bluuebunny/crossref_metadata_2025 huggingface dataset:
Solutions of the Compound Survivorship Assurance Problems
Authors: Makeham William Matthew
Published: August 2016

The problems referred to under the above designation are those which involve the double survivorship contingency of A living after the death of B before C. The following solutions of these cases are quite general, and consequently applicable to any mortality table whatever.

I have taken the problems in the order in which they appear in the work of Mr. Milne, whose notation I have also followed. By means of an obvious relation which exists between the value of the compound survivorship assurance and the corresponding annuity, the whole of these cases are reduced to the single fundamental annuity value combined with the values of simple survivorship assurances. Consequently, if we should ever possess a complete table of the values of the simple survivorship assurance and also of the values of the annuity we shall be able to calculate the numerical value in any case with the greatest facility.
On a Plan for making Conditional the Payment of extra Premium in the case of a Life supposed to be Diseased or more than ordinarily Hazardous
Authors: Younger Samuel
Published: August 2016

The difficulty of dealing equitably with assurers whose prospects of longevity are considered by the medical referee to be below the average of healthy lives of the same age will not, I think, be disputed. The practice of charging a higher premium corresponding to an advanced age, which has hitherto generally been adopted, is, of course, open to the double objection, that the judgment of the medical officer may have been at fault in placing the life in the second or third class at all, and, supposing he were right in his decision thus far, that the premium at which he has assessed the risk may not be in accordance with its precise magnitude. It is true that in after years reductions may be made if the person's health improve; but this plan is not without trouble both to the Office and the assured, and it has the further disadvantage of leaving the latter in a state of uncertainty as to what his future premiums may be.
Some Considerations on the Government Life Annuities and Life Assurances Bill
Authors: Adler Marcus N.
Published: August 2016

The measure by which the Chancellor of the Exchequer proposes to extend the benefits of life assurance and annuities to the nation at large has now been before the public some time. Few Bills have of late attracted so much attention, and have been so earnestly discussed by all classes, as this. But beyond what was elicited at an interview that took place between the Chancellor of the Exchequer and the Actuaries of several Offices, and some passing remarks on the subject made at the last meeting of the Institute of Actuaries, when we had the advantage of hearing Mr. Samuel Brown's excellent paper on Friendly Societies, the public have as yet had no opportunity of hearing the opinions of those, who after all are best able to judge. It now appears, that the Chancellor of the Exchequer is opposed to a general inquiry and the calling for “persons, papers and records” by the Select Committee, to whom the Bill has been referred, and it therefore becomes all the more desirable that the merits of the proposition of the Chancellor should be calmly and impartially discussed by the members of this Institute.
On a Table for the Formation of Logarithms and Anti-Logarithms to Twelve Places
Authors: Gray Peter
Published: August 2016

With the exception of the publication on a more limited scale (and which will be hereafter referred to), of that which is now to be developed, there exists at present no ready and practical method of forming to more than seven places the logarithms of numbers of more than seven or eight figures. The great extent of the tables requisite, if formed on the plan of our present seven-figure tables, is not only likely ever to prove a bar to their construction, but it would also render them too cumbrous, if constructed, to be easily and readily used. In the present method, the end in view is sought to be attained in another way. The principle of the method is the resolution of the number whose logarithm is required, by a direct and easy process, into factors of a peculiar form, the logarithms of which, to the requisite extent, admit of easy tabulation. The logarithms of the factors, then, being taken from the table, their sum is the logarithm of the given number.
On a Method of Graduation applied to the Peerage Mortality deduced by Mr. Bailey and Mr. Day, with Tables founded thereon
Authors: Berridge G. W.
Published: August 2016

In a paper read before this Society on the 29th of April, 1861, and contained in the ninth vol. of the Journal, there is given a table showing the probability of living a year at each age, as evidenced by the males of the families of the peerage existing within the last half century; and it appeared to me that it would be worth while, if only as a matter of curiosity, to put the information there given into a form better adapted to ordinary calculation.
On Government Annuity and Assurance Rates and Regulations
Authors: Adler Marcus N.
Published: August 2016

In briefly commenting upon the papers that have been presented to Parliament on the subject of life assurances and annuities now granted by Government, it is less my object to criticize the measures that have been adopted, than to explain the mode of construction of the several tables of rates that are being acted upon. In consequence of several assumptions and suppositions, these have not been computed in the mode usually adopted by actuaries.
On the reliability of data, when tested by the conclusions to which they lead
Authors: Farren Edwin James
Published: August 2016

It is a well-known feature in the advancement of learning, that the more knowledge becomes prevalent, the less positive do opinions become. Not only in theology does the staggering demand of “What is truth ?” assail us, but we meet with the same impregnable question in all the pursuits of literature and science, obliging us to admit that the truth of yesterday is not always the truth of to-day, and that to-morrow may still be about to offer us some third form.

