# Richard Price and His Observations (1772)

Richard Price is known for two particular achievements that made permanent contributions to the history of actuarial thought and beyond. As we saw above, he presented Bayes’ revolutionary work on statistical inference to the

Royal Society. Secondly, Price produced the Northampton mortality table, which was the market standard for life assurance pricing in Great Britain and North America for nearly a century—over which period the industry finally and substantially blossomed.

His contributions to actuarial thought, however, are even wider than these two singular achievements. From 1768 through to the 1780s, the Equitable consulted him as an actuarial trouble-shooter whenever challenging technical issues arose. When his advice was given, it appears to have been consistently taken and acted upon. More generally, his actuarial writing reveals a deep understanding of both theoretical and practical issues in pricing and reserving for life assurance. His work provides perhaps the earliest examples of a rounded actuarial analysis—quantitative and technical, but only as a means to an end, and with the clear objective of developing pragmatic solutions to real and complex financial problems, implemented with judgement and an awareness of the limitations and complications of business reality and human behaviour.

Like Thomas Bayes, Richard Price was a clergyman, a mathematician, a philosopher and a Fellow of the Royal Society. Price was also actively involved in political debate, arguing passionately in favour of freedom of thought and political reform in areas such as the extension of the voting franchise, the reduction of the national debt and in favour of the revolution in America. Politically, he was a prototypical liberal Victorian. Intellectually, he was a classic late Enlightenment figure: an intense man who was passionate about rational thinking and reasoned debate, and who obtained the highest regard of influential men across a diversity of intellectual circles. Price died in 1791, aged 68.

The philosopher and historian Ian Hacking ranks Price’s Northampton tables highly amongst the historical mortality tables. They were ‘perhaps the first statistical results to be taken seriously ... The Institute ofActuaries did not do anything much better until 1869.’^{[1]} The first version of the Northampton table was published by Price in 1772 (along with tables based on population data for Norwich and London) in the second edition of his *Observations on Reversionary Payments. ^{[2]}* This book addressed a wide selection of topics relating to life contingencies (and a number of financial topics outside the field of life contingencies). The construction of mortality tables is addressed in Essay IV of the book, which, in keeping with the style of the times, had an exhaustively descriptive title: ‘On the Proper Method of constructing Tables for determining the rate of human mortality, the number of inhabitants, and the values of Lives in any Town or District, from the Bills of Mortality in which are given the numbers dying annually at all ages’.

Halley’s Breslau table had remained the standard reference table from its publication in the 1690s until the arrival of Price’s tables. No practical use was made of Halley’s table by life assurers, but it was used by other writers such as de Moivre in their research on life contingencies. Halley’s paper did not include an explicit description of his assumptions and methodology. Price was more forthcoming and provided a detailed discussion of exactly what steps he believed should be taken in transforming raw data on deaths into a finished mortality table.

In the earlier discussion of Halley’s tables, we noted that Price had specified how mortality rates could be derived from ages at death when the underlying population is at a stable level and there is no immigration or emigration (recall that the technical challenge here arose from the lack of information about the number *alive* at each age, and this therefore needed to be inferred from the data on numbers of deaths). In his essay, Price showed how mortality rates could be derived when there is a stated level of net immigration or emigration, setting down the following general rule:

From the sum of all that die annually, after any given age, subtract the number of annual settlers after that age; and the remainder will be the number of the living at the given age.^{[3]}

He noted that the effects of immigration would be particularly significant in London, and he analysed its impact on estimated mortality rates by calculating rates from the London Bills of Mortality 1759-1768 data with and without his immigration adjustment. Naturally, the modelled impact of immigration must be a function of assumptions about the ages at which immigration occurs. Price noted that deaths significantly exceeded reported births in London, even though London was generally reckoned to have increased in population over the period. The available birth data of the period was somewhat unreliable and he expected it would underestimate the true levels of births, but he still conjectured that around one quarter of the annual deaths were of immigrants to the city. He assumed that all immigrants would be aged 20 at the time they entered.

Based on these assumptions, the mortality rates for ages less than 20 would be *understated,* as the no-immigration assumption would result in an *overestimation* of the numbers alive at ages up to 20 (recall that the estimation of the numbers alive at age *x* is based on the total numbers annually dying aged more than x). With the assumption that all immigration occurs at age 20

and none occurs beyond that age, no adjustment for immigration would be implied for mortality rates from age 20 onwards.

Figure 2.3 compares the no-adjustment and with-adjustment results obtained by Price for ages up to 20.

The chart shows that mortality rates at very young ages had, until this time, been significantly understated by calculations using the Bills of Mortality (such as those produced by Dodson for use by the Equitable). This did not have much direct consequence for the pricing of life contingencies as most policies were written on lives older than twenty years of age. It did, however, have implications for the assessment of life expectancy at birth and for government health policies. Great Britain underwent an unprecedented period of population growth in the second half of the eighteenth century. The population is estimated to have grown from around 6.3 million to 9.2 million between 1751 and 1801, and most of the growth over this period is believed to have occurred form 1780 onwards.^{[4]} A sharp fall in infant mortality rates is generally believed to have been a significant factor in this population growth. Price’s estimates certainly highlight how shockingly high infant mortality rates were at the start of this period—according to his calculations, most of London’s newborn babies of this time would not survive to see their fifth birthday.

