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.

Perhaps I owe to the Institute an apology for bringing forward so late in the day this second part of my paper. When first I resolved to enter on the investigation of which I have now the honor to submit a second series of results, the ground, at least in so far as the experience of the 20 Offices was concerned, was practically unoccupied; for although Mr. Sprague had written on the mortality prevailing among assured lives, yet, as he had not touched on financial matters, I conceived that there remained ample room for another worker. It was only when I was on the point of sending in the part of my paper which has already been printed in the Journal, that I heard of Mr. Berridge's labours; and not till the other day, when the calculations for this second part were all but finished, did I become aware that Mr. Sprague intended to deal with the question of the premiums for the insurance of recently selected lives. Had I known that these gentlemen were engaged very similarly to myself, I should probably have been deterred from my task: but seeing that in ignorance of their movements it has been completed, perhaps the results will not be considered valueless.

Some years ago I had the honour of reading before this Institute a paper which, though mainly devoted to a study of surrender values, treated such surrender values as purchases by life offices of reversions. On that occasion I stated that sound British companies could well afford to repurchase their own acceptances at 6 or even at 5 per-cent. Subsequent experience has done much to confirm me in the opinion then expressed, but I did not then, nor do I now assert that offices should on such terms purchase other reversions than their own policies; and I trust it will not be thought wasting your time, if I preface this paper by enumerating a few considerations which seem to indicate that the present market price of reversions gives, comparatively, a return none too high to an investor. Such market rate I take to be 5 per-cent in the case of an investor who does not stand, or, what is the same thing, arrives at his price as if he did not intend to stand, any risk of loss, whether through longevity of life tenant or death of reversioner.

I may say at the outset, that the object of the present paper is not to discuss the merits of Mutual Assurance, as distinguished from the other systems of Life Assurance, but to consider whether there are not special circumstances connected with that system which demand the attention of the professional actuary.

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.

I cannot better introduce the subject of this paper than by a quotation from Mr. Morgan’s Preface to his Tables of the Experience of the Equitable Society. He writes as follows :—

“In a body of lives of the same age, all selected as healthy from the general mass of mankind, it is obvious that the rate of mortality must be considerably less for the first ten or twenty years after selection, than amongst those from whom they are thus chosen. As, however, these selected lives advance in age, their general health and the rate of mortality amongst them will naturally approximate to the common standard. This approximation cannot be accurately estimated, if the observations be taken from a blended mass of lives, selected at the age of twenty, for instance, which have attained the age of fifty, and of lives selected at the age of fifty. The rate of mortality will for the former class be less, for the latter greater, than that which occurs in either class separately.

What is the best method to use in the graduation of a mortality table, although not a new question, is one which always must have a practical interest for the profession; and as the various methods of graduation which have been devised from time to time can all boast of their respective admirers, the subject is one which cannot easily be disposed of at present. In the following paper it is not proposed to open up any discussion on the general principles involved, nor is it proposed to go into any detail as to how far these general principles have been carried out in the various well known methods of graduation. The purpose of the paper is of a somewhat different nature.

The folloing results wer obtaind in the course ov the years 1870 and 1871, the calculations having been carryd on at intervals, as my professional engagements allowd ov the time ov myself and my assistant being devoted to the matter. It was my intention to carry the investigation still further, by forming graduated mortality tables for entrants ov varios ages and deducing annuity values therefrom; but this intention not having been carried out, I think it better to lay the results in their present state before the readers ov the Jurnl, insted ov delaying them longer in the hope, which may perhaps never be realized, ov more complete.

All our empirical knowledge depends not only upon the exactness, trustworthiness, and frequency of the observations on which it is founded, but quite as much upon the possibility of drawing general conclusions from the observations. The great variety of form in which the results of observation appear, even independently of the diversity of the nature and the method of observation, will easily explain why the range of the conclusions to be drawn from the observations, differs widely according to the field of the science to which they refer. While the natural philosopher must be satisfied with the mere knowledge of a fact in a field which has not previously been investigated, he goes in other realms of science far beyond this first step of observation; he obtains an explanation about the causes of the phenomenons and the connection of the forces and the laws which govern the events that come under his observation. Such explanations cannot in general be drawn directly; they require a number of intermediate steps in which the observed result has to be critically examined and analyzed into its single elements. The proceeding to be adopted for this purpose depends upon the nature and form of the observation, and it would be a mistake to suppose that we can follow the same method for all kinds of observation.

Some apology is needed, perhaps, for proposing to rediscuss a subject which has been handled so often, and by so many distinguished actuaries. But I am probably addressing gentlemen who have not had occasion to become familiar with all that has been written or said on the subject, and others to whom it may usefully be presented from a fresh point of view; and there are some important practical questions too on which the profession seem scarcely yet agreed, which will admit of further consideration. I shall not attempt to notice the various papers which from time to time have been contributed by actuaries towards the solution of the questions I am to consider. For many of the views I am to express I am necessarily indebted to those who have preceded me; and the aim of this paper is less to introduce what is new, than to enforce, and illustrate if possible, sound and correct opinions. I am anxious to set the subject in a light which will make it intelligible to others than actuaries, and to discuss principles rather than formulas.

The strict analogy which exists between these two classes of annuities has doubtless received the consideration of the majority of students, but I am not aware that it has ever in print had that attention which it deserves; and at any rate our usual text books hardly even refer to it. Although it is shown, in the short but suggestive article of the late Prof. De Morgan (vol. iv, p. 277), in the most general way, that years, or periods of time, may be interchanged with any other kind of status whatsoever, whether certain or contingent; yet, as successive statuses are almost exclusively spoken of, the less complex cases are liable to escape notice. Under these circumstances, the following brief notes may not be thought superfluous.

If, a few years ago, a paper had been offerd to the British Association on the subject ov the insolvency of life insurance companies, it woud probably hav been met with the objection that it was ov too theoretical a character. No life insurance company in the contry had ever been knoen to becom insolvent, and ther was no reason to anticipate that such a catastrofy woud ever occur. The well knoen remark ov the late Prof. De Morgan miht probably hav been quoted, that ther is nothing at all comparabl in point ov security with a well establisht and prudently conducted life insurance company.

The influence of Occupation upon health and longevity, is, I think, a subject deserving of even more attention than has hitherto been given to it by the Actuarial and Medical Professions. As the benefits conferred by Life Assurance are now almost universally recognized, they are being taken advantage of by classes of the community which, until the last few years, did not seem fully to appreciate them. It is therefore of no little consequence to have, as far as possible, a clear understanding of the relative risks attaching to different occupations and to the various branches of industry.

In the first part of this paper, it has been shown to what extent the reserve under a single policy is affected by various methods of graduating the mortality experience on which such reserve is based. We will now endeavour to obtain some notion of the financial effect upon the total reserve made by an office at a periodic valuation of its liabilities.