few particulars which we judged worthy of the author's revision, we proceed to lay before our readers a few interesting and instructive extracts; and the rather, because it is not probable that we shall be soon called upon to resume the subject. Our first quotation may be of use to corporate bodies, who allow of the renewal of leases for lives, and afterwards for a term certain. Many of the estates belonging to the Corporation of Liverpool are held on the tenire alluded to in these examples; and till lately they were in the constant habit of renewing their leases on the following terms : viz. One year's purchase for adding one life dropt, Three years purchase for adding two lives dropt, and Seven years purchase for adding three lives dropt when the 21 years remain unexpired. In all these cases no regard was paid to the age or state of health of the existing lives in the lease. This practice of demanding an uniform sine for renewing with any life, and without regard to the age or state of bealth of the lives remaining in the lease, betrayed a total want of knowledge on the subject; and was in most cases injurious to the interests of the Corporation. • But, the most singular circunstance attending this subject, was their custom of exchanging, for the sum of only one guinea each, lives, not ex. ceeding 50 years of age and in good healthy for lives of any and in estates of any yearly value !!! A practice which could hardly be supposed ever to have existed in so enlightened a place as Liverpool. The Corporation at length, suspecting that their mode of proceeding was incorrect in principle, referred the matter to a Committee, who directed it to be laid before me for my opinion: and agreeably to their request I calculated a set of Tables for their use, founded on the principles detailed in the preceding examples. • As it is probable that mauy other Corporate bodies are still pursuing the same incorrect and absurd practice of leasing their estates, I have been more particular in these examples.' pp. 418-419. It may be proper to add, that the tables here mentioned, and many others, tending to facilitate this branch of computation, may be found in Mr. Baily's 'Tables for the purchasing and renewing of leases.' Our next quotation exhibits the solution of a question, which may frequently prove oť considerable utility, but which has not we believe been answered publicly before. “ If a person were to make an assurance at any of the Offices or his own life for a single year, and to repeat this at the end of every successive 'year to the utmost extremity of life, the annual payment (for such assurance would be continually increasing till his death. But, if he made the assurance on the whole continuance of his wife, and contracted with the office to pay the value of such assurance by equal annual pay; ments during his life' (as is usually the case ), it is evident that such annual payment ought to be greater than the premium required for an assurance for a single year at his present age, but less than the premium required for a simi. lar assurance at the more advanced periods of life. Hence, it appears that if a person, who was originally assured for the whole term of his life, should be desirous (either through inability, or any other motive) of renouncing his claim upon the office, and of cancelling his policy, he ought to have some part of those annual payments returned to him or, in other words, a compensation ought to be made him for that excess in the annual payments which he has been advancing to the Society. The object of the fol. lowing question is to determine the amount of that remuneration. • QUESTION XXXIV–To find the sụm that ought to be giyen to a person, who is assured for the whole term of his life, for a given sum, in order that he may renounce his claim thereto. Solution. Subtract the equal anaual payment, 'which he has been giving since the assurance commenced, from the equal annual payment which ought to be given for the assurance of the given sum on the life at its present age; multiply the remainder by the value of an annuity (increased by unity*) on the life at its present age: the product will be the sum required. • Example. A person now aged 50, who has been paying 21-790, or £21 :15:10,+ annually for the assurance of £1000 at his death, is desirous of discontinuing the same; what sum ought to be given to him, by the Office, as a compensation for so doing; interest being reckoned at 3 per cent, and the probabilities of living as at Northampton? I • The annual premiums which ought to be given for the assurance of £1000 on a life aged 50 is, by Table Ll, equal to 45-300; and the difference between this and 21-790 is equal to 23.510; which being multiplied by 13.436 (or unity added to the value of an annuity on a life aged 50, will produce 315.880, or £315:17:7, for the answer required. pp. 456—459. The following extracts will serve to shew, how egregiously the public are duped, while they conform to the exorbitant ** This supposes that the policy is cancelled immediately before the