We may approach our subject by considering how we are to define probability. This at once raises a number of questions which may be classified under a few broad headings in the following way. First, is the definition to be subjective? By a subjective definition we shall mean in this paper one that admits the possibility that two persons in the same state of relevant knowledge may reasonably hold different views as to the probability of a given event. Secondly, is probability to be only a theoretical concept, or must our definition be such that a probability can be determined by practical experiments and applied to practical problems? Thirdly, are we to embrace all types of event or only a restricted class, e.g. those that occur or can be imagined to occur on a large number of occasions? Fourthly, shall we define probability as a number or are we to admit the existence of a non-numerical probability? And if we stipulate that it is a number, must it be a rational one?

Although the subject of probability is of such vital importance to insurance, no great detailed study was given to it until the sixteenth century. This was nearly two thousand years after the world's first insurance venture, that of Antimenes on behalf of Greek slave-owners against loss of their slaves (the early Greeks had also a good system of marine insurance). As in many fields of study, the early students of probability met with a great deal of opposition, which must have extended well into the nineteenth century, judging from the evidence of de Morgan's book of 1838.

When I was invited to address the Society on some legal subject of my own choice, it was natural perhaps that my thoughts should go back to my first appearance before the Society some 25 years ago. My subject then was ‘Children's Deferred Assurances’, which was appropriate from a personal point of view—I had just become a proud father! My present subject is ‘Estate Duty’, and I am not asking anyone for a q factor to show me how appropriate it may be from the same standpoint!

It is bold for anyone outside the Estate Duty Office to speak on estate duty; but I have deliberately chosen a dull and uninteresting subject because of its recent added importance to all who are concerned with life policies and annuities. The maximum rate of duty under the Finance Act, 1894, which first imposed it, was 8%. The maximum rate to-day is 80 %. Actuaries and lawyers must clearly do something about it—if they can—for otherwise there will be still less of estates left to be frittered away among the beneficiaries!

In a recent pension fund valuation, which involved splitting the fund between two groups of members in proportion to the net liabilities, the writer found that he had to calculate a large number of annuity values for a variety of ages between 60 and 80 at several rates of interest with guarantee periods ranging from 1 to 10 years. In view of the nature of the valuation a broad adjustment was not good enough, and the problem arose of how to reduce the calculations to a minimum and yet retain an acceptable degree of accuracy in the approximation. It is thought that an account of the method which was evolved may be of interest and of use to readers of the Journal.