Developments in Mortality Modelling, 1808-1881
Chapter 2 noted that Richard Price’s Northampton table has been described as a mortality benchmark that stood for the next 100 years and that according to Hacking ‘the Institute of Actuaries did not do anything much better until
1869’. The truth is more nuanced. The next significant development in mortality modelling to follow Price was made by Joshua Milne in 1815. Milne was then actuary at the Sun Life Assurance Society, one of the many new life assurance offices established in the early 1800s following the success of the Equitable.
Milne’s two-volume tome A Treatise on the Valuation of Annuities and Assurances on Lives and Survivorships largely consisted of mathematical bookwork on the valuation of various forms of annuities and assurances. But the inclusion of his Carlisle mortality table made it a historic actuarial publication. The Carlisle table was, like Price’s Northampton table, based on observations from local Bills of Mortality data (the Carlisle table was based on data from 1779 to 1787 inclusive). However, Milne’s Carlisle data was superior to the data used in any prior mortality table in one key respect: it included population data categorised by year of age (it was also categorised by sex, and whilst Milne did analyse and discuss some of the differences in mortality behaviour observed for the two sexes, he did not publish separate male and female mortality rates in his Carlisle table). The ‘Exposed to Risk’ element of the mortality calculation therefore did not need to be estimated with the use of assumptions about population stability and immigration/emigration patterns. It could now be directly observed. This naturally supported a more refined and accurate estimation of experienced mortality rates.
The Carlisle data had another advantage over Northampton: it was somewhat less out of date. The Northampton data was based on mortality data from 1741 to 1770, whereas the Carlisle table used data from 1779 to 1787. Life expectancy had improved at a significant rate over the second half of the eighteenth century, particularly for infants. Milne noted that, even over his observation period, infant mortality rates were rapidly changing, largely due to the impact of social health policy initiatives such as smallpox inoculation. The Carlisle data included some data on cause of death, and Milne noted that the number of deaths from smallpox in the Carlisle Bills was 90 in 1779 and only 141 for the remaining eight years from 1780 to 1787. Dr Heysham, a local doctor who prepared some of the Carlisle data noted that in 1779 that ‘several hundreds were inoculated in the neighbourhood of Carlisle, and it is a pleasing truth that not one of them died’. Almost all of these smallpox deaths would have occurred at age five or under. In the Carlisle data, only 712 deaths of age five and under occurred over the full nine-year data period. If the smallpox death rate had continued at its 1779 rate, more than 700 would have died from smallpox alone. This reduction in smallpox mortality therefore had a substantial impact on the overall infant mortality rates of the period.
Figure 3.1 compares the Carlisle and Northampton tables.
Fig. 3.1 The Carlisle and Northampton tables
The Carlisle table produced lower mortality rates than the Northampton table for all ages from birth to aged 80. This reduction in estimated mortality is attributable to two effects: the later data period of the Carlisle table and the improvements in life expectancy that occurred over the second half of the eighteenth century; and Price’s assumption of a stable population from Northampton in estimating Exposed to Risk, when in fact its population was rising (as noted above, Milne did not require any such assumption for Carlisle due to the availability of population data).
The Carlisle table was undoubtedly an improvement on Price’s Northampton table, and it was widely recognised as such in the actuarial and academic circles of the period. For example, Professor Augustus De Morgan, an academic mathematician and influential writer on life contingencies (and the great grandson ofJames Dodson!), wrote of the Carlisle table in 1838 ‘they are to be considered the best existing tables of healthy life which have been constructed in England’. Yet life offices never applied it to the pricing of life contingencies to the degree that they used Price’s table. This was at least partly because, as we shall see later, life offices increasingly focused on the use of mortality data directly observed on assured lives rather than the general population. Meanwhile, at the Equitable, an aging William Morgan, increasingly set in his ways, would never abandon his uncle’s Northampton table. As we shall see in the next section, Morgan’s intransigence was set to become an increasing source of contention and controversy within actuarial circles and beyond.