In what time will 5237. amount to 1,0877. 5s. 7d. at 5 per cent. compound interest? Divide 1087 2794, &c., by 523, and the quotient will be 2.0789, &c., which under 5 per cent. in Table I. is opposite to 15 years, the time required. If it had been required to find the time in which a given annuity, improved at a certain rate of compound interest, would have increased to some given sum, the question would have been answered by dividing, as above, the given sum by the annuity; and looking for the quotient (not in Table No. I., but) in Table No. III., under the given rate per cent., it would be found on a line with the time required. Thus, A. owes 1,00071. and resolves to appropriate 101. a year of his income to its discharge: in what time will the debt be extinguished, reckoning compound interest at 4 per cent.? 1,000 divided by 10 gives 100, the number in Table No. III. under 4 per cent., and nearest to this quotient is 99-8265, &c. opposite to 41 years, the required time. Had the rate of interest been 5 per cent., the debt would have been discharged in somewhat less than 37 years. This example is given by Dr. Price (Annuities, 6th ed. vol. ii. p. 289.); and on this principle the whole fabric of the sinking fund was constructed. Of the abstract truth of the principle there cannot, indeed, be a doubt. But every thing depends on the increasing sums annually produced being immediately invested on the same terms; and this, when the sum is large, and the period long, is altogether impracticable. Let it next be required to find an annuity which, being increased at a given rate of compound interest during a given time, will amount to a specified sum: in this case we divide the specified sum by the amount of 11. for the time and rate given, as found in Table III., and the quotient is the answer. Thus, What annuity will amount to 1,0871. 5s. 7d. in 15 years at 5 per cent. compound interest? Opposite to 15 years in Table III., and under 5 per cent., is 21-5785, &c., the amount of 11. for the given time and rate; and dividing 1087-2794, &c., by this sum, the quotient 50 387, &c., or 50l. 7s. 9d., is the annuity required. Deferred Annuities are those which do not commence till after a certain number of years; and reversionary annuities, such as depend upon the concurrence of some uncertain event, as the death of an individual, &c. The present value of a deferred annuity is found by deducting, from the value of an annuity for the whole period, the value of an annuity to the term at which the reversionary annuity is to commence. -Thus, What is the present value of an annuity of 50l. to continue for 25 years, commencing at 7 years from the present time, interest at 4 per cent. ? According to Table No. IV., the value of an annuity of 11. for 25 years at 4 per cent. is 15-62207,995, and that of 11. for 7 years is 6.00205,467, which being deducted from the other, leaves 9.62002,528, which multiplied by 50 gives 4811., the answer required. Supposing the annuity, instead of being for 25 years, had been a perpetuity, it would have been worth 1,2501., from which deducting 3001. 2s., the value of an annuity for 7 years at 4 per cent., there remains 9497. 18s., the value of the reversion. For a selection of problems that may be solved by Tables of annuities certain, see Smart's Tables, pp. 92--100, 2. Life Annuities.-After what has been stated in the article on INSURANCE (GENERAL PRINCIPLES OF), respecting Tables of mortality, it will be easy to see how the value of a life annuity is calculated. Supposing, to revert to the example given before (p. 73.),— that it were required to find the present value of 17., the receipt of which is dependent on the contingency of a person, now 56 years of age, being alive 10 years hence, taking the Carlisle Table of mortality, and interest at 4 per cent.: Now, according to that Table, of 10,000 persons born together, 4,000 attain to 56, and 2,894 to 66 years of age. The probability that a person, now 56 years, will be alive 10 years