Changes in Pure Premium Policy-Values consequent upon variations in the Rate of Interest or the Rate of Mortality, or upon the introduction of the Rate of Discontinuance

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The volumes of the Journal contain numerous theoretical investigations dealing with the changes produced in Policy-values by alterations in the basis of valuation, and a number of interesting and important results have been obtained, of which the most valuable have been embodied in the Text-Book, Part II, Chapter xviii, Articles 39–72. These investigations are almost entirely based upon analytical transformations of algebraical expressions for the Policy-value. Fruitful as this method has been, and elegant as are many of its processes and results, it yet labours under certain serious disadvantages. From the nature of the method it can be applied only to the limited class of benefits—practically confined to ordinary Whole-Life and Endowment Assurances, with uniform Premiums and Benefits—which admit of the Policy-values being expressed in an algebraical form involving only one class of function, e.g., ax, Ax or Px: so that relations which are really of very wide generality remain undemonstrated, except for those particular classes of Assurance. Again, demonstrations based upon a particular mathematical formula for the Policy-value are not easily modified to meet changes of conditions which, though slight in themselves, may render the fundamental formula completely inapplicable: while abstract analytical demonstrations are both harder to grasp in the first instance, and harder to reproduce or adapt when subsequently required, than are demonstrations based upon more concrete considerations.