In J.I.A. Vol. XLIV, pp. 261–83., Lidstone developed, as far as terms involving μ. and δ, the adjustments required in a valuation in order to allow for premiums payable more frequently than once a year. Practical working approximations to these adjustments were also given, and were summarized in a table at the end of the paper, which for convenience of reference is here reproduced (the symbol OS is used to denote the instalments of premium falling due between the valuation date and the next ensuing policy anniversary).

For many years it has been fashionable to assume that a published Net Premium Valuation is merely a piece of window-dressing designed to reproduce the results obtained by an internal Bonus Reserve Valuation carried out for the information of the office only. It may be as well, therefore, to start by giving some of the reasons which make it desirable to re-examine a few of the problems which arise in a Net Premium Valuation, viz.:

(1) The Examiners require our students to have a thorough knowledge of the Net Premium Valuation Method, both as regards the technique of its application and the results which it can be expected to produce.

(2) A Net Premium Valuation probably forms the main instrument for measuring the surplus to be distributed to policyholders in Industrial Funds.

(3) Even if the Net Premium Valuation were no more than window-dressing, the published results have to agree with those obtained by the internal Bonus Reserve Valuation and a thorough understanding of the method is necessary to ensure this.

The opinion is expressed sometimes that of recent years less interest has been taken by the profession generally in Sessional meetings of the Institute, and in the preparation of original contributions for the Journal. It is urged that actuarial subjects have been fully covered by earlier writers, and that in consequence a narrowing field for active interest exists.

The object of this paper is to examine the statistics available and, if possible, to decide whether the views stated above are justified. For this purpose two main ‘yard-sticks’ will be used.

Punched-card machinery is in fairly common use in Life Offices and in the following paper I shall assume familiarity with the more usual machines, i.e. punches and verifiers, sorters and tabulators, together with ancillary machines such as reproducers, collators or interpolaters, and interpreters. For those who are not familiar with these machines several explanations have appeared already in print, a recent example being The development of the punched card method by K. J. Hedley, read before the Actuarial Society of Australasia in the latter part of 1946 and appearing in the Transactions of that body. The purpose of the present paper is to explain with some detailed examples of the method of application how the addition of a multiplying punch to a punched-card installation adds greatly to the power of that installation to perform calculations of certain types important to Life Offices, and how those calculations can be performed with great economy of manpower as compared with existing methods. I do not pretend to make an exhaustive survey of the uses of this remarkable machine.

The conclusion reached from a consideration of Table 1, that ‘in a perfectly stationary fund…the bonus for which the premium scale is loaded will emerge from a net premium valuation whatever margin is allowed between the experience and the valuation rate of interest’, is difficult to accept without a more detailed investigation.

At first sight it appears to be supported by the incontrovertible argument that if the premiums are loaded to allow for a £2 % bonus and the experience is exactly that assumed in the calculation of the premiums, the profit made must be sufficient to pay a level bonus of £2 % per annum irrespective of the valuation basis. In a stationary fund, therefore, it might be argued that the valuation basis cannot affect the surplus brought out each year.

The purpose of this note is to develop some of the formulae of the classical theory of risk by a uniform treatment in which the “deaths’ column of the life table is considered as a discontinuous frequency distribution. The paper thus forms an example of the application of standard statistical processes to actuarial theory, and also supplies demonstrations of certain of the results quoted without proof in the survey by E. Lukacs. The aim has been to provide an elementary introduction both to an actuarial subject long neglected in this country, and to statistical methods of general utility.

The paper owes its origin to a stimulating series of talks on the Theory of Risk, given by S. Vajda to the statistical Study Group of the Society. The particular approach adopted and the treatment of composite policies are thought to be original.

A point estimate may provide, in logic or in practice, an unsatisfactory answer to some problems, and the additional requirement then arises for estimating limits within which a parameter is thought to lie. An elementary example of such an ‘interval estimate’ occurred in Example 2(c) of § 1, and we may recall the custom of quoting the measurement of a physical constant in such a form as g = 32 ± 2 units—which is an assertion, made according to some convention, that the true value of the quantity g lies between 30 and 34 units. We consider below some of the criteria which have been suggested for interval estimation, some methods of satisfying them, and a few of the results to which they lead.

f(x), f′(x),…, f(n+1)(x) are one-valued, finite and continuous (v≤x≤w).

