Life Contingencies at the End of the Seventeenth Century

The sale of life annuities is known to have occurred since late Roman times and perhaps even earlier. The Roman juror Ulpian produced a table of annuity prices in the third century AD[1]: a life annuity would cost a twenty year-old 30 times its annual income, whilst a 60 year-old’s annuity would cost seven. The need for these annuity prices was not driven, however, by an active market in annuities. Rather, it was common for bequests to take the form of life incomes, and a capital value for these income streams was required to assess estate duty.[2] Further, there is nothing to suggest that a statistical approach to mortality modelling was used in the derivation of these prices. In Tudor England, government life annuities were sold at ?7 for ?1 of annual income (an annuity rate of around 14 %), irrespective of age. Interest rates at the time were around 12 %,[3] suggesting this annuity pricing was not particularly generous.

Following the glorious revolution of 1688, King William III was in much need of funding for his European wars, and the English government looked to raise ?1 million through the issuance of life annuities in 1692. Interest rates had fallen significantly since the Tudor annuity issuance: between the sixteenth and late seventeenth centuries, English interest rates had dropped from around 12 % to 6 %. Despite this substantial fall in interest rates, the 1692 annuity issuance was offered at exactly the same price as the one of the 1540s! Once again, the annuities were priced at a multiple of seven times annual income with no variation by age or sex of the nominated life.

Halley’s Breslau table implied that the government’s annuity pricing was extremely generous. He included a table of life annuity prices using an interest rate of 6 %, a reasonable assumption for the long-term risk-free interest rate prevailing at the time.[4] The government also sold 99-year fixed-term annuities at a price of 15.5 x annual income,[5] again consistent with the 6 % interest rate assumption. Using his table and this interest rate assumption, Halley produced an annuity price for a ten year-old of 13.4 x annual income; for a 40 year-old a price of 10.6; and for a 60 year-old a price of 7.6. Despite the apparent generosity of the government pricing, the life annuities sold poorly—less than ?110,000 of the targeted ?1 million was raised from the sale of annuities on 1,002 lives.[6] But many of the investors who did participate were well aware of the opportunity created by the age-insensitive annuity pricing: more than half of the nominated lives were under eleven years of age, and the annuities were therefore obtained at almost half of the fair price implied by Halley’s Breslau table.

Full records of the mortality experience of these annuitants have not been found, but it is known that 503 of the 1,002 lives were alive in 1730; 175 lives were still alive in 1749 and that the last survivor died in 1783.[6] This mortality experience is quite similar to the ten year-old in Halley’s table. The government borrowed at a realised interest rate of 12 % when they could have borrowed at a rate of 6 % by issuing long-dated gilts.

The historical literature offers different interpretations of the influence that Halley had on the government’s annuity pricing policy. Some modern historians have suggested that his annuity pricing recommendations were rejected by the government.[8] But Walford[9] noted that the government subsequently

Government annuity prices and Halley

Fig. 1.2 Government annuity prices and Halley

increased its single life annuity prices in a 1703 Act of Parliament from a multiple of annual income of seven to nine, and Francis attributes this revision to the influence of Halley’s paper.[10] The conventional contemporary historical position is that the early mortality tables had no discernible impact on the practice of insurers or governments in their pricing of life contingencies.[11] Figure 1.2 compares Halley’s annuity prices with the government prices of 1692 and 1703.

Irrespective of its direct impact on contemporary government annuity pricing policy, Halley’s paper introduced a new standard of method for mortality modelling and the Breslau table remained the benchmark mortality table until well into the second half of the eighteenth century.

We saw above how, despite the emergence of an actuarial approach to annuity pricing, at the end of the seventeenth century the English government did not feel compelled to move to an annuity pricing approach that priced annuities as a function of age (or indeed any other variable). Virtually no life insurance business existed in the private sector at this time. The growth of this industry would gather pace in the following decades; but in the final decade of the seventeenth century, there were only one or two examples of businesses that were transacting life contingencies.

The Mercers’ Company, established in London, was one such example. It launched a form of reversionary annuity product in 1698. This product paid a widow an annual income of ?30 for the remainder of her life following the death of her husband for every ?100 of single premium. This annuity rate did not vary as a function of the age of either the husband or the wife. Mercers was ultimately unable to pay the annuities promised by this product. It reduced the size of the promised annuity payments by one third in 1738 and finally required an Act of Parliament in 1747 to facilitate a partial government bail-out.

Halley’s Breslau table was published several years prior to the launch of the Mercer reversionary annuity product. It is natural to consider what annuity rates were implied by the Breslau table and how they compared to the practice of the Mercers’ Company. The Breslau table implies that the fair annuity rate varies substantially as a function of the age of husband and wife. For a husband and wife both aged 40, it implies that the Mercer product’s promised annuity income was actually lower than the fair value—at 6 % interest and before allowance for expenses, the Breslau table implies an annuity income of around ?45 per ?100 of single premium. But for a husband aged 60 and wife aged 30, an annuity income of only ?20 can be provided for according to the table. So the general level of the pricing is not obviously inconsistent with Halley’s table. But the fair pricing of the product is highly sensitive to the ages of the husband and wife (and, in particular, to the difference in their ages). The decision not to vary the pricing by age of husband or wife therefore heavily exposed the company to selection effects.

Today we can only speculate as to whether it was this selection risk that was the root cause of the business’s failure. The experience with the 1693 government annuity issuance, where most of the nominated lives were under eleven years of age, suggests enough of an understanding of annuity pricing existed for investors to take advantage of this selection opportunity. Many decades later, in 1772, Richard Price, of whom we will hear much more later, wrote of the fall of the Mercers’ Company: ‘The rapid fall of interest of money; their admitting purchasers [i.e. husbands] at too advanced ages; and, particularly, their paying no regard to the difference of age between husband and their wives, must have contributed much to hurt them.’[12]

The Mercers’ Company also offered one-year life assurance policies that paid a lump sum on death. The lump sum, however, was not a fixed amount. The total premiums (plus income less expenses) received from the cohort of lives purchasing insurance for the year were divided at the end of the year amongst the policies of those who had died. No mortality risk was transferred to the insurer, the company simply acted as an administrator of the direct pooling of mortality risk amongst the policyholders.

In summary, at the start of the eighteenth century, the theoretical probabilistic and actuarial developments of the preceding 50 years had not yet significantly impacted on the life contingencies pricing practices of insurers or governments. Relatively little mortality risk transfer took place in this era, and the examples we do have are mainly of life annuities which resulted in significant losses or failures for the recipients of the risk (whether government or private sector company).

  • [1] Hacking (1975), Chapter 13, p. 111.
  • [2] Ogborn (1953), Phillips in Discussion, p. 196.
  • [3] Chapter IX, Homer and Sylla (1996).
  • [4] Homer and Sylla (1996) p. 127.
  • [5] Francis (1853), Chapter 3, p. 55.
  • [6] Leeson (1968), p. 1.
  • [7] Leeson (1968), p. 1.
  • [8] Daston (1988), p. 139.
  • [9] Chapter IV, Walford (1868).
  • [10] Francis (1853), p. 59.
  • [11] For example, see Daston (1988) Section 3.4, Hacking (1975), p. 113.
  • [12] Price (1772), p. 105.
 
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