The older class of writers felt no such difficulty. They conceived they were in possession of certain abstract truths or data that might be considered as fundamental axioms, and that by these axioms might all conclusions whatsoever be tried.
On the Results of the Operations of the Gotha Life Assurance Bank for the first Twenty-five Years of its existence, particularly with respect to the Mortality amongst the Lives Assured. (Concluded from p. 337, Vol. V)
Authors: Hopf Rath G.
Published: August 2016

In order to be able more accurately and under different points of view to observe the law of mortality which prevails amongst the assured, I have, besides the usual observations from one calendar year to another, on which the above calculations are founded, arranged other observations, in which the assured are classified from one year of assurance to another. The date of the first policy issued on the life of a person is the period at which, in this register, he yearly advances from one year of age to another. The payment of the premium and continuation of the assurance in the Gotha Office is regularly made for one year: however, it is permitted (but only to persons assured for the whole term of life) to pay their premiums also at half yearly rates; but the second half yearly payment is only considered as delayed, and those who desire to withdraw have no claim to any return for the premiums they have paid until they have also made the second half yearly payment. The consequence of this is, that, with very few exceptions, persons withdraw only at the end of a year of assurance. All the assured persons enter at the beginning of a year, die in the course of one, or withdraw at the end of one. They are therefore, with the exception of the year of their death, always full years in the circle of observation. This arrangement requires no correction for admission and withdrawal, as was necessary in the former case, and affords therefore a greater accuracy. Besides, tables arranged in this way offer the means of being able to consider separately all persons assured at a certain age, and to compute the mortality for these persons by themselves, and likewise to determine the mortality for each period of the assurance from its commencement.
On the Valuation of Property held for Life and in Reversion; and on the due Apportionment of it, when so held on the same Life, between the Tenant for Life and the Remainder-man
Authors: Jellicoe Charles
Published: August 2016

In a paper published some time back in the Journal of the Institute, an endeavour was made to show the true principles upon which the value of securities depending on an isolated human life should be determined. Discarding the methods once so generally practised, and in accordance with which such securities were confounded with those depending on the average duration of many lives, it was urged that they should be regarded in a distinct point of view, and as affording the means of investments to be made not subject to any contingency whatever, but as securing to the holder a given rate of interest so long as he retains them, and the return of the capital invested when such interest shall cease to be paid or to accrue.
Some Suggestions respecting Fire Insurance Statistics
Authors: Miller Thomas
Published: August 2016

The efforts made by Fire Insurance Companies to collect accurate particulars of the state of their business, and the profit or loss on different classes of risks, evince at once their appreciation of correct statistical information and the difficulty of obtaining it. Most Offices of any standing keep registers in which risks are divided into distinct classes—such as cotton mills, corn mills, farm stock, dwelling-houses, carpenters' shops, &c. &c.; and in one set enter under their proper heads the sums insured, premiums received, and number of risks, so that they can tell from inspection how many corn or cotton mills, or other risks, they insure each year, the total sums insured on them, and the amounts of premiums received: while in another set they enter under their proper heads the number of corn or cotton mills, or other risks, in which loss has occurred, and the losses paid—so that by inspection they can see what losses have been sustained on each class. The difference between the premiums received and the losses paid, leaving expenses of management out of view, constitutes a surplus or deficiency from which they judge of the profit or loss on each class of their insurances.
Continuation of a Paper on a Method of Distributing the Surplus among the Assured in a Life Assurance Company
Authors: Sprague T. B.
Published: August 2016

In the present paper I propose to give some numerical illustrations of the working of the method of which the formulæ were given in the last Number of this Magazine, and to add some further results which have been subsequently arrived at.

In order to apply the formulæ to numerical cases, it is necessary to assume some law according to which the successive bonuses proceed. It is obvious, that whatever law is assumed must be to a great extent arbitrary, as it is of course quite impossible to say what will be the state of a particular Office after the lapse of a number of years. But it seems probable that, on an average, an Office may return to the assured in the form of bonus 20 per cent. of the premiums paid, and it is accordingly assumed in the calculations that each quinquennial cash bonus declared on a policy is equal in amount to one year's premium. This will probably not be far from the truth where the method of division of profits adopted in the Office is one that leads to nearly constant cash bonuses to each policy: such as the method proposed by Mr. Jellicoe, which divides the profits in proportion to the loading of the premium, accumulated at compound interest for the interval between two successive bonuses. But in cases where one of the more common methods of division is adopted, which lead to a series of increasing cash bonuses on each policy, the above illustration will not be applicable, and the conditions of the problem will be greatly changed. The calculations are made by means of the “Experience” mortality, assuming interest at 4 per cent.
On the Inadequacy of existing Data for determining the Rate of Mortality among Select Lives
Authors: Spens William
Published: August 2016

I. While so much improvement has recently taken place in the arrangement and construction of various tables for facilitating calculations founded on existing data, very little has been done in the way of investigating and correcting the data themselves; and it is feared that the question of the rate of mortality among select lives is still involved in the greatest doubt and obscurity.

