In J.I.A. Vol. XLIV, pp. 261–83., Lidstone developed, as far as terms involving μ. and δ, the adjustments required in a valuation in order to allow for premiums payable more frequently than once a year. Practical working approximations to these adjustments were also given, and were summarized in a table at the end of the paper, which for convenience of reference is here reproduced (the symbol OS is used to denote the instalments of premium falling due between the valuation date and the next ensuing policy anniversary).

The purpose of this note is to develop some of the formulae of the classical theory of risk by a uniform treatment in which the “deaths’ column of the life table is considered as a discontinuous frequency distribution. The paper thus forms an example of the application of standard statistical processes to actuarial theory, and also supplies demonstrations of certain of the results quoted without proof in the survey by E. Lukacs. The aim has been to provide an elementary introduction both to an actuarial subject long neglected in this country, and to statistical methods of general utility.

The paper owes its origin to a stimulating series of talks on the Theory of Risk, given by S. Vajda to the statistical Study Group of the Society. The particular approach adopted and the treatment of composite policies are thought to be original.

The rules of deductive logic are inculcated throughout our intellectual education. They are universally accepted. They provide firm standards of judgement for many aspects of academic study, research, and of our daily work. It is from some points of view perhaps a matter of regret—and from others of challenge–that their scope is not all-embracing, and that a wide range of problems, both of practical affairs and of intellectual inquiry, is beyond their jurisdiction. These problems, of which the subject-matter of this essay is one, involve the process known as induction, or inductive logic. Rules, standards of judgement, do exist in this field, but few have won universal acceptance and many are the subject of vigorous philosophical dispute.

The accurate calculation of the net single premium at rate of interest i for an assurance to cover the amount outstanding under a Building Society Mortgage subject to a higher rate of interest j is a fairly lengthy process. However, as a fairly heavy contingency loading is usually added, the following simple method gives results sufficiently accurate for most practical purposes.

For a loan of initial amount unity repayable over n years by level annual instalments in arrear the single premium calculated at rate of interest j on a life age x is given by the expression

The sum assured is payable at the end of the year of death and is equal to the amount of the loan then outstanding before payment of the instalment then due.