hence, is consequently, 2,894 ; and the present value of 17., to be received certain 10 years hence being 0.675564/., it follows, that if its receipt be made to depend on a life 56 years of age, attaining to 66, its value will be reduced by that contingency to = 0·488771., or 9s. 94d. If, then, 4.000 2,894 X 0-6755641. we had to find the present value of an annuity of 17. secured on the life of a person now 56, we should calculate in this way the present value of each of the 48 payments, which, according to the Carlisle Table, he might receive, and their sum would, of course, be the present value of the annuity. This statement is enough to show the principle on which all calculations of annuities depend; and this also was, in fact, the method according to which they were calculated, till Mr. Simpson and M. Euler invented a shorter and easier process, deriving from the value of an annuity at any age, that of an annuity at the next younger age. There is a considerable discrepancy in the sums at which different authors, and different insurance offices, estimate the present value of life annuities payable to persons of the same age. This does not arise from any difference in the mode of calculating the annuities, but from differences in the Tables of mortality employed. These can only be accurate when they are deduced from multiplied and careful observations made, during a long series of years, on a large body of persons; or when the average numbers of the whole population, and of the deaths at every age, for a lengthened period, have been determined with the necessary care. It is to be regretted, that governments, who alone have the means of ascertaining the rate of mortality by observations made on a sufficiently large scale, have been singularly inattentive to their duty in this respect. And until a very few years since, when Mr. Finlaison was employed to calculate Tables of the value of annuities from the ages of the nominees in public tontines, and of individuals on whose lives government had granted annuities, all that had been done in this country to lay a solid foundation on which to construct the vast fabric of life insurance had been the work of a few private persons, who had, of course, but a limited number of observations to work upon. The celebrated mathematician, Dr. Halley, was the first who calculated a Table of mor tality, which he deduced from observations made at Breslaw, in Silesia. In 1724, M. De Moivre published the first edition of his tract on Annuities on Lives. In order to facilitate the calculation of their values, M. De Moivre assumed the annual decrements of life to be equal; that is, he supposed that out of 86 (the utmost limit of life on his hypothesis) persons born together, one would die every year till the whole were extinct. This assumption agreed pretty well with the true values between 30 and 70 years of age, as given in Dr. Halley's Table; but was very remote from the truth in the earlier and later periods. Mr. Thomas Simpson, in his work on Annuities and Reversions, originally published in 1742, gave a Table of mortality deduced from the London bills, and Tables founded upon it of the values of annuities. But at the period when this Table was calculated, the mortality in London was so much higher than in the rest of the country, that the values of the annuities given in it were far too small for general use. In 1746, M. Deparcieux published, in his Essai sur les Probabilités de la Durée de la Vie Humaine-a work distinguished by its perspicuity and neatness,-Tables of mortality deduced from observations made on the mortuary registers of several religious houses, and on the list of the nominees in several tontines. In this work, separate Tables were first constructed for males and females, and the greater longevity of the latter rendered apparent. M. Deparcieux's Tables were a very great acquisition to the science; and are decidedly superior to some that are still extensively used. Dr. Price's famous work on