The Emergence of with-Profits, 1756-1782

The previous section discussed how the work of Bayes, Laplace and others during the decades following 1760 provided a fundamental breakthrough in mathematical statistics. Contemporaneously, another small group of men were revolutionising what life assurance could be and how it could be delivered in a sustainable and equitable way. Richard Price uniquely spanned both these sets of men and their ideas, and for this he must be regarded as one of the most important people in the history of actuarial thought. But before Price, first came James Dodson.

James Dodson and Whole-of-Life Assurance (1756-1772)

James Dodson, born in England in 1710, was a friend and pupil of Abraham de Moivre. It is reasonable to assume that his interest in life contingencies arose from de Moivre’s influence. Dodson published on mathematics, including topics in annuity valuation, at a level that saw him inducted as a Fellow of the Royal Society in 1755. He worked as a schoolmaster in London and supplemented his income by consulting on financial matters relating to annuities. The story, perhaps apocryphal, goes that he was refused a life assurance policy by the Amicable in 1755 due to their then-policy not accept lives over age 45 (a policy that was necessitated by their use of a flat 5 % premium rate for annual life assurance for all permitted ages at the time). This motivated Dodson to advocate developing rational age-dependent life assurance pricing. In 1756, he published a treatise, First Lectures on Insurance, espousing his ideas and he initiated a project to create a life assurer that would implement them. This project ultimately led to the foundation of Equitable Life, though he did not live to see it as he died prematurely in 1757, aged only 47 (perhaps somewhat ironically, Amicable therefore did well to avoid insuring his life!).

Though Dodson was clearly an able mathematician, his permanent contribution to actuarial science is not in mortality modelling or statistics, but in his conception of a new life assurance product that could be appealing to the emerging salaried and prudent middle class. Up until this time, the only form of life assurance written was short-term (usually one-year) assurance. As noted in Chap. 1, the demand for this business was very small, and was largely limited to insuring the lives of business creditors and to speculative gambling activities related to the mortality prospects of the celebrities of the age. Dodson proposed a new form of life assurance product: instead of buying life assurance for one year, the policy would offer insurance over the entire remainder of the policyholder’s life. Instead of a short-term insurance policy that would only pay out in circumstances of extreme misfortune, Dodson envisioned a policy that would pay out the fixed sum assured with certainty. The only variable was the timing of the payment. In his Lectures, he went on to show how this whole-of-life policy could be paid for by regular even premiums. As mortality rates tend to increase with age, the regular premium in the early years of the contract would be greater than the regular premium payable for a single-year term assurance contract. This excess funded a reserve, which was essentially a long-term savings element of the contract. The reserve would be used to partly fund the claims that would arise many years later, when the regular premiums payable in those years would not be sufficient to fully meet the cost of that year’s cover. This conception of a whole-of-life policy transformed life assurance into a form of long-term savings vehicle.

By paying for the contract with regular premiums, the policyholder was providing a life annuity to the insurer. Dodson, as an expert in annuity pricing, showed how the regular premium could be set such that their present value equated to the present value of the sum assured benefit for a policyholder of a given age, according to a given mortality table and interest rate. Dodson developed some example premium calculations using a mortality basis derived from the London Bills of Mortality from 1728 to 1750. In 1728, the London Bills had started to include age of death in its data (though even then it was recorded only by decade rather than year of age).

Significantly, Dodson argued that the pricing basis for setting the whole- of-life premium rate should be set by referring to the worst mortality that was experienced in any given year in the data sample, rather than the average. Thus he derived two sets of mortality rates: a ‘mean deaths’ table and a ‘greatest deaths’ table. Dodson used the bills to show that the average number of deaths over the sample period was 26,207, whilst the worst year experienced 32,169 deaths (this occurred in 1741, where severe weather resulted in high food prices and near famine in parts of England). Thus the ‘greatest deaths’ basis had mortality rates that were almost V higher than the ‘mean deaths’ basis. Dodson also noted that even the ‘mean deaths’ basis would likely provide some margin over the experience of a well-run life assurer:

As the Bills of Mortality contain the deaths of all kinds of people healthy and unhealthy and as care will be taken not to insure those lives which are likely to be soon extinct therefore in all probability fewer of the persons insured will die in proportion to their number than those who are not insured, which will also contribute to the gain of the corporation since the premiums are proportioned to the Bills.16

The use in the pricing basis of the ‘greatest deaths’ rates experienced in any one year of a 22-year period may appear today as a somewhat ad hoc and noisy way of setting a pricing margin. But he provided an explicit rationale for this margin, based on a distinction between those policyholders who would bear the risks and those who would not. In his conception of a mutual life assurer, there would be two classes of policies: one that would underwrite the guarantees and participate in the profits of the business; and another that would not participate in the risks or profits of the business. Furthermore, at this time limited liability was only applicable by exception to companies that were granted it by royal charter. So the participating policyholders would have unlimited liability, and in the event that claims were due which could not be met by the assets of the corporation, they would face a call on their personal assets.

Dodson’s view was that a non-participating policyholder should be charged more than a participating policyholder for the same sum assured, so that the participating policyholder was offered a compensatory expected return for the risks they were underwriting. His proposal was therefore that the participating business be priced using ‘mean deaths’ whereas the non-participating premium basis would use the ‘greatest deaths’ table: ‘if the persons who shall desired to be insured without being liable to such a call should be rated in proportion to the greatest number of deaths that have happened within our knowledge as I think they ought, then the latter ought to pay near И part in the premium more than the former’.[1] For those aged 40-50, Dodson’s ‘greatest deaths’ table would result in a non-participating pricing basis that was quite similar to the 5 % assumption used by the Amicable.

