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stating that De Wit's treatise was very little known in this country, and had "no sensible influence on the subsequent progress of the science." The reasons for this will appear in our notice of De Wit. Leibnitz appears to have been one of the first who drew the attention of Europe to this report of De Wit's; but it was Mr. Fredk. Hendriks who discovered the document itself.

In 1674 there was issued in Lond. a scheme of Tontine annuities, of which the following is an outline: "An advertisement and demonstration concerning the improvement of moneys to the great benefit and advantage of all persons of what nation, sex, age, degree, or quality soever, willing to advance any sum or sums, according to the method hereinafter mentioned, propounded to the Right Honorable the Lord Maior, Aldermen and Commons in Common Council assembled." The sum to be advanced was not less than £20 upon one life; but several advances might be made upon the same life. The prospectus further said: "This extraordinary gain being not only lawful, but very advantageous, there can be no other way proposed whereby, in laying out so small a sum as £20, there can be produced so great an increase as by survivorship will most certainly accrue to many persons, and especially to the longest liver of this rank." We, however, hear no more of the project.

In 1677 there was pub. by Michael Dary: Interest Epitomized, both Compound and Simple: "very useful for every one that lendeth or borroweth ; and for purchasing and selling annuities or pensions, and leases in reversion. Whereunto is added, a short appendix for the solution of adfected equations in numbers by approachment: performed by logarithms.

In 1681, a paper by Adam Martindale was published in the Phil. Trans., under the following title: Twelve Problems in Compound Interest and Annuities resolved.

The following, taken from a work on church and college leases, pub. at Cambridge in 1686, Tables for renewing of leases and purchasing lives [LEASES], gives a curious indication of the conjectural methods then commonly adopted in life valuations, and also confirms what we have already seen regarding the small values put upon lives:

The way of purchasing by lives was commonly to reckon one life as a lease of seven years, two lives as a lease of fourteen years; and three lives as a lease of twenty-one years. But this way seeming unequal, there is another way, which is more agreeable to reason; and it is this, viz., for every life to decrease one year; as, if one life be reckoned as a lease for ten years, then two will be as a lease for nineteen, and three as a lease for twenty-seven years, etc. So if you reckon one life as a lease of nine years, then two will be a lease of seventeen, three as a lease for twenty-four, etc. So if one single life be reckoned as a lease of twelve years, then two will be as a lease of 23, three as a lease of 33 years, etc. After some further exemplifications, there follows a Table which (as Mr. Farren has already pointed out) appears to constitute the first species of life annuity table offered for public use in Gl. Brit. The following is the Table, and the necessary explanation thereto : A Table for the purchasing of Lives.

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The repetition of 7 p.c. (2nd division of Table) may on first perusal appear merely a misprint, but on considering the fanciful construction of the Table, the cause of the duplicature will become apparent. In the first division one life is assumed as equal in duration to 10 years, in the second, 9 years are assigned to it: consequently the two divisions, though in connexion with similar rates of int., represent different values. It seems scarcely possible to afford to the modern reader a more obvious example of the inconsistencies of the methods formerly prevailing.-Farren.

The authorship of this work was very generally and for some years attributed to Sir Isaac Newton, no doubt from the fact of his having given a certificate of his belief in the accuracy of its Tables. It is now known to have been written by one Mabbot, of King's College, Cambridge.

The most unpractised reader will at once perceive that until distinctions of age were

Number
of Lives.

Number
of years.

Years.

Quarters.

Months.

Decimal

parts.

recognized as influencing the relative annuity values, all life estimates were too imaginary and inconsistent to be even worthy of the name of calculated results.-Farren.

In 1692 the first attempt was made by the English Government to raise money by means of life annuities; and hence there arises the first mention of life annuities in the English Statute book. This was by 4 Wm. & Mary, c. 3, known as the "Million Act." Its object was to raise one million sterling, "to carry on the war against France," by means of tontine annuities, for the int. upon which £100,000 p.a. was to be set apart until A.D. 1700, and then £70,000, but in the event of the entire million not being so subscribed by a given date, those who did sub. were to have, in lieu of the tontine advantages, an annuity of £14 in respect of every £100 subs., during the remainder of their own or their nominee's lives. There were no provisions or restrictions as to age. Even on these terms however, only £881,493 12s. 2d. could be raised.

