This paper explores the benefits and limitations of a valuation framework as a management tool within a general insurance operation. Two models are presented, one a model of the firm and the other an option valuation model, which together create a robust framework that enables management to analyse how different decisions would affect both the overall firm value and its distribution amongst investors. The model of the firm assists in understanding how key factors such as the momentum of a general insurance portfolio and the allocation of scarce resources affect the value of the firm. The second model, an option framework for corporate liabilities, highlights the critical distinction between the value of the firm and the value of investors' claims on the firm.

The Arithmometer of M. Thomas (de Colmar) has been already brought under the notice of the readers of this Journal by General Hannyngton, in a remarkably lucid and suggestive paper, which will be found at p. 244, vol. xvi. General Hannyngton, in his paper, explains the manner of working the machine, and gives examples of some of its applications to the construction of actuarial tables, with hints as to others. These afford an idea of the very striking adaptation of the machine to the formation of such tables; and they cannot fail to have excited the interest of many of the readers.

The new method described in Mr. King's recent paper, “On the Construction of Mortality Tables from Census Returns and Records of Deaths”, marks such a great advance on that employed in the construction of the official English Life Table, that it occurred to me that it might be worth examining whether the numerical application could not be further simplified by the use of central differences. Everett's formula (J.I.A., xxxv, 452) lends itself admirably to the construction of tables by subdivision of intervals. It involves only even central differences of each of the two middle terms of the series between which the interpolation has to be made, and, as was pointed out by the author in communicating his formula to the Journal, “each sum of three terms does double duty, serving both for “the preceding and the succeeding interval. In an extended “computation, the number of ‘sums of three terms’ to be “calculated is accordingly practically identical with the number “of intervals, and the labour of calculation is only about half “what it appears to be on the face of the formula.”

The problem before us is that of fitting between consecutive pairs of a series of points a series of partial interpolation curves, which shall have the same slope and curvature at their points of junction. In what follows, the central difference notation is that introduced by Dr. W. F. Sheppard in his paper on “Central Difference Formulæ” (Proceedings of the London Mathematical Society, xxxi, 449).

It has been represented to me that, in view of the influential position occupied by M. Violeine, the quasi official and authoritative character of his book, and the currency likely to be thereby imparted to the erroneous principles employed by him in the treatment of problems of the class to which the first of those I have discussed belongs;—I say it has been represented to me that in these circumstances it might be well to be a little more explicit as to his manner of proceeding. In my discussion of the problem in question I did enough, I believe, to shew that M. Violeine's solution is certainly erroneous. I admit however that I did not succeed to my own satisfaction (nor apparently to that of others) in precisely hitting the blot in it, and, as it were, reducing it to its elements. I think that from the consideration I have since given to the matter, I am now able to point out the fallacy by which it is vitiated, and I therefore willingly accede to the suggestion that has been made to me.

It is now nearly a decade since the author had the honour of submitting a paper on the mortality of assured lives in India, and the manner of its reception by the members of the Institute has emboldened him to make a second venture. Particular impetus was given to the writing of this paper by the fact that, in the discussion that followed the reading of the earlier paper, several speakers observed that they would be interested to know the results of an analysis of the mortality prevailing in the large subcontinent of India, subject to varying climates and other conditions of life, according to geographical situation, and an attempt has now been made to give this information as revealed by the statistics of the “Oriental”. The words, “the more investigations were made into mortality in various parts of the world and in various types of business the better”, with which Sir (then Mr) William Elderton closed his observations on the earlier paper, are typical of the attitude of the Institute towards papers on mortality investigations, particularly as applied to countries like India, which are becoming very important from a life assurance point of view. Information concerning the mortality to which the various parts of the country are subject is, however, still very meagre.

Interpolation and other formulae involving advancing differences are frequently developed, or at least conveniently reproduced from memory, from the familiar operational identities

where

Several writers, notably Messrs. Higham and Sprague, have treated of the Mortality Experience of Life Offices, basing their investigations upon various data; and it may be thought that there is no necessity to pursue the subject further. The question is, however, one of great interest and importance, both in its relation to abstract science, and in its bearing on the actuarial management of companies; and as it possesses collateral branches which have hitherto received but little attention, it occurred to me that my time would not be wasted in attempting to elicit new facts, and illustrate old ones, from the abundant materials now at our disposal.

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.

This paper has the strictly practical objective of devising procedures for managing Equity portfolios to the best advantage.

First, Modern Portfolio Theory, (MPT) which has been developed over the last 35 years with just this objective, is critically examined; from a study of the history of MPT and of its philosophy, principles and practices, the authors conclude that this discipline makes no contribution whatever to improving the performance.

Various definitions of mathematical probability are described and commented upon: (i) the classical formulation, based on the ratio of favourable to total “equally likely” aspects of a system; (ii) formulations based on relative frequency in a sequence of trials, or on the limit of relative frequency as the number of trials is increased indefinitely; (iii) von Mises' refinement of this, in which the sequences obey special postulates ensuring “randomness”; (iv) modern revisions of the classical definition, “equally likely” aspects being replaced by an aggregate, and ratio of aspects by relative measure of a sub-aggregate; (v) the approach of Keynes, in which probability is the “logic of uncertain inference,” developed by a symbolic algebra; (vi) the approach of Jeffreys, by axioms and conventions of procedure, based on the principle of indifference and the use of the rule of inverse probability. It is suggested that, whereas in formulations based on relative frequency, frequency cannot be held to afford any explanation of the facts of frequency, in the a priori formulations, on the other hand, any postulates put forward are still very far from explaining to inquiring minds the commonly observed phenomena of so-called chance. It is further suggested that in actuarial applications the nexus between axioms of probability and causes of mortality is especially obscure, with consequent need of caution in adopting and interpreting the more refined modern methods.

In England and Wales prior to 1956 annual estimates for the infant mortality rate q0 were obtained as the quotient of an observed number of deaths and an exposure calculated by reference to the related births. Since 1956, however, infant mortality rates have been measured as the ratio of the numbers of deaths and births in a given calendar year (cf. reference 4, Part I, supplement to Table 24).