II. It is not proposed in the present paper to go farther than to show that the rate of mortality, during the first year of selection, of select assured lives is so materially different from what it has hitherto been represented, as to lead to the inference that the data from which the erroneous deduction has been made cannot be true data for the ascertainment of the value of selection. To investigate the rate of mortality of select lives at separate ages, I conceive to be of the utmost importance for the elucidation of truth, and the proper direction of sanatory inquiries; but I do not consider that sufficient data at present exist for the determination of this, and these can only be obtained by a united inquiry. I shall be very happy if the present observations have any effect in hastening such an investigation, which sooner or later must be entered upon.
On the Rate of Mortality amongst Europeans and their Descendents residing in the Island of Jamaica
Authors: Marshall J.
Published: August 2016

As already stated by the writer, in a former paper on the subject, “In the absence of any large collection of facts and observations in Jamaica, from which such a table could have been deduced, he has necessarily been compelled to proceed upon probability and approximation, according to the experience and opinions of persons whose age and long residence in the community, or whose peculiar means of information, would give weight and authority to their statements; and, as a consequence, to any system which might be based upon them.
An Investigation into the proper Method of Determining the Amount of an Annuity forborn and improved at Interest, during the existence of a given Life
Authors: Hardy Peter
Published: August 2016

The late Mr. William Morgan, in the second edition of his Principles and Doctrine of Assurances and Annuities on Lives, published in the year 1821, has, in the 7th Problem of that work, investigated “the method of determining the amount of an annuity forborn and improved at compound interest during the continuance of a given life.” This problem he has subsequently discussed in an algebraical form, in Note V., page 154; and he has moreover given a table (Table VII.) showing the sum to which an annuity for-born and improved at 4 per cent, compound interest will amount, on the extinction of a given life, computed according to the foregoing method, from the Northampton Table of Observations.
On the Causes of Fires in London during the Twenty-four Years from 1833 to 1856 inclusive; with some Remarks on the Deduction of correct Rates of Premium for Fire Insurances
Authors: Fothergill Charles George
Published: August 2016

With the view of turning to some farther account the information afforded by the periodical reports of fires in the metropolis furnished by the London Fire-Engine Establishment to the Offices supporting it, and extending now over a term of twenty-four years, I have drawn out some results in a tabular form, and have attached to them a few comments more suggestive than at present available; premising that these results are necessarily of limited value, in consequence of the indefiniteness of the data in many essential respects. For Mr. Braidwood’s kind assistance I take this opportunity of expressing my obligations; and especially I am indebted to Mr. W. M. Browne, Secretary of the Westminster Fire Office, for the readiness with which he has placed the results of his experience at my disposal.
On the Data collected by the Council of the Institute, with a view to determine the Rates of Premium for the Assurance of the Lives of Persons residing in Foreign Climates or engaged in Pursuits attended with Extra Risk
Authors: Jellicoe Charles
Published: August 2016

It will no doubt be in the recollection of many of our readers, that some few years ago attempts were made by the Council of the Institute to collect, from the Offices willing to contribute the results of their experience, such information connected with the extra risks undertaken by them as might afford the means of determining, if not exactly, at least with some approach to accuracy, the proper rates of premium to meet them. The Council made application to all the existing Companies; but did not succeed, unfortunately, in obtaining returns from more than fifty-two of them. These fifty-two Companies, however, were at the pains to supply as many as 6,154 cases—each of them of a special nature, as regarded unusual hazard to life arising from residence in foreign climates, or from particular pursuits; but so infinitely varied in their conditions and circumstances were they, that it became immediately obvious that considerable difficulty would arise as to grouping them into classes which should be sufficiently well defined to yield results of a useful and practical character. If, on the one hand, the conditions of the risk were kept within limits sufficiently restricted, the numbers in each class were all but insignificant; and if, on the other, attempts were made to remedy this last inconvenience, the groups comprised risks which were altogether heterogeneous.
On the Progress of Fire Insurance in Great Britain, as compared with other Countries
Authors: Brown Samuel
Published: August 2016