Α

Great efforts were made to float this loan, and there was pub., apparently by authority, the following Table predicting the course of death of the entire 10,000 nominees. copy of the table still exists in the British Museum; and we think it sufficiently remarkable to be given here entire, with the following explanation, which, in the original, is given at the foot of the Table:

This Table designed for the encouragement of contributors to advance monies upon the funds and terms expressed in the Act of Parl. newly passed, for granting to their Majesties certain rates and duties of excise upon Beer, Ale, etc., and being applicable to the first proposal therein mentioned, upon terms of yearly payments during life, with advantage of survivorship, was calculated upon this supposition, that the whole sum of Ten Hundred Thousand Pounds may be advanced, and consequently as many nominees as there are shares or 100 advanced, viz. 10,000. The following example will show the use of the table. Look in the left-hand column for the number of years desired. Suppose 22; you will find in the columns answerable to it that at the end of 22 years, there are 5034 nominees dead, and 4966 living, and that each contributor (whose nominee is then alive) will receive that year for interest £14 15. for each share or £100 by him advanced.

A table shewing how many out of 10,000 Lives or Nominees will be likely to die in any number of years proposed during the term of 99 years, and the yearly interest due, etc.

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The first consideration which strikes us in reference to this Table is, upon what data was it founded? the next, who supplied it? Was it Halley? It comes suspiciously near to the advent of his Table, as we shall presently see. If not, it can no longer be maintained that he invented the first life table. Or is the preceding Table purely conjectural ? Its application was certainly conjectural, inasmuch as the ages of the nominees not being restricted in any manner, could not be known. But then, as we have shown ages had not at this period been taken into account. Halley, however, knew the necessity and the value of the element of age!

We have seen that the entire million was not sub. under the Act of 1692; accordingly in 1693 another Act was passed -5 Wm. & Mary c 5-for granting life annuities also at the rate of 14 p.c. p.a. on the sum of £118,506 5s. 10d. required to make up the orig. amount. This granting of life annu. at 14 p.c. was virtually estimating the value of an annu. on a single life at 7 years' purchase. But the actual experience of this batch of annu. has been tested by means of the investigations of Mr. John Finlaison [see 1829], and was found to have ranged from 17 years' purchase at age 10, down to 4 years' purchase at age 75.

In 1693 we reach the name of Dr. Halley, who (subject to the preceding query) constructed the first mort. Table, and thus placed the valuation of life annuities upon a scientific basis. He read a paper before the Royal Society: An estimate of the degrees of mort. of mankind, drawn from curious tables of the births and funerals at the city of Breslaw, with an attempt to ascertain the price of annuities upon lives; wherein he showed how the prob. of life and death, and the values of annu. and assu. on lives, might be determined by the aid of Mort. Tables. He states that till then all such matters had been dealt in by imaginary valuation!

Dr. Halley having defined the chances of life, proceeds to illustrate the application of the table to the determination of the value of life annu., and apparently selected 6 p.c. int. as the rate most current in his own day.

He first indirectly advances that the worth of an annuity not dependent upon life would be represented by adding together the several discounted values of the successive payments. But in a life annuity, as each payment would only be claimable upon the persons living to the respective dates, so each payment would prospectively be worth such a proportion only of the discounted value as there was a proportionate chance of the persons living to the date in question. Now the chances of a person of any particular age living to the respective dates of payment were easily determinable, as already mentioned, by considering the relative number of yearly survivors, indicated by the table of mort

The value of the life annuity was therefore dependent: firstly, upon deriving from int. tables or calculations, the discounted value of each yearly payment; and, secondly, upon taking such a proportion only of each discounted value as there might be a proportionate chance of the payment being respectively claimed. Such separately reduced values being added together, would then constitute the total value of the life annuity.

Age.

Years'
Purchase.

TABLE-6 p.c.
Age.

Years' Purchase.

I

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10'57

5

13'40

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13'44

9'91 9.21

15

20

13933
12.78

7.60

25

12.27

65

6.54

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8.51

This method of valuing life annuities by ascertaining the value of each future payment separately is obviously and demonstrably true in the present day, as in Dr. Halley's own time; but inasmuch as the labour of such subdivision for every age would be necessarily very great, the reader will not be surprised to learn that less elaborate methods have since been discovered. Dr. Halley himself remarked upon the operose requirements of his formula, but very frankly owned that after several trials he had not succeeded in devising any less laborious method. He had, however, the perseverance to calculate the results for every fifth age as embodied in the accompanying table, which, from its constituting the

first annuity table for different ages, seems no less interesting as the primary step in the art of life-finance, than the preceding table was in that of life-measurement.

