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The preceding table must be read in the light of the following explanation : By the English Life Table (No. 1), the present value of a life annu. of £1 at the age of 21 is £20 19s. 9d. on the life of a male; £21 4s. 3d. on the life of a female. The present value of an annu. of £1000 a year on the life of a man would be 1000 times £20 195. 9d., or nearly £20,986; and in the same way the value of an annu. of £10, £20, £35, or any other number of pounds, may be found. £100 will purchase an annu. of £4 155. 4d. on the life of a man aged 21, and £4 145. 3d. on the life of a woman at that age. £1000, which is 10 times £100, will purchase an annu. of 10 times the amount, or £47 13s. on the life of a man, and £47 3s. on the life of a woman. The parties who granted annu. at these rates would gain nothing, as the values in the tables are the nett values: they might lose if the lives were well selected. Upon these accounts a certain arbitary per-centage must be put on the " present values," and taken off the "annuities."

In this year also, by 7 & 8 Victoria, c. 83, the Act of 1833, for enabling trustees of Savings Banks to grant annu., was amended to the extent that an annu. of £30 (instead of £20) might be granted upon any one life. Husband and wife might each have an annu. This, and the Act of 1833, were repealed in 1853.

Up to 1850 the annu. granted under the two Acts just named, were 5648, and of these 3385, or three-fifths of the whole number, came in after 1844.

In 1845 Dr. Farr pub., in 8th Report of R. G., what he termed the New Northampton Table. It was based upon the same data as Dr. Price employed in the orig. table; but that data was "corrected" in the light of the more exact information we now possess as to the circumstances of the pop. of Northampton at the period embraced in the registers upon which Dr. Price relied; and it therefore purports to be not only a new but a true Table of the Mort. of Northampton. It is instructive to note the annu. values deduced from this new and true table as contrasted with that which came to be so extensively employed for the purposes of annuity valuation at a former period.

Under the head of NORTHAMPTON Table (New), more extended and complete comparisons will be given. In 1846 Mr. Charles Rann Kennedy, Barrister-atLaw, pub. A Treatise on Annu. In the same year

Mr. Edward Scury pub. Life Assu. and Annu. Tables.

True Table. Dr. Price's T.

3 p.c.

3 p. c.

Age.

Years'
Purchase.

Years' Purchase.

O

17:08

12.27

IO

23.39

20.66

20

21'41

18.63

30

19:08

16.92

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In 1847 Mr. W. E. Hillman pub. a little book on the Theory and Practice of Assu., and therein he suggested, that as it was shown that the duration of life was affected by various diseases, whether persons suffering from such diseases should not have the benefit of the consequent reduced value of annu. on their lives. The subject will be further discussed under DISEASED LIVES, INS. OF.

Mr. Charles Ansell said before a House of Lords Committee in 1848 that most of the ins. offices had discontinued granting annu. at all: they had been found unprofitable. Many of the offices referred parties to the Gov. Mr. Finlaison said before the same Committee that there was nearly one million p.a. paid by the National Debt Office for annu.; but he thought the number of annuitants had reached its maximum-it had been going on for 40 years-the old ones died off about as fast as the new ones came on.

There are various sos. regis. under the Friendly Sos. Acts which grant annu., and which are not popularly known as annu. sos. Thus in the returns made by Friendly Sos. for the five years ending 31st Dec., 1850, there were included as being in the City of Lond. 6 sos., the oldest being founded in 1835, having an aggregate of 10,876 members, (the members of the 2nd largest of the 6 were not returned,) and total assets of £717,441 145.; while in other parts of Lond. 9 other sos. were returned, being, however, nearly all small ones.

In 1850 Mr. Thomas Weddle contributed to the Phil. Mag. a paper On Annu. and Assu. on Successive Lives. In vol. ii. of Assu. Mag., Mr. Peter Gray says, "The subject is treated with that gentleman's usual ability and elegance."

