occur in practice usually possess no great difficulty, and the employment of the Commutation Tables is very convenient. In the example alluded to just now of an endowment with returnable premiums, we have a case where part of the benefit consists of an increasing assurance, and in Mr. Gray's Tables and Formulæ, p. 106, you will find instances of increasing and decreasing annuities.

Let us return, now, to some other points of more immediate practical importance. There are few questions of more importance to actuaries than that of the values of policies, inasmuch as upon this hinges, as it were, the whole question of life insurance. In every Office, valuations have to be periodically made of the liability of the Office under its existing policies, and this I need scarcely tell you is simply finding the values of the different policies, and adding such values together, the total giving the estimated liability of the Office under its policies. A knowledge of the method of finding the values of policies is alone all things essential to an actuary, and in its entirety involves very considerable practical experience and skill. With the refinements of the subject we have nothing to do; but those of you who may desire hereafter to become better acquainted with the subject may refer to Mr. Tucker's paper in vol. x p. 312; Mr. Sprague's papers, vol. xi p. 90, and vol. xv p. 411, and Mr. Meikle's paper, vol. xi p. 241. Mr. Sprague's paper, vol. xi p. 90, contains nearly all that we shall require as to the theoretical part of the subject, and Mr. Manley's Prize Essay, in vol. xiv p. 249, affords us valuable information as to the result of employing particular mortality tables in an Office valuation; you will gain from the former a remarkably clear idea of the theoretical value of a policy, and the latter brings before you some of the difficulties of the subject, such for instance as the connection between the values of policies and the mortality table from which they are constructed. There are many remarkable points connected with the values of policies. As a familiar instance we very well know that the value of a policy decreases as the rate of interest increases, but a strict proof of this properly is by no means a simple matter. All these difficulties arise from the same cause, that the values of p in the expression for the value of an ordinary annuity given above, viz.,

ax=Px, 1v+Px, 2v2+ &c.,

do not proceed according to any known law. We cannot therefore in the above instance tell what difference in the value of the annuity will result from a given change in the value of v, i.e., from a given change in the rate of interest.

Let us now return for a few moments to a point mentioned

some time ago. We have seen that the values of annuities and assurances as ordinarily calculated and tabulated are the values of annuities upon the last payment made is, that at the end of the year preceding that in which death occurs, and the values of assurances assume in like manner that the assurance is payable at the end of the year in which death occurs. In practice, however, neither of these assumptions necessarily holds, annuities may be complete, i.e., a proportionate part payable for the year in which death occurs, and in assurances the assurance may be payable immediately death occurs, or within a short time after. So, that, the values of an annuity and of an assurance as ordinarily given are less than the annuity and assurance of practice. A correction therefore has to be applied to the tabulated values, and it is desirable that you should be acquainted with the methods by which these corrections are obtained. This branch of our subject is of considerable interest and difficulty, and will afford you capital practice in the reasoning processes on which the theory of life insurance is based. You will find this point, and the kindred one of annuities and assurances, where interest and the payments of the annuity are made more than once a year, ably and fully discussed in Mr. Sprague's series of papers on Annuities in vol. xiii pp. 188, 201, 305, 358; xiv p. 36; and also in Mr. Woolhouse's papers in vol. xi pp. 61, 301, xii p. 136, and xv p. 95.

It may be said that many of such questions as those just enumerated have only a theoretical importance, and practically are of little consequence. But I would urge in reply that it is by no means a scientific method of procedure to determine à priori what points are of importance and what are not. The proper method should be to investigate as completely as possible the value of any corrections to be applied to the general formulas, and then to ascertain by actual calculation whether such corrections may safely be neglected. There are, however, several such investigations which must be admitted to be of great utility in the practice of life insurance. Thus, for instance, in making a valuation of the liabilities of an Office, we may require to ascertain the average present value of the current premiums at each present age. Now the next premium in individual cases will be due at periods of time ranging from those just due to those due just a year hence, and Mr. Woolhouse in his paper on Continuous Annuities and Assurances, already quoted, shows that the proper annuity value to take in calculating the value of the future premiums is ā.

