Tuesday, 20 August 2013

Faith, Hope and Charity in the genesis of actuarial science

Before the crisis caused by the South Sea Company, Daniel Defoe had written in 1719 that he “hoped that mathematicians might cure the reckless of their passion for cards and dice with a strong dose of calculation” Gigerenzer [1989, p 19], and after the Bubble he returned to the theme in Complete English Tradesman in which he observed
A tradesman’s books are his repeating clock, which upon all occasions are to tell him how he goes on, and how things stand with him in the world: there he will know when it is time to go on, or when it is time to give over; and upon his regular keeping, and fully acquainting himself with his books, depends at least the comfort of his trade, if not the very trade itself. If they are not duly posted, and if every thing is not carefully entered in them, the debtor’s accounts kept even, the cash constantly balanced, and the credits all stated, the tradesman is like a ship at sea, steered without a helm; he is all in confusion, and knows not what he does, or where he is; he may be a rich man, or a bankrupt—for, in a word, he can give no account of himself to himself, much less to any body else. Defoe [1726, Chapter I]
Defoe seems to be endorsing Locke’s empiricism, with a nice dose of Puritan Prudence, with respect to finance. It was not English tradesmen who would demonstrate the usefulness of Defoe’s advice, to employ mathematics built on Locke’s philosophy, but Scottish clergymen.
When the Scottish Church was reformed in 1560, so that married men could be ministers, it became necessary to provide for ministers’ widows and their children. In 1672 the Scottish Parliament passed the ‘Law of Ann’ which legislated that widows were entitled to half a year’s stipend from their husband’s parish, with half of that going to any children. By the 1730s this one-off payment was regarded as insufficient, and the Church of Scotland should set up a ‘general fund’ from which pensions for widows could be drawn and in 1742 a scheme was proposed by two Edinburgh ministers, Robert Wallace, born in 1697 in Kincardine, and Alexander Webster, born in 1708 in Edinburgh.
Despite the reputation that Scottish Presbyterians sometimes have, Wallace and Webster appear to be convivial characters. Wallace was a founder member of a university ‘debating’ (drinking) club, the Rankenians, and it was said of Webster that it was “hardly in the power of strong liquor to affect Dr Webster’s understanding of his limbs” Ferguson [2008, p192]. Both clergymen were popular preachers, in fact Webster was considered so eloquent that a friend asked him approach a wealthy young lady, Mary Erskine, concerning the friend marrying Mary. Webster was a little to good at persuasion and Mary said to him “You would come better speed, Sandy, if you spoke for yourself”. Alexander and Mary lived happily ever after Dow [1992, p 27]. (Mary’s sister was the mother of James Boswell, Samuel Johnson’s friend.)
Wallace and Webster were also skilled mathematicians, and Wallace was part of a group which founded the Philosophical Society in 1735, which then became the Royal Society of Edinburgh in 1783. This group was led by the mathematician Colin Maclaurin who was regarded as the United Kingdom’s leading mathematician after Newton died in 1727. In 1742 Maclaurin wrote a Treatise of Fluxions in defence of Newton’s ideas around rates of change, which Newton called fluxions Boyer and Merzbach [1991, p 397] and underpinned the physics in his Principia. He wrote his Treatise in order to put Newton’s ideas on a firm basis, not because of any doubt coming from scientific circles but because they had been attacked by Bishop George Berkeley in 1734 in a tract called The Analyst. Berkeley was a Tory, an opponent of the Whigs Defoe and Locke, who “had been nettled on having a sick friend refuse spiritual consolation” Boyer and Merzbach [1991, p430] because the astronomer, Edmund Halley, had convinced him that Christian dogma was untenable. The subtitle to The Analyst was
Or a Discourse Addressed to an Infidel Mathematician Wherein It Is Examined Whether the Objects, Principles, and Inferences of the Modern Analysis are More Distinctly Conceived, or More Evidently Deduced, than Religious Mysteries and Points of Faith. “First Cast the Bean Out of Thine Own Eye: and Then Shalt Thou See Clearly to Cast Out the Mote Out of Thy Brother’s Eye.” Boyer and Merzbach [1991, p 430]
The problem was, Bishop Berkeley had a valid mathematical criticism of fluxions, which Maclaurin needed to address, and would not be resolved for another hundred years.
In response to the problem of funding widows’ pensions, in 1741 Webster wrote to all the presbyteries (parishes) of the Church of Scotland asking for statistics on ministers’ deaths and their dependants, alive and dead, for the period March 1722 to March 1742. Meanwhile, Wallace tackled the theory behind the problem using information provided by that “Infidel Mathematician” Edmund Halley. In 1693 Halley had been asked by the Royal Society to analyse data relating to births and deaths from the German city of Breslau (Wroclaw) for the period 1687 to 1691 which Leibnitz had provided Poitras [2000, p 197]. This data enabled Halley to construct a more precise life-table than Graunt’s, which had estimated the ages of deaths, and using this data, Halley solved the problem of valuing an annuity independently of Jan de Witt.
In 1743 Wallace compared the data gathered by Webster to that in Halley’s work and realised the ministers were less likely to die young than the general population of Breslau. Wallace then calculated what premiums, paid by the ministers when they were alive, could yield annuities, pensions, to the widows. Wallace calculated the premium rates for four different classes of pensions as Hare and Scott [1992, p 59]
Widows’ Annual Annuity
Ministers’ Contribution
£2 . 12s . 06d
£3 . 18s . 09d
£5 . 05s . 00d
£6 . 11s . 03d
The next set of calculations Wallace undertook were concerned with the cashflows in the early years of the fund. It would take time for the fund to build reserves from which to pay out to the widows and Wallace needed to make sure the fund did not fail within a few years of its establishment, and the fund would be founded on the basis of a one-off ‘marriage tax’, charged to all married ministers who joined the fund.
Wallace assumed that all 930 married ministers would join the scheme, all be it in different classes, and 800 ministers would pay the marriage tax in the first year. This would establish the fund with £7,560 and after the annuity payments, would be worth £5,900 at the end of the first year Hare and Scott [1992, p 64]. Wallace then asked Maclaurin to check his figures, more to give them authority than because he had any doubts, though Maclaurin did spot and correct an error. The trio, based as much on their faith in mathematics as their faith in God, were confident that the fund was structured in such a way that it would not fail.
Wallace travelled to London to get the Scottish MPs at the United Parliament to pass an Act establishing the Fund in March 1744 and it was a great success. Up until 1778 the fund’s capital never deviated by more than 5% from estimates calculated in 1748 Dunlop [1992, p 18]. Wallace moved on to be an important figure in the Church of Scotland but did not involve himself with its further development, which was managed by Webster. Maclaurin was not so fortunate, he was active in the opposition to the Young Pretender and fled south to England to avoid the rebels and the stress of the flight is believed to have caused his death in summer 1746.
The Scottish Ministers’ Widows’ Fund was the first fund covering insurance liabilities, either in general insurance or in life insurance, to be managed on a mathematical basis Hare and Scott [1992]. The Scottish Ministers’ Widows’ Fund was copied in the Presbyterian’s Ministers Fund of Philadelphia in 1761 and the following year the English Equitable Company was founded, the oldest public life insurance company until its failure in 2000.
The story of the establishment of the Scottish Ministers’ Widows Fund is a story of the triumph of British Empiricism. However, it can be more than that, it can be seen as an expression of the ‘Christian’ (misnamed they were evident in pre-Christian Rome and Greece) virtues of Faith, Hope and Charity. The fund was clearly Charitable, in the sense that the Church is concerned for the welfare of the Widows of its dead Ministers. Faith is the ability to believe without seeing, and the Latin root is fides captures the concept of trust. In this case the Minister’s had Faith in the data, as advocated by Defoe. Today Science rests on Statistics, which provides a formalisation of when we can trust data: it creates ‘knowledge’ out of data. While Faith looks backwards, Hope looks forwards and in mathematics is represented by Probability through the operation of Expectation. While the link between Probability and the virtue Hope is obscure in English, it is explicit in French using the word espérance, ‘hope’, when referring to mathematical expectation. The root of espérance is in the Roman goddess Spes, usually depicted with a cornucopia and flowers, and is linked to the Greek daimon Elpis, which was the spirit of hope left in Pandora’s Box.
Deirdre McCloskey, in her examination of the Bourgeois Virtues, challenges Hume’s fact/value dichotomy McCloskey [2007, p 265]
Does it contain any abstract reasoning concerning quantity or number?, No. Does it contain any experimental reasoning concerning matter of fact and existence? No. Commit it to the flames: for it can contain nothing but sophistry and illusion
McCloskey asks
No theology: no natural law: no ethics. We are only to have mathematics on the one hand and on the other the “sciences”
My point is that while the establishment of the Scottish Ministers’ Widows’ Fund is mathematical and scientific, it was also moral, and by separating fact from value we lose something of the meaning of the Fund.


