The records of papers read before the Students' Society or published in its Journal contain few references to the system whereby surplus is allotted to policyholders primarily in the form of reduction of premium. It is thought that this system would form a suitable subject for discussion and would be of interest to members as well as of value to those students who are concerned in their studies with Valuation and Surplus.

The description of the system contained in Mr R. K. Lochhead's book issued in the Consolidation of Reading Series is necessarily brief, and it may be convenient to enlarge on this by tracing the development of the system in one of the offices which has transacted this class of business for upwards of one hundred years dating from the early 1800's.

Although the first essay on the theory of risk (41) was published in 1786, this theory has been for the most part developed during this century. Moreover, the original presentation was rather difficult and the formulae inconvenient for the practical computer. Only recently has the classical theory of risk attained a satisfactory and seemingly final form. Few text-books, none of them in the English language, discuss these developments; it may therefore be useful to give an exposition of these results. As a rule no proofs will be given, instead reference will be made to the relevant literature.

This survey will be restricted to the classical theory of risk, which deals essentially with the individual insurance. It is possible to approach the problem of risk from an entirely different angle, considering not the individual insurance but all the policies in force. This leads to the collective theory of risk (27, 29, 15, 40), which is based upon the theory of homogeneous random processes. Only the classical theory will be considered here, the inclusion of the collective theory of risk would considerably extend the scope of this paper, and in any case many of the relevant papers are in English.

The orthogonal transformation used in a previous paper(1) to derive the distribution of x2 has its counterpart in a double orthogonal transformation which can be applied to a variable y dependent on another variable x in order to deduce the known tests of significance in the simple regression problem. The general theorem is due to R. A. Fisher (2), and the particular transformation now used is a special case of the problem considered by Vajda (3) in the multivariate case. Attention is also directed to papers by Bartlett (4) and Elfving (5). In the present problem the use of two successive transformations is of interest in showing what happens, and in providing a relatively elementary proof of the known results.

The statistical fate of a group of individuals who are subject to a process of ‘wastage’ forms the theme of an important class of actuarial problems. Usually the rates of the several individual sources of wastage are relatively complicated functions of such variables as age, duration, or rank, and the full apparatus of the service table must be employed. In other cases, however—including some of practical importance—these functions degenerate, in fact or in supposition, into constants, and a simpler approach accordingly becomes possible. By considering the operation of wastage—which includes such events as death, breakdown of health, resignation, and discharge—as equivalent to the financial operation of discounting, a formal identity can be created between certain ‘manpower’ problems and corresponding problems involving compound interest; and numerical work on the former is much simplified by the use of tables of common compound interest functions.

The simplicity of L. Solomon's solution to the manpower problem discussed in the preceding article is due partly to the use of compound interest functions but mainly to the fact that transfers to semi-skilled status and thence to the skilled classification each occur at a single point, after two and five years respectively, from entry. When dealing with the corresponding problems of, say, office staffs who are recruited upon leaving school or university and may serve until age 60 or 65 the conditions are very different. Transfers from one category to another, in this case promotions from grade to grade, generally occur at varying rates over a range of ages. Further, the work cannot be simplified by the use of compound interest functions, the full service-table technique must be employed.