When your Secretary asked me some months ago if I would talk to you about research I made some enquiries and was told that many of you would like to undertake some kind of research work but found it hard to think of suitable subjects. Even if you had subjects, how were you to set to work and what were the possible pitfalls? I am doubtful if I can be of much help to you, and in one sense each man who wants to do any original work must do it in his own way and must find his own subject. The ideal subjects are those of which no one else has ever thought. There is an infinite number of them, and they have been “waiting on the tedious shores of Lethe millions of ages” for “existence and a name” till you bring them to life. To any one of you who may find and solve such a problem “I am not worthy” to speak for he has that touch of inspiration that we poor mortals call genius, being perhaps fearful of using the word “divinity,” but even such an one may like to try his prentice hand before he embarks on his great adventure.

The following notes are for the most part based upon the actual valuation of a registered friendly society and an attempt is made to describe the various operations involved and the points which arise for consideration, from the time the actuary (or valuer as he is commonly known in friendly society circles) is approached, to the time he signs his report and presents it to the society. I have, however, digressed occasionally from the main subject of these notes in order to express my views on certain general points which frequently arise in the valuations of friendly societies. It should be understood that these notes do not necessarily embody official opinion.

The specimen society is a typical “centralised” rural society (i.e. without branches) of about 1,000 members, their occupations being described as domestic, horticultural and agricultural.

The object of these notes is to give a brief introduction to the treatment of proposals for life assurance which are not acceptable at ordinary rates.

In theory, the extra premium charged for a certain type of case should be based upon the abnormal mortality which is expected. Little, however, is known about the rates of mortality applicable to any class of extra risk, while in practice extra premiums involve a comparatively small amount of cash and cause much trouble, particularly to the outside staff. The imposition of extra premiums is therefore restricted as far as possible, and a tendency towards their virtual abolition would probably be much accentuated, but for fears that to do so might involve an undesirably large influx of second class business.

It may not unfairly be said that the earliest approximate valuation method—Woolhouse's—was born out of its time. Presented as it was nearly fifty years ago to a world whose views on such matters were rather different from our own, it attracted but little attention, and appears to have given rise to no subsequent developments. Indeed, the author himself seems to have thought rather slightly of the suggestion, and in the brief discussion which followed the reading of his paper, emphasised distinctly that he regarded the method merely as an auxiliary check. The truth is, the idea was altogether too simple, and so has suffered half-a-century of neglect whilst other methods (some of which have received impressive algebraic treatment and given rise to a quite extensive literature), dating from independent investigations made many years after Woolhouse's, have been devised, tested, acclaimed and have passed into current use. And so we come to the latest pronouncement on Woolhouse's method, the almost ex cathedra sentence passed by Mr. Lochhead in “Valuation and Surplus“ (page 52) that the “method is extremely simple, but is not very powerful.”

The law which governs life assurance contracts, their formation and their execution is, to a very large extent, the same law as that which governs mercantile contracts generally. The titles to life policies are affected by the law of contract, of mortgage, of partnership, and so on. There are, however, some legal principles and statutory provisions which affect life assurance alone, and among them are the following sections in the Married Women's Property Acts, 1870 and 1882.

Married Women's Property Act, 1870, Section 10.

A married woman may effect a policy of insurance upon her own life or the life of her husband for her separate use, and the same and all benefit thereof, if expressed on the face of it to be so effected, shall enure accordingly, and the contract in such policy shall be as valid as if made with an unmarried woman.

It may be well to recapitulate briefly the questions asked at the recent Census, and the statistics which were published, showing the results of the previous Census. The questions asked on the 1931 householder's schedule comprised name, relationship to head of household, usual residence, sex, age, marital condition, birthplace, nationality, occupation, industry, and size of house. The volumes based on the 1921 Census dealt with occupations, industries, workplaces, dependency and orphanhood, ages and marital conditions, education, birthplace, nationality and housing, and general, sectional and occupational mortality, the subjects being dealt with both nationally and, in a condensed form, for each administrative county. For simplicity, attention is confined in this paper to the Census in England and Wales.

