The following remarks do not purport to contain any original matter but are merely an attempt to outline the routine followed in the valuation of a pension fund and to indicate certain problems which must be dealt with by the Actuary undertaking the valuation.

The method of valuation is in the nature of an exercise in the use of the salary scale to give actual increases and not rates of increase.

For the purposes of illustration a fund providing the following benefits is taken as an example.

The first Income Tax proper was the Income Tax Act of 1799 imposed by Pitt as a more or less temporary measure intended to finance the war with France.

Subsequent amending Acts were passed in 1803, 1805 and 1806. In 1816—just after Waterloo—the Tax was abolished and remained in oblivion for 26 years. In 1842 it was resuscitated by Sir Robert Peel, who was responsible for the Income Tax Act of 1842, and the Tax has remained in force ever since as a permanent and increasingly important part of our financial system. Of the Acts subsequent to 1842 the most important is the Act of 1853 when Gladstone left his mark on the Tax in several important particulars.

Up till 1860 the annual Act by which the Tax is imposed was known as the Income Tax Act. From that time onwards—in order to make it more difficult for the House of Lords to exercise their constitutional right to reject money bills—the Tax, along with several others, was incorporated into one Act called the Customs and Inland Revenue Act. In 1894 the title was changed to the Finance Act, by which name it has been known ever since.

In paragraph 27 of the Parmoor Report (Cmd. 614/1920) it is stated that an investigation into the methods by which different Industrial Assurance Companies ascertained their actuarial liabilities revealed the fact that among nineteen such Institutions no fewer than eighteen different methods were found to be in use. That the methods in use were practically as numerous as the Offices adopting them was a fact worthy of being recorded, and one which would appear to require some explanation, but I submit that, surprising as such a revelation may be, it would have been even more remarkable had the investigation disclosed that all these Offices used a similar Valuation basis. The views of the Parmoor Committee on this very important matter are summed up in the following extract:

‘The business of all the (Industrial) Companies is practically identical, their Tables of Rates are strikingly similar to one another and the adoption by all of an identical method of valuing the liabilities might reasonably be expected.’

It is usual to begin a sermon with a text and if I were making a sermon to-night my text would be from a favourite poet who, unconsciously and unsuspectingly, I fear, described the ideal actuarial department when its total staff is three and by implication described the ideal methods of examining actuarial problems.

Since games of chance have always exercised a fascination for people of all countries and at all times it is not to be wondered at that the theory of probability was first evolved from the dice-table. Possibly the earliest problem was to find the different probabilities of the various throws which can be made with three dice: this occurs in a document published in the fifteenth century, and the problem was taken up again by Cardan about a hundred years later. The first serious investigation into the laws of chance was, however, due to Pascal, who may be considered as the founder of the science. Pascal's attention was first drawn to the subject by a gambler who put to him problems relating to the game of ‘hasard.’ One particular problem seemed to Pascal of the utmost importance and led to a series of discussions between Pascal and Fermat, another equally eminent mathematician. The problem was the ‘Problem of Points’ and may be stated thus: Two players want each a given number of points to win a game; if they separate without finishing the game, how should the stakes be divided? Many variations of this were considered, even those where the players were not of equal skill, and although the problems would not be difficult of solution in light of modern knowledge, there was considerable divergence of opinion as to the correct methods to be applied.

Probably all of you have played cricket and football, and if I were to say that you could learn those games by reading about them in books you would laugh me to scorn. The furthest you would go would be to admit that having played a game to some extent you might get useful hints from books or conversation. Some of you might even feel that in a few cases the hints do as much harm as good by tending to reduce originality which, after all, counts for a good deal in any game, and by introducing an element of surprise may in games, as in war, win when a stereotyped method is doomed to failure. It is a steep descent from these high altitudes to the subject we are discussing tonight, but I suggest that what is true of games is also true, to a large extent, of graduation. You will never learn how to graduate by reading about it. Take an experience, that is, the exposed to risk and the deaths (or the week's sickness or whatever the subject implies) or the rates of mortality, and see what they look like and appreciate the hideous unevenness and spend time in trying out of your own head to ‘make the rough places smooth,’ and after that begin your reading.