Annuities, the first edition of which was published in 1770, contributed powerfully to direct the public attention to inquiries of this sort; and was, in this respect, of very great utility. Of the more recent works, the best are those of Mr. Baily and Mr. Milne, which indeed, are both excellent. The latter, besides all that was previously known as to the history, theory, or practice of the science, contains much new and valuable matter; and to it we beg to refer such of our readers as wish to enter fully into the subject. The Table on which Dr. Price laid the greatest stress, was calculated from the burial registers kept in the parish of All Saints in Northampton, containing little more than half the population of the town. There can be no doubt, however, as well from original defects in the construction of the Table, as from the improvement that has since taken place in the healthiness of the public, that the mortality represented in the Northampton Table is, and has long been decidedly above the average rate of mortality in England. Mr. Morgan, indeed, the late learned actuary of the Equitable Society, contended that this is not the case, and that the Society's experience shows that the Northampton Table is still remarkably accurate. But the facts Mr. Morgan disclosed in his View of the Rise and Progress of the Equitable Society (p. 42.), published in 1828, are quite at variance with this opinion: for he there states, that the deaths of persons insured in the Equitable Society, from 50 to 60 years of age, during the 12 years previously to 1828, were 339; whereas, according to the Northampton Table, they should have been 545! And Mr. Milne has endeavoured to show (Art. Annuities, new ed. of Ency. Brit.) that the discrepancy is really much greater. The only other Table used to any extent in England for the calculation of life annuities, is that framed by Mr. Milne from observations made by Dr. Heysham on the rate of mortality at Carlisle. It gives a decidedly lower rate of mortality than the Northampton Table; and there are good grounds for thinking that the mortality which it represents is not very different from the actual rate throughout most parts of England; though it cannot be supposed that a Table founded on so narrow a basis should give a perfectly fair view of the average mortality of the entire kingdom. In life insurance, the first annual premium is always paid at the commencement of the assurance, and the others at the termination of each year so long as the party assured survives. Hence, at the beginning of the assurance, the whole of the annual premiums payable for it exceed the value of an equal annuity on the life by one year's purchase. And, therefore, when the value of an assurance in present money is given, to find the equivalent annual premium during the life, the whole present value must be divided by the number of years' purchase an annuity on the life is worth, increased by 1. Thus, for an assurance of 100% on a life 40 years of age, an office, calculating by the Carlisle Table of mortality, and at 4 per cent. interest, requires 53-4467. in present money. Now according to that Table and rate of interest, an annuity on a life just 40 years of age is worth 15.074 years' purchase, so that the equivalent annual premium is 4X1=3.3251., or 31. 6s. 8d. The annual premium may, however, be derived directly from the value of an annuity on the life, without first calculating the total present value of the assurance.—( -(See Mr. Milne's Treatise on Annuities, or the art. Annuities in the new edition of the Ency. Britannica.) 53.4461. In order to exhibit the foundations on which Tables of life annuities and insurance have been founded in this and other countries, we have given, in No. V. of the following Tables, the rate of mortality that has been observed to take place among 1,000 children born together, or the numbers alive at the end of each year, till the whole become extinct, in England, France, Sweden, &c., according to the most celebrated authorities.