1. Kennedy & Howroyd (1956) have discussed the application of the Lagrangian multiplier technique to actuarial problems. This technique permits the analytical solution of constrained optimization problems, where both the function to be optimized and the constraints must satisfy stringent analytical conditions. In particular, the first derivatives of all functions must exist.

When constraints comprise inequality as well as equation restrictions, as is the case in linear and non-linear programming, then the conditions required for the Lagrangian multiplier technique do not hold. It was therefore found necessary to develop a new body of techniques, known as mathematical programming techniques, for the solution of constrained optimization problems involving inequality constraints.

The costing technique known variously as Controlled Funding, Stabilized Costing and Aggregate Costing, is a comparatively recent development in the field of Life Office group pension schemes. Before explaining and discussing the methods used, it may be helpful to outline the principal types of Life Office group pension scheme, and to examine the more traditional methods of costing employed. It should be made clear at the outset that the term ‘group pension scheme’ relates to schemes insured by means of group deferred annuity contracts and approved under section 388(1) or section 379(3) of the Income Tax Act, 1952—endowment assurance schemes and group life assurance schemes are not within the scope of this paper.

It is probably fair to say that during the last decade problems of internal administration have faded into insignificance besides the legal problems which now arise out of a life office's relationship with its policy-holders—in particular in regard to estate duty matters.

Before examining the position in detail I would like to make some general remarks regarding the burden of estate duty on life assurance policies today. Rates of duty are progressive, as the table on page 399 shows, and bear most heavily on the large estates. Therefore, not surprisingly, a good deal of interest and effort has centred in recent years in reducing this burden.

This report summarizes the conclusions reached and difficulties encountered by Group E in the discussions held over the past 4 years. It is thus an amplification and extention of the report made to a meeting of the various computer Study Groups held on 27 May 1957.

The Group consists mainly of actuaries employed by ordinary life assurance offices (some composite and some purely life). The Chairman is a member of the Mathematics Division of the National Physical Laboratory, Teddington. The Group has been joined from time to time by representatives of computer manufacturers and other visitors.

The best known and most common type of option is that where an investor pays money (option money) for the call—that is for the right to buy shares at the current price in 3 months time.

This is best explained by an example. Suppose share A stands at 50s. (market price 49s. 10½d.–50s. 1½d.) and the option rate is 4s.

Then the investor pays:

Option money 4s.

Commission (as for buying a share at 50s. 1½d.)

This payment gives him the right to buy the share at the ‘striking price’ on any ‘declaration day’ within the term of the option.

1. During the early 1950's the annual numbers of births in England and Wales varied little from an average of 675,000, but since then there has been a marked rise and in 1960 no fewer than 783,000 births were registered. This steep increase was unforeseen in official population projections, including the range of projections published by the Royal Commission on Population, and is likely to have serious repercussions in some quarters. For example, it will in due course result in an increase in the numbers of children of school age to well above the numbers on which plans for school building and teacher training were based.

2. In a paper presented to the Society a year ago (J.S.S.16, 85) the authors showed that the increase in births was attributable in part to an increase in marriage rates and in part to an increase in fertility rates, but the latest data available were those relating to 1957 when there were 723,000 births. No explanation could be offered for the further increase and, in deciding upon the number of births to be assumed in the early years of the projection contained in the paper, it was assumed arbitrarily that the numbers would decline from 750,000 in 1959 to 710,000 in 1963, the latter being the number calculated on the marriage and fertility bases thought suitable for the long-term projection.

The following method of solving a certain type of problem in permutations and combinations, and in probability, is so simple that it must have been used before. On the other hand, the writer has never seen it expounded in the following form, and it does not seem to be well known to students. Whilst, therefore, no originality is claimed, this note may be found of interest to readers; and they may care to compare the length of the solutions to the examples below with those to be found in the text-books employing more conventional methods.

On being asked to prepare something on Friendly Societies for the Students' Society it seemed reasonable to attempt two things—one, to put them more into perspective in modern conditions, the other, to see what they might be able to teach the actuary professionally. The following notes are the result: it is hoped that, besides forming a useful basis for the discussion, they will themselves have something worth while to contribute to the subject.

Perhaps the most important characteristic of a Friendly Society is its registration under the Friendly Societies Acts (the relatively rare occurrence of an unregistered society is ignored); the restrictions of these acts have therefore to be accepted for the sake of their advantages. The constitution is a book of rules and the government is in the hands of the trustees—in whose names the assets are registered—and a committee appointed by the members at a general meeting. The rules cover all such matters as objects, eligibility, benefits, contributions, duties of officers, changes of rules, of name, nominations, dissolutions, investments and so on as required by good sense and the Acts.