In considering the question of surrender values and paid-up policy values, it must be borne in mind that the conditions affecting different offices vary to a great extent and that, consequently, no hard and fast bases can be given.

(i) The surrender values and paid-up policy values must be reasonable from a competitive point of view.

(ii) The values of one class of policy must be comparable with those of other classes.

(iii) The surrender values and paid-up policy values must be interdependent inasmuch as at the limiting age of the table they are equal.

(iv) Simple calculations only must be involved, the results being obtained as far as possible by the inspection of prepared tables.

(v) A reasonable profit must be reserved for the office.

(vi) The heavy expenses incidental to the issue of a policy must be met, and hence no surrender value should be allowed in ordinary cases until, say, two full years' premiums have been paid.

The following is a short account of the various records used and of the entries necessitated by the purchase of, and subsequent dealing with, Stock Exchange investments. In other words, it endeavours to trace so much of their history as is written in the books of that department of an insurance company which deals with Stock Exchange securities, which may be called for convenience the Investment Department.

Such a department would not keep the final books of account which provide the items for the Balance Sheet and other returns, all such information being obtained from the accountants' section. It would, however, decide upon the method of treatment of the many problems which arise in connection with security transactions.

It has been said that the chief object for which insurance offices exist is to pay claims. Whether this is true or not it is certain that the payment of claims is an important part of the business of life offices. Before, however, claims can be paid it is necessary to ascertain who is the proper person to receive payment and in view of the troubles in which an office may be involved if payment is made incorrectly it is essential that those who aspire to official positions in life offices should have some knowledge of law.

As under the rules of this Society this paper must be read in 20 minutes it will only be possible to deal with a few of the points which arise, but it is hoped that it will give a bird's-eye view of the subject and so enable those who are about to consider it more fully to do so with advantage.

In the following notes English law is referred to in the absence of any statement to the contrary.

This Lecture was given with the object of explaining the preliminary steps to be taken in ascertaining the mortality experienced by a body of lives, commencing with the extraction of particulars from the records of individual cases and leading up to the point where the rates of mortality are calculated from the raw material (usually referred to as the unadjusted rates of mortality) before any smoothing process is adopted for the purpose of preparing monetary Tables based on the experience. The lecture was given with particular reference to the Mortality of Annuitants 1900–1920, the first report on which had just been published.

It is proposed to deal in somewhat greater detail than is customary with the reasons why we are entitled to separate symbols of operation from the quantities upon which they operate, combine them together as if they were ordinary symbols of quantity and interpret the results in terms of other operations.

As a preliminary it may be well to say a word or two on the three primary laws of algebra, the law of distribution, of commutation and the index law and what these imply. Chrystal, in the first chapter of his Algebra, starts from the primitive notion of numbers (i.e. positive integers) and shows that the essential laws according to which these whole numbers combine are these three primary laws. He then shows how the meanings and properties of negative and fractional numbers are fixed by the condition that the meanings which are given to them are to be such as will give consistent results when we combine such numbers with other numbers according to the three primary laws. An illustration from the index law will perhaps make the point clearer.

There are three conditions which must be borne in mind in the calculation of office premiums.

Necessity for adequacy. Every premium charged must be sufficient to cover the risk and the expenses which are incurred in connection with the particular policy in question, and also to provide a reasonable contribution towards the general expenses and to leave a margin for profit. The fact that several offices charge very low rates for whole of life without-profit policies seems to indicate that the necessity of complying with this condition is a question on which some difference of opinion exists.

Influence of competition. The rates should compare well with those charged by other companies. An office which charges exceptionally low rates at certain ages or for certain classes of assurance will find that it is issuing a large number of policies in these sections of the business, and that people are going elsewhere for the classes of policy for which its rates are high.

