34 p.c.

1 2 3 4








9 10 15 20 25 30 40 50 60




TABLE IV.-Amount of an Annuity of of life annuities depends will be more One Pound.

fully explained in the articles PROBA

BILITY and MORTALITY. Let us suppose Years. 3 AC.

4 p.c. 5 p.e.

100 persons, all of the same age, buy a 1.000 1.000 1.000 1.000 life annuity at the same office. Let us 2.030 2.035 2.040

also suppose it has been found out, that of 3091 3: 106 3.122

4.215 4.246 4.310 100 persons at that age, 10 die in the first 5.309 5.362 5.416 5.526 year, on the average, 10 more in the se

6.802 cond year, and so on. If then it can be 7.662 7.779 7.898 9.142 8.892

relied upon that 100 persons will die 9.052 9.214 9.549 10.583 11.027

nearly in the same manner as the average 11.464 1•731 12.006 12.578 of mankind, or at least that in such a 18.599 19.296 20.024 21.579

number the longevity of some will be 26.870 28.280 29.778 33.066 36.459 41.646 47.727

compensated by the unexpected death of 47.575 51.623 56.085 66.439 others, the fair estimation of the value of

84.550 95.026 120.800 a life annuity to be granted to each may 112.797 130.998 152.667 209 348

be made as follows:- To make the ques163.053 196.517

353 5 4 70 230.594 288.938 364.290 588.529

tion more distinct, let us suppose the bar

gain to be made on the 1st of January, In this Table we see what would be 1844, so that payment of the annuities is possessed by the receiver of an annuity due to the survivors on new-year's day of at the end of his term, if he put each each year. Moreover let each year's anyear's annuity out at interest so soon as nuity be made the subject of a separate he received it. For example, an annuity contract. The first question is, what of 11., in 40 years, at 5 per cent., amounts ought each individual to pay in order to 120.8l., which includes 40l. received that he may receive the annuity of il., if altogether at the end of the different he survives in 1845. By the general law years, and 80.8l. the compound interest of mortality, we suppose that only 90 will arising from the first year's annuity, remain to claim, who will, therefore, rewhich has been 39 years at interest, the ceive 90l. among them, the remaining 10 second year's annuity, which has been 38 having died in the interval. It is suffiyears at interest, and so on, down to the cient, therefore, to meet the claims of last year's annuity, which has only just 1845, that the whole 100 pay among been received. When the annuity is them, January 1, 1844, such a sum as payable half-yearly or quarterly, its pre- will, when put out at interest (suppose 4 sent value is somewhat greater than that per cent.) amount to 90l. on January 1, given in the preceding Table. For the 1845. This sum is 86.6541., and its hunannuitant, receiving certain portions of dredth part is •866511., which is, therehis annuity sooner than in the case of fore, what each should pay to entitle himyearly payments, gains au additional por- self to receive the annuity in 1845. There tion of interest. Since 4 per cent. is 2 will be only 80 to claim in 1846, and, per cent. half-yearly and i per cent. therefore, the whole 100 must among quarterly, and since every term contains them pay as much as will, put out at 4 twice as many half-years as years, and per cent. for 2 years, amount to 801. This Your times as many quarters, it is evident sum is 73.9681., and its hundreth part is that an annuity of 1ool. a-year, payable •739681., which is, therefore, what each nalf-yearly, at 4 per cent., for 10 years, is must pay, in order to receive the annuity, the same in present value as one of 50l. if he lives, in 1846. The remaining per annum, payable yearly, at 2 per cent., years are treated in the same way, and For 20 years. Again, 1001. a-year, pay- the sum of the shares of each individual able quarterly for 10 years, money being for the different years, is the present at 4 per cent., is equivalent to an annuity value of an annuity for his life. We of 251., payable yearly for 40 years, money must observe, that in the term value of an being at 1 per cent.

