The subject of Graduation has recently been brought somewhat prominently before the profession in connection with the exposition of various methods that have been proposed. While the bases of these individual methods have been set forth in a way leaving nothing to be desired, the precise relation of each to the entire problem of the adjustment of probabilities derived from statistics, does not appear to have been so clearly marked out; and therefore, in the following observations, it is proposed to keep prominently in view the conditions under which one method is to be preferred to another.

An intelligent outsider, desirous of becoming acquainted with the actuarial theory of graduation, might, if he overlooked the necessarily limited application of particular systems, derive from his researches an impression of disagreement among the various authorities, which would have a rather bewildering effect.

The question ov the calculation ov premiums for the insurance agenst issue to a man at present marryd but without a family, has been on two ocasions bro't under the notis ov the Institute by Mr. Archibald Day. On 31 January 1859 he redd his paper On the determination ov the rates ov Premium for asuring agenst Issue (Journal viii, 127), in which he explaind his proces of calculating, by means ov statistics furnisht by the Registrar General's returns, the present value ov £1 payabl at the end ov the year in which a husband shal contract a second marryge. On 24 March 1862 he redd a stil more important paper, On the statistics ov first and subsequent Marryges among the Familys of the Peerage, considerd specially with reference to the calculation ov Premiums for Asurances agenst Issue, in which the same method was employd to obtain more accurat premiums from the more apropriat statistics furnisht by the peerage familys. The general problem is a very complex one involving several distinct contingencys. Mr. Day delt very succesfully with som ov these contingencys, but his treatment ov the subject was confessedly incomplete, and he himself speaks ov it as giving only a ruf aproximation. He omitted altogether from the calculations the circumstances that the insurance is not payabl until the deth ov the husband after contracting the second marryge, and that ther is a probability ov his second marryge being unfruitful. I hope on som future ocasion to take up the general question and carry the investigation ov it one or two steps beyond the point at which Mr. Day left it, but on the present ocasion I shal deal only with one element involvd in the calculation, namely, the probability ov the remarryge ov a widoer.

That the Institute of Actuaries has, during the one-third of a century of its existence (1847-79), been of real utility to the Life Offices—directly by the training of experts, and by the compilation of the Actuaries Experience Tables; and indirectly by lending practical aid to the development of the science of life contingencies—is a fact which will readily be admitted by all who are conversant with its history. That another function may he added to its utility it is my purpose on the present occasion to show.

The business of life insurance has, during the present generation, undergone a very complete transformation. The change began in the year 1844 with the passing of the Joint Stock Companies' Registration Act, 7 & 8 Vict., c. 110. Under this Act facilities were given for the formation of Joint Stock Insurance Companies such as had never before existed. The life insurance associations existing at that date were institutions of good repute; were, in fact, regarded as money-making projects of the highest class, and it was certain, therefore, that promoters would turn their attention in this direction. The failures which had resulted from the Joint Stock Company Epidemic of 1824-5 had passed out of remembrance; the course was clear for a new spurt in the same direction. Hence, a large number of new life offices were, in fact, founded; and in their struggle for existence, an amount of competition was engendered, which (while productive of good in widening the area of the insuring class, by bringing the practice down to the middle, and, gradually, to the lower ranks of the population) has eventuated in the evils whereof I here propose to speak, and concerning which it is my design to attempt a remedy.

In a former paper, I had occasion to make some reference to the question of Local Medical Examiners, in tracing the changes in the practice of life insurance during the present century; but I then purposely avoided entering upon a detailed consideration of the points involved, believing these to be worthy of special attention whenever a suitable occasion should arise.

The important part which local medical examiners are called upon to take in the every-day practice of life, and some other important branches of, insurance business, is too well understood in this Institute to require any specific mention. It is enough for my momentary purposes, to say that a most sacred trust is reposed in them; and it will fall within the limit of these purposes to consider in some detail, on this occasion, the manner in which this trust is on the whole performed.

The first difficulty which arises in the treatment of a question of this character, is the fear of doing injustice, by implication or otherwise, to a body of gentlemen, a considerable proportion of whom, I trust, do conscientiously perform their important functions. But on this point it has to be understood, that on those who do discharge their duties honorably and efficiently there can be no adverse criticism. Our remarks, then, in the nature of the case, can only apply to those who are regarded as not acting up to this high but essential standard; and as to these latter, I venture to think that—in very many cases certainly—the true position of matters has but to be placed in a clear light, in order to lead to very speedy and decided improvement.