The system of fire insurance has been practised in Great Britain, either by private societies, public companies, or corporate bodies, for more than 180 years. In an article on fire insurance in Vol. XII., Part II., of the Encyclopædia Britannica, 8th edition, recently published, Mr. F. G. Smith draws attention to the various attempts which were made to interest the Corporation of the City of London in the subject, and to the proposals which for this purpose were submitted to the Court of Common Council between the years 1669 and 1680, and more especially to one from Mr. Deputy Newbold, which appears to have been the most acceptable. The delay which occurred before the report of the Committee was presented, enabled other private individuals to originate a scheme for fire insurance; and by the advertisements of the day it appears that they offered to insure against damage by fire, brick houses at 6d. in the pound, and timber houses at Is. in the pound—rates which showed the little knowledge which at that time prevailed upon the subject, provided sufficient business could be reckoned upon to obtain an average of the risks. On the 13th October, 1681, the Court of Common Council decided to effect fire insurances on houses within the city and liberties, and engaged a sufficient fund and undoubted security by the Chamber of London, on lands and ground rents, to provide for the fulfilment of their contracts. The war that ensued between those private Societies which had been the first in the field, and the city insurers, gave rise to much amusing pamphleteering and advertisements in the Gazette. In 1681, 1682, and 1683, the journals of the Court of Common Council record the signing of many policies, and refer to the discussions and arguments of their opponents; but the city authorities appear to have been soon weary of the scheme, for, by a resolution of the 13th November, 1682, the Court decided to relinquish the business, to cancel the existing contracts, and to return the money accepted for them.
On the rates of interest for the use of money in ancient and modern times. Part II
Authors: Hodge William Barwick
Published: August 2016

On a former occasion I had the honour to lay before the Institute a historical sketch of this subject, from the earliest period of authentic records down to that of the legal establishment of the practice of taking interest, in the reign of Elizabeth.

The facts then adduced showed that very high nominal rates of interest were obtained in the ancient world, and during the middle ages; but such rates are, by no means, to be taken as measures of the profits made upon loans, which must always have been seriously diminished, and often entirely swallowed up, by the exactions and spoliations to which the lender was exposed, and the difficulties he experienced in enforcing his claims, from defective laws, and the prejudices entertained against him by the community.
On the Principles of Marine Insurance
Authors: Morrison Richard
Published: August 2016

The subject of marine insurance is one which occupies, at the present time, a large amount of attention, both in the mercantile and the non-commercial world. The latter regard the formation of Companies for that object as a profitable means of investment; and the former on account of the necessity for their employment and encouragement. No prudent merchant or shipper would neglect the precaution of insuring his goods, at least against the risk of total loss, if not against the ordinary perils of the sea, which cause damage, for which the insurers render themselves liable by the charge of an additional premium.
On the Summation of a Compound Series, and its application to a Problem in Probabilities
Authors: Terrot Bishop
Published: August 2016

The series proposed for solution in the following paper is—

The law of this series is manifest. Each term is the product of two factorials—the first consisting of p, and the latter of q, factors; and in each successive term, the factors of the first factorial are each diminished by one. and those of the latter increased by one.
On the Laws of Mortality and Sickness of the Labouring Classes of England
Authors: Edmonds Thomas Rowe
Published: August 2016

The mortality, according to age, of the general population of England in all localities, was first satisfactorily determined for the period of seven years, ending with the year 1844. The observations and results were published by the Registrar-General in the year 1849; and in the following year I was allowed the opportunity of exhibiting inThe Lancet(vol. i. 1850, pp. 297–330) the principal results in new combinations, with my conclusions therefrom. The knowledge thus obtained of the mortality of the general population (with distinction of town from country districts) is valuable for testing the correctness of independent observations made on the mortality of classes of the labouring population, such as are contained in “Friendly Societies,” whose mortality and sickness it is now proposed to discuss.
The interest in assurances effected by one person upon the life of another
Authors: Bailey Arthur H.
Published: August 2016

Sir,—The letter of your correspondent “Verus,” in the last Number of theMagazine, induces me to trouble you again upon this subject; especially as, if the views there expressed are entertained by any number of your readers, it is evident that so much difference of opinion exists on this question as to render further discussion desirable.
An Examination of the Objections urged against the plan of Decimal Coinage proposed by the Royal Commissioners and by the Select Committee of the House of Commons
Authors: Jellicoe Charles
Published: August 2016