annuity it is always implied that the first The principles on which the calculation annuity becomes payable at the expiration of a year after the payment of the high a mortality at all the younger and purchase-inoney

middle ages of life, and, consequently, too The value of a life annuity depends, low a value of the annuity. The second therefore, upon the manner in which it is is from the Carlisle Table, formed by Mr. presumed a large number of persons, si- Milne, from observations made at Carinilarly situated with the buyer, would lisle. It gives much less mortality than die off successively. Various Tables of most other Tables, and, therefore, gives these decrements of life, as they are called, higher values of the annuities ; but it has have been constructed, from observations since been proved to represent the actual made among different classes of lives. state of life among the middle classes, in Some make the mortality greater than the century now ending, with much others; and of course, Tables which give greater accuracy than could have been a large mortality, give the value on the supposed, considering the local character annuity smaller than those which suppose of the observations from which it was demen to live longer. Those who buy an- rived. The third table is that connuities would, therefore, be glad to be structed by Mr. Finlaison, from the obrated according to tables of high mor- servation of the mortality in the governtality or low expectation of life ; while ment tontines and among the holders of those who sell them would prefer receive annuities granted by government in reing the price indicated by tables which demption of the national debt, and differs give a lower rate of mortality. In insu- from the former two in distinguishing the rances the reverse the case: the shorter lives of males from those of females. the time which a man is supposed to live, Most observations hitherto published the more must he pay the office, that the unite in confirming the fact, that females, latter may at his death have accumulated on the average, live longer than males, wherewithal to pay his executors. We and in the annuities now granted by gonow give in Table V. the values of an- vernment, a distinction is made accordnuities according to three of the most ce-ingly. The mean between the values lebrated Tables.

of annuities on male and female lives, TABLE V.-Present Value, or Purchase

according to the Government Tables,

agrees pretty nearly with the Carlisle money, of a Life Annuity.

Tables, the rate of interest being the Northampton. Carlisle. Gov.M.Gov.F.

same. Age. 3 p.c. 4 p.e. 5p.c. 3p.c. 4 p.c. 5 pc. 4p.c. 4 p.c. For the materials of Table V. we are 0 12.3 10.3 8.9 17.3 14.3 12:1

indebted the works of Dr. Price, on 5 20.5 17.2 14.8 23.7 19.6 16.6 19.3 20.0 10 20:7 17.5 15:1 235 19.6 16.7 18.8 1907

Reversionary Payments ; of Mr. Milne, 15 19.7 16.8 14.6 22.6 19.0 16.2 18.0 19.1 on Annuities and Insurances ; and to Mr. 20 18.6 16.0 14.0 21.7 18.4 15.8 17:3 1866 Finlaison's Report to the House of Com25 17.3 15.4 13.6 2007 17.6 15.4 16.9 18:1 30 16.9 14.9 13.1 19.6 16.9 14." 16.4 17.5

mons on Life Annuities; to all of which 35 15.9 14.0 12.5 18.4 16:0 14.1 15.7. 16.9

we refer the reader. The tables are of 40 14.9 13.2 11.8 17.1 15.1 13.4 14.9 16.2 course very much abridged. 45 13.7 12.3 11.1 15.9 14:1 12.6 13.8 15.3

To use the Table V., suppose the value 50 12.4 11.3 10.3 14.3 12.9 11.7 12.4 14.2 55 11.2 10.2 9.4 12.4 11.3 10.3 11.0 12.8

of an annuity of 1001. a-year, on a life 60 9.8 9.0 8.4 10:5 97 8.9 9.7 11.3 aged 35, is required, interest being at 4 65 8.3 7.8 7.3 8.9 8.3 7.8 8.2 9.6

per cent., which is nearly the actual 70 6.7 6.4 6.0 71 6.7 6.3 6.8 7.9 75

value of money. We find in the column 5.2 5.0 4.7 5.5 5.2 5.0 5.4 6.3 90. 1.8 3.6 3.5 4.4 4.2 4.0 3.8 4.9

marked 4 per cent., opposite to 35, under 95 2.6 2.5 2.5 3.2 3:1 3.0 2.3 3.8 the Northampton Tabl 14.0, under the 90 1.8 1.8 1.7 2.5 2.4 2.3 1.3 2.1

