The present effort does not profess to be complete. It deals with the main problems that come before the Consulting Actuary in connection with Staff Pension Funds, but does not go into the details of what may be called the fancy benefits which are occasionally met with. The solutions of the general problems here given will, however, indicate the methods to be applied in attacking other, and sometimes more complex, questions, which, when the principles have been grasped, will not present serious difficulty.

Following on the reading of Mr. King's paper “On the Comparative Reserves of Life Assurance Companies” (J.I.A., xxxvii, 453), there appeared in the Insurance Record an interesting correspondence, in which certain developments of that paper were asked for. As these promised to lead to results of considerable interest and to throw much light on the working of bonus systems, I communicated with Mr. King, and it was with his concurrence that the present investigation was undertaken. I desire now to express my thanks to him for his kindness in placing at my disposal his manuscript tables, which greatly shortened the work of certain parts of the investigation.

I have heard that a former distinguished President of this Institute once publicly stated that “Doctors rush in where Actuaries fear to tread.” It is also said that “at forty a man is either a physician or a fool”: but the presidential saying seems to suggest that a man may be both. With such a warning ringing in my ears, I yet venture to read before this learned Institute a paper on a thorny subject. I am doing so, first because that subject can be dealt with only by the united labour of actuaries and doctors; and second, because I am sure of the consideration and forbearance of the Fellows in any matters wherein I travel with halting gait.

The last two lines on p. 141 of the present volume are misplaced, and should be transposed to the top of the same page. This error has been corrected in the later issues of the Journal for April 1905.

The method of valuation generally adopted in practice is that known as the Prospective Method; by it, the value of a policy is obtained by a consideration of the future, and is the discounted value of the payment to be made, less the discounted value of the payments (if any) to be received; discounted, that is, in each case, for both mortality and interest.

The subject of Widows' and Orphans' Pension Funds occupies but a very small space in the pages of the Journal of our Institute. Indeed, until Mr. Manly's classic paper was written in 1903 (J.I.A., xxxviii, pp. 101 et seq.) there had appeared no paper from which the student could glean any information on this branch of our work.

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.

Insurance Companies are sometimes asked to lend on a prospective interest under the will of a living person. In most cases, such an expectancy would of course be quite out of the question as a security; for besides other circumstances that may arise to defeat it, there is the possibility that the person who has executed the will may change his mind, or get rid of all his property in his lifetime. But in certain circumstances the chance of realizing such an expectancy may be materially increased. For example, the person who has made the will may become lunatic and incapable of dealing with his property or making a testamentary disposition of it. The property would then be placed in the custody of a trustee or the Court, and, subject to a proper allowance for his maintenance, would be kept intact till his death or recovery. And, if there is strong evidence that there is practically no chance of the lunatic recovering his reason, the prospective interest can almost be regarded as an ordinary contingent reversion to fall in on the death of the lunatic, if the beneficiary survive him. In such circumstances, the prospective beneficiary may desire to raise a loan on his interest; and life offices are sometimes approached in this connection. Though, as the writer shows in the following remarks, such an expectancy cannot form a mortgage security, even when the chances of its realization are as favourable as those mentioned above, it may be useful to examine the chief points that come up for consideration in a proposal of this sort, and the reasons for which it should be declined.