annual payment becomes due: but if immediately after, we must mula tiply the remainder, above alluded to, by the value of an annuity on the given life, without the addition of unity. + This is the annual payment for the assurance of 81000 on a life aged 20, as appears by Table LI. • The rate of interest and probabilities of Life, 'in such computations, ought to be the same as those adopted by the Office, at which the policy is effected. • This case may be stated in another manner, as follows. The Society may be considered as indebted to the assured in the present value of an assurance of 81000 on a life aged 50; which is equal to 608.66, or 8608:13: 2. And the assured may be considered as owing to the Society, the present value of all the annual payments of 221:15:10, duriog the remainder of his life; the first of which pay: ments is supposed to be made, immediately, therefore the value of all those payments will be equal to 21.790 multiplied by 13-436; which produces 292.78, or £292: 15:7. Consequently the interest of the assured in his policy will be equal to the difference between $608 : 13:2, and £292:15:7; that is, equal to 6315:17: 7, as found by the example in the textil terms of all the Life Assurance Companies in the kingdom; and consequently, to evince the necessity of some revision taking place, in the tables by which these societies think proper to grant assurances, unless they are willing to resign all preiensions to fair and honourable dealing. • By means of the general solution here given, may be determined all questions relative to the value of such sums as ought to be given for the Endowments of Children. Thus, suppose a person has a son aged 11, for whom he wishes to secure 2100 on his coming of age : the sum which he ought to pay down for the assurance of the same (reckoning interest at 5 per cent, and the probabilities of living as according to M. de Parcieux ) is equal to a multiplied by 61.391; which produces 56.744, or £56", 14 : 10 for the answer required. • In the table of rates published by the Globe Assurance Company, and by the Provident Institution", the sums demanded for the Endowments of Children are in general full as much as (and from the age of 9 years and upwards are even more than) the present values, at 5 per cent, of £100 certain to be received at the end of the given term without any contingency. For instance, £62:11:2 is required in ready money by the Globe, and 462 by the Provident, for the payment of £100 on the event of a child, aged 11, arriving at the age of 21 years : whereas either of these sums put out to interest at 5 per cent would amount to more than $100 at the end of that period, without the liability to loss in case the child should happen to die before that time!!! No person can, I think, be so blind to his own interest as to risk his money in this absurd way.' pp. 360, 361. These examples will show the method of proceeding in all similar cases : and for the information of the reader 1 shall here subjoin a table of the sums demanded by the different Assurance Companies for the assurance of $100 for one year on a single life at the several ages therein mentioned: to which I shall add the fair value that ought to be given for the same, according to the probabilities of life as observed by M. de Parcieux, and reckoning interest at 4 per cent. Northampton. De Parcieux. 4 per cent. per cent • From which it appears that the several Assurance Companies require, in most cases, half as much again as ought to be given ; and in some cases nearly double the sum that should be given for the assurance. And ** * The Other Companies have not published their rates for the Endowments of Children: but, from the similarity of most of the rates at the various offices, we may conclude that there is no great difference on this point.' same though some compensation ought to be allowed for the expenses incurred in carrying on the business of the Office, as well as a proper remuneration for the services of those who conduct it ; yet it is evident that these sums are greater than onght reasonably to be taken : particularly when it is considered that those w/o insure at any of the Offices, for a term of years only, have not much prospect of 'deriving any advantage from the profits of the concern.' pp. 440~442. « If we take the case of Contingent assurances (that is, of an assurance made on a given life, or the contingency that it dies before another) we shall find still greater cause for censure. For, independent of the rates being computed from the lowest probabilites of life and at the lowest rate of interest, they are also deduced from a rule given by Mr Simpson which produces a result that is oftentimes more than one third of the true valde too much, even when computed from the Northampton table, at 3 per cent interest ! -A single instance will confirm this also. • A person 10 years of age, is desirous of assuring the sum of £100 on his life, on the contingency that he dies before another person aged 60. The sum which would be demanded by all the Assurance