* The rate of mortality * The greater part of this Table was originally published by Dr. Hutton in his Mathematical Dictionary, art. Life Annuities. Mr. Baily inserted it with additions in his work on Annuities; and it at Carlisle, represented in this Table, is less than that observed any where else: the rates which approach nearest to it are those deduced from the observations already referred to, of M. Deparcieux, and those of M. Kersseboom, on the nominees of life annuities in Holland. In order to calculate from this Table the chance which a person of any given age has of attaining to any higher age, we have only to divide the number of persons alive at such higher age, given in that column of the Table selected to decide the question, by the number of persons alive at the given age, and the fraction resulting is the chance. We have added, by way of supplement to this Table, Mr. Finlaison's Table (No. VI.) of the rate of mortality among 1,000 children born together, according to the decrement of life observed to take place among the nominees in government tontines and life annuities in this country, distinguishing males from females. The rate of mortality which this Table exhibits is decidedly less than that given in the Carlisle Table; but the lives in the latter are the average of the population, while those in the former are all picked. The nominees in tontines are uniformly chosen among the healthiest individuals; and none but those who consider their lives as good ever buy an annuity. Still, however, the Table is very curious; and it sets the superiority of female life in a very striking point of view. Tables VII. and VIII. give the expectation of life, according to the mortality observed at Northampton and Carlisle; the former by Dr. Price, and the latter by Mr. Milne. The next Table, No. IX., extracted from the Second Report of the Committee of the House of Commons on Friendly Societies, gives a comparative view of the results of some of the most celebrated Tables of mortality, in relation to the rate of mortality, the expectation of life, the value of an annuity, &c. The coincidence between the results deduced from M. Deparcieux's Table, and that for Carlisle, is very striking. And to render the information on these subjects laid before the reader as complete as the nature of this work will admit, we have given Tables (Nos. X.-XV.) of the value of an annuity of 17. on a single life, at every age, and at 3, 4, 5, 6, 7, and 8 per cent., according to the Northampton and Carlisle Tables; we have also given Tables of the value of an annuity of 17. on 2 equal lives, and on 2 lives differing by 5 years, at 3, 4, 5, and 6 per cent., according to the same Tables. It is but seldom, therefore, that our readers will require to resort to any other work for the means of solving the questions that usually occur in practice with regard to annuities; and there are not many works in which they will find so good a collection of Tables.-We subjoin one or two examples of the mode of using the Tables of life annuities. Suppose it were required, what ought a person, aged 45, to give, to secure an annuity of 50%. a year for life, interest at 4 per cent., according to the Carlisle Table? In Table No. XI., under 4 per cent., and opposite 45, is 14.104, the value of an annuity of 17., which being multiplied by 50, gives 705·2, or 705/. 4s., the value required. According to the Northampton Table, the annuity would only have been worth 6147. 38. The value of an annuity on 2 lives of the same age, or on 2 lives differing by 5 years, may be found in precisely the same way. Some questions in reversionary life annuities admit of an equally easy solution. Thus, suppose it is required to find the present value of A.'s interest in an estate worth 100%. a year, falling to him at the death of B., aged 40, interest 4 per cent., according to the Carlisle Table? The value of the perpetuity of 100%. a year, interest per cent., is 2,500l.; and the value of an annuity of 100%. on a person aged 40, interest at 4 per cent., is 1,5071. 