In the same year Mr. Samuel Brown read before the Inst. of Act. a paper On the General Method of Approximation to the Values of Annu. and Assu. for long terms of years, depending on one or two Lives. The paper is one of considerable interest, but of too scientific a character to be quoted in these pages. It will be found in Assu. Mag.i., p. 20. In the same vol. will be found a paper by the late Mr. Peter Hardy: On the Values of Annu., which are to pay certain given Rates of Int. on the Purchase-money during the whole term of their continuance, and to replace their Orig. Values, on their expiration at certain other given Rates. The writer says:

Notwithstanding the very large amount of leasehold property which in the course of every year is bought and sold in this country, and notwithstanding the extensive transactions-of almost hourly occurrence in the public market, in Gov. and other temporary annu., the subject of the rate of int. which any given purchase will yield the buyer is very imperfectly understood, even by the most deeply interested in the inquiry, unless they happen to be at the same time well versed in actuarial computations. It is not unfrequently imagined by a buyer, that if he purchase a leasehold property or a temporary annu. at a price corresponding with the value of an annu. at a given rate of int. (say 5 p.c.), he has made a purchase which will pay him 5 p.c., or which, in other words, will enable him to spend 5 p.c. p.a. on his outlay, and at the same time replace his cap. undiminished at the expiration of the term. This is a grave error, and very frequently leads to serious inconvenience. If a purchaser buys an annu. for a term of years, according to a 5 p.c. table, it is absolutely essential that the surplus of the annu. over and above the int. on the purchase-money should be ann. reinvested in some fund which will also yield a clear 5 p.c.; otherwise, the buyer's expectation to replace his cap. at the expiration of the term will not be realized.

This proposition is then fairly demonstrated; and a TABLE is furnished showing the present value of an annu. of £1 p.a. for a given number of years certain, supposing the purchaser thereof to take out of the annu. 5 p.c., 6 p.c., or 7 p.c. p.a., as an available int. on his purchase-money or cap. advanced, while he is only enabled to re-invest the surplus of the annu. beyond the available int. so as to make 3 p. c., 34 p.c., 4 p.c., and 5 p.c. thereof. In the same vol. will be found a paper compiled from materials supplied by Mr. W. T. Thomson: Mort. amongst Lives selected at the ages 75 to 81 for Gov. Annu. We have already referred to this subject in the present art. under date 1829. We shall speak of it more at large under Gov. Annuitants.

In the same vol. will be found some important correspondence on points involved in the preceding articles.

In 1851 there was pub., as the joint production of Mr. Peter Gray, Mr. Henry Ambrose Smith, and Mr. William Orchard: Assu. and Annu. Tables according to the Carlisle Rate of Mort. at 3 p.c. The editors of the Assu. Mag. say, "This work is of much greater value to actuaries than its mere title would seem to imply" (vol. ii. p. 194). In the same vol. there is a communication from Prof. De Morgan: On a Method of checking Annu. Tables at different Rates of Int. by help of one another. In this year Mr. J. H. James pub. A Treatise on Fire and Life Assu, Annu., and Rev. Payments, etc., etc. In 1853 an Act, 16 & 17 Vict., c. 45, To consolidate and amend the laws, and to grant add. facilities in relation to the purchase of Gov. annu. through the medium of Savings Banks, and to make other provisions in respect thereof, was passed. The former Acts were repealed in order that they might be consolidated and amended. Nearly all the provisions of the former Acts were reintroduced, and there were a few new ones. The age at which life annu. might be granted was reduced to 10; the amount of annu. to be not less than £4 or more than £30; husband and wife might each have an annu. Deferred annu. might be granted upon the condition of money paid for same being returnable in event of death before annu. has commenced. Or annu. from £1 to £30 might be granted on condition of not being returnable. In cases of periodical payments, if payments were not continued, a smaller annu. might be granted. Deferred might be converted into immediate annu. Commissioners might decline to grant annu. on "sufficient grounds." The Treasury should direct what tables to be used. Quarter's payment in add. to arrears to be paid at death. Annu. not assignable, except in case of bankruptcy. Annu. to be free of taxes, and to be deemed personal estate.

Mr. J. W. Stephenson pointed out in vol. x. of Assu. Mag., p. 44, that the tables pub. in conformity with this Act "were not computed in the usual way, but apparently on some principle wholly different from it." Of this he gives examples. We do not remember to have seen any explanation.