The object of this paper is to draw attention to a process of checking the accuracy of the valuation results of any financial year from the preceding year's valuation results (the attained ages as at valuation date increasing by one year) by means of the elements which connect both valuations. These elements are the interest required to maintain the reserves, the net premiums, the reserves of the policies coming in and going out during the financial year, and the expected death strain. In the appendix to this paper the checking process has been applied to the revenue account for the year ending 31st December, 1921, given in Appendix B of Mr. C. Carpmael's paper in J.S.S. Vol. II, No. 2. The determination of the profit or loss from each source has been made on two different assumptions—namely, the one adopted by Mr. Carpmael, that the income and outgo are uniformly distributed over the year allowing half-a-year's interest at valuation rate and the other adopted in the North American gain and loss exhibit of the convention edition by which no allowance is made for half-a-year's interest for the two cases in which the policies are assumed to come in and go out at the beginning of the valuation year as done by Mr. Carpmael in his paper, and in the middle of the valuation year.

A very large proportion of the life assurance policies in force in the British Isles give the assured the right to participate in the profits of the issuing office. Where a policy has been effected in recent years, it is almost certainly the case that the assured at the time of making his proposal, was given some indication of the bonuses which he might expect to be allotted to his policy. Sometimes an intending assured merely learns the latest rate of bonus declared, or the rate of interim bonus for the time being in force, by reading the prospectus of the issuing office, while on other occasions figures given verbally by an agent, or in a leaflet or letter describing the proposed policy, contain an allowance for future bonuses. It is almost invariably the case that some information along these lines is given and, no matter how guardedly it is done, the assured regards the figures which he has received as the office's estimate of the bonuses which will in due course be allotted to his policy.

For the purposes of this paper the term “curve-fitting” is not synonymous with “graduation.” It is restricted to graduation by means of an explicit algebraic or trigonometrical formula: graphic and summation methods are excluded. Even so the subject is unlimited, and this paper cannot provide more than an explanation of a few general principles, and a brief exposition of some of the problems of special interest to the actuary.

Our starting point is this: we have decided to fit a curve (vx) to a series of crude figures (ux) and the problems that confront us are:—

(a) What formula are we to ascribe to vx?

(b) How are we to choose the constants in vx in the best possible way?

(c) How are we to test the goodness of fit and to compare the results of different graduations?

In considering the question as to whether an assurance can be granted to a particular proposer, the Chief Medical Officer and the Actuary of the Company have available the Medical and other reports of the life proposed. From the information given in those reports they have to form an opinion as to the present state of health and likelihood of longevity, taking into account all the factors before them. In some cases it does not require a doctor nor yet an actuary to decide that the policy can be issued at ordinary rates, but in other cases there will be some degree of impairment, varying from the case where a spin of the coin might be as fair a method as any, to the case which is obviously uninsurable at any rate. It is the border line cases which cause the most inconvenience. A certain proposer is slightly overweight, so we do not like the look of him, but this may be counterbalanced by favourable chest and abdominal measurements. Then, again, there may be a lack of longevity in the family history, but on the other hand he may only require a short term endowment assurance.

Everyone has, theoretically, the choice, or option of being or declining to be assured. In order to prevent discrimination being exercised against him it is usual for the assurer to demand some evidence of health before issuing an assurance at his advertised rate of premium, and this method of preventing discrimination by rejecting proposals below a certain standard results in the phenomenon of temporary selection. That the assurer is not primarily concerned with equity may be seen by considering group assurance. The assurance is of different value to the different members of the group but as they have no option, or a very restricted one, this does not militate to the disadvantage of the assurer.

This type of policy was introduced into British Life Office practice some three years ago and rapidly proved most attractive; so much so that a leading office announced that it obtained over £2,000,000 new Sums Assured under this class of contract within a year of issuing the prospectus. Details of the schemes of no less than forty-six offices are available. On the academic side, the examiners have shown their appreciation of its possibilities by setting questions concerning calculation of premiums, valuation, etc., in the examinations of both the Institute and the Faculty of Actuaries.

Sufficient time has now elapsed since the commencement of this type of business for the problems involved to force themselves into notice. Certain of these problems are dealt with in the following note.

Prior to 1915 profits arising from monies invested by Life Offices in the purchase of Reversionary Securities were untaxed. Rapidly increasing expenditure forced the Chancellor of the Exchequer to break new ground and the Life Offices were invited to show cause why this apparent anomaly of an untaxed source of revenue should be allowed to continue. In due course there remained only to be determined the basis of taxation of these profits.

