John Graunt, Johan de Witt and Their Prototypical Mortality Tables (1662-1671)

Amongst the small group of men who made profound contributions to probability and actuarial thinking in the seventeenth century, John Graunt stands out for his relative mediocrity. He was not a world-leading mathematician of his generation (Pascal, Fermat), nor was he an Astronomer Royal (Halley), nor an international statesman of historical significance (de Witt). Graunt appears to us as a middling merchant, a son of a tradesman without formal academic training or any trace of earlier exposure to scientific analysis, armed only with a peculiar interest in the London Bills of Mortality.

The London Bills of Mortality recorded the numbers of deaths and christenings in the parishes of London. They were produced weekly and continuously from 29 December 1603. The Bills first appeared in 1592, but were shortly discontinued before being resumed in 1603. Both 1592 and 1603 were years of heavy mortality arising from the plague. Part of the government’s motivation for the production of the Bills appears to be to ease the panic induced in the London population by showing that the impact of the plague on human mortality was not as severe as rumoured.^{[1]}

Graunt’s paper ‘Natural and Political Observations made upon the Bills of Mortality’ was published in 1662. It foreshadowed the use of statistical analysis of population data to steer government social policy that would emerge 150 years later with the development of government statistical offices in Britain, France, Germany and Prussia. For example, Graunt observed that the statistical rate at which beggars died from starvation was very low, and suggested that, as they were already living off the wealth of the nation, they might as well be paid a guaranteed wage to keep them off the streets. With the use of statistical observations to form such policy arguments, it is arguable that Graunt could be viewed more naturally as the antecedent of Condorcet, Quetelet and nineteenth-century ‘social mathematics’ rather than an as an early pioneer of actuarial science.

As a source of statistical data to estimate mortality rates, the Bills of the seventeenth century had serious limitations. The data provided no direct information on the distribution of the population by age (so there was no total ‘exposed to risk’ available from the data). Even more fundamentally, the information included in the records of deaths did not show the age of the deceased. Inevitably, any table of mortality rate as a function of age derived from this data would require some giant assumptions and extrapolations. Undaunted, Graunt made several such assumptions to produce what can be recognised as the first mortality table based on explicit statistical analysis.

Though the seventeenth-century London Bills of Mortality did not include the age of the deceased, in the 1620s some parish records did start to include a cause of death. The format of these records varied from parish to parish. Some simply recorded whether the deceased had died of plague or not. By 1632, the parish of Westminster used a total of 63 categories to attribute the causes of death (including, for example, ‘bit with a mad dog’). The most common causes were ‘Consumption’ and ‘Fever’, and ‘causes’ that were actually references to the age of the deceased: ‘Aged’ and ‘Chrisomes,^{[2]} and Infants’. This data was recorded by ‘Searchers’, a form of local government official. They were generally not medically trained and, for social and cultural reasons, may not have been entirely unbiased in their reporting of the cause of death. Graunt felt compelled to make rough adjustments to their records— for example, he decides that ‘we shall make it probable, that in years of plague, a quarter-part more dies of that disease than is set down’.

The cause of death data was the key to the deductions Graunt made in developing his mortality table. The table is not a major focus of his paper and Graunt is not explicit in his workings. He identified the causes of death that were most associated with death at a young age, and from this he estimated that slightly more than one third of deaths occured by the age of six. He made the assumption that, of every 100 births, there would be one survivor by age 76. In the intervening 70 years, he assumed there was an equal probability of dying in each decade. This implies an annualised mortality rate of around 8 % over the first six years of life and around 5 % for the next 70 years.^{[3]}

Graunt’s analysis was conjectural and heuristic rather than statistically rigorous. His focus was on providing sociological insight, not on making improvements in the pricing of life contingencies. Nonetheless, his work on the analysis of population mortality data broke new quantitative ground and provided a starting point for the actuarial thinking that would later be pursued by some of its greatest names. Whilst the publication of his paper had no discernible impact on what limited life insurance business was practiced at the time, it did highlight the latent potential in existing population records and the possibilities of statistical analysis. His paper was reviewed by William Petty in a Paris journal in 1666. In 1667, France started to collect statistical data of a similar form to that found in the London Bills. The influence and impact of Graunt’s work was confirmed when he was made a Fellow of the Royal Society at the behest of King Charles II, who is said to have commended him to the sceptical and snobbish fellows with the remark ‘that if they found any more such tradesman, they should admit them all’.^{[4]}

Johan de Witt’s life contrasts starkly with the humble background of John Graunt. De Witt was born in 1825 as the son of an influential politician, and by the age of 28 he had obtained the position of grand pensionary—roughly equivalent to prime minister—of the States of Holland, and held it for the following nineteen years. The Dutch Republic was one of the most significant European powers throughout this period, and de Witt established himself as one of Europe’s pre-eminent statesmen of his time. His political career, and ultimately his life, was cut short in 1672 when 120,000 French troops of Louis XIV invaded Holland. De Witt resigned but was nonetheless assassinated, along with his brother, by a mob supportive of his political rival Prince William of Orange, the future King William III of England, Ireland and Scotland (William II in Scotland).

