The object of the actuary, in applying the science of vital statistics to pecuniary transactions, being to determine the values of payments dependent upon conditions of human survivorship, and therefore necessarily deferred for periods of longer or shorter duration, it is obvious that the rate of interest at which money may be increased is an element of his calculations, nearly if not quite as important as the probable risk of mortality that may affect the lives involved. The labours of actuaries have been generally directed to determining with precision the laws of mortality, rather than towards the present more humble branch of inquiry–the usefulness of which, however, must always become apparent with the earliest professional experience. Believing that an extensive and well digested collection of facts relating to the rate of interest would lead to some valuable and important conclusions as to future fluctuations, I trust that the present attempt to contribute to such a result will not be without interest for the members of the Institute, although I make no pretension to give a complete view of the whole subject, still less to discuss the important and intricate questions arising out of it; my object being merely to offer a general historical sketch, leaving the full prosecution of the inquiry to others, possessing more leisure and better qualifications for the task. Apart from its professional, the subject is one of historical interest. Men's pecuniary affairs afford unfailing indications of their characters, habits, and inclinations; and it has been truly remarked by a philosophical writer, that “a right measure and manner in getting, saving, spending, giving, taking, lending, borrowing and bequeathing, would almost argue a perfect man.” What is true in this respect of individuals is true to nearly the same extent of communities; and hence the conviction, becoming every day more general, of the importance of financial, for the illustration of general history.

The paper which I have the honour to submit is so elementary in its details that I have to claim the indulgence of the meeting; but, as the practical application of the simple formulæ to which I shall have occasion to refer may be interesting, I trust I may not unworthily occupy a short portion of your time.

A few months ago it occurred to me that tables to facilitate the process of multiplication, based upon the identity

would be of extensive use in actuarial and other calculations, and would effect a considerable saving of labour—to an extent even greater than that acquired by the use of logarithms. This led me to give attention to the subject; and after a trial on a small scale, I deemed it of sufficient importance to construct for my own use a series of tables of the fourth parts of the squares of all integer numbers up to 100,000.

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.

The Duke of Wellington, as Master-General of the Ordnance, having given orders in 1825 that practical architecture should form part of the course of instruction of the junior officers attending the Royal Engineer Establishment at Chatham, under my direction, which had hitherto been confined to their duties in the field, and a Professor in every respect competent having been selected for that duty, I was induced to enter into the details of measuring and estimating buildings and other works myself—which I had never done before, because, like other officers of my corps, who studied the other branches of practical architecture and engineering with pleasure, I had, when employed in garrison duties, left those details to the clerks of works, and to the overseers and foremen of the department.

The values which it has been heretofore customary to tabulate for actuarial purposes, in connection with specified rates of mortality and interest, are those of annuities on single and joint lives at all ages; and from these, by operations more or less complex, and with the aid of subsidiary tables, the actuary has had to form such other values as might be required in the solution of any particular problem in hand. Of the values thus requiring to be formed, those of the single and annual premiums for assurance occur perhaps more frequently than any others; and hence the desirableness of simplifying as much as possible the operations by which these are deduced from the corresponding annuities. To effect this simplification was the object of Mr. Orchard's work, in which, as will be admitted by all competent to form an opinion on the subject, the author has been completely successful. The arithmetical operation heretofore necessary has been entirely superseded, and the required values are found simply by inspection.

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.

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.

To ascertain with precision the loss of life occasioned by war is an object of so much importance in every point of view, whether national or social, that whatever apology may be required for the manner in which the present subject is treated, none can be necessary for bringing it forward.

That so little should have been done towards the elucidation of the question, is no doubt partly attributable to the difficulty of obtaining accurate information with respect to it, but in a greater degree, perhaps, to the reluctance to dwell upon it felt by the public.

In this paper Mr. Meikle states it to be his object to simplify and popularize what has been written by Mr. Gompertz and Mr. Farren on the subject in question. He says, “In the construction of tables of mortality, the results arising out of the statistics, viz. the logarithms of the number alive at the beginning of each year, or the logarithms of the probability of living one year, generally proceed so irregularly from age to age that it is deemed expedient to graduate and reduce them to a more consistent and harmonious form. For this purpose it is thought that the characteristic features of the table which is being developed will be as equally retained in any set of equidistant terms of the series as they are now exhibited in the rough and irregular items of the table. Selecting, therefore, the terms of the series at ages 10, 20, 30, &c., or at ages 10, 20, 40, &c., and making these the fixed points of a new series, it is required to interpolate for the values at the intermediate ages.”

I. The object of the first part of this memoir was to determine the normal course of mortality as affected by age alone, without reference to other circumstances.