There are several cases in which the factors of a number may be of use in ordinary mathematical processes; as for example, in the calculation of logarithms. This, although not the raison d’être of such a table, is so important an application as to deserve special notice, for by its means the number of numbers whose logarithms are known is greatly extended. For example, Abraham Sharp’s table contains 61-decimal Briggian logarithms of primes to 1,100, so that the logarithms of all numbers whose greatest prime factor does not exceed this number, may be obtained by simple addition. Similarly, Wolfram’s table gives 48-decimal hyperbolic logarithms of primes up to 10,009. Thus a factor table forms a very valuable complement to many-place logarithmic tables, the range of which must necessarily be comparatively limited ; and even in the case of large prime numbers, or numbers exceeding the limits of the factor table, it affords a very convenient method of calculating logarithms.

In Part I ov this paper I gave (vol. xxi, page 246) a tabl ov the finally ajusted probabilitys ov deth in the first five insurance years, for quinquennial ages at entry from 20 to 75. I gave also specimen mortality tabls for each ov those ages at entry, so ajusted as to join on, after the laps ov five years, to the Institute HM(5) Tabl; also values ov anuitys at 3 and 4 per-cent during the first five insurance years for lives entering at the same ages; and, lastly, commutation tabls calculated at 3 and 4 per-cent interest. I hav since found that ther is a misprint in the tabl ov the probabilitys ov deth, as q[35]÷3 shoud hav been printed ·010177 insted ov ·010077. This, however, is a mere misprint and dos not afect the other tabls. (It wil be notist that, insted ov riting q35+3, as in the first part ov this paper, I now always enclose the age at entry in square brakets, thus [x].)

In order to make the new tabls practically useful, it is necesary to hav results, not merely for quinquennial ages at entry, but for every age. The first step to be taken for this purpos is the calculation ov the probabilitys ov dying in the first five insurance years, for every age at entry. These calculations wer made by means ov the formula explaind in my paper in the number ov this Jurnl for last July, in which I gave (see page 280) the actual work ov interpolating between q[30]+3 and q[40]+3. But for the two quinquennial intervals at the beginning and at the end ov the series, namely, from age 20 to 30 and 65 to 75, I uzed constant fifth differences insted ov fourth differences as explaind in my paper. Whenever the calculated probability exceeded that acording to the HM(5) Tabl, the latter was substituted.

This paper had its origin a year or two ago in an investigation entered upon with an object which has since fallen into abeyance. At that time, it being desired to obtain a thoroughly well-graduated table of mortality giving a favourable view of human life, attention was directed to the Carlisle Table, and sundry attempts were made to remove its irregularities.

Many of the experiments then tried would not now be of any public interest, but the effort to use Mr. Makeham's formula brought out a few facts which it is perhaps worth while to record; and, at any rate, whatever may be thought of the present notes, the authors do not consider themselves to be guilty of any impropriety in bringing forward for discussion once more the practical employment of Mr. Makeham's expression for the law of human mortality.

They are the more emboldened to suggest this subject for debate, in that Mr. Walford, at a late meeting, announced that the formula was found in America to give perfectly satisfactory results when applied to a table formed from the experience of assured lives on that continent.

The first step in the original investigation was to graduate the Carlisle Table (as published by Mr. Milne) by means of Mr. Woolhouse's formula; and as the resulting curve was not entirely free from roughness, the process was repeated. The changes introduced by the second graduation were insignificant, but the table produced was one of very great smoothness, although there was a wave-like form observable in the succession of the values of some of the functions.

A Number of suggestions made at various times by different persons for subjecting Fire Insurance Statistics to systematic treatment were reviewed by Mr. Cornelius Walford some time since in an Article “On the scientific application of Data to the purpose of deducing Rates of Premium for Fire Insurance”, which is published in the April 1878 number of the Assurance Magazine. The present paper is an attempt to lead the investigation into another course, and its purpose is to enunciate a Theory of Statistics in dependence on which algebraic formulae may be constructed and arithmetical calculations made of the values of risks. The particular line it will take is that of averaging quantities, but it could be developed in other directions if that were desirable.

Proceeding, without further preface, to the consideration of the subject, we observe :—

That the readiness with which materials can be ignited by any particular agency varies—

1. In consequence of their constituents being different, as Wool contrasted with Cotton.

2. From the same kinds of substance being in different forms, as timber in logs, deals, shavings, sawdust, or splinters.

3. From substances existing in different states, as flax lying loose in repose, and flax tossed about and being cleaned in a heckling-house.