As a youngster, de Witt received tutelage from some of Holland’s leading thinkers of the time, from which he obtained a lifelong interest in mathematics. Between plotting geopolitical strategic alliances and fighting wars with England and France, he also devoted some time to the theory of annuity pricing. His interest in actuarial science was not driven entirely by mere mathematical curiosity. The Netherlands was advanced in the sophistication of its approach to raising government funding and by the mid-seventeenth century it had a long-established practice of using perpetuities and life annuities to issue state debt. By contrast, England at the time had no facilities for raising funded government debt until the final years of the century (it had experimented with issuing life annuities in the first half of the sixteenth century, but never in significant volume).^{[5]} De Witt was concerned that the state was paying too much for its life annuity funding relative to the cost of perpetual annuity funding, and he proceeded to produce the first rigorous analysis of annuity pricing to make his point.

As a graduate of law and mathematics, de Witt was well equipped to apply mathematical expectation as a form of legal equity. In a similar vein to Graunt, de Witt made some sweeping assumptions about mortality rates. He weaved together Graunt’s approach to basic mortality modelling with the mathematical expectations concept of valuing aleatory contracts pioneered by Pascal, Fermat and others such as Huygens, so as to develop prices for life annuities.

In 1671 de Witt wrote a series of letters to the Estates General (the body responsible for Dutch government funding) where he set out his analysis of annuity pricing. He argued that with perpetual debt achieving a price of 25 times its annual interest, life annuities should be priced higher than the fourteen times annual income at which they were being sold at the time of writing.

He first suggested that an uncertain cashflow should be valued at its expectation: ‘The value of several equal expectations or chances, a certain sum of money or other objects of value pertaining to each chance, is found to be exactly determinable by adding the money or other objects of value represented by the chances, and by then dividing the sum of this addition by the number of chances: the quotient or result indicates with precision the value of all these chances.’^{[6]} His demonstration of the reasonableness of this assumption borrows liberally from his compatriot Huygens and showed how it was consistent with the result obtained from a series of equitable exchanges of aleatory contracts.

In a similar heuristic style to Graunt, de Witt assumed that mortality rates were uniform between the ages of three and 53. An important clarification is required when describing what de Witt meant by a uniform mortality rate assumption. He meant that, if we consider a starting pool of people of a given age, a constant number from that pool would die every future year. The size of the pool shrinks as people die, and so the mortality rate at which the survivors consequently die is always increasing. Thus, de Witt supposed that, if we start with a pool of 128 lives aged three, two would die every year over the next 50 years, leaving 28 remaining alive aged 53. This implies that the mortality rate for a 53 year-old is more than four times higher than for a three year- old.^{[7]} Such a progression in mortality rates would seem intuitively desirable to modern eyes, but it does not occur in de Witt’s table by explicit design. De Witt did not appear to appreciate the increase in probability of death that the specification of a constant number of deaths from a fixed starting pool implied—the modern conception of a mortality rate had simply not yet been realised. In contrast to Graunt, de Witt was happy to work with fractional lives in his projected pool: he assumed that between ages 53 and 63 people die from the remaining pool (of 28) at a rate of 4/3 per year; between 63 and 73

they die at a rate of one per year; and between 73 and 80 they die at a rate of 2/3 per year. All 128 lives are therefore extinguished by age 80.

Unlike Graunt, de Witt offered no empirical basis for these mortality assumptions. De Witt had some correspondence on these assumptions with Johannes Hudde, a contemporary mathematically minded Dutch politician who was mayor of Amsterdam during the period. Hudde developed an empirical mortality analysis from the records of the annuities sold by the Dutch government between 1586 and 1590. This qualifies as the first mortality experience analysis of annuity business. He grouped the data by age of the annuitant when they purchased the annuity and tabulated the number of years for which each annuitant received their annuity payment. De Witt was satisfied that this data was consistent with his assumptions, noting that the data suggested that the proportion of 50 year-olds dying by age 55 was 1/6, and of 55 year-olds dying by age 60 was 1/5 (although de Witt’s assumptions implied both these proportions would be very close to U).

Armed with the above mortality decrements and the 4 % interest rate derived from the perpetual annuity price, de Witt then undertook the arithmetic computation of mathematical expectations discounted by the time value of money in order to calculate the price of an annuity for someone aged three on purchase. He found that the annuity price was sixteen, thus supporting his initial statement that the price of fourteen, at which the annuities were currently being sold, was too low. De Witt then went on to argue that the selection effect ‘of choosing a life, or person in full health, and with a manifest likelihood of prolonged existence’ should significantly increase the price of the annuity further. This is notable as perhaps the first published discussion of the impact of selection on the price of life contingencies.

De Witt’s work can be viewed as a synthesis of the probabilistic and valuation thinking developed by Pascal, Fermat and Huygens together with the pragmatic mortality modelling introduced by Graunt. Its originality lay in demonstrating how these emerging ideas could be used to rationally obtain fair prices for annuities, and how such pricing must consider practical effects such as anti-selection. The application of the then-recent developments in probabilistic thinking to the valuation of life contingencies had begun.

[3] The mortality rates implied by Graunts survival table increase after age 56, but this arises due to rounding of the number of remaining survivors from a pool that begins with the arbitrary total of100 (‘for mendo not die in exact proportions nor in fractions’) rather than by explicit design.

[4] Francis (1853), Chapter 1, p. 11, Hacking (1975), Chapter 12, p. 106.