Johan de Witt - The first calculation on the valuation of life annuities
An important contribution to insurance mathematics
As mentioned before, de Witt's work can be seen as the first mathematically sound application of probability theory to insurance calculations. Before the 17th century, forms of annuities could be found throughout Europe, dating back to ancient Rome. Hereditary law known as Lex Falcidia ensured standards on the distribution of properties, but valuating annuities after the death of the holder was a more intricate matter (Kopf, 1927). To deal with this, two tables were widely used - Emilius Macer's table, and Ulpianus' table (Kopf, 1927). These were both examples of very simplistic efforts toward the use of mortality rates, and more approximate than accurate. Still, due to a lack of further mathematical advancements, the tables were indeed widely used, in some areas in Italy even up until the early 19th century (Kopf, 1927).
Later uses of annuities in Europe were mainly to finance war budgets by hypothecating property in exchange for annuity payments.
Historical contributions to annuity valuation prior to de Witt mainly came from the construction of mortality tables more advanced than the Macerian or Ulpianian ones. It lies in the nature of these tables that they form a time-consuming matter, as records of deaths need to be kept over a few generations to be able to estimate mortality rates (Hogendijk, 2010). A time with few influences on mortality such as epidemies or wars should be chosen, and one needs to acknowledge generalisation problems due to demographic and geographic selection biases. A large contribution to mortality tables ought to be attested to the Church. In the 16th century, many parishes and cities began keeping records of marriages, burials and christenings (Kopf, 1927). The 1538 established ecclesiastical records in England, and the shortly afterwards introduced Bills of Mortality in London, were important sources for information on death rates and causes (Poitras, 2000).
The Bills of London were used by mathematician John Graunt to create one of the first death tables in 1662 in the "Natural and Political Observations Upon the Bills of Mortality of the City of London" (Hogendijk, 2010). The motivation for the table was to estimate the number of men fit for war at the time. This marks a crucial contribution to insurance calculation since death tables are a necessary component for the valuation of annuities.
This table was also known to the three mathematicians introduced earlier: de Witt, Huygens, and Hudde. Hudde constructed a mortality table after the example of Graunt, but states that he arrives at results "quite different to those of Graunt" (Hald, 2003, p.126). De Witt corresponded with both Huygens and Hudde on topics evolving around mortality rates, mortality tables, and the valuation of annuities of one or more lives (Hald, 2003). Hudde also helped to proof-read and verify de Witt's methods in the Waerdye, and in his function as Burgomaster of Amsterdam introduced the first widespread age-dependent annuity scheme (Kupers, 2014). Much of the correspondence between the three mathematicians has been lost.
The next important figure was Edmond Halley, who was long - and by many still is - seen as the first to valuate life annuities, since the Waerdye had been forgotten until the 19th century. In 1694 - 18 years after de Witt - he published "An Estimate of the Degrees of Mortality on Mankind, drawn from the curious Tables of the Births and Funerals at the City of Breslaw; with an Attempt to ascertain the Price of Annuities upon Lives", in which he presents a life table derived from the death records of Breslaw (Poitras, 2000). He used the table to calculate the value of life annuities in a very similar fashion to de Witt, whose work he did not know (Hogendijk, 2010). Other than de Witt, he used 5-year age intervals, making his calculations more intricate than de Witt's. Importantly, he outlines principles which should be used for the construction of mortality tables, making a significant contribution to actuarial mathematics (Kopf, 1927). An advantage that Halley had was the use of Stevin's compound interest table and mortality rates found from actual data, which made his calculations more accurate than de Witt's (Kopf, 1927).
In 1725, French mathematician Abraham de Moivre simplified Halley's calculations, who had been less concerned about the practical applications of his formulae (Poitras, 2000). De Moivre added the assumption that after the age of 12, the mortality table follows a linear function, which leads to a quicker, yet less accurate, calculation (Hogendijk, 2010). De Moivre's simplifications made annuity valuation more applicable, which was shown by its wide use by practitioners (Poitras, 2000).
This work influenced the efforts of Nicolaas Struyck, a Dutch mathematician. He constructed his own mortality tables to be used for the valuation of life annuities, realising that life annuity registers are better generalizable to their valuation than for example Halley's data of widely non-annuity holders (Poitras, 2000). Significantly, Struyck was the first to realise that male and female personas have different life expectancies, which influences the value of annuities for men and women. He constructed two tables, separated by gender, and published them in 1740 in his "Aanhangsel op de Gissingen over den Staat van het Menschelyk Geslagt en de Uitreekening der Lyfrenten" (Hogendijk, 2010). In his calculations for the annuity valuation he discussed the work of de Witt, but relied on simplifications similar to those of de Moivre.
Two years after Struyck's publication, in 1742, Thomas Simpson published the "Doctrine of Annuities and Reversions", which provided calculations for valuating annuities at any age n derived from age n+1.
The 18th century saw the rise of many insurance organisations, such as the British "Amicable Society for Perpetual Assurances" of 1705. This Society, however, did not adjust premiums according to age, and Dr. James Dodson - influenced by the works of de Moivre - founded the "Equitable Life Assurance Society of London" in 1756 to incorporate age-dependent schemes (Kopf, 1927). Dr. Richard Price was another mathematician working for the "Equitable", who used de Moivre's work to establish principles for the practicing of modern life insurance and actuarial work (Buer, 2005).
For a long stretch of history, life insurance and annuities had been dealt with in a dilettantish way, not in any way founded on actual data or mathematical considerations. This is likely to have resulted in losses to states and insurance organisations or holders equally. The efforts of the above-mentioned figures - and the works of many more unmentioned - marked the birth of the actuarial science and created the base for today's insurance industry.