It is now somewhat more than twelve months ago that the decimal coinage question was brought under the notice of this Institute, and that a resolution was thereupon unanimously come to approving of the plan recommended by the Committee of the House of Commons. Since that decision, public attention has been repeatedly called to the subject; and although various plans have been proposed and discussed, founded on a different basis from the one thus recommended, it must, I think, be conceded, that none of them have made any serious impression on the public mind, but that, on the contrary, the one proposed by the Committee has made some if not considerable progress in the course of the sifting which the question generally has undergone.
On the Limitation of Risks; being an Essay towards the Determination of the Maximum Amount of Risk to be retained by a Life Insurance Company on a Single Contingency
Authors: Sprague Thomas Bond
Published: August 2016

It is my object in the present paper to discuss the principles which should guide the actuary in fixing the maximum amount of risk to be retained by a Life Insurance Company upon a single contingency.

Those principles are by no means limited to life insurance, but extend with suitable modifications to other kinds of insurance, as fire, marine, hailstorm, health, and accidental death insurance. They are also applicable to the risks of mercantile transactions, and in particular to the risks incurred by a bank in discounting its customers' bills. There can be little doubt that some recent failures of banks have been caused principally by the neglect of these principles; in other words, the manager has failed to put a proper limit on the risks undertaken, i.e., has trusted individuals to an imprudent extent, and the loss of heavy sums consequent on their failure has proved disastrous to the bank.
General Average (Concluded from p. 51)
Authors: Morrison Richard
Published: August 2016

In the Report before alluded to, this case is especially noticed as affording evidence of the concessions made by foreign countries to obtain uniformity. The practice abroad has hitherto conformed to the principle recognized universally, as to the propriety of admitting as general average the damage done by voluntary stranding which was attended by the preservation of the ship and cargo. They still retain, however, as a subject of contribution, any damage which may arise from running a ship ashore to avoid capture; but it is clear that the same objection cannot be raised as in the case of a vessel in danger of foundering; for she may be, under the former circumstances, perfectly seaworthy, and therefore there would not be the difficulty of determining the exact amount of damage done to the cargo or the ship.
On the construction of tables by the method of differences. Section IV
Authors: Gray Peter
Published: August 2016

Irrational Functions, strictly speaking, are functions whose numerical values cannot be expressed by finite fractions. In the present connexion the term irrational has a wider signification. It is used to denote any function whose numerical value cannot, in general, be expressedexactlywithin the limits as to decimal places, to which we restrict ourselves in the table under formation. As here employed therefore the term designates not only transcendental functions, (as exponential, logarithmic, circular, &c.,) and algebraical irrational functions, (which are such as contain fractional powers of the variable in either numerator or denominator, or both,) but also algebraical fractional functions, which are such as contain integer powers of the variable in both numerator and denominator, or in the latter only. And it is to the formation of tables of the values of functions such as these that our attention is now to be directed.
Further Considerations on M. Violeine's Solution of a Problem on the Rate of Interest in Loans repayable by Instalments
Authors: Gray Peter
Published: August 2016

It has been represented to me that, in view of the influential position occupied by M. Violeine, the quasi official and authoritative character of his book, and the currency likely to be thereby imparted to the erroneous principles employed by him in the treatment of problems of the class to which the first of those I have discussed belongs;—I say it has been represented to me that in these circumstances it might be well to be a little more explicit as to his manner of proceeding. In my discussion of the problem in question I did enough, I believe, to shew that M. Violeine's solution is certainly erroneous. I admit however that I did not succeed to my own satisfaction (nor apparently to that of others) in precisely hitting the blot in it, and, as it were, reducing it to its elements. I think that from the consideration I have since given to the matter, I am now able to point out the fallacy by which it is vitiated, and I therefore willingly accede to the suggestion that has been made to me.
On the Determination of the Average Risk attaching to the Grant of Assurances upon Lives
Authors: Kanner M.
Published: August 2016

An Insurance Office, which, in consideration of fixed premiums undertakes the payment of a sum on the death of the assured, strictly speaking lays a wager with each of the assured, in which the stakes are proportional to the probabilities of the happening of the two contrary events—his living and his dying. Every assurance of a sum payable at death admits of being divided into partial assurances for a fixed period, as a year. Let the sum assured be S and the probability of death within a year p, then pS will be the stake of the assured, or the premium; and (1−p)S the stake of the office, i.e. the amount which the office would lose in case of the death of the assured within the year. The two stakes together therefore amount to the sum assured.
On the Method of Constructing Tables of Mortality
Authors: von Baumhauer M. M.
Published: August 2016

