The achievement of smoothness is one of the main purposes of graduation. Smoothness, however, has never been defined except in terms of concepts which themselves defy definition, and there is no accepted way of measuring it. This paper presents an attempt to supply a definition by constructing a quantitative measure of smoothness, and suggests that such a measure may enable us in the future to graduate without prejudicing the process by an arbitrary choice of the form of the relationship between the variables.

It is a common experience in many fields of investigation to find that the standard methods which have been developed and used in a given field prove inadequate in a particular application. In these circumstances it is often found that no further progress can be made along lines hitherto considered conventional, because some fundamental question has not been explicitly posed and answered by the standard methods. In life assurance a feature of existing methods of approximate valuation, which has not received prominence, is their inability to measure satisfactorily the size of the possible error involved. A wide variety of approximate methods have been in use, and some have been found to give consistently ‘good’ results, but the question of what is meant by ‘good’ has been left unanswered. The background of this paper is that the authors, faced with this situation when tackling an essentially practical problem, were virtually forced to break away from traditional methods and seek a new tool.

The original intention of this paper was to discuss the impact of income tax on annuity business only. In practice it was found impossible to discuss the impact on annuity business without at the same time discussing the impact on Life assurance business. A company transacting Ordinary life assurance, Pension annuity and General annuity business is not assessable in respect of three separate businesses; for the purpose of income tax the company conducts only one business, i.e. ‘Life Assurance and Annuity business’. Attention is drawn to this point at the outset because it is considered most important. A student coming freshly to the subject should not be misled into thinking that separate taxable units are involved because in the Income Tax Act, 1952 and the Finance Act, 1956 such terms as ‘annuity fund’ and ‘profits arising to an assurance company from pension annuity business’ are used. It may be noted that in this paper the term ‘fund’ has been avoided wherever possible, because, without precise definition on each occasion that it is used, it is liable to be misleading. It would be impracticable to discuss in one paper all the problems of a composite company which writes all classes of insurance business both at home and overseas. It is, therefore, proposed to consider only the position of a proprietary company which writes Ordinary life assurance and annuity business in the United Kingdom, values its liabilities annually and is taxed on an interest less expenses basis.