Define

Then*

u (x) is one-valued, finite and continuous (v ≤ x ≤ w).

by writting

During the past 25 years much attention has been given by prominent statistical writers, especially J. Neyman and E. S. Pearson, to building up the theory of the Testing of Statistical Hypotheses. This theory and the practical tests which have been developed from it are of great interest to actuaries, those employed in normal actuarial spheres as well as those employed in purely statistical roles. The leading exponents of the theory, however, are mathematicians of no mean standing and the subject has been approached from a mathematical angle. From the point of view of the actuary or student whose study of mathematics has not proceeded beyond that of the Institute's examination syllabus this is unfortunate, as the consequent complexity of the approach masks from him the essential simplicity of the basic ideas underlying the theory.

The rules of deductive logic are inculcated throughout our intellectual education. They are universally accepted. They provide firm standards of judgement for many aspects of academic study, research, and of our daily work. It is from some points of view perhaps a matter of regret—and from others of challenge–that their scope is not all-embracing, and that a wide range of problems, both of practical affairs and of intellectual inquiry, is beyond their jurisdiction. These problems, of which the subject-matter of this essay is one, involve the process known as induction, or inductive logic. Rules, standards of judgement, do exist in this field, but few have won universal acceptance and many are the subject of vigorous philosophical dispute.

The standard deviation of the distribution of sickness is a statistic that appears to have been ignored by British Actuaries, with the exception of Sir G. F. Hardy who derived an approximation in his well-known Essay (J.I.A. Vol. XXVII, 1889, p. 245). By considering the graduated results of the A.O.F. Experience 1871–75 he concluded that the mean deviation of expected sickness was approximately 2√(expected weeks of sickness).

This note has arisen from some investigations being made into the statistics of non-life business, from which it appeared that the results obtained could be used to place the derivation of the standard deviation of expected sickness on a more satisfactory basis, and it is being put forward in the hope that some member with access to the necessary statistics will be able to make a more detailed study of this problem.

It is intended in the following paragraphs to give an outline of a very general statistical method which originated in agricultural experimentation and which has since been used in many other fields, including that of mortality investigations. We will start with a special case, that of regression analysis involving only one independent variable. Well-known formulae will be developed with the aid of orthogonal polynomials and it will be shown how this tool can be used for generalizations. A new theorem referring to these polynomials, and its significance for the analysis of variance, will also be mentioned.

In 1923, lack of capital in Germany led to a system of financing house-building which later conquered large parts of central and south-eastern Europe. The ‘Gesellschaft der Freunde’ (in translation ‘Society of Friends’, but entirely unconnected with the body commonly known under that name in this country) was the first Continental Building Society that worked on the following principle:

Members entering the society pay an entrance fee E and undertake to pay periodically thereafter amounts a per annum. All payments of a member are accumulated at a certain rate of interest and credited to his individual account. Whenever there is enough money in the pool to pay an amount of 1, say, then the member next on the list, in the order of entry, receives this amount and becomes a debtor for the difference between unity and the accumulated amount of his previous payments. He undertakes to pay from then on b, say, per annum until all his payments E, a and b, reach unity, when he is released from all obligations and ceases to be a member of the community.

In the discussion on the paper ‘The Variance Ratio test in Statistics’, given to the Students' Society on 13 December 1946, and printed in J.S.S. Vol. VI (1947), p. 172, the desire was expressed for a proof by relatively elementary methods of the distribution of the variance ratio. As such a proof must first derive the distribution of X2 modified into that of a variance estimate, three distributions are involved (in addition to that of the sample mean). The various steps are given in this paper. A note is added on the probability integral of the variance ratio distribution.

The accurate calculation of the net single premium at rate of interest i for an assurance to cover the amount outstanding under a Building Society Mortgage subject to a higher rate of interest j is a fairly lengthy process. However, as a fairly heavy contingency loading is usually added, the following simple method gives results sufficiently accurate for most practical purposes.

For a loan of initial amount unity repayable over n years by level annual instalments in arrear the single premium calculated at rate of interest j on a life age x is given by the expression

The sum assured is payable at the end of the year of death and is equal to the amount of the loan then outstanding before payment of the instalment then due.

In practice it is desired occasionally to calculate the value of an annuity-certain at a rate of interest decreasing steadily from the outset. If the rate of interest for the first year is ix and decreases by h per annum throughout the term n of the annuity, the value required is

say, where

This expression is somewhat laborious to evaluate.

The values of the returns of past and future contributions on death before retirement, such contributions being accumulated to the date of death at rate j, are given respectively by

(TPC)xPCFx and (AC)xFCFx

where (TPC)x=total past contributions at age x accumulated with (compound) interest at rate j to the valuation date,

(AC)x = actual annual contributions payable at valuation age x.

Consider a continuous probability distribution of a variable x measured from its mean in units of its standard deviation, and suppose that deviations from the mean are taken irrespective of sign; let px represent the ordinate applicable to x (this ordinate is, of course, the sum of the original ordinates at + x and — x). We then have

, and we write .