Let us suppose, for the purpose of calculating premium rates, that interest rates will not fall below those of the straight lines from their present value io to in in n years' time (in < io), thereafter remaining constant at in. On the basis of this assumption we derive formulae for the (minimum) effective rates of interest on single and annual premium contracts of term m years (m ≤ n, or m > n) and give examples in which gross interest rates fall from 8%, 9%, 10%, 11%, to 4½% in 40 years' time, and other examples. We conclude with a discussion of modifications to the term of the contracts which enable our results to be used for pure endowments, with-return endowments, endowment assurances, deferred annuities and other policies in life assurance and pensions business, and we mention some objections to our methods.

Our thanks are due to C. M. Stewart, F.I.A., F.S.S., for suggesting some improvements to this note.