Carlisle 16.0, and under the Government 95 • 2 .2 2.9 2.7 2.6 .6 1.0

Tables 15•7 or 16.9, according as the life The first of these is calcnJated from the is male or female. These are the numNorthampton Table, formed by Dr. Price, ber of pounds which ought to buy an from observations of burials, &c., at annuity of il., according to these several Northampton. As compared with all authorities; and taking each of them 100 other Tables of authority, it gives too | times, we have :

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Northampton Table

14001. aged 25 and 55, in which the difference Carlisle Table


of age is 30 years. In the Carlisle Table Government Table (males) 15701. opposite to 25, the younger, and under 30, Government Table (females) 16901. the difference, we find 10.3; and 8.8 in

We cannot suppose that the annuity the Northampton. For the value of an could be bought for less than would be annuity of 100l., the first tables give, required by the Carlisle Tables.

therefore, 1030l., and the second 8801. To find the value of an annuity on a The value of an annuity on the longlife whose age lies between two of those est of two lives, that is, which is to be given in the table, the process must be payable as long as either of the two shall followed which has been already ex- be alive to receive it, is found by adding plained in treating of annuities certain. together the values of the annuity on the

An annuity on two joint lives is one two lives separately considered, and subwhich is payable only so long as both the tracting the value of the annuity on the persons on whose lives it is bought are joint lives. For the above species of analive to receive it.

nuity puts the office and the parties in

precisely the same situation as if an anTABLE VI.—Present Value, or Purchase- nuity were granted to each party sepa

money, of an Annuity of One Pound on rately, but on condition that one of the two Joint Lives.

annuities should be returned to the office Carlisle.—4 per cent.

so long as both were alive, that is, during Age. 0. 10. 20. 30, 40. 50. 60. 70.

their joint lives. For example, let the 0 8.9 12.3 1167 10.9 9.9 8.6 6.6 4.7 ages be 25 and 55 as before, and let the 5 16.8 16.5 15.6 14.4 12.9 10.5 7.8 5.0 Carlisle Table be chosen, interest being 10 17.0 16.3 15.2 13.8 12.0 9.2 6.5 4:1 at 4 per cent., we have then :15 16.3 15.5 14.3 12.9 10.5 7.9 51 30 20 15.6 14 7 13.6 11.8 9.0 6.4 4.1 2.4 TABLE V.-Annuity at age 55

11.3 25 14.8 13.8 12.5 10.3 7.8 5.0 3.0 2.6


17.6 30 13.9 12.9 11.4 8.8 6.3 4.0 2.3 1.6 40 12:1 10.9 8.6 6.2 3.9 2.3 1.6


28.9 50 10.1 8.1 6.0 3.9 2.3 1.6

TABLE VI.-Joint Annuity, 55 & 25 10.3 6.9 5.3 3.6 9:1 1:5 70 4.4 3.1 1.9 1.5


18.6 80 2.4 1.6 1.3

The value, therefore, of an annuity of Northampton.-4 per cent.

il. per annum on the survivor is 18.61. Age. 0.

10. 20. 30. 40. 50. 60. 70. The value of an annuity which is not 1 8.3 10.8 10.1


8.6 7.5 6.1 4.4 to be payable till either one or other of 5 13.6 13.5 12.6 11.7 10.5 8.9 7.0 4.6

two persons is dead, and which is to con10 14.3 13.4 12.6 11.5 10.1 8.3 6.0 3.5 15 13.4 12.6 11.8 10.6 9.1 7.1 4.7 2.3

tinue during the life of the survivor, is 20 12.5 11.9 10.9 9.6 8.6 6.8 3.4 1.7 | found as in the last case, only subtracting 25 11.9 11.2 10.2 8.8 6.9 4.6 2.4 0-2 twice the value of the joint annuity, in30 11.3 10.5 9.3 7.8 5•7 3'4 1.7