Dr. Halley, in connexion with this Table, observes :

This shows the great advantage of putting money into the present fund lately granted to their Majesties, giving 14 p.c. p.a., or at the rate of 7 years' purchase for a life; when young lives, at the usual rate of int., are worth above 13 years, purchase. It shows likewise the advantage of young lives over those in years; a life of ten years being almost worth 13 years' purchase, whereas one of 35 is worth but 11.

This appears to us to lead to the inference that Halley may have been concerned in the preparation of the preceding scheme.

Dr. Halley then considers the chances appertaining to two or more lives, and shows that the same principle of investigation already applied to single lives should be extended to the more compound cases; for the separate yearly values being ascertained and added together, would similarly constitute the total value of such annuities; care, however, being taken that each yearly value was previously compounded of the chances of the different lives under consideration, according to the terms of the particular case to be solved. -Farren.

In 1693 Leybourn's Panarithmologia was pub., and it contained tables of annuity values, and of compound int., with much other interesting information concerning marine ins., usury, leases, etc., but no mention is made of life annu.

Some money was borrowed in 1694 on annu. for lives, under the authority of 5 Wm. & Mary, c. 20. The terms were 14 p.c. for one life, 12 p.c. for two lives, and 10 p.c. for three. "Natives or foreigners" might contribute. Such terms were in the highest degree extravagant; particularly as no attention was paid to difference of ages. Assuming no frauds had been committed on the Government, these terms yielded the lender 10 p.c. in the case of the two lives, and 9 p.c. in the case of three. But there is reason to believe that serious frauds were committed. The amount of the annuities at first was about £22,800. In 1762 (68 years afterwards) they had only been reduced by deaths to £9215; while 20 years later (1782) they remained at £8027. See 1718.

It was in the preceding Act that mention is first made of annu. granted upon more than one life.

In 1694, also, by 5 & 6 Wm. & Mary, c. 5-An Act to enable such persons as have estates in life annu. payable by several former Acts therein mentioned, to purchase or obtain further or more certain interests in such annuities; and in default thereof for admitting other persons to purchase or obtain the same, for raising moneys for carrying on the War against France-it was provided that the person who had purchased annu. under the Acts of 1692-3, might exchange their life annu. for others of 96 years certain, upon paying into the Exchequer £63 in add. to the £100 originally paid.

It is thus seen that the difference between 14 p. c. for life, and 14 p.c. for 96 years, was sold for £63, or 4 years' purchase. In other words an annu. of £50 could be secured for 99 years by payment of £675. Such an annu. is now worth 1500, or considerably more than double. But Adam Smith, reviewing the transaction and the period, says that "such was the supposed instability of the Government that even these terms procured few purchasers."

By the 7 & 8 Wm. 3, c. 2 (1695), the opportunity of embracing the advantages of the last-named Act were extended to a longer period; and for 5 years' purchase, or £70, persons not such original purchasers might obtain an annuity of 14 for 96 years.

The purchasers appear to have well understood their own interests: the instability of Government would affect life annu. as much as long annu.; but the life annu. at 7 years' purchase was by far the best bargain, for the int. of money being 6 p.c., the life annu. was worth, at Halley's estimate, 13 years' purchase (13'4 at the age of 10), and an annu. for 96 years was worth only 16% years' purchase. The value of a life annu. of £100 was £1300, which was obtained for £714; and the new offer to such a purchaser was, that, if he would advance £450 more, he should obtain an annu. worth £1660; by accepting the offer he would have gained £496 on £1164; by rejecting it, his profit was £586 on £714.

These schemes of the Government drew much attention to the subject of life annu., and induced several projects to be set on foot on very fallacious foundations. We must notice several of these.

On the 24th Oct. 1695, a scheme was set on foot for making a fund for granting Annuities for Lives, etc. The same was proposed to be worked in connexion with a then existing inst. called the "Million Bank." In the proposals it was provided that any person might subscribe money or land for granting annuities for terms of years, or for I, 2, or 3 lives, at 10, 12, and 14 years' purchase, renewable at any time for a reasonable fine. The money subscribers were to have 6 p.c. p.a. for their money; and the subscribers for land were to be accommodated with money at 3 p.c. p.a., and the remainder of the profit to be divided between the land and the money. Any person might subscribe to purchase annuities at the aforesaid rates, and the subscribers of money might, if they thought fit, transfer the same to the account of annuities for a term of years, or for 1, 2, or 3 lives, as they liked best. But that the money or what is purchased therewith should in the first

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