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

Scotland. Rural, Town, and Rural, Town, and City Districts. City Districts.

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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

13'413

55

10.995

8.475

6.724

5'574

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

Mort. of the Gov. L. Annuitants, being at once a revision and continuation of the Obs. of his late father, brought down to 1854. Of the main results of this Report we shall speak at large under head of GOVERNMENT ANNUITANTS T. OF MORT. It will be useful for the purposes of comparison to introduce the following abstract here:

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The quinquennial periods fall under a different arrangement in the following T., for reasons which will be explained under Gov. ANNUITANTS.

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In 1861 Mr. H. W. Porter, M. A., contributed to the Assu. Mag. (vol. 9, p. 277), in the form of a letter, an able review of some of the more important points considered in or suggested by Mr. Finlaison's R. We shall speak of some of these points under head of CENSUS AND POP. Mr. Peter Gray communicated some remarks upon a formula used in relation to L. Annu. And there is a reprint of a paper contributed by Prof. De Morgan to the Phil. Mag. in 1839 : On the rule for finding the value of an Annu. on 3 Lives.

In 1863 Mr. Andrew H. Turnbull pub. Tables of Compound Int. and Annu.: yearly, half-yearly, and quarterly payments in decimals and currency; with rules for determining the amount of principal and int. in any payment of annu.; and for the construction of tables showing the same.

Age.

In 1864 the ENGLISH L. T. No. 3, was pub. The variations from the preceding T. (Nos. 1 and 2) are very slight. The table is so constructed as to show the value of annu. on the life of a male, or of a female; and on two lives,-namely, 2 males; 2 females; I male and I female, at all the various combinations of age. They are all calculated at 3 p.c. int. The following is an abstract of the T. of annu. values on single lives.

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Male.
Years'
Purchase.
21.0913
20'0143
18.8105

17'4744
16.0102

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A difference in favour of female life prevails at all ages.

*0744

Also in 1864, by 27 & 28 Vict. c. 43, was introduced Mr. Gladstone's Gov. Ins. and Annu. Scheme, which created so much alarm amongst the ins. offices before it came into operation; and has been heard of so very little since. The scheme did not really come into force until 17th of April, 1865. We consider the measure a sound and good one; calculated to be of great benefit to those for whom it was especially intended; and not likely in any important degree to interfere with the bus. of the ordinary ins. offices. The full details of the scheme will be given under GOVERNMENT INS. SCHEME. We may remark that while the English Life T. No. 3, is used for the life ins. branch, Mr. Finlaison's T. of 1829 continues to be used for annuities. The reasons for this have already been indicated, and will again be referred to.

In 1865 Prof. De Morgan communicated to the Inst. of Act. a paper entitled: On a Problem in Annu., and on Arbogast's Method of Development. The paper is printed in vol. 12 of Assu. Mag., but it is purely mathematical.

In 1866 Mr. William Lewins pub. Hist. of Savings Banks, and Post Office Annu, and Ins.

In the U.S. life annu. are by no means a popular investment. In the year 1867, 44 of the principal ins. offices only issued in the aggregate 19 annu., involving an ann. charge of 9197 dols.; and in the four previous years, although a greater number had been granted, they were still sufficiently few to show that no interest is taken in this branch of business. The following is an abridgment of an art. upon life annu. which appears in the Ency. Americana:

Many such annu. are granted for public services. Of this description are the pensions granted by the Acts of the Congress of the U.S. to surviving officers and soldiers of the Army of the American Revolution. Many such are granted by every Gov., and, as these do not arise from a specific contract, and are not usually subjects of purchase [the Acts of Congress, especially, by which those of the U.S. are created, contain provisions to prevent their sale and transfer from the original grantees], their precise value is not often a subject of investigation. But life annu. are often created by contract.. When granted by a Gov., they are generally one mode of raising loans; when created by a contract with a private corp. or co., their object usually is to give the annuitant the use, during his life, not only of the income of his cap., but of the cap. itself. If a person, having a certain cap., and intending to spend his cap., and the income of it during his own life, and leave no part to his heirs, could know precisely how long he should live, he might loan his cap. at a certain rate during his life, and by taking every year, besides the int., a certain amount of the capital, he might secure the same annual amount for his support during his life, in such manner that he should have the same sum to spend every year and consume precisely this whole cap. during his life. But since he does not know how long he is to live, he agrees with the Gov. or any annu. office, to take the risk of the duration of his life, and to pay him a certain annu, during his life in exchange for the capital which he proposes to invest in this way. The probable duration of his life, therefore, becomes a subject of computation: and for the purpose of making this calculation, tables of longevity are made, by noting the proportions of deaths, at certain ages, in the same country or district. A table of this sort was made by Professor Wigglesworth, of Cambridge University, and pub. in the Trans. of the American Academy. This table is very much used in estimating the value of life-estates and annu. in the U.S.

