All business is, to a greater or lesser extent, concerned with the taking of risk. This paper attempts to explain some of the main approaches that have been adopted in pursuing the concept of risk from a probabilistic viewpoint. The word ‘risk’ itself has been used fairly indiscriminately by writers, and thus gets given a number of different senses, but it has attained a clear and precise definition at various times and in different types of literature. Indeed we can distinguish three main avenues of development. The first is the pure economic concept, the second the actuarial approach, and the third the more recent linkage of risk analysis with the general approach to decision-making under uncertainty.

In the Diary for 1850 the following; mathematical query was proposed by the Rev. T. P. Kirkman, of Croft, near Warrington:—

Two answers were printed in the Diary for 1851; one by the proposer and another by Messrs. Samuel Bills, of Hawton; Thomas Jones, of Chester; Thomas Wainman, of Burley; and W. H. Levy, of Shalborne.

The Paper discusses the general problem of appreciation and depreciation, with special reference to the relation between maturing assets and maturing liabilities, and describes a method of calculating the bonus-earning power of different classes of policy. This method is used to test the effect on bonus-earning power of two types of changes in interest rates:–

1. (a) a change, such as that caused by the War Loan conversion of 1932, which takes effect gradually; and

2. (b) a subsequent change, due to a rise in the rate of income tax, which has an immediate effect on the income from existing as well as investments.

In the Introduction to his famous Théorie Analytique des Probabilités, Laplace, after an eloquent summary of the progress of Astronomy from the days when every rare celestial phenomenon was regarded as a portent of human misery, to the verification in April 1759 of the perihelion passage of Halley's Comet, proceeds to say :—

“The regularity which Astronomy shows us in the movement of Comets exists, beyond doubt, in every phenomenon of nature. The curve described by a simple molecule of air or of vapour is regulated in a manner as certain as the planetary orbits: there is no difference between the two cases save our ignorance.

As far as we are aware there have been no life insurance demutualizations in the U.K. in recent times, that is conversions from mutual to proprietary form. There has been movement in the other direction: in the 1960's a number of U.K. proprietary companies converted to mutual form to escape the unwelcome attentions of the corporate raiders of the time.

In the U.S. and South Africa however, the two countries which give us the demutualization examples in this paper, there have been in recent years a number of significant demutualizations. In the next chapter we give a sketch of the background to demutualization in these countries.

In Section 3 we explore why a mutual might wish to undergo such a lengthy, costly and probably painful process. The motivation will vary from country to country, depending on the particular economic and institutional background. We have tried to be as general as possible.

The problem of finding the rate of interest corresponding to a given annuity-value, has engaged considerable attention at various times; and many formulas have been proposed for obtaining approximate solutions by the aid of existing tables of annuities. Most of these will be found in Mr. Sutton's paper on the Rate of Interest yielded by Foreign Loans (J.I.A., xix, 77). Of the 5 formulas there given, (2) is probably the most generally useful, (3), (4), and especially (5) (Prof. De Morgan's formula), involving a good deal of calculation. The object of the present short paper is to suggest 2 new formulas, which give very accurate results, and at the same time require but little work in their practical application.

Let a be the value of an annuity for n years certain, at a rate of interest i, (which it is required to find), and let a1, a2, a3, be the tabulated annuity-values for the same term at the rates i1, i1 + h, i1 + 2h, where a lies between a1 and a2, and consequently i between i1 and i1 + h.

Until within a very recent period the doctrines of probability have been applied to scarcely any great public purpose except the Assurance of human life. In that particular department of knowledge, a number of labourers have been for many years busily occupied in gathering the facts, endeavouring to classify events, and trace some general laws. In later years, researches have been extended into the effects of different classes of diseases upon health and longevity. Still more recently, the calculations of fidelity risks, of railway and other accidents, of bankruptcy, and of other circumstances affecting the person of an individual, or his commercial dealings, have been made with a view of proving their capability of being estimated for Insurance.

It has been remarked on many occasions that whilst there is a plethora of material which has been contributed to the Institute and the Students' Society on the actuarial valuation of liabilities, the subject of valuation of assets seems to have been consistently overlooked by all save one or two authors. We believe that a compelling reason for this apparent indifference to the subject may be found in the trend of interest rates within the experience of the actuaries who introduced the valuation techniques which a student may now tend to take for granted.

Consider Table 1, compiled from information given in a paper by C. M. Douglas, F.F.A., on ‘The Statistical Groundwork of Investment Policy’. During the period 1833–1912 the average annual rate neither exceeded £3. 11s. od.% nor fell below £2. 9s od.%, and in general the yield drifted slowly downwards.

In the course of an investigation I have lately made, into the rate of mortality that has prevailed among the female lives insured in the Scottish Equitable Life Assurance Society, I have tried various methods of arranging the facts furnisht by the books of the office; and I think it may be useful to persons engaged in similar investigations, if I describe these methods, and state the conclusions at which I have arrived as to the best course of procedure.

This is the preamble to the petition presented to the Secretary of State's office of the British Parliament in April 1757 by the promoters of ‘The Corporation for Equitable Assurances on Lives’, the seed from which modern life insurance has sprung. It is undoubtedly the clearest statement made as to the purpose and justification of the life insurance industry and stands as a challenge to those engaged in it.

Mr. Sprague, in his report on Mr. Finlaison's Tables of Mortality among Government Life Annuitants, states as the result of his study of the subject of graduation that the most satisfactory system of adjustment is the graphic method; and he supports this opinion by referring to the irregularities which he regards as a consequence of the use of Mr. Woolhouse's method of adjustment. I do not doubt that Mr. Sprague would conceive and correctly draw a curve to satisfy all the conditions of good graduation; but in my own hands the graphic method would be a failure, and there may be others who doubt if they have skill to draw the curve and patience to amend it by the successive trials which Mr. Sprague himself finds necessary.