In the same year was passed the Succession Duty Act-16 & 17 Vict., c. 51—which contains elaborate schedules of the values of single and joint annu. for lives and for years, based upon Finlaison's T., males, 4 p.c. This Act supersedes the Tables given in the Legacy Duty Act, 36 Geo. III., c. 52 (see SUCCESSION DUTY ACT). The difference between the two T. in the value of annu. is very remarkable. Take the case of a £100 annu. on one life :

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In 1853 also the 12th Ann. Report of the Reg.-Gen. was pub., and contained the ENGLISH LIFE T. No. 2.

This was

The

constructed from the deaths during the years 1838-44; the pop. of 1841 serving for the basis in this as in No. I. results of No. 2 T. were confined to Males only. The annu. values deduced therefrom are given at 3 rates of int. ; and are shown in the margin.

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24'2042

20.3828

17'5339

15

23 2225

19'7109

17'0564

20

22.2334

19'0263

16.5663

25

21'2703

18.3612

16.0955

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In vol. iii. of Assu. Mag. pub. in 1853, will be found an important practical suggestion by Mr. Peter Hardy, viz., whether the approximate value of an annu. on 3 joint lives would be more accurate, if it were obtained by means of the curtate expectation of two joint lives, instead of being ascertained, at present, by means of the value of an annu. on the same joint lives, at the particular rate of int. involved in the calculation." The subject, however, is so technical as to be beyond our present range. In the same vol. Mr. C. W. Merrifield, of the Privy Council Office, communicated "a method for obtaining an expression of the rate of int. in immediate annu.," and Mr. Jas. Meikle, whose name will become familiar to the reader of these pages, communicated formulæ for obtaining the value of a L. annu. at one rate of int. from the value at another given rate.

In 1854, by 17 & 18 Vict., c. 90, all the Acts relating to the inrolment of annu. were repealed, in conjunction with the repeal of the Usury Laws. It is probable that the great majority of life annu. granted during some years previous were really void for the want of inrolment. Thus, after more than three centuries of restrictions from these Usury Laws, the dealing in life annu. became free and unfettered-with this general exception, that the Courts of Equity will always interfere to set aside an "unconscionable bargain," especially if made under any aspect of fraud, or with a minor, or person mentally incapacitate.

It may seem strange at first sight, but it is unquestionably true, that since dealings in life annu. have been freed from all restrictions, the number of such transactions has very greatly diminished. But we must look to the true reason of this, and undoubtedly it is due to the repeal of the Usury Laws, which had a direct tendency to encourage the very description of transactions they were supposed to suppress. The attempt to enforce one common rate of int. for loans on every varying description of security was an act of unmitigated folly. Any person attempting to borrow on a security demanding a higher rate of int. than the legal rate, as on a life int., a leasehold estate, or a rever., was compelled to obtain the advance under cover of a life annu. And these transactions, when reasonably conducted, were not without some special advantages. To the borrower on a redeemable annu. there was the great advantage of being certain not to have the money called in by the lender, combined with the equally great convenience of being able to pay off the principal whenever he pleased. He had to consider how much he could afford to pay for these advantages. To the lender there was the advantage of securing in general a permanent investment of his money at a higher rate of int. than he could obtain by way of mortgage, and without sinking his principal, unless he chose to waive the ins., in order to increase the amount of his annu. This mode of raising money has the further advantage, that, in the various ramifications of property incident to a highly civilized state of society, it frequently affords the only means by which a parent without cap., but with a good and certain yearly income, can raise a sufficient sum to secure the advancement of his children in the world.-Kelly.

It was right, therefore, that all restrictions should be removed (see 1855).