stead of that value itself. In the preced40 9.8 8.8 7.5 5.6 3.3 1.7 7:0 5.3 3.2 1.7

ing case it is 8.31. For this case only 60 4.9 3.1 1.6

differs from the preceding, in that the 2.8 1.5

annuity is not payable while both are 80 2.1 1.3

alive, that is, during the joint lives. ConThe preceding table gives the results sequently the value in this case is less of the Carlisle and Northampton Tables than that in the last, by the value of an on the value of this species of annuity, annuity on the joint lives. interest being at 4 per cent. The first The value of an annuity to be paid to column shows the age of the younger life, A from and after the death of B, if the and the horizontal headings are not the latter should happen to die first, is the age of the elder life, but the excess of the value of an annuity on the life of A, age of the elder life above that of the diminished by the value of an annuity younger. For example, to know the on the joint lives of A and B. For the value of an annuity in two joint lives, situation is exactly the same as if the




8.1 6.2




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office granted an annuity to A, to be re- | granting of a single annuity, or of a few turned as long as both should live. The annuities, as a commercial speculation, ages and Table being as before, and the would deserve no other name than gamlife on whose survivorship the annuity bling, even though the price demanded depends being that aged 25, we have:-- should be as high as that given in any

tables whatsoever. TABLE V.-Annuity at age 25

In the preceding tables, we would again TABLE VI.—Joint annuity, 25 & 55

remark, that our object has been simply Difference

to furnish the means of giving a mode

rately near determination of a few of the whence the value of the required annuity most simple cases. We should strongly of 11. is 7.31.

recommend every one not to venture on The following Table, extracted with important transactions without profesabridgment from Morgan on Insurances, sional or other advice on which he can deduced from the Northampton Table, depend, unless he himself fully underwith interest at 4 per cent., gives the stands the principles on which tables are average sum to which the savings of an

constructed. The liability to error, even individual may be expected to amount in using the most simple table, is very at the end of his life, improved at com

great, without considerable knowledge of pound interest from the time he begins the subject; and most cases which arise to lay by :

in practice contain some circumstances TABLE VII.- Pribable Amount of One peculiar to themselves, which have not

and could not have been provided for in Pound laid by yearly, and improved to the end of Life.

the general rules.

The following references to works on Age. Amt. Age. Amt. Aye. Amt. Age. Amt. this subject may be found useful :0.137.8 79.2 50 . 29.5 75

ANNUITIES ČERTAIN. 1. Smart's Tables 159.1 66.0

4.8 of Interest, 8c., London, 1726. There 54.6 60 18.5

is an edition published in 1780, which is 65

90 .2.0 70. 10.3

said to be very incorrect. The values

for the intermediate half-years given in That is to say, according to the North- this work are not correctly the values of ampton Tables, if a person were, at the the annuities on the supposition of halfage of 26 (that is, a year after 25), to yearly payments ; in other respects it is begin laying by 1001. a year at interest, to be depended upon. 2. Corbaux, Doehe might expect the amount at the end trine of Compound Interest, gc., London, of his life to be 79.21. for each pound | 1825. 3. Baily, Doctrine of Interest and laid by yearly; or 79201. Or, to speak | Annuities, London, 1808. Smart's Tables more strictly, if 100 persons were to do are republished in this work from the this, they might expect that the average correct edition. Works on life-annuities amount of their savings, reckoning the generally contain principles and tables for accumulations at their deaths, would be the calculation of annuities certain. 79201. each. As we have already ob- LIFE ANNUITIES. 1. Price, Observaserved, the mortality of the Northampton tions on Reversionary Payments, &c.

