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 term certain. Many of the estates belonging to the Corporation of Liverpool are held on the tenure 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 fine for renewing with any life, and without regard to the age or state of health 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 circumstance attending this subject, was their custom of exchanging, for the sum of only one guinea each, lives, not exceeding 50 years of age and in good health, for lives of any other age, 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 of 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 on 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 iife, and contracted with the office to pay the value of such assurance by equal annual payments 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 similar 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 sum that ought to be given 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 annual 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?‡ The annual premiums which ought to be given for the assurance of £1000 on a life aged 50 is, by Table L1, 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 multiply 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 £1000 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. The This case may be stated in another manner, as follows. Society may be considered as indebted to the assured in the present value of an assurance of £1000 on a life aged 50; which is equal to 608.66, or £608:13: 2. And the assured may be considered as owing to the Society the present value of all the annual payments of 21:15:10, during the remainder of his life; the first of which payments 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 8292:15:7; that is, equal to 315: 17: 7, as found by the example in the text.' 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 pretensions 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 £100 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 9 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 862 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 I 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. 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.' 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 ought reasonably to be taken: particularly when it is considered that those who 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. 60. 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 value 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 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 £16:0 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 same 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 ninepence 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. Simpon'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, |