In the 4th vol. of Assu. Mag. (1854), there appears a paper by Prof. De Morgan: On the Demonstration of Formula connected with Int. and Annu. The paper is purely scientific. In the same vol. is a paper by Holmes Ivory, then one of the Vice-Presidents of the Inst. On the Method of approximating to the Values of Deferred and other Life Annu., when payable Half-yearly and Quarterly. The writer says:

:

The difference between the value of an immediate life annu. when payable yearly, and that of the same annu, when payable half-yearly or quarterly, has, as is well known, been variously estimated by our leading authorities. According to Dr. Price and Mr. Morgan, a fifth of a year's purchase will be generally more than a sufficient add. if the value of the annu. is desired payable half-yearly, and three

tenths of a year's purchase if the value of the annu. is desired payable quarterly (Price, vol. i., p. 246, 6th and 7th eds.). Mr. Baily again, following Simpson, lays it down as a rule, that the value of annu. payable yearly must be increased nearly one-fourth of a year's purchase, in order to show the value of the same annu, payable half-yearly; and that they must be increased nearly three-eighths of a year's purchase in order to show the value of the same annu, payable quarterly "(Francis Baily, c. x., sec. 355). The same rule is also given by Mr. Milne (ch. viii., sec. 485).

The writer, while explaining the cause of the difference of these estimates, adopted Mr. Baily's rule.

By 18 & 19 Vict., c. 15, secs. 12 and 14, 1855, it is provided-in order that it may be ascertained by search what life annu. or rent-charges may have been charged upon lands by their owners that any annu. or rent-charge granted after the passing of that Act (and not given by marriage settlement or by will) for one or more life or lives, or for any term of years, or greater estate determinable on one or more life or lives, shall not affect any lands, tenements, or hereditaments, as to purchasers, mortgagees, or creditors, until a memorandum containing the name and the usual or last known place of abode, and the title, trade or profession of the person whose estate is intended to be affected thereby, and the date of the instrument whereby the annu. or rent-charge is granted, and the amount of the annual sum to be paid, shall be left with the senior Master of the Common Pleas at Westminster, who shall forthwith enter the particulars aforesaid in a book in alphabetical order by the name of the person whose estate is intended to be affected by the annu. or rent-charge, together with the year and the day of the month when every such memorandum or minute is so left with him.

In the 5th vol. of Assu. Mag. (1855), there are various papers on the subject of life annu. Mr. Edwin H. Galsworthy contributed a formula, to find the amount of an annu. increasing or decreasing by a constant quantity; also to find the present value of an annu. increasing or decreasing by a constant quantity. Mr. Charles Gabriel Shaw contributed another ingenious" formula, for a like purpose. Mr. James Meikle contributed a formula, to determine the rate of int. in a life annu., the tables of mort. and age being given. Mr. Charles Jellicoe a short formula in elucidation of these. Mr. Peter Hardy contributed an 'elegant " formula for an approximate value of annu. at simple int.

66

In the same vol. there is a reprint from the Trans. of the Cambridge Phil. So. of a paper by Sir J. W. Lubbock, Bart., in 1828, On the Calculation of Annu., and on some questions in the Theory of Chance; and another paper from the same source, by the same author, read 1829: On the Comparison of Various T. of Annu.

In 1856 Mr. Wm. Orchard pub., Single and Ann. Prems, for Annu. In this same year also were pub. the remaining copies of Mr. Griffith Davies' Treatise on Annu., etc., (first pub. 1825,) in a more complete form than orig. Mr. William Braid made a communication to the Assu. Mag. (vol. vi. p. 109) On Annú. and Assu. Tables for 3 Lives.

In 1857 Mr. Charles M. Wallich, author of the well-known T. bearing his name, read before Sec. F. (Economic Science and Statistics) of the British Asso. a paper on Annu. for Lives. The author said:

The table of the values of annu. on lives, as hitherto constructed, only show the rate of int. which a purchaser may make on the money employed, and replace the cap., provided he can reinvest the surplus income, beyond the int. on the purchase-money, at the same rate of int. As it is often important to know the price which should be paid for such annu., in order that the purchaser may enjoy a certain higher rate of int. on the money invested, whilst the circumstances of the times only permit a much lower rate to be calculated upon for the future investments, I have constructed the following formula, which will show the purchase-money to be paid for an annu. under such circumstances.

Mr. Willich then furnishes the following

Table of Annu. according to the Carlisle T. of Mort. at 3 p.c.; also the price which should be paid for an annu. so that the purchaser may obtain 5 p.c. int. on the money invested, while the reinvestments to replace cap. are estimated to be made at only 3 p.c.: Tables of this character are of great value in monetary transactions.

Age.

Value of Annuity
at 3 p.c. Carlisle.

