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) per sons 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 15074X1=3.3251., or 3l. 68. 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.) 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 Tabie? Accord In Table No. XI., under 4 per cent., and opposite 45, is 14.104, the value of an annuity of 11., which being multiplied by 50, gives 7052, or 7051. 4s., the value required. ing to the Northampton Table, the annuity would only have been worth 6147. 3s. The value of an annuity on 2 lives of the same age, or on 2 lives differing by may be found in precisely the same way. 5 years, 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 1007. 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 4 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. 8s., 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 Car lisle 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 X 50 125-650, or 125/. 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 21, 3, 31, 4, 4, 5, and 6 per Cent., at the End of every Year, from 1 to 70. Years. 1234567890 2 per Cent 3 per Cent. 3 per Cent. 4 per Cent. 4 per Cent. 5 per Cent. 6 per Cent. 1.06000,000 1.09202,500 1:10250,000 1-12360,000 1-14116,612 1·15762,500 1-19101,600 1.19251,860 1-21550,625 1-26247,696 1.24618,194 1-27528,156 1-33822,558 1-30226,012 1.34009,564 1-41851,911 1·02500,000 1:03000,000 1:03500,000 1 04000,000 1.04500,000 1·05000,000 1-40710,042 1-50363,026 1-47745,544 1-59384,807 1.55132,822 1-68947,896 1-85194,492 1-77219,610 11 1.31208,666 1:38423,387 1.45996,972 1.53945,406 1.62285,305 14 1-41297,382 1.51258,972 1-44829,817 1-55796,742 16 1-48450,562 1-60470,644 1.73398,604 1-87298,125 2-02237,015 17 1.52161,826 1-65284,763 1-79467,555 1.94790,050 2-11337,681 1-70243,306 1-85748,920 2-02581,652 2-20847,877 240661,923 2-85433,915 1.75350,605 192250,132 2-10684,918 2:30786,031 2.52695,020 3-02559,950 1.80611,123 198978,886 2-19112,314 2-41171,402 2-65329,771 3-20713,547 1.86029,457 2:05913,147 2-27876,807 2.52024,116 2-78596,259 3.39956,360 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 2:46471,555 2-75216,635 3:07152,376 3-81974,966 24 1.80872,595 2-03279,411 2-28332,849 2-56330,417 2-87601,383 3-22509,994 4:04893,464 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 3-28200,956 3.73345,632 4-82234,594 28 1-99649,502 2-28792,768 2-62017,696 2-99870,332 3-42969,999 3.92012,914 5:11168,670 29 2:04640,739 2.35656,551 2.71187,798 3-11865,145 3:58403,649 4:11613,560 5.41838,790 30 2-09756,758 2-42726,247 2-80679,370 3-24339,751 3.74531,813 4-32194,238 574349,117 4.53803,919 6-03810,064 2.15000,677 2.50000,035 2.90503,148 3-37313,341 3.91385,745 32 2-20375,694 2:57508,276 3-00670,759 3.50805,875 4:08998,104 4-76494,147 6-45338,668 2-25885,086 2.65233,524 3.11194,235 3.64838,110 4.27403,018 2:31532,213 2-73190,530 3.22086,033 3.79431,634 4-46636,154 2-37320,519 2.81386,245 3.33359,045 3.94608,899 4-66734,781 36 243253,532 2-89827,833 3:45026,611 4.10393,255 4.87737,846 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 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 2-75219,043 3.35989,893 4.09783,381 4-99306,145 6.07810,094 7.39198,815 10-90286,101 2-82099,520 346069,589 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 44 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 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 3-27148,956 4.13225,188 5.21358,898 6:57052,824 8-27145,557 10-40126,695 16-39387,173 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 3.61111,235 4.65088,590 5.98271,327 7-68658,871 9.86386,463 12-64280,826 20-69688,534 3-70139,016 4.79041,247 6.19210,824 7.99405,226 10:30773,853 13-27494,868 21.93869,846 8-31381,435 10-77158,677 13-93869,611 23.25502,037 8-64636,692 11-25630,817 