The value of an annu. will depend, also, in some degree, upon the responsibility of the Gov. or co. agreeing to pay it. An annu. guaranteed by a pledge of real estate is worth more than one of the amount resting upon the mere promise of a gov. or private co. Accordingly, for the purpose of raising money upon better terms, that is, selling the annu. for a greater present value, some of the Gov. of Europe have occasionally pledged their domains or the income of certain taxes, to secure the payment of the annu.-Encyclopædia Americana.

In 1867 Mr. Sprague pub. in Assu. Mag. a paper On the Value of Apportionable Annu.; or of Annuities in which a proportionate part is payable up to the day of death. The paper is almost entirely mathematical. The author says:

In former papers I have investigated the value of an annu. on a single life, when payable half-yearly, quarterly, etc. It still remains to consider the values of annu. on the joint duration of two or more lives, or on the life of the last survivor of several lives under similar circumstances. It will, however, be more useful to consider first the increase in the value of annu, on a single life, when a proportionate part of the annu. is payable up to the day of death, instead of the annu. ceasing, as is generally assumed, with the payment that precedes the date of death. [ANNUITY APPORTIONMENT ACT.]

In 1868 Mr. Huie pub. an excellent little work on Widows' Funds, giving, amongst much useful information, tables of the values of annu. to the widows from first marriage of Bachelors, according to the experience of the Scottish Schoolmasters Widows' Fund, and the Scottish Ministers Widows' Fund; also values of annu. to widows from remarriage of widowers; and values of annu. from death of widower or widow until youngest child attains 21; and values of annu. on death of a man whose wife is still living, payable if he die last, and leave a family. It is very difficult for those unfamiliar with the working of these Widows' Funds to comprehend at first sight the ramifications into which they extend. We shall speak of these in some detail under head of WIDOWS' FUNDS.

In the same year Mr. J. H. James pub. Practical Treatise on Life and Fire Assu., Annu., and Rev. Sums, and Leases for Terms and for Lives; illustrated by tables deduced from general and rural mort. [RURAL MORT. TABLE.]

In 1868 Mr. Makeham contributed to the Assu. Mag. a paper On the Theory of Annu. certain, and the same is pub. in vol. xiv.

The following T. shows the amount of Stock trans. and money paid to the Commissioners for the Reduction of the National Debt, for L. annu. under the Acts 10 Geo. IV. c. 24; 16 and 17 Vict. c. 45; and 27 & 28 Vict. c. 43; and the amount of the annu. granted for the same during the under-mentioned periods.

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This requires some explanation. In the first place, as the year (most foolishly) terminates on 5th Jan., the figures of necessity relate almost entirely to the preceding year, and should be so read. In the next place, the "deferred annu." have, since the passing of the 27 & 28 Vict. c. 46 (1864), been carried to a separate account.

It appears that about £1,000,000 of the public revenue is disbursed every year in the form of L. annu., and that about £600,000 on the average is ann. invested in this mode with the Gov. The eds. of the Assu. Mag., reviewing the preceding figures, say:

The amount of these investments fluctuates considerably; and curiously enough, the fluctuations are a very fair index of the state of the money market at the time. Thus in the 15 years under review, it will be observed that the greatest amount of annu. was granted in 1858, the year immediately succeeding the suspension of the Bank Charter Act in Nov. 1857; and the least in 1864, a period of reckless speculation, the reaction from which is clearly discernible in the largely increased transactions of the following year:

In 1869 there was pub. the (NEW) EXPERIENCE MORT. T. deduced from the "Mort. Experience of Life Assu. Cos., collected by the Inst. of Actuaries." This, as we have already intimated, we shall call EXPERIENCE TABLE No. 2; under which title a full account of the data upon which it is constructed will be given. There were no money tables deduced from this T. pub. at this date. See 1871.

In the Assu. Mag. (xv. p. 95), 1869, will be found a paper by Mr. Woolhouse, On an Improved Theory of Annu. and Assu. It is not any part of our design to reproduce even the substance of modern papers so easily obtainable as those in the Assu. Mag., although we have to note their existence for the purpose of complete chronological sequence. We may quote one passage which will give an indication of the scope of the present paper:

The new theory might appropriately be called the continuous method. According to the principles laid down, all moneys invested, in place of receiving yearly increments of int., are considered to be continuously growing. This is undoubtedly the only true way to assimilate our computations with actual facts, since moneys as they come to hand are invested in various securities, and at all seasons, and should therefore not be assumed to bear int. at estab. and immovable periods, but should be regarded as realizing the accumulations from the exact dates at which they are received. Also, lives, instead of being subjected to successive yearly decrements, are, in like manner, properly considered to be diminishing continuously. It is no longer necessary to be dependent upon such gratuitous suppositions as that the deaths which take place during any year shall be equally distributed throughout the year. The inaccuracy introduced by their adoption would be made very conspicuous were we to deal with quinquennial in lieu of ann. intervals, since no experienced calculator would for a moment tolerate the hypothesis of deaths being equally distributed throughout a quinquennial period, and yet the principle is the same for one year as for five.

