In my previous sketches upon the subjects embraced in the above title, whilst, on the one hand, I was desirous to avoid burdening them with any apparently superfluous details, on the other hand, I clearly foresaw that the task would remain of following out, in at least another article, some needful explanations or justifications. To my former readers I therefore inscribe the following notes, in the anticipation of their being found not wholly unserviceable in the view just adverted to.

In this paper, Mr. Meikle advocates the making of valuations by Assurance Companies, not solely in accordance with the actual mortality which may have prevailed, but on the basis of that which it was at first assumed would prevail. He says, “There is inherent, in every table of net annual premiums, a rate of mortality and rate of interest, which it is assumed will be those experienced by the Office; and books and accounts should be so formed as to show at any time to what extent the assumed rates of mortality and interest have come up to the rates experienced, and, by comparing the expected receipts and disbursements with the actual receipts and disbursements, a correct view would be given of the true position of the Office.” He then proceeds to point out what he considers the best methods of estimating the “expected mortality,” and of registering it and the actual mortality from year to year, and gives tables by way of illustrating such methods.

Considering the repeated discussions which have taken place on the question of the Income Tax, it is not a little surprising that the true understanding of it has not made greater progress. After all that has been said to demonstrate the injustice and unequal pressure of the tax as at present levied, the perseverance of some portion of the press, and of one or two persons of considerable reputation as regards their understanding of such matters, in the opposite opinion, is quite unaccountable. It will be seen by the Second Report of the Committee of the House, just published, that some still maintain that the taxing a temporary annuity at the same rate as a perpetuity is perfectly fair and equitable, because, as they are at great pains to show, the present value in each case is mulcted in the same ratio. It is, at all events, satisfactory to find that these authorities so far admit that “present value” is the proper thing to tax. They maintain, in fact, that in these cases a property tax and an income tax have an identical operation; and as they approve of the one, they must clearly give their sanction to the other. But, unfortunately, the identical character of the two does not hold good, in spite of the assertion of these gentlemen to the contrary; they do not observe that in the case of the temporary annuity the wave of taxation having once passed over, must, to make their argument good, leave it ever untouched thereafter, let the capital out of which it arises reappear in what shape it will: otherwise it is clear that such capital is liable to be taxed over and over again.

In a short paper read before the Institute at the early part of this year, after alluding to the erroneous methods commonly adopted for the determination and division of surplus in Life Assurance Companies, I intimated my intention to investigate the nature and character of those methods, and to endeavour to demonstrate the mischievous and injurious consequences resulting from them. In the present paper I propose to commence this undertaking, trusting that the undeniable importance of the subject will be considered a sufficient apology for my somewhat frequent introduction of it at these meetings.

I. Wiener Allgemeine Wittwen und Waisen Pension Institut. (General Annuity Fund for Widows and Orphans in Vienna.) –This establishment, founded in 1823 under the patronage of Prince Schwarzenberg, has for its object to insure annuities to the widows and children of the subscribing members who happen to die at least three years after their admission into the Society. The widows are pensioned until their death; the children until the age of 20.

When the entire ownership of renewable leaseholds for lives or years is vested in one person, common prudence dictates the setting aside of a portion of the annual income towards the payment, in the one case of the premiums of a policy upon the lives of the cestuis-que-vie, or, in the other case, to accumulate at compound interest, so as to form a fund for the future renewal, the amount reserved being proportionate to the length of the term. When, however, such a property has been settled, and successive interests have been carved out of it, if there is no separate fund provided for the purpose, questions arise between those entitled for life and in remainder, whether there is in the particular case any obligation to renew at all, and, in the event of renewal, by whom the expense is to be borne. In such cases the first point to be considered is, what are the provisions of the instrument by which the leaseholds are settled—whether they prescribe renewal expressly or by implication; and if so, whether they point out the mode by which the expenses of the renewal are to be paid: for (1), property of this description may be given in such a mode as to indicate that it was not intended to impose any obligations to renew; and (2), the donor or settlor may so model his dispositions as to throw the burden upon all or any of those in whose favour he has created limitations of the estate; or (3), he may expressly charge the renewals upon the annual income, to the intent that they may become as it were incidents to the estate, prior to any beneficial enjoyment by the objects of his bounty; or (4), upon the corpus, with the intent that the estate should be settled subject to the subordinate direction that it is to undergo a perpetual diminution, with a view to its being otherwise preserved.

One of the most important problems of social statistics, after the questions of food and labour, is to determine the ratio of mortality and sickness.

The general mortality having been a subject of investigation during a long period, and the collection of facts and deductions therefrom being essential to the operations of the various respectable and long-established Assurance Companies, has at length been determined with sufficient exactness.

The merits and demerits of the present mode of assessing the Property and Income Tax have been so frequently made the subject of popular discussion, and are in themselves questions of such national importance, that I feel no apology to be necessary for again bringing the subject under the notice of the Institute of Actuaries.

