The principal difference between the accounts of an insurance ecompany and those of an ordinary trader lies in the fact that the trader buys before he sells, while the insurance company sells before it buys, that is to say, the insurance company first receives the money on any transaction, and afterwards pays out. From this it arises that the trader keeps account of his liabilities by book-keeping and ascertains his assets by taking stock, while the insurance company keeps account of its assets by book-keeping and ascertains its liabilities by taking stock. There are certain classes of trade, chiefly brokerage, where there is either no stock, or a stock not needing stock-taking, in which the trader can keep account both of his liabilities and of his assets by book-keeping, and has thus a far better knowledge than he otherwise would of the sources and the progress of his profits. The purpose of this paper is to suggest that an insurance company, including even a life office, may keep accounts of both assets and liabilities by book-keeping methods, and may thus have clear information as to the sources and progress of its profits.

It has been said that there is nothing so delusive as figures. especially in the form of statistics; and to a very large extent this is true. There is a real science of statistics which very few are acquainted with; and the consequence is that enormous labour is often expended on the collection of “Facts” which are not only useless but misleading. So-called “Facts” are often used as a crushing answer to some statement which may be perfectly true; and sometimes with effect, simply because the materials are not at hand to measure correctly the value of the statistics used. The most common offence is that of attempting to draw general conclusions from partial or imperfect observations. For instance, what is the value of the statistics of a coroner respecting the proportion of children murdered, out of the number which come under his observation, when the millions who do not come under his observation are not taken into consideration? Yet such statistics were gravely used to prove that two millions of people insured their children's lives with a view to murdering them, and that therefore such insurances ought to be suppressed. If any weight were to be attached to such statistics we might have it gravely stated, on the evidence of a coroner, that nearly every one died a sudden or violent death.

The realm of England was claimed by the Conqueror and his successors on the throne. On the theory of possession by conquest, the feudal system of tenure was built and developed. The Sovereign granted land to his soldiers and favourites, or left it in possession of the ancient holders, at his own pleasure. In return, those who enjoyed the property undertook various obligations to the lord paramount, such as military services. The grants were at first made for the lifetime of the holders; but ere long the rights and duties were extended to the succeeding heirs. Landowners holding thus of the king conveyed portions of their estates to their retainers, in return for services to be rendered. These services were usually either of a military or an agricultural character. With the advance of time, the progress of civilization, and the development of the arts of peace, the latter mode of compensation for the use of land gradually superseded the former; and after the Restoration the feudal burdens were abrogated by law. Since the abolition of these uncertain incidents of land tenure, the obligations subsisting among the holders of the several rights and interests pertaining to lands and tenements have been reducible to a money standard, with at least an approximation to justice and accuracy.

The custom of letting land for agricultural or building purposes i s a very ancient one, the earliest records of such a custom being those which have been preserved in the Institutes of Justinian. From these records we discover that under the Roman Empire there were in use four distinct forms of leasehold tenure. The most important tenure was that known as Emphyteusis—a Greek word meaning planting—which was introduced in the 2nd century, and arose out of the practice of making perpetual leases of provincial lands belonging to the State and acquired by the rights of war. This tenure was afterwards adopted by private proprietors, who, through inability to devote sufficient time or trouble to the cultivation of their property, preferred to let it to a suitable tenant called the Emphyteuta, who agreed to pay an annual rent for the use of the land. Of a similar nature to Emphyteusis was the tenure called Superficies, by which a landed proprietor conceded to another person an area of ground for the erection of buildings without parting with the ownership of the soil. A third class was that known as Precarium, which, being equivalent to a tenancy at will, could be determined by the grantor at his pleasure. Lastly, there was a general system of short leaseholds, the usual term being five years, at the end of which the lease might either be renewed or be considered as one from year to year. In all leases the tenant was bound to pay the burdens attached to the holding of the land and to deliver the receipts to the owner (1).