The theory of the construction of Tables of Mortality offers no difficulty when applied either to the members of a Society, or to a collection of individuals, who can be traced so that all the necessary particulars as to them are known. But in the population of a country or a town, several incidental causes, such as epidemics and migrations, exercise an influence; some of them on the total numbers of the population, and others on the members living at certain ages. The mobility of a population or the constant change taking place, from one residence to another, or from one country to another, prevents us from following each individual or each group of individuals from the moment of birth to that of death.
A Budget of Paradoxes (Continued from vol. xiv., page 121)
Authors: De Morgan A.
Published: August 2016

Mr. James Smith will, I have no doubt, be the most uneclipsed circle-squarer of our day. He will not owe this distinction to his being an influential and respected member of the commercial world of Liverpool, even though the power of publishing which his means give him should induce him to issue a whole library upon one paradox. Neither will he owe it to the pains taken with him by a mathematician, who corresponded with him until the joint letters filled an octavo volume. Neither will he owe it to the notice taken of him by Sir William Hamilton, of Dublin, who refuted him in a manner intelligible to an ordinary student of Euclid, which refutation he calls a remarkable paradox easily explainable, but without explaining it. What he will owe it to I proceed to show.
On the Construction of a Graduated Table of Mortality from a Limited Experience
Authors: Chandler S. C.
Published: August 2016

The discussion of the methods of dealing with the mortuary statistics of a life insurance company has been hitherto almost entirely neglected by American actuarial writers. This is not a little singular, considering the attention bestowed on other subjects of not greater difficulty or importance. Viewing the large and rapidly-growing body of data already accumulated by American companies, the problem is not of less concern to us than to our English and German brethren, to whom, notwithstanding, we are indebted for the principal contributions to this branch of actuarial science.
Reduction of Formulæ for Annuities and Assurances Investigated by Mr. Sprague on the Common hypothesis of Equal Decrements in each Year of Life
Authors: Woolhouse W. S. B.
Published: August 2016

In 1869 Mr. Sprague read before the Institute an interesting paper “On the value of Reversionary Annuities payable half-yearly, quarterly, &c., according to the conditions which prevail in practice.” This valuable paper is given in the Journal, vol. xv, page 126. The conditions which prevail in practice, as here for the first time introduced and mathematically investigated by Mr. Sprague, are the following:—For a yearly reversionary annuity on a life x after the death of another life y, the first payment of the annuity is supposed to be made just one year after the day of the death of y, provided that x be then living; and for a like annuity payable by m instalments in each year, the first payment of an instalment is supposed to be made just of a year after the day of the death of y. Also, in each case, it is further supposed that whenever x shall die, after surviving y, the annuity shall be completed up to the day of the death of x by the supplementary payment of the proportion of the annuity estimated in respect of the fractional interval that may then have elapsed since the last yearly payment or instalment was received.
On the Integral of Gompertz's Function for expressing the Values of Sums depending upon the contingency of life
Authors: Makeham W. M.
Published: August 2016

Gompertz's Function for expressing the probability of a person surviving any given period, x, is . For the purpose of adapting this expression to the entire period of adult life I have added the factor ∈−ax. The two expressions become identical in form when the factor ∈−δx representing the effect of discount is introduced, and both may therefore be correctly designated by the name of “Gompertz's Function.”
On a general Formula for the value of present or future Benefits, whether free or burdened with charges; and on the application of the Formula to determining the Surrender Values of Life Policies
Authors: Macfadyen James R.
Published: August 2016

With the exception perhaps of extra premium, there is, it seems to me, no actuarial point in which the practice of assurance offices is so empirical as that of the allowance to be given for the surrender of a policy. The usual custom, so far as I am aware, is to find the value of the policy (generally by the table of mortality and interest used in calculating the reserve) and from that value to deduct a certain proportion, handing over the remainder to the policyholder. The amount of this deduction is purely arbitrary, and were it not for the equalizing effects of competition, would show very startling diversities; since, even with this counteracting effect, the differences in practice can hardly help striking anyone who has examined the amounts given as surrender value by the various offices interested in large reassurance cases which from any reason do not follow the rule of the parent company. In fact, making any return whatever under discontinued policies is the result of that struggle for existence called competition; and like all things growing from this root, the practice has been moulded much more by the pressure of circumstances than by theoretical considerations. Consequent on this method of growth we find extraordinary and indefensible anomalies imperatively calling for rectification. Further it seems to me, looking at the matter as a purely practical question and not one of actuarial fitness at all, there is no point more needful of examination. As things are at present, nobody can say positively whether as its proportion of the value the assurance office retains too little or too much. If the former, the sooner a practice resulting in loss is reformed, the better.
On Mathematical Statistics and its application to Political Economy and Insurance. (Continued from page 369.)
Authors: Wittstein Theodor
Published: August 2016