88., which deducted from 2,500l. leaves 9921. 12s., the present value required. A person, aged 30, wishes to purchase an annuity of 50%. for his wife, aged 25, provided she survive him; what ought he to pay for it, interest at 4 per cent., according to the Carlisle Table? The value of an annuity of 17. on a life aged 30 is 16-852; from which subtracting the value of an annuity of 17. on 2 joint lives of 25 and 30, 14-339, the difference, 2.513 × 50 125.650, or 1257. 13s., the sum required. = For the solution of the more complex cases of survivorship, which do not often occur in practice, recourse may be had to the directions in Mr. Milne's Treatise on Annuities, and other works of that description. To attempt explaining them here would lead us into details quite inconsistent with the objects of this work. was published with the column for Carlisle added, in the Report of the Committee of the House of Commons on Friendly Societies. K 2 TABLES OF INTEREST AND ANNUITIES. Table showing the AMOUNT of £1 improved at Compound Interest, at 2, 3, 31, 4, 44, 5, and 6 per Cent., at the End of every Year, from 1 to 70. Years. 24 per Cent. 3 per Cent. 3 per Cent. 4 per Cent. 4 per Cent. 5 per Cent. 6 per Cent. 4 5 6 7 8 9 10 1 1·02500,000 1.03000,000 1.03500,000 104000,000 1.04500,000 105000,000 2 1-05062,500 106090,000 1.07122,500 1-08160,000 -09202,500 1.10250,000 1-12360,000 3 107689,062 1.09272,700 1-10871,787 1-12486,400 1.14116,612 1-15762,500 1-19101,600 F-10381,289 1-12550,881 1.14752,300 1.16985,856 1-19251,860 1.21550,625 1-26247,696 1-13140,821 1-15927,407 1-15969,342 1.19405,230 1.18868,575 1-22987,387 1-21840,290 1-26677,008 1.24886,297 1.30477,318 1.36289,735 1-42331,181 1-48609,514 1-28008,454 1.34391,638 141059,876 1-48024,428 1.55296,942 1.06000,000 1.18768,631 1.21665,290 1-24618,194 1.68947,896 1-79084,770 11 1.31208,666 1.38423,387 1-45996,972 1-53945,406 1-62285,305 12 1-34488,882 1.42576,089 I:51106,866 1-60103,222 1-69588,143 13 1.27528,156 1-33822,558 171033,936 1-89829,856 1-79585,633 201219,647 1-37851,104 1.46853,371 1.56395,606 1.66507,351 1.77219,610 1-88564,914 2-13292,826 14 1-41297,382 1.51258,972 1-61869,452 1.73167,645 1-85194,492 1.97993,160 2.26090,396 1-44529,817 1-55796,742 1-67534,883 1-80094,351 1.93528,244 2-07892,818 2-39555,819 1-48450,562 1.60470,644 1.73398,604 1-87298,125 2-02237,015 2-18287,459 2-54035,168 1.52161,826 1.65284,763 1-79467,555 194790,050 2-11337,681 2-20201,832 2-69277,279 1-55965,872 1-70243,306 1.85748,920 2-02581,652 220847,877 2-40861,923 2-85433,915 1.59865,019 1.75350,605 1.92250,132 2-10684,918 2-30786,031 2-52695,020 3-02559,950 1-63861,644 1-80611,123 1.98978,886 2.19112,314 2-41171,402 2-65329,771 3-20713,547 15 16 17 18 19 20 23 24 21 1-67958,185 1-86029,457 2.05943,147 2.27876,807 2-52024,116 2-78596,259 3-39956,360 22 1-72157,140 1-91610,341 2-13151,158 2-36991,879 2-63365,201 2-92526,072 3-60353,742 1-76461,068 1-97358,651 2-20611,448 246471,555 2-75216,635 3-07152,376 3.81974,966 1-80872,595 2-03279,411 2 28332,849 2-56330,417 2-87601,383 3-22599,994 404893,464 25 1-85394,410 2-09377,793 2-36324,498 2-66583,633 3.00543,446 3.38635,494 4-29187,072 26 1.90029,270 2.15659,127 2-44595,856 2-77246,979 3-14067,901 3-55567,269 4-54938,296 1.94780,002 2.22128,901 2.53166,711 2-88336,858 3-23200,956 3.73345,632 4-82234,594 1-99649,502 2-28792,768 2.62017,696 2.99870,332 3-42969,999 3.92012,914 5-11168,670 2.04640,739 2-35656,551 2-71187,798 3.11865,145 3.58403,649 4-11613,560 5-41838,790 2-09756,758 2-42726,247 2-80679,370 3-24339,751 3.74531,813 4-32194,238 5-74349,117 27 28 29 30 31 2-15000,677 2:50000,035 2-90503,148 3.37313,341 391385,745 4-53803,949 6-08810,064 32 2-20375,694 2-57508,276 3-00670,759 3.50805,875 4-08998,104 4.76494,147 6-45338,668 33 2-25885,086 2.65233,524 3.11194,235 3.64838,110 4-27403,018 5:00318,854 6.84058,988 34 2.31532,213 2-73190,530 3.22086,033 3-79431,634 4:46636,154 5-25334,797 