There is one other point upon which I wish, in conclusion, to

say a few words. You will remember that when dealing with questions of compound interest, we discussed the case of a man purchasing an annuity certain for a term of years, and paying such a price for the said annuity as would pay him a certain rate of interest upon his invested capital, and likewise enable him to replace that capital at a certain other given rate of interest. Now the theory of life insurance affords us a case remarkably analogous to that just mentioned; and it will be of interest to consider in what respects the two transactions differ from one another. In the first place it is necessary that in order to reproduce his capital when the life annuity expires he must effect an insurance upon the annuitant's life. It is quite clear, too, that inasmuch as he receives no payment of the annuity at the end of the year in which the annuitant dies, while on the other hand the premium has been paid by him at the beginning of each year instead of at the end; then if S denote the amount paid for the purchase of the annuity, the capital to be reproduced by insurance at the end of the year in which the life dies is S+ amount of one payment of annuity, so that we have the equation

or

(Px+d) (S+m)=amount of yearly payment of annuity=m.

where

d=discount on 1 for a year

Pannual premium to insure 1 on the death of x.

We see now, then, exactly how the purchase of a life interest differs from that of an annuity certain. The term of the investment depends upon the duration of an individual life instead of a term of years certain. The amount of capital invested is something more than the purchase money of the annuity, and the life insurance premium takes the place of the surplus annuity for accumulation which is to reproduce the capital invested at the expiration of the annuity. There are other cases, such as the purchase of contingent annuities and reversions, which are more complicated than the above, but which may be shown in a similar way to bear an analogy to that we have already had to deal with; but as these questions come more properly under the subjects of the Third Year's Examination we need not consider them. At the same time, they afford a capital insight into what has become a very important branch of the subject. Mr. Jellicoe, years ago, gave a series of papers in the Assurance Magazine, vol. ii p. 159, vi p. 61, viii p. 310, in which he very lucidly explained the reasoning by which

the different formulas are obtained, and since then Mr. Sprague has treated the subject in a very able manner in his papers recently published in the Journal, vol. xiv p. 417, xv p. 126. For your present course of reading, however, you will not find it necessary to give much of your time to the study of them.

It may not be out of place to conclude this Lecture with a few general remarks. At the beginning of our studies we found that all pecuniary questions contingent upon human life, being those which may be said to fall more properly within the province of an actuary, involved two distinct elements-interest on money, and probabilities of life. The first of these is fully developed in the Doctrine of Compound Interest, and all our information about the latter is given in the form of mortality tables. Having discussed each of these elements in detail, we have now, in this, the Third Part of our course of study, seen how the values of the different pecuniary interests depending upon human life are deduced by combining the two elements of interest and mortality in the proper manner. Incidentally, too, we have discussed the theory of logarithms, and the construction of auxiliary tables, both of which are questions of great practical importance to the actuary; and both of which the Council of the Institute has very properly decided should form a part of the subjects for the Second Year's Examination. In going thro' the various subjects, I have little doubt that many points have arisen about which even now your convictions are by no means so clear as you would like them to be. You must not be dissatisfied with yourselves, altogether, on that account. In reading new subjects, and acquiring new ideas, it is almost always a work of time before we can be said to have well digested the knowledge acquired; and in this respect the theory of life insurance is no exception. Nothing, however, will do more towards assisting you towards this end than the solution of examples, and in this you will find the questions of previous years with their solutions, as given in the Journal from time to time, of very great service. See solutions in vol. i p. 123, by Mr. Porter, and in x p. 45, xiii p. 253, xiv p. 147, by Mr. Sprague, and the solutions for 1868 by Mr. Peter Gray and Mr. R. P. Hardy, in vol. xv p. 232.

ERRATUM.

Dr. T. N. Thiele points out the following:

Vol. xvi p. 314, in formula (2), for a ̧ ̄1⁄2o read a3⁄41⁄2o.

INDEX TO VOL. XVI.