What we name the law of mortality is not a simple law, but a compendious expression, by which we denote the operation of various laws, physiological, physical, and moral. Of these, the physiological laws are so uniform in their operation that they impress certain characteristic features upon the law of mortality, according to age, which are observed amidst all the diversities which it exhibits under varying circumstances, physical and moral.

The progressive rates of mortality, from birth to extreme old age, in a given number of any population, are known from the proportionate numbers who die at the different ages. This is the only mode of estimating correctly the expectation or conceived “security” of life at any given age—viz., by showing the chance which an individual has of living a certain number of years or shorter periods.

In a former Number we took occasion to direct attention, in general terms, to the unnecessarily high rates of premium charged for life assurance in India; we propose, in the present article, to present our readers with some further observations on the same subject.

Everybody who is in the habit of reading the newspapers must have been, if not enlightened, at least amused, by the claims to public support put forth in the advertisements of rival Insurance Companies. Ignorance of the theory and practice of life assurance is, even in England, profound and very nearly universal; but here this ignorance is in no small degree aggravated by the obscurity which is supposed to prevail on the subject of European mortality in India, and by the ambiguous and contradictory statements which continually meet our eye in newspapers and on the covers of magazines.

The application of the doctrine of chances to questions connected with the money values of contingencies depending on human life is now become so common, in this and many other countries, that we may he apt to forget its origin and the process by which it arrived at its present degree of improvement. Although the history of this calculus may be well known to some members of this Institute, there may be others whose time is, perhaps, too occupied to allow of more than the mere study of the calculations and formulæ necessary for the daily routine of a Life Assurance Office, and who may not be aware of the number of other subjects to which, at an early period, the calculus of probabilities was suggested to be applied. I hope, therefore, that in taking a rapid glance at the changes and improvements effected since the first crude ideas on the subject, I may not only be able to excite an increased interest in the science, but induce a further extension of its usefulness.

A Specific Policy is one which insures stated sums on particular risks, and guarantees the assured against any loss thereon within the limits of the amounts insured. It differs from anAverage or Floating Policy in several respects—the latter, amongst other privileges, having only to bear a share of any loss compared with the whole loss, similar to the proportion which the amount it insures bears to the whole value of the property insured at the time of the fire.

The object of this paper is to give a brief exposition of the different possible methods or classes of methods of equitably dividing the net profits of a Mutual Life Assurance Office among the members. By an equitable division must be understood one which is fair not only between persons of different ages entering the Office at the same time, but also between persons entering the Office at different epochs of its existence,–one from which the members of different ages will, relatively to one another, receive the same benefit from the profits during the earlier and growing years of the Office, and during years when the amount of its business is declining, as during its stationary period. I assume the premiums actually paid to be composed of the net premium calculated from a correct life table, and of a charge of 25 per cent. on the net premium added to guard against unfavourable contingencies, and that this additional charge is really the source of all the profit divisible among the members; or at least, that whatever profits arise from any other source may be fairly apportioned in the same proportions as appear equitable on the supposition that the charge is the sole source of profits. The addition to the premiums for defraying the expenses of the Office, for the sake of simplicity, I leave out of account. In consideration of the payments of the successive net premiums, the assured is guaranteed a sum at death such that the present prospective value of that sum is equal to the present value of the successive net premiums. Similarly, in consideration of the successive charges on the premiums, he should have the prospect of receiving a sum or sums either at death or at any times before that; not guaranteed, it is true, to any fixed amount, but variable with the circumstances of the Society, such that the present value of such sums or bonuses shall be equal to the present value of the successive charges. That this equation between the present values ought to subsist is as clear in the case of the charges and bonuses as in that of the net premiums and sums assured. If the system of division of profits be such that this equation always subsist, it must be fair as respects each member; and any Office adopting a system in which it does not subsist is committing as great an injustice towards its members as one which calculates the premiums on false life tables.

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.

As the reasonings of the actuary, when extended beyond the mere official routine of a Life Assurance Office, depend upon the proper collection of data relating to population, and the other important elements of the condition and progress of a country, we need make no apology to our readers for presenting to them the following very interesting supplement to the valuable Report of Dr. Farr on the Statistical Congress at Paris in 1855:—

“There are seven great powers in the world.

“England, France, Turkey, and Austria, have existed as great powers for several centuries. Prussia, Russia, and the United States of America, have entered this class within the last hundred years.

It is well known to those who have studied the theory of chances. that where the magnitude of a risk may be divided into an indefinite number of parts, it is possible, by this multiplication of smaller parts, to confine the probability of the occurrence of any given excess or deficiency in the result, above or below its mean value, within any given limits; and it is of great practical importance to ascertain both how far the subdivision ought to be carried with regard to insurances, and what are the inconveniences to be apprehended from the admission of occasional deviations from the general rule.