7-25102,528 35 2.37320,519 2.81386,245 3.33359,045 3.94608,899 4-66734,781 5-51601,537 7-68608,679 36 2-43253,532 2.89827,833 3-45026,611 4.10393,255 4-87737,846 5-79181,614 8-14725,200 37 2-49334,870 2.98522,668 3-57102,543 4.26808,986 5-09686,049 6-08140,694 8.63608,712 38 2-55568,242 3.07478,348 3.69601,132 4-43881,345 5-32621,921 6-38547,729 9.15425,235 39 2.61957,448 3-16702,698 3-82537,171 4-61636,599 5-56589,908 6-70475,115 9.70350,749 2-68506,384 3.26203,779 3·95925,972 4.80102,063 5.81636,454 7-03998,871 10-28571,794 40 41 42 43 44 45 52 2-75219,043 3.35989,893 4-09783,381 4-99306,145 6-07810,094 7-39198,815 10.90286,101 2-82099,520 3.46069,589 4.24125,799 5.19278,391 6-35161,548 7-76158,755 11.55703,267 2.89152,008 3:56451,677 4.38970,202 5-40049,527 6-63743,818 8:14966,693 12.25045,463 2.96382,808 3-67145,227 4:54334,160 5.61651,508 6-93612,290 8-15715,028 12-98548,191 3-03790,328 3-78159,584 4-70235,855 5.84117,568 7-24824,843 8-98500,779 13-76461,083 46 3-11385,086 3-89504,372 4-86694,110 6-07482,271 7.57441,961 9-43425,818 14:59048,748 47 3-19169,713 4:01189,503 5.03728,404 6-31781,562 7-91526,849 9-90597,109 15-46591,673 48 3-27148,956 4.13225,188 5-21358,898 6-57052,824 8-27145,557 10-40126,695 16:39387,173 49 3-35327,680 4-25621,944 5-39606,459 6.83334,937 8-64367,107 10-92133,313 17-37750,403 50 3-43710,872 4-38390,602 5.58492,686 7-10668,335 9-03263,627 10-46739,978 18:42015,427 51 3-52303,644 4-51542,320 5-78039,930 7-39095,068 9-43910,490 12-04076,977 19-52536,353 361111,235 4.65088,590 5.98271,327 7-68658,871 9-86386,463 12-64280,826 20-69688,534 53 3-70139,016 4.79041,247 6-19210,824 7-99405,226 10-30773,853 13-27494,868 21.93869,846 54 3-79392,491 4.93412,485 6-40883,202 8.31381,435 10-77158,677 13-93869,611 23-25502,037 55 3-88877,303 5.08214,859 6.63314,114 8-64636,692 11-25630,817 14-63563,092 24-65032,159 56 3.98599,236 5.23461,305 6-86530,108 8-99222,160 11-76284,204 15-36741,246 26-12934,089 57 4-08564,217 5:39165,144 7.10558,662 9-35191,046 12-29216,993 16-13578,308 27-69710,134 4-18778,322 5.55340,098 7.35428,215 9-72598,688 12-84531,758 16-94257,224 29-35892,742 4-29247,780 5.72000,301 7.61168,203 10-11502,636 13-42335,687 17-78970,085 31-12046,307 4-39978,975 5 89160,310 7.87809,090 10-51962,741 14-02740,793 18-67918,589 32-98769,085 61 4-50978,419 6-06835,120 62 4-62252,910 6.25040,173 63 4-73809,233 6-43791,379 64 4-85654,464 6-63105,120 65 4-97795,826 6.82998,273 66 5-10240,721 7.03488,222 58 59 60 8.15382,408 10-94041,251 14-65864,129 19-61314,519 34.96695,230 8-43920,793 11-37802,901 15-31828,014 20-59380,245 37-06496,944 8-73458,020 11-83315,017 16-00760,275 21-62349,257 39-28886,761 9 04029,051 12-30647,617 16-72794,487 22-70466,720 41-64619,967 9-35670,068 12-79873,522 17-48070,239 23-83990,056 44-14497,165 9-68418,520 13-31068,463 18-26733,400 25-03189,559 46-79366,994 5-22996,739 7-24592,868 10.02313,168 13.84311,201 19-08936,403 26-28349,036 49-60129,014 68 5-36071,658 7-46330,654 10-37394,129 14:39683,649 19-94838,541 27-59766,488 52-57736,755 69 5-49473,449 7-68720,574 10-73702,924 14-97270,995 20-84606,276 28-97754,813 55-73200,960 70 5-63210,286 7-91782,191 11-11282,526 15-57161,835 21-78413,558 30-42642,553 59-07593,018 67 II. Table showing the PRESENT VALUE of £1 receivable at the End of any given Year, from 1 to 70, reckoning Compound Interest, at 21, 3, 31, 4, 4, 5, and 6 per Cent. Years. 2 per Cent. 3 per Cent. 3 per Cent. 4 per Cent. 4 per Cent. 5 per Cent. 6 per Cent. 67683,936 *62741,237 *64460,892 •59189,846 67556,417 •64392,768 -74355,589 *70137,988 .66178,330 62159,705 58966,386 ⚫55683,742 *49696,936 <.72542,038 68095,134 69046,556 *64186,195 19 *62552,772 •57028,603 *52015,569 61027,094 *55367,575 *50256,588 ⚫47464,242 26 *52623,472 •46369,473 21981,003 -26784,832 •20736,795 -28384,606 •22146,318 ⚫17299,843 •13530,059 ⚫10594,225 ⚫08305,117 ⚫05121,544 *04831,645 *07532,986 *04558,156 69 •18199,242 •13008,628 ⚫09313,563 *06678,818 ⚫04797,039 03450,948 01794,301 *03286,617 *01692,737 |