Fig. 2.3 **Price's mortality rates derived from London Bills of Mortality 1759-1768 data, with and without adjustment for immigration**

Price then identified Northampton and Norwich as towns that had kept Bills of Mortality which included ages at death for many decades. He applied his methodology, including his adjustment for immigration, to both these datasets to produce tables for each town. He also compared these results with Halley’s Breslau table. The mortality rates of these two tables, together with Price’s London table and Halley’s Breslau table, are compared below in Fig. 2.4.

The consistency of these tables is quite striking, and Price noted, with an unbridled satisfaction, ‘there is a striking conformity between all the three Tables [Norwich, Northampton, Breslau], which gives them great weight and authority’. The London table produced noticeably higher mortality rates than the other three tables and this was unsurprising: John Graunt had noted a century earlier the relatively lower mortality rates associated with country living.

As we might expect from Price given his history of involvement with statistical inference, he gave some consideration to the potential sampling error in his data. He did not attempt to calculate an explicit standard error for his mortality rates (no such concept existed at this time). Instead, he used an empirical approach noting that whilst 30 years of data were used in each of the Northampton and Norwich tables, the results for each were very similar when any ten-year period within the 30 years was used instead. Based on this analysis, he concluded: ‘These Tables, therefore, are founded on a sufficient

Fig. 2.4 Price's mortality tables for London, Northampton and Norwich and Halley's Breslau table number of observations; and it appears, that there is an invariable law which governs the waste of human life in these towns.’^{[5]}

Price’s *Observations* is most renowned for these mortality tables. But there was much else in the book that had an important intellectual impact on the developing actuarial thinking of the time. *Observations* was based on Price’s experiences as an advisor to a number of embryonic or would-be insurance companies that were considering how to price a variety of forms of annuity contract. The book begins with an exhaustive discussion of his experiences in this area, giving annuity pricing examples for a number of variations of reversionary and deferred forms of annuity in single premium and regular premium forms. In these examples, he computed exact valuations rather than relying on the sort of approximations that de Moivre had developed. In the intervening 40 years since de Moivre’s *Annuities on Lives,* methods of arithmetic computation had become somewhat more efficient, making the exact calculations more feasible, if still highly laborious. Price included an essay on de Moivre’s annuity pricing approximations that warned of the limited accuracy of his joint life pricing formula. In particular, Price showed how this could lead to substantial proportional errors when valuing a reversionary annuity (which is valued as the *difference* between the single life annuity and the joint life annuity).

*Observations* also included discussion of a number of annuity products that had been launched by life and annuity businesses since the 1750s. In these discussions, he repeatedly found that the products were substantially underpriced and concluded it unlikely that policyholders would receive their full promised benefits. This treatment was applied to the reversionary annuity products of the London Annuity Society and the Laudable Society, and more widely: ‘There are in this kingdom several institutions for the benefit of widows, besides the two on which I have now remarked; and in general, as far as I have had any information concerning them, they are founded on plans equally inadequate.’

The actuarial management of the life assurance companies was considered too. Whilst the Royal Exchange and London Assurance were long-established writers of life assurance with guaranteed sums assured, Price focused on the Amicable, which wrote significantly higher volumes of business, though, as was discussed earlier, without offering guaranteed sums assured. Price strongly criticised this mutual benefaction model of life assurance that Amicable provided: ‘It is obvious that regulating the dividends among the nominees by the number of members who die every year is not *equitable;* because it makes the benefit which a member is to receive to depend, not on the value of his contribution, but on a *contingency;* that is, the number of members that shall happen to die in the same year with him.’^{[6]} He also noted that charging all members the same premiums irrespective of their age was not equitable. He went on to contrast these imperfections with the Society for Equitable Assurances on Lives and Survivorships (of which he was a paid consultant) and provided some public advice to its managers:

- • First, on the need for rigorous underwriting to avoid anti-selection: ‘I have already more than once observed that those persons will be most for flying to these establishments, who have feeble constitutions, or are subject to distempers, which they know render their lives particularly precarious; and it is to be feared that no caution will be sufficient to prevent all danger from hence.’
^{[7]} - • Then on adverse variation in experience: ‘The calculations only determine probabilities; and agreeably to these, it may be depended on that events will happen on the whole. But at particular periods, and in particular instances, great deviations will often happen; and these deviations, at the commencement of a scheme, must prove either very favourable, or very unfavourable.’
^{[7]} - • And finally, with impressive prescience given the tumultuous debates that would embroil the Equitable in the coming decades, on the prudent distribution of surplus: ‘the encouragement arising from the possession of a large surplus (should) be led to check or stop the increase of its stock by enlarging its dividends too soon, (otherwise) the consequences might prove pernicious.
^{[9]}

These pieces of advice highlight the completeness of Price as an actuarial thinker. He strongly advocated a reasoned approach to pricing based on the rigorous analysis of statistical data, and he furnished the profession with much by way of methods and results of such analysis. But he also understood the practical aspects that must attend the successful and sustainable actuarial management of a life assurance business, and he anticipated some that had not yet arisen at the time of his writing. He used both these technical and practical insights to actively direct the development of new actuarial tech?niques at the Equitable, and these would be a critical part of that business’s huge success in the following decades. Added to these contributions is his vital role as the midwife of Bayes’ revolutionary work on statistical inference. All this taken together, Richard Price stands as a titan of actuarial thought. If an actuary is someone who provides rational and rigorous advice on the sustainable long-term financial management of life contingencies business, then Price was arguably the first actuary, and inarguably one of the most important and influential actuaries in the profession’s history.