

At the time Simpson (Reference Simpson1742) appeared, Thomas Simpson was still in the early stages of an academic career that was to produce a number of seminal advances, primarily in mathematics, earning the eponym Simpson’s Rule for a method of approximating an integral using a sequence of quadratic polynomials.Footnote ¹⁸ His appointment as the head of mathematics at the Royal Military Academy at Woolwich—a position that provided Simpson with sufficient security to pursue academic interests—was not to occur until 1743. Being the son of a weaver with little formal education, Simpson had for some years made a living as an itinerant lecturer teaching in the London coffee houses. Around this time, certain coffee houses functioned as “penny universities” that provided cheap education, charging an entrance fee of a penny or two to customers who drank coffee and listened to lectures on topics specific to that coffee house (see, e.g., Poitras Reference Poitras2000, pp. 293–297). Popular topics were art, business, law, and mathematics. It is well known that de Moivre was a fixture at Slaughter’s Coffee House in St. Martin’s Lane during this period.

In support of activities at the penny universities and continuing at Woolwich, Simpson started producing a successful string of textbooks, beginning with Simpson (Reference Simpson1737), a text on the theory of fluxions (Blanco Reference Blanco2014). The next book, Simpson (Reference Simpson1740), bears a strong similarity to de Moivre (Reference de Moivre1718; see also 2nd ed. 1738, and 3rd ed. Reference de Moivre1756a). In both Simpson (Reference Simpson1740) and Simpson (Reference Simpson1742), grateful references to de Moivre appear in the preface but no references in the body of the text despite obvious similarities in format and some content. The incensed reaction to Simpson in de Moivre (Reference de Moivre1725, with corrections 1743) is sometimes viewed as a response to Simpson’s perceived plagiarism, though this is a generous interpretation, as the practice of borrowing without attribution was common. A precise explanation requires information that is unavailable. The seemingly damning evidence that Simpson was also accused of plagiarism by some others can, initially, be attributed to his desire to build a reputation required to sustain his livelihood and, over time, to a lack of sympathy for formal academic acknowledgment gradually gaining foothold in the eighteenth century. Even after appointment at Woolwich and election to the Royal Society (in 1745), Simpson continued to be accused of plagiarism, though not by persons with the academic stature of de Moivre.Footnote ¹⁹

In contrast to the seminal theoretical work on joint (and single) life annuities accomplished by de Moivre, Simpson’s contributions on joint life annuity valuation were decidedly more applied. Hald (Reference Hald1990, p. 511) identifies three important applied contributions: “(1) a life table based on the London bills of mortality; (2) tables of values of single- and joint-life annuities for nominees of the same age based on this life table; and (3) rules for calculating joint-life annuities for different ages from the tabulated joint-life annuities.” This assessment would appear to be based largely on content of Simpson (Reference Simpson1742) and ignores the practical usefulness of Part VI in Simpson (Reference Simpson1752), “The Valuation of Annuities for single and joint Lives, with a Set of new Tables, far more extensive than any extant.” Judging from references to Simpson, de Moivre, and Halley in the seminal Richard Price (Reference Price and Morgan1771), it was the practical content of Simpson’s numerous worked problems and examples that had the most relative impact. While recognizing the importance of de Moivre’s arithmetically declining survival rate approximations, Price (Reference Price and Morgan1771) dedicates Essay II to the unacceptable errors that assuming “geometrically declining life probabilities” has on the calculated values for joint life annuities on two and three lives.

Even casual reading of Simpson (Reference Simpson1742) reveals the close connection to de Moivre (Reference de Moivre1725). The presentations are similar and various results are more general versions of the same problems. For example, corollaries III and V of Problem I give the solution for geometrically declining and arithmetically declining survival rates, respectively. Corollary III covers five or more joint lives and Corollary V applies up to three joint lives, extending results in de Moivre (Reference de Moivre1725). This said, Simpson does not mimic de Moivre’s analytical approximation agenda. Results are adapted to practical ends and derivations are often either sketchy or hard to follow. To see the implications of this, consider the approximation $ E{\left[{}_x{A}_y\right]}_{\mathrm{S}} $ for the joint survivor annuity on two unequal lives given in terms of two equal lives stated by Simpson (Reference Simpson1742, p. 50) without proof:

$$ E{\left[{}_x{A}_y\right]}_S=E\left[{}_x{A}_x\right]+\frac{2E\left[{}_x{A}_x\right]\left(E\left[{}_y{A}_y-E\left[{}_x{A}_x\right]\right]\right)}{E\left[{}_y{A}_y\right]}\hskip1.92em x<y $$

This “fictitious” solution, which does not appear in de Moivre (Reference de Moivre1725), reduces the unmanageable practical problem of preparing tables for the value of joint life annuities on two unequal lives to be solved using a manageable table for joint annuity values on two equal “real” lives, tables that Simpson provides. Significantly, contributions of both de Moivre and Simpson to joint life annuity valuation had a profound impact on the seminal contributions to life insurance and pensions plans by Richard Price and, to a lesser extent, James Dodson.