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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

Published online by Cambridge University Press:  18 August 2016

George James Lidstone
George James Lidstone
Affiliation:
Alliance Assurance Company, Limited
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Extract

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.

Type
Research Article
Information
Journal of the Institute of Actuaries , Volume 39 , Issue 2 , April 1905 , pp. 209 - 252
Copyright
Copyright © Institute and Faculty of Actuaries 1905

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References

page 210 note * This phrase, which is perhaps not in itself a very accurate one, is used in accordance with long-established custom to denote a train of reasoning which deals directly with the things which form the subject matter of the problem, and not with abstract symbols or formulae representing those things.

page 211 note * See Remarks on pp. 250, 251.

page 213 note * The “excess” may be either positive or negative, and the word is used as an abbreviated expression for the algebraical difference between the accented and non-accented functions.

page 214 note * This will be seen at once if it be considered that the [Excess Strain less Excess Interest] represents the loss that will be shewn if the Valuation be made on the Normal-Basis but the Experience be in accordance with the Special-Basis. The present value of such loss must be equal to the present value of the uniform difference between the Normal-Basis Premium and the Special-Basis Premium which will just work out the assurance, with neither profit nor loss, on the Special (Experience) basis. A purely mathematical demonstration of the relation is given in Appendix B to the Author's Paper“On the Distribution of the Divisible Surplus of a Life Assurance Company”, J.I.A., xxxii, 73, et. seq.: vide Equation (12) page 108, where it must be borne in mind that R” equals the Reserve on the Experience basis, represented in this Paper by V', we must have at the date of entry R0=R”0 , or the present value of the loss equals the present value of the Excess Premium.

page 215 note * The diminution will not usually commence immediately after the zero point because the Variation Fund is accumulating with interest and benefit of survivorship, and for some years this may more than counterbalance the reduction due to elements of contrary sign.

page 224 note * This result has been taken by the Author from Question 17 in the Examination Paper for Part III of the Institute Examinations, 1904.

page 225 note * When Tables of Policy-values are available, the criterion (1 + i)∆V › = ‹ ∆V 1 will be found more convenient.

page 231 note * Table XXIII appended to Messrs. Bacon and Ackland's Paper (J.I.A., xxxviii, 605) shows an apparent exception for duration 0, but the explanation appears to be (1) that a halved lapse rate is not consistent with a halved force of lapse, and the difference is very marked for duration 0; (2) that the Surrender-value there dealt with was not a fixed proportion of the Normal Reserve.

page 235 note * See Remarks on pp. 248, 249.

page 238 note * See Remarks on pp. 247, 248.

page 241 note * See Remarks on page 251.

page 244 note * See Remarks on pp.250, 251.

page 248 note * It is here assumed that no Surrender-values are payable.