Mr. Geoffrey Marks, who represented the Life Offices in the negotiations with the Revenue Authorities, was instrumental in obtaining a valuable concession in respect of contracts completed prior to 5th April 1915.

The immense growth of Industrial Assurance in Great Britain during the last 80 years is a matter of common knowledge. A plain record of the technical developments which have from the first formed an unobtrusive background would make an interesting story. We should read of the great cost of financing expansion, of the prolonged and ultimately successful struggle to raise the reserves to a satisfactory actuarial standard, and finally, of surpluses sufficiently large to cause actuaries and others furiously to think in arriving at satisfactory methods of distributing them.

There is neither space nor time to tell the story here. We must begin by accepting the fact that relatively large surpluses are now commonly revealed on the valuation of Industrial Assurance funds and that various problems have to be solved in deciding upon the best methods to be adopted in dealing with them.

A great deal of interest was aroused by a paper entitled “The Numerical Rating System,” which was submitted to this Society by Mr. C. F. Wood a few years ago. In that paper Mr. Wood showed how the numerical ratings are used in arriving at the final terms, standard or otherwise, on which policies are issued. A basic rating for build ranging from 95 upwards, is first determined and to this basic rating are added ratings for any adverse features that are reported, these additional ratings varying from as few as 10 points for minor impairments such as remote history of pleurisy or kidney disease, slightly elevated blood pressure and absence of longevity in family history, to heavy ratings of 100 or more for very serious impairments such as mitral regurgitation and other heart impairments, glycosuria and recent history of pulmonary tuberculosis. Having allotted the appropriate numerical ratings to each adverse feature, the total is obtained and from it are deducted certain small allowances for plan and good family history if the case is entitled to them.

The title to this paper covers a very wide field and unless each of the subjects was dealt with somewhat briefly, the paper would be of inordinate length.

The general principles on which a valuation should be made are the same for Industrial as for Ordinary Assurance business. The methods adopted differ, however, one of the main differences being the system of grouping the policies in “select” form, either by calendar year of issue, sub-grouped by age at entry, or by age at entry sub-grouped by year of issue, the former being possibly the more popular and the one described in these notes. One of the original reasons for this form of grouping was the system of increasing the sums assured after a policy had been five years in force and again after it had been ten years in force, and in consequence the duration of the policies had to be taken into account in the valuation.

The student approaching the subject of summation formulae is constrained to follow the historical order in which it develops in the Journals of the Institute and the Students' Society. Not only is his path made harder by the difficulty of finding a suitable starting point, but the development he must follow is not always the logical one.

The purpose of the present note is primarily to provide the student with a starting point. If a few of the more important conclusions of the subsequent reading are anticipated herein the author feels that the student's appreciation of the original papers by Hardy and Lidstone will be none the less keen for having already navigated some of the wider channels.

It should be added that except in the method of preparation no originality is claimed for this note.

The definition of the term “probability” is almost as illusive as that of the term “electricity.” Much as the average person knows that the effect of an electric current is to produce certain phenomena with which he is familiar, so he understands in a general manner what is implied by the words “the probability that an event will take place.” The term is employed colloquially in many instances in which the quantitative idea is missing. “ Kain is probable today”; “He will probably be late”; and so on. Frequently, however, some measure of probability is implicit. For example, if the odds against a specified horse are quoted as being longer than the odds against another horse, it is understood that the chances in favour of the second horse are greater than those in favour of the first. It is to be supposed that it is more probable that the second horse will win than will the first, and a certain amount of arithmetical work has been necessary in order to obtain the figures for the odds.

These notes assume the reader knows the general principles on which bonus reserve valuations are made.

The essential idea is to value as a liability the amounts the office will be liable to pay as sum assured, bonus and expenses, setting of f as an asset the amounts the office will receive—the office premiums. The valuation must be based on really accurate estimates. We must use the mortality we think will prevail and the interest we think can be relied on in respect of existing business. Admittedly bonus is uncertain—a valuation of future bonus does not guarantee it—but the valuation must be as accurate as possible. The bonus allowed for will be realised if the conditions assumed prevail in future.

The table of mortality is usually a standard table to which the experience of the office may be regarded as an approximation, but need not necessarily relate to the same class of life. The essential qualification of any table for use must be that it is a real approximation to the likely future experience of the office for which it is used.