14-63563,092 | 24-65032,159 54 3-79392,491 4.93412,485 6.40883,202 3-88877,303 5.08214,859 6-63314,114 56 3.98599,236 5.23461,305 6-86530,108 8-99222,160 11-76284,204 15-36741,246 26 12934,089 4.08564,217 5.39165,144 7-10558,662 9-35191,046 12-29216,993 16-13578,308 27-69710,134 60 4-39978,975 5 89160,310 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 7.35428,215 9-72598,688 12-84531,758 16-94257,224 29-35892,742 7-61168,203 10-11502,636 13-42335,687 17-78970,085 31-12046,307 7-87809,090 10.51962,741 14-02740,793 18-67918,589 32-98769,085 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 66 5-10240,721 7-03488,222 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 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 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. *76214,478 72212,126 *68494,571 *64958,093 -61619,874 ⚫58467,929 52678.753 74355,589 *70137,988 66178,330 *62159,705 .58966,386 ⚫55683,742 *49696,926 13 72542,038 *68095,134 63910,415 60057,409 *56427,164 53032,135 46883,902 70772,720 *66111,781 35448,483 *28895,922 •23577,910 •19257,493 *15744,026 .31583,886 28054,294 •22780,590 •18516,820 •15066,054 ⚫33740,376 27237,178 22010,231 •17801,635 •14417,276 -32917,440 26443,862 21265,924 17119,841 *13796,437 -32114,576 25673,652 •20546,787 16461,396 13202,332 31331,294 *21925,877 •19851,968 30567,116 *24199,880 19180,645 *29821,576 23495,029 *18532,024 50 -29091,221 *22810,708 ⚫17900,337 III. Table showing the AMOUNT OF AN ANNUITY of £1 per Annum, improved at Compound Interest, at 21, 3, 31, 4, 41, 5, and 6 per Cent. at the End of each Year, from 1 to 70. Years. 1234567890 2 per Cent. 3 per Cent. 3 per Cent. 4 per Cent. 44 per Cent. 5 per Cent. 6 per Cent 1:00000,000 2:06000,000 3:18350,000 4:37461,600 3-13702,500 3-15250,000 1-00900,000 1·00000,000 1.00000,000 1-00000,000 1.00000,000 1-00000,000 9-38001,362 11 12-48346,631 12-80779,569 13-14199,192 13-48635,141 13-84117,879 14-20678,716 14:97164,264 12 13-79555,297 14-19202,956 14-60196,164 15-02580,546 15-46403,184 15-91712,652 16-86994,120 13 15-14044,179 15-61779,045 16:11303,030 16-62683,768 17-15991,327 17-71298,285 18-68213,767 14 16-51895,284 17.08632,416 17 67698,636 18-29191,119 18-93210,937 19-59863,199 21-01506,593 15 17-93192,666 18:59891,389 19-29568,088 20-02358,764 20-78405,429 21-57856,359 23-27596,988 16 19-38022,483 20-15688,130 20-97102,971 21-82153,114 22-71933,673 23-65749,177 25-67252,808 20-86473,045 21-76158,774 22-70501,575 23-69751,239 24-74170,689 25 84036,636 28-21287,976 22-38634,871 23 41443,577 24-49969,130 25-64541,288 26-85508,370 28 13238,467 30-90565,255 19 23-94600,743 25-11686,844 26-35718,050 27-67122,940 29-06356,246 30-53900,391 33-75999.170 20 25-54465,761 26-87037,449 28-27968,181 29-77807,858 31-37142,277 33-06595,410 36-78559,120 27-18327,405 28-67648,572 30-26917,068 31-96920,172 33-78313,680 35-71925,181 39-99272,668 28-86285,590 30-53678,030 32-32890,215 34-24796,979 36-30337,795 38-50521,440 43 39229,028 30 58442,730 32-45288,370 34-46041,373 36-61788,858 38-93702,996 41-43047,512 46-99582,769 32-34903,798 34-42647,022 36.66652,821 39-08260,413 41-68919,631 44-50199,887 50-81557,735 34 15776,393 36-45926,432 38-94985,669 41-64590,830 44-56521,014 47-72709,882 54:86451.200 36 01170,803 38-55304,225 41-31310,168 44-31174,463 47-57064,460 51-11345,376 59-15638,272 37-91200,073 40 70963,352 43-75906,024 47-08421,441 50-71132,361 54 66912,645 63-70576,568 39-85980,075 42-93092,252 46-29062,734 49-96758,299 53-99333,317 58-40258,277 68 52811,162 41-85629,577 45 21885,020 48-91079,930 52-96628,631 57-42303,316 62-32271,191 73-63979,832 43 90270,316 47-57541,571 51-62267,728 56 08493,776, 61-00706,966 66-43884,750 79-05818,622 46-00027,074 50-00267,818 54-42947,098 59-32833,527 64-75238,779 70-76078,988 84-80167,739 