Table is greater than the fact, and the edited by W. Morgan, London, 1812. average accumulations would be greater, (Seventh Edition.) 2. Baily, on Life from young ages considerably greater, Annuities and Assurances, London, 1810. than those shown in the preceding table. 3. Milne, On the Valuation of Annuities

We have seen that the security of the and Assurances, &c., London, 1815. 4. method for estimating the value of life Morgan, on the Principles of Assurance, annuities depends upon the presump- Annuities, fc., London, 1821. 5. Davies' tion that the average mortality of the Tables of Life. Contingencies, London, buyers is known. This average cannot 1825. 6. Finlaison, On the Evidence and be expected to hold good, unless a large Elementary Facts on which Tables of Life number of lives be taken. Therefore, the Annuities are Founded. Printed by the





80 85

25 30 35 40

. 44.9 45. 36.6

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5 10. 137.9 15 . 114.1



20 •

House of Commons, 31st March, 1829. , immediately following the time of the 7. Gompertz, Estimation of the value of death of the proprietor of heritable proLife Contingencies, in Philosophical perty, allowed to the heir that he may Transactions, 1820.

make up his mind whether he will acANNUITY, SCOTCH. The 53 Geo. cept the succession with the burden of III. c. 131, does not extend to Scotland. his predecessor's debts. Within that In that part of the country a fixed sum time he cannot be compelled to adopt an per annum paid periodically, though se- alternative unless he has expressly or cured on heritable property, is called an virtually resigned the privilege. The annuity. Such an annuity is generally practice is adopted from the title of the secured for life, and it may either be Pandects, * De jure deliberandi,' xxviii. created by reservation in a transfer of the tit. 8. The term of a year was fixed by absolute property of the lands, thus con- a constitution of Justinian, Cod. vi. tit. stituting a burden on the new proprietor's 30, $ 19. title, or it may be granted by the abso- ANTI-LEAGUE. (LEAGUE.] lute proprietor, the annuitant making his APANAGE ( Apanagium, Apanamentitle real, as in the case of an absolute tum), the provision of lands or feudal estate in land, by an “infeftment.” Pro- superiorities assigned by the kings of visions to widows and children may be France for the maintenance of their thus secured. This species of security on

younger sons. land is to be distinguished from an an- Some of the proposed etymologies of nual-rent right, which has a reference to the word apanage are mentioned by Richa capital sum, and was generally the form elet, Dictionnaire de la Langue Françoise. in which the payment of the interest of The prince to whom the portion was money lent on heritable security was assigned was called apanagiste, or apanmade a real burden on the lands before ager; and he was regarded by the ancient the more effective security was devised of law of that country as the proprietor of making a redeemable disposition of the all the seigniories dependent on the apanlands themselves to the creditor. The age, to whom the fealty (foi) of all subannual-rent right had its origin in the ordinate feudatories within the domain laws against usury. The taking of in- was due, as to the lord of the “dominant terest on a sum borrowed was illegal, but fief.” an irredeemable annuity was not affected Under the first two races of French by the law; and thus the lender was kings, the children of the deceased king invested with a perpetual estate in the usually made partition of the kingdom land. The form used for this purpose among them; but the inconvenience of was afterwards, as above stated, brought such a practice occasioned a different in to aid of the heritable bond, but it is arrangement to be adopted under the now seldom employed. When the obligor | dynasty of the Capets, and the crown of an annuity became bankrupt, there descended entire to the eldest son, with was until lately no statutory provision no other dismemberment than the severin Scotland for ranking the annuity ance of certain portions of the dominions creditor, i.e. for enabling him to prove. for the maintenance of the younger The Court of Session was in use to inter- branches of the family. Towards the pose equitably to allow the annuitant to close of the thirteenth century the rights draw a dividend on the value of the an- of the apanagiste were still further cirnuity. By 2 & 3 Vict. c. 41, SS 40 and cumscribed; and at length it became an 41, provisions similar to those of the 6 established rule, which greatly tended to Geo. IV. c. 16, SS 54 and 55, relative to consolidate the royal authority in that the claims of annuitants against the bank- kingdom, that, upon the failure of lineal rupt estate of the principal debtor, and heirs male, the apanage should revert to against sureties, were applied to Scot- the crown. land.

The time at which this species of proANNUS DELIBERANDI, in the vision was first introduced into France, law of Scotland, is the term of a year the source from which it was borrowed,

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