Some forty years after the ground-breaking correspondence of Pascal and Fermat, another excellent mathematician dabbled in the emerging subjects of probability and the valuation of life-contingent claims. Edmond Halley, his interest piqued by the British government’s plans for a significant issuance of life annuities, presented a paper to the Royal Society in 1693 on mortality modelling and annuity pricing.^{[1]} It is little more than a footnote in the published work of the famed Astronomer Royal, but the paper fundamentally advanced the methods of mortality modelling and what we might now call actuarial statistics.

Over the preceding 40 years, several seeds of actuarial thought had been planted. Probability as a discipline of applied mathematics had been tentatively established and had attracted some of the greatest mathematical minds of the generation. This calculus of probabilities had been synthesised with contemporary legal doctrine to provide an intellectual basis for the use of mathematical expectations in valuing future uncertain cashflows, such as those arising from life annuities. Graunt’s analysis of the empirical data provided by the London Bills of Mortality illuminated the possibilities of what might be inferred from the analysis of existing datasets. De Witt had taken similar mortality assumptions to those of Graunt and had applied them to develop prices for life annuities.

Up until this time, however, empirical mortality data had not been used to provide rigorous or granular insights into how mortality rates varied by age. Graunt and de Witt’s mortality tables had relied on sweeping assumptions and simplifications and were not based on what would be recognised today as statistical fitting to a dataset. This was largely because of the limitations in the empirical data. In particular, the lack of data on both age at death and the numbers alive at each age severely limited the opportunity to conduct a straightforward statistical analysis of probability of death by age.

In his Royal Society paper of 1693, Halley first identified the shortcomings in the London Bills of Mortality data used by John Graunt. He noted that the population numbers of each age were unknown; that the ages of the people dying were unknown; and that the population size was unstable due to immigration flows. Halley, apparently with the assistance of Leibniz,^{[2]} located bills of mortality for the German town of Breslau for the years of 1687-1691. The Breslau bills were the only such records known of at the time that recorded age at death. Halley also argued that the town of Breslau was a small, sleepy place that did not experience much immigration or emigration compared to the growing metropolis of London. Thus, two of three defects in Graunt’s London data were arguably addressed by the Breslau bills.

Halley used this superior mortality data to produce the first mortality table with a granular description of how mortality rates varied as a function of age.

However, his paper is silent on the quantitative steps he took to transform the five years of raw data into this mortality table. Specifically, he does not explicitly describe what steps he took to work around the remaining defect of the data—the unknown total number alive at each age. Halley states that he assumed a stable population. He most likely used this assumption to infer the numbers alive at each age from the numbers that died: if the population is stable, the numbers alive at any given age must be equal to the sum of all those dying at that age and older. This observation allows the mortality rates to be inferred without any explicit data on the numbers alive. The approach is well-explained by Richard Price^{[3]} some 80 years after Halley’s paper:

In every place that just supports itself in the number of its inhabitants, without any recruits from other places; or where, for a course of years, there has been no increase or decrease, the number of persons dying every year at any particular

age, and above it, must be equal to the number of living at that age____From this

observation it follows, that in a town or country, where there is no increase or decrease, bills of mortality which give the ages at which all die, will show the exact number of inhabitants; and also the exact law, according to which human life wastes in that town or country.

Figure 1.1 compares the three seventeenth-century mortality tables discussed so far. Halley’s table has a regularity that is absent from that of both Graunt and de Witt, whose mortality rates are subject to jumps that result from arbitrary assumptions. Halley’s access to improved data permitted an advancement in methodology that resulted in a mortality table recognisable to modern eyes. To the extent there was such a thing, it remained the standard reference mortality table until the second half of the eighteenth century.

Halley’s paper went on to apply his mortality table to the valuation of life annuities for both single and joint lives (and, in keeping with the time, these include novel geometric representations!). In a similar vein to De Witt, Halley concluded that government annuity prices appeared cheap relative to longterm government bonds. Halley noted that with the long-term interest rate at 6 %, his table implied a fair annuity rate of less than 10 % for a 40 year-old, yet the government’s annuity pricing would offer an annuity income of 14 %.

Halley’s motivation for his mortality and annuity investigation arose from the English government’s plans to imitate the Dutch and make more use of life annuities as a means of government borrowing. We now turn to the English government annuity issuance that occurred in the year of the publication of

Fig. 1.1 Three seventeenth-century mortality tables

Halleys paper, and consider it alongside the other practices that were taking place in the life contingencies business at the end of the seventeenth century.