We have now, in conclusion, to deduce from the foregoing investigations the proper formulas for deducing a mortality table from observations arranged as in § 17. The preceding notation will not be strictly adhered to.
On a Problem occurring in connection with Entailed Estates in Scotland
Authors: Sprague T. B.
Published: August 2016

The position of an entailed proprietor in Scotland (technically called an heir of entail in possession) differs in many important respects from those of a tenant for life and a tenant in tail in England. One fundamental difference is, that the entailed proprietor in Scotland has all the rights of a landed proprietor except those which are taken from him by the deed constituting the entail, whereas the tenant for life in England has only those rights which are expressly conferred upon him by the deed under which he holds. There is a still more important difference as regards the power to disentail the estate. Before the passing of the Rutherfurd Act (11 & 12 Vic., c. 36) Scotch entails were perpetual. The entail might sometimes be set aside after a suit in the Courts, in consequence of some defect or informality—perhaps very slight—in the deed constituting it; or an entail might expire, in consequence of the total failure of heirs; but under no other circumstances could the entail be got rid of.
On the Hourly Distribution of Mortality
Authors: Lawson Robert
Published: August 2016

Several interesting observations have recently been made regarding the existence of maximum hours of mortality and the allied subject of recurrent variations in the activity of physiological function. During the present year an important contribution to the literature of the subject has been made by Dr. James Finlayson, of Glasgow, who in a couple of papers published, the one in the Transactions of the Philosophical Society of Glasgow for 1873-4, and the other in the Glasgow Medical Journal for April 1874, has supplemented and summarized Schneider's searching examination into the statistics bearing on the subject, tabulated the results of Mr. West Watson, a Glasgow predecessor in the study of vital problems, and collected the scattered statistics contained in the important contributions to the literature of this interesting enquiry. In an independent summary, showing the hours at which the greatest number of deaths occur in several Glasgow institutions, Dr. Finlayson determines the comparative cumulative mortality during successive hours of the same day, and during groups of hours collocated on account of some marked contrasts in meteorological conditions and physiological activity. As far as regards deaths from chronic diseases, the results obtained by Dr. Finlayson are confirmatory of those of Caspar and the summarized statistics of Schneider. They show a notably increased mortality during ante-meridian hours, as compared with those of the afternoon and evening, and especially a determinate maximum between 4 A.M. and noon. With regard to acute diseases, the figures of Dr. Finlayson show that in them the morning rise is nearly if not altogether equalled by a second rise in the extent of hourly mortality occurring in the afternoon. He accounts for this divergence from the general results, by referring to the influences exerted by the post-meridian rise in temperature which characterizes that group of diseases, as a means of determining in them the modification of their hours of maximum mortality.
Solution of a Problem in Apportionment
Authors: Sprague T. B.
Published: August 2016

“A father aged 60, and his son aged 30, who hav successiv “life interests in a landed estate, jointly borrow the sum ov £10,000 “from an insurance company, at 5 per-cent interest, upon mortgage “ov their life interests and a policy for £11,000, upon the life ov “the son, effected at the annual premium ov £2. 3s. per-cent. a “private arrangement being made between the borrowers, by which “the father is to pay the interest and premium during his life, and “the son is to pay them subsequently. In what proportion shoud “the £10,000 be divided between the father and the son?”
On Assurances against the Risk of ‘Invalidity’ or Permanent Inability to Work
Authors: Lazarus Wilhelm
Published: August 2016

Life Assurance provides for the family of the deceased in case of premature death; deferred Annuities provide for old age; but both institutions leave uncovered the risk of premature inability to work. Invalidity Assurance, including the benefits of a deferred Annuity, would be the real complement to Life Assurance. This truth is so deeply felt in Germany, that a good many institutions, employing a large number of officers, workmen, and labourers; many mills, and particularly the Railway Companies, long since directed their attention to the providing for their officers in case of their being invalided. How were they to calculate the annual contribution, how to make the valuation of their liabilities?
On the Theory of Probabilities
Authors: Herschel John F. W.
Published: August 2016