48 15027,751 52-50275,852 57-33450,247 62-70146,868 68-66624,524 75-29882,936 90-88977,803 50-35403,445 55 07784,128 60-34121,005 66-20952,743 72-75622,628 80-06377,033 97-34316.471 52-61288,531 57-73017,652 63 45315,240 69-85790,853 77-03025,646 85-06695,937 104-18375,460 54.92820,744 60-46208,181 66-67401,274 73-65222,487 81-49661,800 90-32030,734 111-43477,987 57 30141,263 63-27594,427 70-00760,318 77-59831,387 86-16396,581 95-83632,271 119-12086,666 59-73394,791 66-17422,259 73-45786,930 81-70224,642 91-04134,427 101-62813,884 127 26811,866 62-22729,664 69-15944,927 77-02889,472 85-97033,628 96-13820,476 107-70954,579 135 90420,578 39 64-75297,906 72-23123,275 80-72490,604 90-49914,973 101-46442,398 114-09502,308 145 05845,813 40 67-40255,354 75-40125,973 84-55027,775 95-02551,572 107 03032,306 100-79977,423 154-76196,562 24 32 34 41 70-08761,737 78-66329,573 88-50953,747 99-82653,635 112-84668,759 127-83976,294 165 04768,356 72-83980,781 82-02319,645 92-60737,128 104-81959,780 118-92478,854 135-23175,109 175 95054,457 75-66080,300 85-48389,234 95-84862,928 110 01238,171 125-27640,402 142-99333,864 187-50757,724 44 78.55232,308 89-04840,911 101-23833,130 115-41287,698 131-91384,220 151-14300,558 199-75803,188 81-51613,116 92-71986,139 105-78167,290 121-02939,206 138-84996,510 159-70015,586 212-74351,379 84-55403,443 96-50145,723 110-48403,145 126-87056,774 146-09821,353 168-68516,365 226-50812,462 87-66788,529 100 39650,095 115-35097,255 132 94539,045 153-67263,314 178 11942,183 241-09861,209 90-85958,243 104-40839,598 120-38825,659 139-26320,607 161 58790,163 188-02539,292 256-56452,882 94-13107,199 108 54064,785 125-60184,557 145-83373,431 169-85935,720 198-42666,257 272-95840,055 50 97-48434,879 112-79686,729 130-99791,016 152-66708,368 178 50302,828 209-31799,570 290 33590,458 51 100-92145,751 117-18077,331 136-58283,702 159-77376,703 187-53566,455 220-81539,548 308-75605,886 104-44449,395 121-69619,651 142-36323,631 167-16471,771 196-97476,946 232-85616,52€ 328-28142,239 53 108-05560,629 126-34708,240 148-34594,958 174-85130,642 206 83863,408 215-49897,352 348 97830,773 54 111-75699,645 131-13749,488 154-53805,782 182-84535,868 217-14637,261 258-77392,220 370 91700,620 115-55092,136 136 07161,972 160-94688,984 191-15917,362 227-91795,938 272-71261,831 394-17202,657 56 119-43969,440 141-15376,831 167.58003,099 199-80553,994 239-17426,755 287-34824,922 418-82234,816 123-42568,676 146-38838,136 174-44533,207 208-79776,154 250-93710,959 302-71566,168 444-95168,905 127-51132,893 151-78003,280 181-55091,869 218-14967,200 263 22927,953 318 85144,477 472-64879,039 131 69911,215 157 33343,379 188-90520,085 227-87565,888 276-07459,710 335-79401,700 502-00771,782 135-99158,995 163-05343,680 196 51688,288 237-99068,524 289-49795,397 353-58371,785 533-12818,089 140-39137,970 168-94503,991 204-39497,378 248-51031,265 303-52536,190 372-26290,375 566-11587,174 144 90116,419 175 01339,110 212-54879,786 259-45072,516 318 18400,319 391-87604,893 601-08282,404 149-52369,330 181-26379,284 220-98800,579 270-82875,416 333-50228,333 412-46985,138 638-14779,349 15-1-26178,563 187-70170,662 229-72258,599 282-66190,433 349-50988,608 434-09331,395 677 43666,110 65 159-11833,027 194-33275,782 238-76287,650 294-96838,050 366-23783,096 456-79801,115 719-08286,076 164-09628,853 201·16274,055 248 11957,718 307-76711,572 383-71853,335 480-63791,170 763-22783,241 169-19869,574 208-19762,277 257-80376,238 321-07780,035 401 98586,735 505-66980,729 810-02150,235 174-42866,313 215-44355,145 267-82689,406 334-92091,236 421 07523,138 531 95329,765 859-62279,249 179-78937,971 222-90685,800 278 20083,535 349-31774,886 441 02361,679, 559-55096,254 912-20016,004 70 185-28411,421 230-59406,374 288-93786,459 364-29045,881 461-86967,955 588-52851,066 967-93216,964 1 69 |