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Value of an Annuity if 5 p.c. is to be enjoyed while the reinvestments are only to be made at 3 p.c.

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In 1857 several papers appeared in the Assu. Mag., vol. vii., bearing upon life annu. Thus, in a paper upon the writings of the late Dr. Thomas Young: A Formula for expressing the Decrement of Human Life. The subject is treated of in a highly scientific Mr. Peter Hardy gave a paper: An Investigation into the proper method of determining the amount of an Annu. forborne and improved at Int. during the existence of a given life. Mr. Thomas Carr offered some obs. on the first-named paper. Mr. Adler supplied a formula for an approximate value of annu at simple int.

manner.

In 1857 Mr. F. G. P. Neison published the 3rd edition of his Contributions to Vital Statistics; and therein is contained Tables of Mort. deduced separately from (1) Rural Districts; (2) Town Districts; (3) City Districts, of England and Wales; and then (4) the preceding three combined; and (5) combined T. of Rural, Town, and City

Districts for Scotland. Annuity values based upon these tables are given, of which the following is an abstract:

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In the orig. T. the values are carried to 5 decimal places. We have considered 3 sufficient for our purposes. [CITY LIFE.] [FRIENDLY SOS. MORT. T.] [RURAL LIFE.] [SCOTLAND.] [TOWN LIFE.]

In 1858 Mr. Wm. Curtis Otter contributed to the Assu. Mag. (vols. 7 and 8) two papers On the Calculus of Finite Differences, and its application to Problems in the Doctrine of Compound Int. and certain Annu. The papers are entirely scientific. In 1858 Mr. J. H. James pub. An Enquiry as to the Duration of Life in Rural Districts, etc. In this work was contained, in a somewhat crude form, what is called the RURAL LIFE TABLE, and the annexed abstract of annu. values, 4 p.c., deduced therefrom:

The T. appears more in detail in Mr. James's Practical Treatise on F. and L. Assu., etc., 1868.

In the 5 years ending 1859, there were granted to 4362 annuitants, at the National Debt Office, £262,963 of annu. The annu. therefore averaged about £60 each. They were applied for in the proportion of I male to 2 females. The mean age of the annuitants was about 63 years. The amount of the annu. granted in the 30 years 1829-59 was £1,967,000; and it is estimated that the substitution of the Gov. for the Northampton T. had in that period saved the country nearly £3,000,000!

Age.

Years' Purchase.

I

19.397

15

18.551

25

17.183

35

16.048

45

13413

55

10.995

65

8:475

75

6.724

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In 1859 Mr. Jardine Henry pub. The Values of Annu. on single and two joint lives, at 3, 4, 5, and 6 p.c. p.a., for every combination of age and sex, founded upon Mr. John Finlaison's Report of 1829 (see Actuarial Tables).

In the same year M. Fedor Thoman pub. Theory of Compound Int. and Annu., with Logarithmic Tables.

In 1859 the Assu. Mag. (vol. 8) contained various papers on life annu. Prof. De Morgan contributes one: On the determination of the Rate of Int. of an Annu. This paper is full of quaint learning. The same writer gives another paper: On a Property of Mr. Gompertz's Law of Mort., in which he treats of the calculating an annu. on 3 joint lives. Mr. W. M. Makeham contributed a paper: On the Law of Mort. and the construction of Annu. T. The author says:

Most writers on the subject of life annu. have had occasion to lament the paucity of tables available for the performance of calculations involving two or more lives. The late Mr. David Jones has done much to supply this deficiency by the pub. of complete sets of tables for two lives at various rates of int.; but beyond this it is extremely improbable that under the present system any considerable progress will be made, owing to the multiplicity of the different combinations when 3 or more lives are concerned, and the consequent magnitude of the task involved in the construction of complete sets of tables for such cases.

The chief object of the following investigation has therefore been to find a formula which should represent with sufficient accuracy the results of obs. on the law of mort.; and which at the same time should be adapted to facilitate the construction of complete sets of tables of annu. involving several lives.

The writer then enters upon the necessary scientific investigation for accomplishing his purpose.

In 1860 Mr. A. G. Finlaison pub. by the authority of Parl. his Report and Obs. on the

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