In his Lectures Dodson produced tabulations of multi-year projections of accumulated assets, premium payments and claim outflows to illustrate the workings of his regular premium whole-of-life policy. Figure 2.2 charts one of these tables. This is a projection of a non-participating product. The projection is made on the ‘mean deaths’ basis but the premium basis used is the ‘greatest deaths’ table.

The regular premium rate for a 40 year-old was calculated to be 4.625 %. The starting annual regular premium income for the cohort of 8,165 lives

Dodson's twenty-year projection of a regular premium whole-of-life policy; Cohort of 8,165 40 year-olds, sum assured of ?100

Fig. 2.2 Dodson's twenty-year projection of a regular premium whole-of-life policy; Cohort of 8,165 40 year-olds, sum assured of ?100

with a sum assured of ?100 each is therefore ?37,763. By the end of Dodson’s twenty-year projection, 4,647 of the 8,165 lives in the cohort have died, leaving a total of 3,518 alive (and hence the total sum assured outstanding is ?351,800), while assets of ?152,392 have been accumulated. Not all of these assets represent a surplus—some of it is required as a reserve for the heavier mortality experience that is expected in the later years of the cohort’s lives. Dodson calculated that, at the end of year twenty, the present value (on the ‘mean deaths’ basis) of the future claims of the remaining 3,518 lives was ?265,917; and that the regular premium annuity stream (again on the ‘mean deaths’ basis) from the 3,518 remaining 60 year-olds had a present value of ?134,412. Thus, a surplus of assets over liability reserve of ?152,392 - (?265, 917 - ?134,412) = ?20,887 had, so far, been generated by the pricing margin implied by using the ‘greatest deaths’ table in pricing.

Interestingly (in the context of actuarial debates to come much later), Dodson presented this prototypical liability reserving calculation by explicitly appealing to the logic of valuing the liability with reference to the cost of transferring the liabilities to a third party:

Now let us suppose that the corporation will contract with some other body, take the insuring of these off their hands by a payment of a sum in hand, being allowed discounts for the money so advanced them by computing the sum for which 100 may be insured for a life of 60 on the manner already planned.[2]

Having established the principle that non-participating business should be priced to generate a surplus for participating policyholders, the question arose of how to distribute this surplus. The equitable distribution of the surplus amongst the participating policyholders is discussed in Dodson’s Lectures. There he argued that the profit should only be distributed to participating policyholders who have already themselves generated a profit for the assurer by paying a total amount of regular premium that exceeded their sum assured, and then in proportion to that excess:

It seems, therefore, unreasonable to divide any part of the profit [from nonparticipating business] among those who have not paid as much as they receive because they have evidently a profit by being paid their claims which profit is very great in first years ... it seems reasonable therefore to divide the profit amongst the persons to have paid more than claims and to do this in proportion to the sums so overpaid.[3]

He suggested this surplus be distributed to the participating policyholders in the form of a cash dividend. He did not consider the possibility of distributing the surplus in the form of an increase in their sums assured.

Dodson argued that it would be unnecessary for the business to hold any initial cash reserve as a contingency for adverse experience. This may seem a peculiar position for a prototypical actuary who understands risk well enough to advocate charging on the basis of the worst single year in 22 years of experience. But he argued that the presence of participating policyholders’ unlimited liability meant it was superfluous: even if a funded reserve was established, it would not absolve the participating policyholders from potential calls on their personal assets, and the security of the non-participating business would still at least partly rely on such calls being made good. It should also be noted that it was typical practice of joint-stock companies of the time to only require a small portion of equity capital to be ‘paid-up’, with the remainder again being funded by future calls on shareholders when the company required the cash. So, in the broader context of the historical corporate practice, this position was reasonably natural.

Dodson’s work was less concerned with technical innovations, such as those developed by Halley and de Moivre over the preceding 60 years. He was motivated by a vision of how such techniques could be applied to the provision of a new form of life assurance that was really quite revolutionary in the context of the life assurance practices of the time. The summary features of his system as set out in First Lectures in Insurance were:

  • • The use of mortality tables to produce age-specific premium rates for life assurance. At the time, a flat non-age-specific premium basis was standard market practice.
  • • The development of a whole-of-life assurance policy, available in single and regular premium forms. At the time, only short-term (usually one-year term) assurance was sold.
  • • A with-profit system where participating and non-participating policies were simultaneously provided. Non-participating policies would be more expensive and would be priced to make it unlikely (relative to experience) that losses would be suffered. Participating policyholders would nonetheless have an unlimited liability exposure to calls on their personal assets in the event of losses exceeding the assets in the fund.
  • • The surplus on non-participating business would be distributed equitably amongst participating policyholders, in proportion to how much profit their own policies had generated for the corporation by virtue of their longevity.

• Contingency capital would be funded by calls on shareholders (the longterm policyholders in a mutual scheme) rather than by cash funding of a contingency reserve.

Everything advocated by Dodson as described above was implemented by Equitable Life within 25 years of the publication of Lectures. The consequent success of the Equitable meant that it would come to be regarded as the model of life assurance practice. But this revolution would occur after Dodson’s death and the significant contributions of some talented others would be required to bring his vision to fruition.

  • [1] Dodson (1756), Chapter 1.
  • [2] Dodson (1756), Chapter 3.
  • [3] Dodson (1756), Chapter 4.
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