The theory of Probabilities has been characterized by Laplace, one of those who have contributed most largely to its advance,—as “good sense reduced to a system of calculation;” and such, no doubt, it is. But it must be especially noticed that there is hardly any subject to which thought can be applied, which calls for so continuous an application of that excellent quality, or in which it i s easier to make mistakes from simple want of circumspection. And, moreover, that its reduction to calculation is attended with difficulties of a very peculiar nature, such as occur in no other application of mathematical analysis to practical subjects, arising out of the great magnitudes of the numbers concerned, which defeat the ordinary processes of arithmetical and logarithmic calculation, by exhausting the patience of the computer, and require special methods of approximate evaluation to bring them within the compass of human industry. These methods form a conspicuous feature of the general subject, and have furnished scope for very extraordinary displays of mathematical talent and invention. That very large numbers will inevitably be concerned in questions where numerous and independent contingencies may take place, and in any order or mode of combination, will be apparent to any one who considers the astonishing fecundity of such combinations numerically estimated, when the combining elements are many. For example, the number of possible “hands” at whist (regard being had to the trump) is 1,270,027,119,200.
On the Rate of Mortality prevailing among Assured Lives, as influenced by the length of time for which they have been assured
Authors: Sprague Thomas B.
Published: August 2016

To say that the subject of the rate of mortality among assured lives and annuitants is at present very imperfectly understood, and that the process usually adopted to deduce the rate of mortality from observations upon these classes, is not a proper one for the purpose, and leads to erroneous and misleading results, may seem at first sight to be very rash assertions; but I believe that upon consideration they will be found strictly correct.
Explanation of a New Method of Adjusting Mortality Tables, with some observations upon Mr. Makeham's modification of Gompertz's Theory
Authors: Woolhouse W. S. B.
Published: August 2016

All tabular numbers drawn directly from observations, especially those appertaining to statistics, are more or less affected by unavoidable imperfections arising from defective information, insufficiency in the number of observations recorded by the experience, and other known or unknown incidental causes. When a consecutive series of tabular numbers is taken and differenced up to a certain order, the existence of the imperfections or errors alluded to, is at once revealed by the differences, which exhibit a conspicuous disturbance of their law of progression. To eradicate the errors and effect a proper adjustment of the original numbers is, however, an operation of peculiar and somewhat novel difficulty, and a critical examination of the empirical expedients, usually resorted to for such purpose, will show that instead of meeting the indispensable requirements of the problem, they only evade its most essential conditions, and thereby impregnate the results with a new series of theoretical errors, which too readily evade detection in consequence of the systematic uniformity of their law.
On the Proper Method of Estimating the Liability of a Life Insurance Company under its Policies
Authors: Sprague Thomas B.
Published: August 2016

Some years ago I read before the Institute a paper bearing on this subject (see Journal, vol. xi., p. 90) in which I examined at considerable length the method of valuation which has been on various occasions so strongly advocated by Mr. Tucker, and which is denominated by him the “reinsurance method.” This term, however, appeared to me unsuitable, and I termed the method the “hypothetical method” of valuation; because the sums assured and the premiums are valued by means of a hypothetical table of the values of reversions and annuities, deduced by an inverse process from the premiums actually charged by the office. In that paper I compared Mr. Tucker's method with the net-premium method of valuation, and gave my reasons for believing the latter to be not only greatly superior to the former, but the method that should be generally employed. Subsequent consideration has however satisfied me that the net-premium method of valuation is also open to very serious objections, and that it is not applicable in all cases; and I propose on the present occasion, first to point out some of these objections, and then to consider briefly what method of valuation should be followed when the net-premium method is inapplicable. The subject however is a very wide one; and the present paper must be considered rather as a contribution to the discussion of a question that is all-important to those who have the responsibility of advising as to the liabilities of Life Insurance Companies, than as claiming to be itself a complete essay on the subject.
On the True Measure of the Death Strain on the Funds of a Life Assurance Society
Authors: Higham Charles D.
Published: August 2016

The practice of making a comparison between the mortality actually experienced by a life assurance society during a given period of time, and that which might have been expected (to use the term in its professional sense) according to the mortality table on which the periodical valuations are based, has now become so general, that no apology is needed for drawing further attention to the subject, if it can be shown that the method usually employed at present is not altogether satisfactory.
On Certain Methods of Valuation
Authors: Gray W. T.
Published: August 2016

The proper method of estimating the liabilities of an office under its policies, is a question of so much importance that numerous essays on the subject have at various times been contributed to the pages of the Journal by some of the most eminent members of the Institute, whilst Mr. Manly, in his Messenger Prize Essay, and Mr. Valentine in his extension of the same, and more recently Mr. Geo. King, in his paper in vol. xx of the Journal, have furnished us with valuable information as to the financial results arrived at by the employment of different data in valuations. The subject has been so extensively treated in these papers, that I should not have ventured to refer to it again, but for the employment recently in some quarters of a method of valuation producing certain anomalies hitherto but little noticed in the pages of the Journal, and I therefore beg to submit for the consideration of its readers the results of an investigation showing the effects of the use of that method on the reserves and profits of an office.