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In the same vol. p. 126, will be found a paper by Mr. Sprague, M.A. On the value of Reversionary Annu. payable half-yearly, quarterly, etc., according to the conditions which prevail in practice. This is purely a scientific paper, the subject being reasoned out with all that mathematical ability for which its learned author is so justly renowned.

In this same year, 1869, there came before the Law Courts a very curious, and perhaps unique case, arising out of the affairs of the Philomaritan (Annuity) So., which had been founded in Lond. in 1765, and now found itself with a considerable surplus beyond its requirements! We shall give details under hist. of that So.

The Life Assu. Cos. Act, 1870 (sec. 8), requires a periodical statement of life and annuity business transacted to be made. See ACCOUNTS.

In 1871 Tables of Money Values, deduced from EXPERIENCE T. No. 2, were completed under the supervision of the Inst. of Actuaries. The vol. at the moment of our writing is not ready for pub.; but through the courtesy of the President and Council of the Institute, we are enabled to furnish our readers with the following abstract of the T. of Annu. Values:

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[The close of this lengthy art. on L. Annu.-prob. the most wearisome to the general reader of any which may appear in our pages; as it has certainly been the most laborious to the writer-seems a fitting opportunity to state two things :-I. That we shall be much obliged to our readers for pointing out omissions-and the more so, if accompanied with references to the materials omitted. 2. That we are open to suggestions for the practical treatment of these historical articles, which must occur again as we reach FIRE INS., LIFE INS., MARINE INS., MORT. TABLES, etc.]

ANNUITIES ON LIVES: PRACTICE.-In the practice of granting annu. on L. considerations arise which have scarcely been glanced at in our historical art. ; a few of these must be briefly noticed.

Some few years since, and nearly all life offices granted such annu. Of later years the practice has been falling into disuse by some of the leading offices. We believe the Economic, National Provident, and Norwich Union have discontinued granting such annu., prob. many others. The last-named office found that it had made a considerable loss by its life annu.

It forms not unfrequently, and we are disposed to think not unwisely, an element of finance on the part of a young L. office to grant L. annu. It seems a ready way of acquiring a portion at least of the cap. required for foundation expenses: but care must of course be taken that this advantage be not purchased too dearly. A reckless system of granting L. annu. would pull down any office, young or old.

An office granting only a small number of L. annu. is always open to the difficulty of not having sufficient lives at risk to secure it a proper average. On the other hand, a young office is not very likely to have annu. lives selected against it, as a matter of speculation.

The rate of int. assumed in the granting of annu. is unquestionably one of the most important considerations. This may be shown by one short example. The value of an annu. of £1 on a life aged 50 is given as follows by 4 Mort. T. at 4 rates of int.

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These differences existing in the value of £1 annu. would reach a very serious magnitude when large sums are dealt with; as for instance 100's and 1000's. The entire question of the bearing of int. on the values of L. contin. will be considered under INT. OF MONEY. Prof. De Morgan considered that certainly not more than 3 p.c. should be assumed in ordinary annu. transactions in Gt. Brit.-Essay on Prob.

A review of the preceding figures also shows that the MORT. T. to be selected as the base for ann. calculations is a matter of very considerable importance-less so no doubt since the Northampton Table has fallen entirely into disuse for such purposes. It has been remarked, and with partial truth, that it would not matter upon what mortality table, either life insurance or annuity rates were based, provided every person who insured his life, sunk in a life annu. with the same office a sum equal to that ins. under his L. pol. But in practice no such equalizing circumstances do or can exist. On the contrary, selection is very commonly made against the office in both directions. Lives below the standard of health seek to insure-lives above the ordinary standard select L. annu. as a profitable investment. In consequence of this state of things, it is regarded only as a matter of prudence that the two branches of bus. should be conducted upon the basis of distinct mort. T.-the Gov. Annuitants T. being in such cases usually selected for the purpose of annu.

An attempt was made some years since, by the late W. E. Hillman, to graduate a system of L. annu. to the varying effect of different diseases upon the human system. We shall refer to this more at large under DISEASED LIVES, INS. OF. We desire to express

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