A valuable paper on the inequitable operation of this tax was, some time ago, read before the Institute by Mr. Jellicoe; and, more recently, an elaborate and important contribution on the same subject was brought by Mr. Farr under the notice of the Statistical Society of London, which paper has given rise to a lengthened and very interesting discussion. An exceedingly temperate and able pamphlet has also been recently published by Mr. John G. Hubbard, the Deputy Governor of the Bank of England, in which that gentleman has discussed, with very considerable ability, the views of Mr. Warburton, and has, in my opinion, very effectually exposed their absurdity.

Next to England, Germany is the country in which the principles of life assurance have had the most successful development. It is true the more recent French Companies, La Caisse Paternelle, L'Equitable, La Caisse des Ecoles et des Families, which also call the assurances granted by them assurances sur la vie, show a greater extension in a shorter time; but what they guarantee to the public are not life assurances in our sense of the word. They are neither assurances granted for previously specified sums, nor is the payment of the claim dependent on the death of the assured. The more modern French life assurances are rather indefinite reversions obtained by paying in any sum the assured pleases, the interest of which (as in the Tontines), according to the mortality found to prevail in the different classes, may prove sometimes greater, sometimes less, and is only divided amongst those who attain the term of life which may have been previously agreed on. The contributions of those who die early, together with the interests thereon, go to augment the dividends to the surviving members. Whilst life assurance amongst the English and Germans is therefore calculated to be on the death of the assured a source of provision for their families, the more recent mode of life assurance amongst the French principally aims at granting to the assured himself, in his lifetime, an annuity or sum of money, which, increasing according as the computations are made for an earlier or later period, affords him the means of extending his business, of completing his education, of securing a dowry on marriage, or a provision in old age. In consequence, partly of the prospect of selfenjoyment of the reserve thus made, partly the possibility of obtaining a high profit for a small investment through a greater mortality in the classes than the tables assumed, this kind of assurance became very much in vogue amongst the French. Recently, however, a clearer understanding of the matter, and the exposure of some deceptions which a few of the Companies or their managers have been guilty of, have in some degree cooled the public favour.

It is a well-known feature in the advancement of learning, that the more knowledge becomes prevalent, the less positive do opinions become. Not only in theology does the staggering demand of “What is truth ?” assail us, but we meet with the same impregnable question in all the pursuits of literature and science, obliging us to admit that the truth of yesterday is not always the truth of to-day, and that to-morrow may still be about to offer us some third form.

The older class of writers felt no such difficulty. They conceived they were in possession of certain abstract truths or data that might be considered as fundamental axioms, and that by these axioms might all conclusions whatsoever be tried.

In the course of the inquiries which led to the paper, read before the Institute of Actuaries, and published in the last number of the Assurance Magazine, on “the Uniform Action of the Human Will,” which was principally illustrated by the regularity in the proportion of marriages between different classes and at different ages, I was very much struck by the law which appears to regulate the relative proportion of the sexes at birth. It has long been observed that, in all countries in which the law of population is not unusually disturbed, more males are born than females. This is the result, not of a few isolated observations, but of more than seventy millions of facts, collected from authentic sources in different kingdoms, and for many succeeding years. The proportion varies from nearly 109 males to 100 females in Russia, down to 104·62 males to 100 females in Sweden; Great Britain occupying the last place but one, about 104·75 to 100. The average births for all Europe may be taken as 106 males to 100 females.

Let a1, a2 . . an be the reciprocals of the roots of the expression 1 − s1x+s2x2 − . . ±sn−1xn−1∓snxn, which call ϕx. We have then ϕx = (1 −a1x) (1 −a2x) . . . (1 −anx), from which the following series of values may be readily deduced, the Σ implying the sum of all the instances of the form placed under it, so that each expression is a symmetrical function of a1, a2 . . an.

The sign is positive when p is even, and negative when p is odd. A single term of the pth expression contains p factors of the form a, and n–p factors of the form 1 – a; and the expression itself is the sum of every term which can be so constructed.

It is a point well worthy of observation, that while England has been for centuries past so eminently practical in every branch of commercial enterprise, there have been but few attempts made until lately, to examine freely the well-worn rules of insurance business with a view to alteration. There has prevailed a general, and not altogether unwise, determination to let well alone, and quietly continue the system, however faulty, which experience had shown could lead to an ultimate and satisfactory profit.

It was hardly to be expected but that an attempt to demonstrate the inconveniences arising from daily increasing competition in the business of life assurance should meet with resistance and reprobation. The large number of persons interested in novel undertakings of the character in question would naturally feel themselves aggrieved at statements which went to prove that such undertakings were mischievous because they could not be successful, and which sought to demonstrate their hopelessness of success by an expose of their actual condition; on the other hand, it is not much to be wondered at, that minds familiar only with a state of affairs so wholly different should regard with anxiety and alarm a succession of enterprises threatening not merely to encroach on their own field of operation, but, by a series of failures, to bring all alike into general suspicion and discredit. As in most other controversies, much allowance is to be made on either side. The interests of the two parties are probably not altogether antagonistic, but they can scarcely fail to come into serious collision unless placed under more carefully devised regulations than at present exist.