Every student of actuarial literature can call to mind the writings of leading experts that deal with the methods to be employed in order to ascertain what pecuniary profit or loss has resulted from the mortality experienced by a life assurance company. In this path of knowledge, Mr. C. D. Higham may be said to have been, if not the first to tread the soil, at least the earliest to write, for the benefit of the community, a record of his travels. Certain it is that since his paper on the “Value of the Death-Strain” (J.I.A., xx, 153), the importance of considering the true amount rather than the nominal amount at risk in such investigations as I have referred to, has been freely recognized and acted upon; and the death-strain as a function in actuarial matters has been much written and talked about.

It is with some reluctance that I venture to address the Institute on the subject of Reversionary Interests, a topic already exhaustively discussed by several members of great experience.

The chief motive which induces me to take this course is the desire to attract further attention to the advantages derived from the method of calculation originally adopted by Mr. Jellicoe (J.I.A., ii, 159) and subsequently laid under contribution by Mr. Bunyon (J.I.A., xviii, 32). This plan of operation, by means of which almost any formula may be derived from one general expression, was recommended by Mr. Bunyon for the following reasons:

(1). The exceptional facility with which any required formula can be written down.

(2). The great advantage secured by keeping the component parts of the formula clearly before the computer; thereby enabling him to modify the several parts without difficulty, in order to suit the case in question.

For instance, allowance may have to be made for loss of interest owing to delay in recovering the capital, and in re-investment of the same. A suitable premium may be necessary to cover a special assurance. Different rates of interest may be desired in possession and in reversion, convertible half-yearly or quarterly. In short, the actual conditions of a problem can be adhered to in the closest possible manner.

It has been, I believe, considered an axiom that payment of Life Assurance premiums should be made in advance, or in other words, that under any sound system of Life Assurance, the premiums paid must never be less than the value of the risk incurred from the outset upon the basis of calculation adopted, and this appears as a fundamental principle in all the published formulas for the determination of the values of any interests involving Life Assurance. The methods that are described in the following pages necessitate a departure from the existing practice in this respect, and it is therefore essential, in the first place, to justify so important a change.

A very important and difficult element in some of the State insurance schemes which have lately been put forward is, the provision of a small annuity to each child of a deceased father until that child attains the age of, say, 12 or 14 years. These may, not inappropriately, be called “family annuities”, and the object of the present paper is to investigate how the premiums for such family annuities may be calculated.

1.—Amongst the various current proposals for the application of the principle of Collectivity to the needs of life, those that relate to the provision for a minimum aid towards sustenance in that period of permanent incapacity known as Old Age, and for the burial of the dead, appear to be the least open to objection on political grounds; whilst the objects in view appeal so strongly to the circumstances of our common Humanity that few are disinclined to lend their assent to this partial recognition of claims that, in their wider extent, they would strongly resist. Fortunately, the question of the cost of these arrangements can be considered apart from all political grounds; and, since the results of its investigation disclose some novel and curious modifications of the formulæ for the ordinary Benefits, it has appeared to me that, by limiting the enquiry to the terms shown on the title to this Paper, the subject might be brought before this Meeting, without any breach of the benevolent neutrality that has so largely contributed to the preservation of our personal harmony amidst the many inevitable divergencies of opinion on subjects of close personal interest to all.

In a paper by the late Mr. Samuel Brown “On the Rate of Mortality and Marriage among Europeans in India” (J.I.A., xi, 1), reference is made to a method used by Griffith Davies to deduce rates of mortality from the records of the Madras Military Fund. It appears that Davies ascertained the average age of the lives in each of certain groups into which the experience naturally divided itself, and obtained the ratios which the actual deaths in the several groups bore to the corresponding number expected according to the Northampton Table at the average age. These ratios were then set out as ordinates corresponding to abscissae representing years of age, and a continuous curve was drawn in the usual way, by reading off which the graduated ratios for each age were obtained, and from these the graduated rates of mortality were at once formed.