TABLE IV.-Amount of an Annuity of 5 p. c. 1.000 2.050 6.802 9.549 Years. 3 pc. 34 p.c. 4 p. c. 1 1.000 1.000 1.000 2 2.030 2.035 2.040 3 3.091 3.106 3.122 4 5.526 8.142 9.214 10.583 11.027 12.578 20.024 21.579 20 25 30 47.575 40 75.401 26.870 28.280 29.778 33.066 47.727 66.439 120.800 152-667 237.991 364.290 209 348 of life annuities depends will be more fully explained in the articles PROBABILITY and MORTALITY. Let us suppose 100 persons, all of the same age, buy a life annuity at the same office. Let us also suppose it has been found out, that of 100 persons at that age, 10 die in the first year, on the average, 10 more in the second year, and so on. If then it can be relied upon that 100 persons will die nearly in the same manner as the average of mankind, or at least that in such a number the longevity of some will be compensated by the unexpected death of others, the fair estimation of the value of a life annuity to be granted to each may be made as follows:-To make the question more distinct, let us suppose the bargain to be made on the 1st of January, 1844, so that payment of the annuities is due to the survivors on new-year's day of each year. Moreover let each year's annuity be made the subject of a separate contract. The first question is, what ought each individual to pay in order that he may receive the annuity of 1l., if he survives in 1845. By the general law of mortality, we suppose that only 90 will remain to claim, who will, therefore, receive 90l. among them, the remaining 10 It is suffihaving died in the interval. cient, therefore, to meet the claims of 1845, that the whole 100 pay among them, January 1, 1844, such a sum as will, when put out at interest (suppose 4 per cent.) amount to 901. on January 1, 1845. This sum is 86.6547., and its hundredth part is 866547., which is, therefore, what each should pay to entitle himself to receive the annuity in 1845. There will be only 80 to claim in 1846, and, therefore, the whole 100 must among them pay as much as will, put out at 4 per cent. for 2 years, amount to 80l. This sum is 73.9681., and its hundreth part is 73968/., which is, therefore, what each must pay, in order to receive the annuity, if he lives, in 1846. The remaining years are treated in the same way, and the sum of the shares of each individual for the different years, is the present value of an annuity for his life. must observe, that in the term value of an annuity it is always implied that the first The principles on which the calculation | annuity becomes payable at the expira In this Table we see what would be possessed by the receiver of an annuity at the end of his term, if he put each year's annuity out at interest so soon as he received it. For example, an annuity of 17., in 40 years, at 5 per cent., amounts to 120-87., which includes 401. received altogether at the end of the different years, and 80-81. the compound interest arising from the first year's annuity, which has been 39 years at interest, the second year's annuity, which has been 38 years at interest, and so on, down to the last year's annuity, which has only just been received. When the annuity is payable half-yearly or quarterly, its present value is somewhat greater than that given in the preceding Table. For the annuitant, receiving certain portions of his annuity sooner than in the case of yearly payments, gains an additional portion of interest. Since 4 per cent. is 2 per cent. half-yearly and 1 per cent. quarterly, and since every term contains twice as many half-years as years, and four times as many quarters, it is evident that an annuity of 100l. a-year, payable nalf-yearly, at 4 per cent., for 10 years, is the same in present value as one of 50l. per annum, payable yearly, at 2 per cent., For 20 years. Again, 100l. a-year, payable quarterly for 10 years, money being at 4 per cent., is equivalent to an annuity of 251., payable yearly for 40 years, money being at 1 per cent. We tion of a year after the payment of the purchase-money. The value of a life annuity depends, therefore, upon the manner in which it is presumed a large number of persons, siinilarly situated with the buyer, would die off successively. Various Tables of these decrements of life, as they are called, have been constructed, from observations made among different classes of lives. Some make the mortality greater than others; and of course, Tables which give a large mortality, give the value o the annuity smaller than those which suppose men to live longer. Those who buy annuities would, therefore, be glad to be rated according to tables of high mortality or low expectation of life; while those who sell them would prefer receiving the price indicated by tables which give a lower rate of mortality. In insurances the reverse is the case: the shorter the time which a man is supposed to live, the more must he pay the office, that the latter may at his death have accumulated wherewithal to pay his executors. We now give in Table V. the values of annuities according to three of the most celebrated Tables. TABLE V.