It is sometimes necessary in disentail proceedings to estimate the probability that a husband, who has been married for several years without having any children, will hereafter have issue, either by his present or by a future wife. The probability of issue by a future wife is a subject comparatively well understood, and it is a recognized part of the business of some insurance companies to grant insurances against the contingency of such issue being left surviving at the death of a married man who has no child. The probability of issue by the present wife has been very little considered, and it is this probability that I propose now to discuss. There are very few statistics that will help us in the matter; in fact, I am only aware of two works in which any such statistics are given. Dr. J. Matthews Duncan gives the following table, in his work Fecundity, Fertility, and Sterility (p. 132, 2nd Ed.) :—

Mr. Charles Ansell, Jun., in his Statistics of Families in the Upper and Professional Classes (page 87) gives a table from which columns (1) to (4) of the table on the opposite page are extracted.

The question of the reserve which a life assurance society should hold, in order to meet the claims under the policies issued by it as they mature, is one of such vital importance that I am led to hope that, notwithstanding the many and valuable papers on the subject which have from time to time appeared in the Journal of the Institute of Actuaries, the following reflections may not be without interest, even though they may not throw much fresh light on the subject. I may premise by stating that the conclusions arrived at, and hereafter referred to, are the result of an investigation undertaken as a matter of curiosity, and carried on in the midst of laborious and urgent official duty; consequently the scheme propounded may appear somewhat crude and ill-defined, and doubtless lacks the completeness which a more mature consideration of the point taken up would possibly have developed. Further, the results given here were arrived at before Mr. Sprague's paper “On the Construction of Select Mortality Tables” (Journal of the Institute, vol. xxi, p. 229) appeared in print; otherwise I should probably not have undertaken the investigation.

What I have aimed at is the discovery of some method of arriving at the requisite reserves for new policies, which shall be somewhat further removed from the realm of pure theory than some others which have been suggested; a method which can be readily applied in practice; and, what is of more importance, one which can be made intelligible to the ordinary man of business.

In my paper On the Construction ov a Combined Marryge and Mortality Tabl (J.I.A. xxi, 423), I shoed how the statistics suplyd by the Scottish Registrar-General may be employd to calculate the marryge rate among bachelors. I wil now use the same statistics to calculate the marryge rate among widos and spinsters. As I stated in the paper abov referd to, the Scottish marryge statistics apear to be furnisht with greater exactnes than the English, and the conclusions drawn from them as to the marryge rate, seem therefore much more trustworthy than the corresponding conclusions drawn from the English statistics.

In a book entitled Principles of the Algebra of Logic, which deals with the Sciences of Necessity and Probability as one whole, I have considered as an example the question in probabilities discussed in vol. xxi (p. 204) of this Journal, namely—“ A says “ that B says that a certain event took place : required the pro- “ bability that the event did take place, p1 and p2 being A’s and “ B’s respective probabilities of speaking the truth.”

When a series ov quantitys proceed by a regular law, ther is no difficulty in interpolating between each ajacent two ov them any number ov terms with any desired degree ov accuracy. The methods to be adopted for this purpos hav been described in varios original and reprinted papers containd in the Jurnl ov the Institute ov Actuarys : (see, for instance, Mr. Woolhouse's paper On Interpolation, Summation, etc., xi, 61 , 301, and xii, 136; Mr. Gray's paper On the Construction ov Tabls by the Method ov Differences, xiii, 61, 149, 293, and xiv, 307; and the translations ov M. Maurice's Essay on Interpolation, xiv, 1; and ov Briggs's explanation ov his Method ov Quintisection, xiv, 73). We somtimes, however, hav to interpolate between quantitys which do not accuratly folio any law. For instance, we may hav calculated premiums for every quinquennial age, and wish to obtain the premiums for the intermediat ages by interpolation. In this case, it wil somtimes happen that the ordinary formulas ov interpolation do not giv satisfactory results, unles we take a very large number ov differences, and then the amount ov labor is offen more than the result is worth. In my paper On the Value ov Anuitys payabl half-yearly, quarterly, etc., I indicated briefly a method ov interpolation which I tho't miht be employd with advantage in such cases (xiii, 322), and it is now my intention to work out the formulas resulting from that method, and giv a practical ilustration ov their use.