Several methods of graduation depend on linear operators–that is, operators of the type …+λ−nE−n+‥+λ0+…+λnEn+… Few attempts have been made to compare these methods, and there is no general theory. There is need for such a theory, for it is easy to show that unless some relation is assumed to hold between neighbouring graduated rates, there is no justification for graduating at all. Any such theory must explain this relation, which cannot be of the simple type that leads to curve fitting.

The paper has two objects : to compare and judge these methods, and to devise a general theory applying to them all.

The first approach is by the classical method–separate examination of fidelity and smoothness.

One of the disadvantages in practice of a “capital” or “two contract” annuity is the necessity for the office to furnish in the annuity-certain bond, for tax purposes, a schedule showing the capital repayments in each of the periodical instalments of annuity. Thus in a 20‐year annuity-certain payable by half‐yearly instalments 40 calculations are required, and these have to be checked and finally adjusted to agree in total with the original purchase price. The labour involved is considerable if such calculations are frequent.

The Paper deals with some of the problems which arise in the valuation of Widows' Funds with special reference to the Widows' Funds of certain professional bodies in Scotland.

An analysis is made of data derived from the Scottish Bankers' Marriage and Mortality Experience 1923–1943 (T.F.A., 19, p. 149) with a view to determining the applicability of the Collective Method to a valuation of the “existing” at the close of that Experience.

The various factors entering into a Widows' Fund valuation are examined in relation to a particular Fund and the importance of the valuation rate of interest illustrated.

The Paper concludes with a discussion of the “reserve for loss on future entrants” often met with in the valuations of such Funds, and of the arguments for and against proposals which have been made for its elimination.

The Paper reviews the growth of the number of actuaries in Great Britain, with particular reference to the last fifty years, and analyses the development in the scope of their employment, both inside and outside the insurance world. Reference is made to the natural hopes and ambitions of the newly qualified actuary, following completion of his long and arduous period of training, and to the responsibilities of a professional organisation.

Consideration is given to the growth of the employment of actuaries in England in industry and commerce, and reasons are suggested for their success in those spheres. Opportunity is taken to bring up to date certain tables first quoted by F. A. A. Menzler in J.I.A Vol. 57 and comment is made on subsequent references in actuarial literature to the problems of expansion in the scope of professional employment.

The second (and greater) part of the Paper includes a series of examples in which a detailed explanation is given of problems which have faced the author in the business with which he is connected–food distribution through a multiple shop organisation–and of the methods by which they have been solved. Whilst the solution in some cases depends upon the direct application of actuarial or statistical technique, emphasis is laid on the value of the form of mental process inculcated by actuarial training rather than on the direct use of technique.

The use of graphical methods is familiar to all actuaries, but the particular form of graph known as a “Nomogram” probably is still little known to most members of our profession. This form of graph does not appear in the set Courses of Reading for the examinations of either the Faculty or the Institute of Actuaries, though the latter body does not now (as it did until recently) specifically exclude “Determinants,” a form of algebraic expression from which nomograms may be developed.

A quarter of a century has elapsed since the publication of complete mortality tables of life annuitants for office use. Statistics showing the results of the Continuous Mortality Investigation have been published from time to time, but, partly as a result of the war, the Joint Mortality Committee have not attempted to construct complete new tables of mortality and monetary functions until the present time.