C. B. Boyer and U. T. Merzbach. A History of Mathematics. John Wiley and Sons, 1991.
D. Defoe. The Complete English Tradesman. Project Gutenburg, 1726. www.gutenberg.org/files/14444.
J. Bremner Dow. Early Actuarial Work in Eighteenth Century Scotland. In A. I. Dunop, editor, The Scottish Ministers’ Widows’ Fund 1743—1993. St. Andrews Press, 1992.
A. I. Dunlop. Provision for Ministers’ Widows in Scotland — Eighteenth Century. In A. I. Dunop, editor, The Scottish Ministers’ Widows’ Fund 1743—1993. St. Andrews Press, 1992.
N. Ferguson. The Ascent of Money: A Financial History of the World. Allen Lane, 2008.
G. Gigerenzer. The Empire of Chance: how probability changed science and everyday life. Cambridge University Press, 1989.
D. J. P. Hare and W. F. Scott. The Scottish Ministers’ Widows’ Fund of 1744. In A. I. Dunop, editor, The Scottish Ministers’ Widows’ Fund 1743—1993. St. Andrews Press, 1992.
D. N. McCloskey. The Bourgeois Virtues: Ethics for an Age of Commerce. University of Chicago Press, 2007.

    G. Poitras. The Early History of Financial Economics, 1478—1776. Edward Elgar, 2000.  


  1. I am a pension actuary in Italy. I am very interested in reading more about the failure of Scottish Equitable in 2000...how it happened that reserves were so underfunded?? In Italy it would be impossible, due to constant vigilance of Insurance Authority. Thank you!