Offices (not even excepting the Equitable)'is €12:18:6 in a single payment or £l : 6:0 in annual payments during the joint lives. But the true value which ought to be given for the same even on the supposition that the rate of interest is no more than 3 per cent, and the probabilities of living the as observed at Northampton) is only £10:13:0 in a single payment, and £1:1:5 in annual payments : consequently the Offices demand about a fourth part of the true value more than (on their own data) is just and equitable. If the value, however, had been deduced from the probabilities of living as observed by M. de Parcieux, and at the rate of 44 per cent interest, it would be £9:3:1 in a single payment, and only eighteen shillings and winepence in annual payments: which makes the Office rate, in this case, nearly one half the true value too much! Mr. Morgan has taken considerable pains to prove that Mr. Simpson's rule for finding the value of these contingent reversions is exseedingly defective : and that it oftentimes leads to conclusions too erroneous to be overlooked. Now, since the true values can in all cases be obtained with so little trouble, it is somewhat singular that the incorrect values, in Table LIII above alluded to, should still be adopted (not only by the Equitable Society, but also by every other Office in London) for the purpose of determining the sums that are required for effecting assurances on the contingency here mentioned. Amongst the numerous societies that have lately been established, is there no one Actuary that has the confidence to propose a new table of the value of such assurances, founded on a true and proper basis; or will the several Companies still persevere in their unjust and illiberal demands? Surely their profits must be sufficiently great by taking the lowest rate of interest, and the lowest probabilities of living as the basis of their calculations ; without adding thereto the unfair advantage arising from the use of inaccurate rules,' pp. 511-513, It is now time to produce one of Mr. Baily's tables by way of specimen, and to offer a few observations, naturally sug, gested by an examination of it. Showing the Number of Persons living and dying at every age, ac cording to the observations made in all Sweden, for 21 years. 011000012300|10000|2090||10000 2195 49 3751 8 ||4097 70113924 78 50 (3666 95111027 75 846 7700 500 7910 518 7805) 509 85 1 2 7200 337) 7392 350 7296 344 51 3571 953952 80||3761 87 3 6863 240|| 7042 250 6952) 245 52 34761 95||3872 853674 90 4 6623 1501 6792 135| 6707 143 53 3381 953787 85||384 90 5 6473 125| 6657| 120 | 6564 122 54 3286 9 3702 853494 91 55 3191 953617 853403 91) 6 6348 1051) 6537| 1051) 6442 105 7 6243 90 | 6432 85 6337 87 56 3096 9513532 85||3312 92) 8 6153 75 6347 706250 73 57 3001 104113447 90|3220 95 9 60781 65 6277 60 6177 62 58 2901 100113357 90|3125 95 10 6013 551) 6217 52 6115 5+ 592801100|3267) 100|3030 100 60 2701 105||3167 110||2930 108 5958 45|| 6165 46|| 6061 45 12 5919 4511 6119 4011 6016 42 61 |2596 1103057) 118||2822 114 13 5868 401| 6079 35|| 5974 38 62 2486 1152939| 120||2708 118 5823 401 6044 35 5936 37 63 123711 115|2819 120||25901 118 15 5788 39 6009]. 35 5899 373 64 2256 115||2699 120||2472 118 16 | 5749 65 2141 115) 2579 120||2354 118 39|| 59741 401 5862 40 17 5710 39|| 5934 40 5822) 40 66 2026 11512459 120||2236| 118 18 5671 44|| 5894 42 5782 42 67 1911 1 201|2339 120121181 121 19 5627 44|| 5852 43|| 5740 43 68 1791 125 2219 120||1997) 1241 20 5583 50 5809 43| 5697 47 69 1666 125 12099 120||1873 124 70 1541 120||1979 130|1749 127 21 5533 501 5766 43 5650 47 22 5493 50 | 5723 43|| 5603 48 71 1416 125||1849 140|| 1622 133 23 5483 551 56801 441 5555 48 772 1291 1201|1709| 150 | 1489 135 24 53781 55 5636 45|| 5507 50 73 1171| 1201559) 160||1354140 25 5323 55 5591 451 5457 50 74 1051 110|1399 150||1214) 130 75 26 5268! 55 5546 50 5407 941 105||1249| 140||10841 121 52 27 55|| 5496 52 | 5355 54 76 836 100|1109.130|| 963 115 28 5158 55|| 5444 55 | 5301 736 90 979 190 848 105 29 5103 561) 5389 55 5246 55 78 6461 851 859 110|| 743 95 30 50491 591 5334 601 5191 59 79 561) 80 749 100|| 648 90) 80 481 75 649 60 | 5132 95 558 31 (4988 60 5274 60 90 32 1 4928 601 5214 65 5072 62 81 4061 70 554 90| 468 841 33 4868 60 | 51491 65 5010 63 82 336 65|| 464 85|| 384 75 34 4808 60 5084 65|| 4947 G3 83 27] ] 60|| 379 80 309 65 35 | 4748 601 50191 60|4884 59 84 211 601 299 75| 244 55 85 161 40 224 551 189 56|| 4825 36 46881 60| 4959 45 58 37 4628 60| 4903 56 | 4767 58 86 121 301 169 40|| 144 35 39 4568 60|| 48471 56|| 4709 58 87 91 22 129 3011 109 27 39 4508 60| 4791 58 4651 60 88 691 17 99 23 82 20 40 4448 65|| 4733 65 4591 65, 89 52 14 76 18 62 151 90 38 12 58 175) 4526 15 47 43831 14 72|| 46681 73 42 4314) 80 | 4593 76|| 4453 78 91 26 91 43 12 33 12 43 4231 80|| 4517 76 4375 778 92 17 7 31 10 211 10 44 4151 801) 4441 751 4297 78 93 10 21 8 11 6 45 40711 801 4366 721| 42191 76 94 4 13 6 5 31 95 1 2 47 3911 801| 4227 65 | 4069 72 96 01 3 2 1 48' 38311 80 41621 65|| 3997) 75 / 97 oll 11 1 01 5213 46 3991 80 4294 67 4143 74 |