-Present Value, or Purchasemoney, of a Life Annuity. Northampton. Carlisle. Gov.M. Gov.F. Age. 3 p.c. 4 p.c. 5p.c. 3p.c. 4p.c. 5p.c. 4p.c. 4p.c. 0 12.3 10.3 8.9 17.3 14.3 12.1 5 20.5 17.2 14.8 23.7 19.6 16.6 19.3 20.0 10 20-7 17.5 15 1 23.5 19.6 16.7 18.8 19.7 15 19 7 16 8 14.6 22.6 19.0 16 2 18.0 19.1 20 18 6 16 0 14 0 217 18 4 15-8 17.3 18.6 25 17.8 15.4 13.6 20.7 17.6 15 3 16.9 18.1 30 16.9 14.8 13.1 19.6 16.9 14 16.4 17.5 35 15 9 14.0 12.5 18.4 16.0 14.1 15.7 16.9 40 14.8 13.2 11.8 17.1 15.1 13.4 14.9 16.2 45 13.7 12.3 11.1 15.9 14.1 12.6 13.8 15.3 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 60 9.8 9.0 8.4 10.5 9.7 8.9 65 8.3 7.8 7.3 8.9 8.3 7.8 8.2 9.6 70 6.7 6.4 6.0 7.1 6.7 6.3 6.8 7.9 75 5.2 5.0 4.7 5.5 5.2 5.0 5.4 6.3 80. 5.8 3.6 3.5 4.4 4.2 4.0 3.8 4.9 95 2.6 2.5 2.5 3.2 3.1 3.0 2.3 3.8 90 1.8 1.8 1.7 2.5 2.4 2.3 1.3 2.1 95 •2 +2 .2 2.8 2.7 2.6 .6 1.0 high a mortality at all the younger and middle ages of life, and, consequently, too low a value of the annuity. The second is from the Carlisle Table, formed by Mr. Milne, from observations made at Carlisle. It gives much less mortality than most other Tables, and, therefore, gives higher values of the annuities; but it has since been proved to represent the actual state of life among the middle classes, in the century now ending, with much greater accuracy than could have been supposed, considering the local character of the observations from which it was derived. The third table is that constructed by Mr. Finlaison, from the observation of the mortality in the government tontines and among the holders of annuities granted by government in redemption of the national debt, and differs from the former two in distinguishing the lives of males from those of females. Most observations hitherto published unite in confirming the fact, that females, on the average, live longer than males, and in the annuities now granted by government, a distinction is made accordingly. The mean between the values of annuities on male and female lives, according to the Government Tables, agrees pretty nearly with the Carlisle Tables, the rate of interest being the same. For the materials of Table V. we are indebted to the works of Dr. Price, on Reversionary Payments; of Mr. Milne, on Annuities and Insurances; and to Mr. Finlaison's Report to the House of Commons on Life Annuities; to all of which we refer the reader. The tables are of course very much abridged. To use the Table V., suppose the value of an annuity of 100l. a-year, on a life 9.7 113 aged 35, is required, interest being at 4 per cent., which is nearly the actual value of money. We find in the column marked 4 per cent., opposite to 35, under the Northampton Tables 14:0, under the Carlisle 16.0, and under the Government Tables 157 or 16.9, according as the life is male or female. These are the number of pounds which ought to buy an annuity of 1l., according to these several authorities; and taking each of them 100 times, we have: The first of these is calcnlated from the Northampton Table, formed by Dr. Price, from observations of burials, &c., at Northampton. As compared with all other Tables of authority, it gives too Government Table (males) 1570l. Government Table (females) 1690l. We cannot suppose that the annuity could be bought for less than would be required by the Carlisle Tables. To find the value of an annuity on a life whose age lies between two of those given in the table, the process must be followed which has been already explained in treating of annuities certain. An annuity on two joint lives is one which is payable only so long as both the persons on whose lives it is bought are alive to receive it. TABLE VI.-Present Value, or Purchasemoney, of an Annuity of One Pound on two Joint Lives. Carlisle.-4 per cent. Age. 0. 10. 20. 30. 40. 50. 60. 70. 60 6.9 5.3 70 4.4 3.1 1.9 1.5 2.4 1.6 1.3 80 Northampton.-4 per cent. Age. 0. 10. 20. 30. 40. 50. 60. 70. 