References to the rate of interest are frequent. In reality, of course, there is no single rate of interest but only a pattern of yields on individual assets or classes of assets. Nevertheless, it is useful as a first approximation to treat the complex of interest rates as a whole, thus bringing into sharper focus the forces that are common to the determination of the yield of all fixed-interest securities, with the exception of the risk of default. It is in this sense that the term ‘the rate of interest’ is used throughout this paper; consideration of the risk of default has been omitted as a variable that can easily be assimilated into the argument at any stage. The paper aims at making a broad survey of the most important forces that influence the rate of interest in a modern industrial economy where capital is for the most part in private hands. Where the assumption that we can regard interest rates as one united family departs too far from reality the more important qualifications that are needed are indicated very briefly.

In several recent papers given to the Institute and Faculty of Actuaries attention has been focused on options in both the assets and the liabilities of a life office. The warnings that have been given about the concealed danger of options and the desirability which has been stressed of making adequate, if not severe, charges therefor are, it is hoped, sufficient justification for introducing a paper confined to the subject of annuity and cash options. The paper is limited to the development of the theory and to supplying tables to assist in the calculation of option costs in practice; questions of policy are considered to be beyond the scope of the paper.

The type of contract which will be considered is a renewable oneyear term assurance on a group of lives which provides, in addition to a capital sum on the death of any life, that if at the end of the year the total claims in the year are less than some fraction, k, of the premiums paid, then a fraction, l, of the difference will be refunded.

Suppose that P is the net and P′ the office premium for n lives assured for a sum of 1 each and that q(r) is the probability that exactly r deaths will occur during the year, then

where c is the integral part of kP′. Since equals the expected deaths, μ say, the equation may be written

where and