These notes are intended to elucidate the treatment of income tax and expenses in the analysis of surplus of a commission-paying office transacting ordinary life assurance and annuity business in the United Kingdom only. The Office is assumed to make an annual valuation for internal purposes, but not necessarily to distribute surplus yearly.

It may first be remarked that the analysis of surplus forms both a check on the accuracy of the valuation and a guide to the amounts and relative importance of the various sources of profit and loss. We are here not concerned directly with the first of these functions and only with two items of the second. It must be realized, however, that the analysis is related to the valuation bases, which may depart materially from the bases underlying the office premiums, particularly at a time when conditions are changing rapidly, or if there is a substantial proportion of very old business on the books.

1. The analysis of the surplus of a life office disclosed by a bonus reserve valuation does not differ in principle from the analysis of the results of a valuation on a net premium basis, but the treatment of bonus requires special consideration. A method of analysis appropriate to a uniform compound reversionary bonus is discussed in the following paragraphs, and a numerical example is given by way of illustration.

2. It is assumed that the valuation formula is

where (S + B) represents the sum assured and existing bonuses,

P is the office premium,

k is the proportion of premium reserved for expenses,

x is the present age nearest birthday,

n is the curtate duration to maturity,

b is the rate of bonus for which a specific reserve is made.

The object of this paper is to attempt to evolve a theoretically correct method of valuing life policies in an office having a ‘level’ Compound Reversionary Bonus system. This method can then be used to examine the existing methods of valuation in detail, and also to trace the connexion between the existing methods. The theoretical method of valuation can be approached by a mathematical analysis on strictly theoretical lines. A Compound Reversionary Bonus method has been used in this paper, but similar principles can be applied in the case of a simple reversionary bonus method.

It is emphasized at this early stage that I am only concerned with the theory of valuations, and the practical points, such as legal requirements, presentation of valuation results and simplicity of valuation machinery, are not considered.

The subject of making an allowance for initial expenses in a net premium valuation is one which has long interested actuaries, and it is usually associated with the name of Zillmer who first wrote an article on it in 1863. The following note, based on practical experience of valuing a young and rapidly growing life fund, may be of interest to students.

The normal method of making such an allowance is, in effect, to adjust the net premium used in the valuation by the equivalent of the initial expense allowance spread over the whole duration of the policy.

In his lecture ‘Computing methods’ (J.S.S. Vol. VIII, pp. 76-7), D. H. Sadler describes a method of inverse interpolation invented by Dr Comrie. This note shows how the method may be used to solve for i the equation = K. The solution of this equation to a sufficient degree of accuracy is a minor but troublesome problem of office practice, and the writer hopes that the method here described will be of practical value. In addition to its simplicity and directness it has the considerable merit that it makes full use of what a calculating machine can do, and, in consequence, requires from the computer very little calculation. It will be seen that, apart from the making of a short difference table, the whole of the work is done on the machine.

Students of the subjects of compound interest and life contingencies are usually baffled by the type of problem in which the value of a premium, say P, can only be ascertained when one knows the number of years, say n, after which there is some definite change in the conditions of the problem. Such is, for example, the valuation of a variable annuity-certain subject to different remunerative and reproductive rates of interest when, in the beginning, the payments under the annuity are too small to meet requisite interest payments. In Chapter IV, article 16, of Todhunter, the student is advised that in such cases the value must in general be sought by trial and error. Similar exhortations perplex the student's mind when he turns to the subject of life contingencies, in which he is frequently required to evaluate the premium for a complicated assurance in which the death benefit is defined by different expressions before and after a period of time which is itself unknown.