6.9 4.6 2.4 0.2 aged 25 and 55, in which the difference of age is 30 years. In the Carlisle Table opposite to 25, the younger, and under 30, the difference, we find 10.3; and 8.8 in the Northampton. For the value of an annuity of 100l., the first tables give, therefore, 10301., and the second 8801. The value of an annuity on the longest of two lives, that is, which is to be payable as long as either of the two shall be alive to receive it, is found by adding together the values of the annuity on the two lives separately considered, and subtracting the value of the annuity on the joint lives. For the above species of annuity puts the office and the parties in precisely the same situation as if an annuity were granted to each party separately, but on condition that one of the annuities should be returned to the office so long as both were alive, that is, during their joint lives. For example, let the ages be 25 and 55 as before, and let the Carlisle Table be chosen, interest being at 4 per cent., we have then : TABLE V.-Annuity at age 55 Ditto 11.3 28.9 10.3 TABLE VI.-Joint Annuity, 55 & 25 The value, therefore, of an annuity of 17. per annum on the survivor is 18.6l. The value of an annuity which is not to be payable till either one or other of two persons is dead, and which is to continue during the life of the survivor, is 8.0 5.8 3.4 1.7 found as in the last case, only subtracting twice the value of the joint annuity, instead of that value itself. In the preceding case it is 8.31. For this case only differs from the preceding, in that the annuity is not payable while both_are alive, that is, during the joint lives. Consequently the value in this case is less than that in the last, by the value of an annuity on the joint lives. 50 8.1 7.0 5.3 3.2 1.7 The preceding table gives the results of the Carlisle and Northampton Tables on the value of this species of annuity, interest being at 4 per cent. The first column shows the age of the younger life, and the horizontal headings are not the age of the elder life, but the excess of the age of the elder life above that of the younger. For example, to know the value of an annuity in two joint lives, The value of an annuity to be paid to A from and after the death of B, if the latter should happen to die first, is the value of an annuity on the life of A, diminished by the value of an annuity on the joint lives of A and B. For the situation is exactly the same as if the L That is to say, according to the Northampton Tables, if a person were, at the age of 26 (that is, a year after 25), to begin laying by 100l. a year at interest, he might expect the amount at the end of his life to be 79.27. for each pound laid by yearly; or 7920l. Or, to speak more strictly, if 100 persons were to do this, they might expect that the average amount of their savings, reckoning the accumulations at their deaths, would be 7920/. each. As we have already observed, the mortality of the Northampton Table is greater than the fact, and the average accumulations would be greater, from young ages considerably greater, than those shown in the preceding table. We have seen that the security of the method for estimating the value of life annuities depends upon the presumption that the average mortality of the buyers is known. This average cannot be expected to hold good, unless a large number of lives be taken. Therefore, the | granting of a single annuity, or of a few annuities, as a commercial speculation, would deserve no other name than gambling, even though the price demanded should be as high as that given in any tables whatsoever. In the preceding tables, we would again remark, that our object has been simply to furnish the means of giving a moderately near determination of a few of the most simple cases. We should strongly recommend every one not to venture on important transactions without professional or other advice on which he can depend, unless he himself fully understands the principles on which tables are constructed. The liability to error, even in using the most simple table, is very great, without considerable knowledge of the subject; and most cases which arise in practice contain some circumstances peculiar to themselves, which have not and could not have been provided for in the general rules. The following references to works on this subject may be found useful: ANNUITIES CERTAIN. 1. Smart's Tables There of Interest, &c., London, 1726. is an edition published in 1780, which is said to be very incorrect. The values for the intermediate half-years given in this work are not correctly the values of the annuities on the supposition of halfyearly payments; in other respects it is to be depended upon. 2. Corbaux, Doctrine of Compound Interest, &c., London, 1825. 