Principle of consistency. The rates charged at differentages, and for different classes of assurance, should be consistent, and the differences between the with-profit and without-profitrates should be reasonable in view of the rate of bonus whichthe office is, in normal times, in the habit of granting.

The following notes are the result of an effort to consider the subject of extra risk from the point of view of the student dealing with the practical side of the case by showing how office practice endeavours to give effect to the actual conditions prevailing in particular cases.

A good many of the risks to be assessed require the assistance of the Medical Officer, especially where the degree and incidence of the extra mortality is concerned. I am of opinion, however, that the actual monetary assessment should be finally decided by the Actuary.

In this connection, there are one or two points which call for criticism and remarks.

A reversion, in its simplest form, is merely the right of one person to receive a sum of money, or its equivalent in securities, on the death of another person now living. In practice this right is often contingent upon the happening of some other event, such as the reversioner surviving the life tenant, and as the existence of such a contingency means that a purchaser might suffer entire loss of capital, it is usual to eliminate it by effecting an appropriate policy of assurance. When this is done the contingent reversion becomes an absolute one, subject to the payment of the appropriate premium.

The present paper has been given the title of ‘The Analysis of profits of a Life Office,’ but it is clear that it will not be possible in the time available to discuss all the points which may arise in practice under this heading, and it is therefore intended to limit the paper to a discussion of the more usual questions which arise in analysing profits where an annual valuation is made by the ordinary net premium method. Whatever be the precise method adopted in valuing the liabilities of a life office, the principles underlying an analysis of the profits will be the same, and once these principles are grasped no difficulty will be found in applying them to any particular case. In the first place, it should be appreciated that the necessity, or at any rate, desirability, of analysing profits arises only through our assuming in the valuation of the liabilities, rates of mortality and interest different from those actually realised in practice.

The present system of decennial censuses has been the method employed ever since the first census was taken in 1801, but there is general agreement amongst those interested in population statistics that ten years is much too long an interval between successive enumerations. A series of censuses at quinquennial intervals would, without doubt, afford a more reliable basis for the solution of many problems in demography. Further, if a permanent census office were set up all the information as regards movements of the population in and out of the country, and to and from various districts in the country, at present supplied to various Government departments, should be given to that office. A closer link between successive censuses would thus be forged, and intercensal populations more readily obtained with much greater accuracy than at present.

Probability questions are particularly difficult to the student, since the methods of solution are varied, and, unless the would-be solver has considerable experience, the best method of attack is not easily detected. The only way in which a satisfactory grasp of the subject can be attained is by working on a variety of problems of different kinds. The student will then find that questions in probability tend to fall into certain groups, but, even when he has arrived at this conclusion, he will discover, also, that the boundaries of these groups are by no means well defined, that a solution may often be reached by quite different lines of reasoning and that, except in the simplest cases, it will need all his experience to make a proper classification of any particular example.

The object of this note is to assist students to work on these lines by affording him examples of various kinds—with hints for their solution. We have deliberately refrained from giving the solutions in extenso, in order to encourage the student to work the examples through and sd obtain the practical experience which is so essential to proper knowledge.

No discussion of Life Office Investments before the Students' Society would be complete without reference to the five canons laid down by Mr A. H. Bailey. Their influence on the investment practice of Life Offices has been almost universal and continuous since they were enumerated in 1862, and when it is reflected that the great majority of Life Offices, although their annual profits are comparatively very small in relation to the very large accumulated funds which they administer, nevertheless emerged from the War period, and the enormous depreciation of securities which it entailed, stronger than ever before, no apology need be made for restating the principles which have been followed with such successful results.

They are as follows:

1. That the first consideration should invariably be the security of the capital.

2. That the highest practicable rate of interest be obtained, but that this principle should always be subordinate to the previous one, the security of the capital.