In the paper which I submitted to the Institute on 28 April 1944 I discussed the most equitable method of treating appreciation or depreciation in the assets of an office distributing its surplus by means of a uniform reversionary bonus. I believe that the conclusions reached were correct within the limits set, but consideration of the discussion convinced me that these limits were too narrow, since they involved the treatment of the problem of appreciation or depreciation without reference to the other factors affecting equity in bonus distribution and excluded any solution based on the use of a system of distribution of profits other than a uniform reversionary bonus. I felt that the methods used in the paper might produce results of interest if applied to the general problem of securing equity when the conditions experienced differ from those assumed in calculating the premiums, and I was encouraged to proceed on these lines by the suggestion made by Mr H. E. Melville, speaking as Presidenti that the time was suitable for such an inquiry in view of the exceptionally wide fluctuations in the experience rates of interest over the past thirty years.

1. The methods of analysis leading to the usual formulae for the value of the benefits provided by a pension fund are contained in the historic papers of H. W. Manly, J.I.A. Vol. XXXVII, p. 193 and G. King, J.I.A. Vol. XXXIX, p. 129. The purpose of this paper is to set forth certain developments of the formulae of G. King from which the balance of forces within the fund may be investigated. It is emphasized that the formulae themselves are but algebraic shorthand for processes of calculation, and the deductions do not arise from the formulae but from the numerical results obtained. These results will vary from fund to fund according to the relative financial importance of the various portions of the benefits granted. Within the limits of this paper it is not possible to deal either with the financial situation of an existing fund or with the varying contributions and benefits met in current practice, and in order to produce results for discussion, calculations have been made on the basis of the benefits and contributions of a standard type of fund.