3. Baily, Doctrine of Interest and Annuities, London, 1808. Smart's Tables are republished in this work from the correct edition. Works on life-annuities generally contain principles and tables for the calculation of annuities certain. LIFE ANNUITIES. 1. Price, Observations on Reversionary Payments, &c. edited by W. Morgan, London, 1812. (Seventh Edition.) 2. Baily, on Life Annuities and Assurances, London, 1810. 3. Milne, On the Valuation of Annuities and Assurances, &c., London, 1815. 4. Morgan, on the Principles of Assurance, Annuities, &c., London, 1821. 5. Davies' Tables of Life Contingencies, London, 1825. 6. Finlaison, On the Evidence and Elementary Facts on which Tables of Life Annuities are Founded. Printed by the 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 accept the succession with the burden of his predecessor's debts. Within that time he cannot be compelled to adopt an alternative unless he has expressly or virtually resigned the privilege. The practice is adopted from the title of the Pandects, 'De jure deliberandi,' xxviii. tit. 8. The term of a year was fixed by a constitution of Justinian, Cod. vi. tit. 30, § 19. ANTI-LEAGUE. [LEAGUE.] APANAGE (Apanagium, Apanamentum), the provision of lands or feudal superiorities assigned by the kings of France for the maintenance of their younger sons.. The prince to whom the portion was assigned was called apanagiste, or apanager; and he was regarded by the ancient law of that country as the proprietor of all the seigniories dependent on the apanage, to whom the fealty (for) of all subordinate feudatories within the domain was due, as to the lord of the "dominant fief." ANNUITY, SCOTCH. The 53 Geo. III. c. 131, does not extend to Scotland. In that part of the country a fixed sum per annum paid periodically, though secured on heritable property, is called an annuity. Such an annuity is generally secured for life, and it may either be created by reservation in a transfer of the absolute property of the lands, thus constituting a burden on the new proprietor's title, or it may be granted by the absolute proprietor, the annuitant making his title real, as in the case of an absolute estate in land, by an "infeftment." Provisions to widows and children may be thus secured. This species of security on 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 formelet, Dictionnaire de la Langue Françoise. in which the payment of the interest of money lent on heritable security was made a real burden on the lands before the more effective security was devised of making a redeemable disposition of the lands themselves to the creditor. The annual-rent right had its origin in the laws against usury. The taking of interest on a sum borrowed was illegal, but an irredeemable annuity was not affected by the law; and thus the lender was invested with a perpetual estate in the land. The form used for this purpose was afterwards, as above stated, brought in to aid of the heritable bond, but it is now seldom employed. When the obligor of an annuity became bankrupt, there was until lately no statutory provision in Scotland for ranking the annuity creditor, i. e. for enabling him to prove. The Court of Session was in use to interpose equitably to allow the annuitant to draw a dividend on the value of the annuity. By 2 & 3 Vict. c. 41, §§ 40 and 41, provisions similar to those of the 6 Geo. IV. c. 16, §§ 54 and 55, relative to the claims of annuitants against the bankrupt estate of the principal debtor, and against sureties, were applied to Scotland. ANNUS DELIBERANDI, in the law of Scotland, is the term of a year Under the first two races of French kings, the children of the deceased king usually made partition of the kingdom among them; but the inconvenience of such a practice occasioned a different arrangement to be adopted under the dynasty of the Capets, and the crown descended entire to the eldest son, with no other dismemberment than the severance of certain portions of the dominions for the maintenance of the younger branches of the family. Towards the close of the thirteenth century the rights of the apanagiste were still further circumscribed; and at length it became an established rule, which greatly tended to consolidate the royal authority in that kingdom, that, upon the failure of lineal heirs male, the apanage should revert to the crown. The time at which this species of provision was first introduced into France, the source from which it was borrowed, |