The ultimate object of the following paper is to elicit a practical and efficient method of resolving, numerically, problems of every description involving an unknown quantity. In order to accomplish this end with the utmost generality, the most important step is to determine the inversion of a given function in a suitable and convenient series, involving the differential coefficients which appertain to an approximate value of the variable.

The Author commences by stating that he has in the course of his investigations found the theory of probabilities in a far from perfect state. In many cases, when he has wished to make use of the theory of probabilities, he has searched in vain for demonstrated formulae capable of immediate application, and he has generally been obliged to begin by working out the formulae independently for himself.

All tabular numbers drawn directly from observations, especially those appertaining to statistics, are more or less affected by unavoidable imperfections arising from defective information, insufficiency in the number of observations recorded by the experience, and other known or unknown incidental causes. When a consecutive series of tabular numbers is taken and differenced up to a certain order, the existence of the imperfections or errors alluded to, is at once revealed by the differences, which exhibit a conspicuous disturbance of their law of progression. To eradicate the errors and effect a proper adjustment of the original numbers is, however, an operation of peculiar and somewhat novel difficulty, and a critical examination of the empirical expedients, usually resorted to for such purpose, will show that instead of meeting the indispensable requirements of the problem, they only evade its most essential conditions, and thereby impregnate the results with a new series of theoretical errors, which too readily evade detection in consequence of the systematic uniformity of their law.

Two valuable contributions to what may perhaps be called the “Finance of Life Assurance” appear in the Assurance Magazine; both by eminent Members of this Institute. The first in 1858 by our present President “On the Investments of the Funds of Assurance Companies;” the second in 1862 by Mr. A. H. Bailey, “On the Principles on which the Funds of Life Assurance Societies should be invested.”

Few who are engaged in the practice of Life Assurance will deny that the prevailing restrictions imposed on, and charges made for foreign travelling and residence are unsatisfactory to all parties concerned. A considerable and increasing portion of the public whose pursuits or inclinations, especially in early life, take them to different parts of the world, are deterred from effecting life assurances on this account. The value of policies as securities is diminished owing to the uncertainty of the extra premiums which may at any time be imposed, and the risk of the assurances being forfeited altogether by causes over which mortgagees have no control. And actuaries feel that this subject is the opprobrium of their profession; for as in general no sufficient data exist whereby the risks incurred may be measured, the extra charges are therefore arbitrary and unscientific, for the most part little better than random guesses made on no intelligible principles. For these reasons it is hoped that any suggestions on the subject, however imperfect, may not be unacceptable, and may be the means of eliciting some useful discussion.

In the construction of the Tables of Policy Values for the Messenger Prize Essay (1868), the operations were considerably facilitated by the use of different modes of working applicable to the different data at hand; an explanation of which forms the subject of the present paper.

In the formation of Tables the great objects to be attained are accuracy and facility of working. These may be acquired in a high degree if it be possible by means of a simple operation to deduce each value in succession from the one which immediately precedes it, in which case the chances of error are reduced to a minimum, and the construction facilitated by the work not requiring to be checked except at distant intervals.

The existing theory and practice of Annuities and Assurances, and the tables ordinarily constructed for the purpose of expediting the various computations relating to the same, are radically founded on annual payments, or upon a series of mathematical values which appertain to certain annual periods that are specifically defined. Indeed, the methods employed are not strictly applicable unless the prescribed condition of annual periodicity is realized or assumed; and this, in fact, is the reason why assurances, as well as annuities, are necessarily supposed to be payable on the exact completion of one of the recurring periods, viz., at the end of the year of age in which the death takes place.

Some years ago I read before the Institute a paper bearing on this subject (see Journal, vol. xi., p. 90) in which I examined at considerable length the method of valuation which has been on various occasions so strongly advocated by Mr. Tucker, and which is denominated by him the “reinsurance method.” This term, however, appeared to me unsuitable, and I termed the method the “hypothetical method” of valuation; because the sums assured and the premiums are valued by means of a hypothetical table of the values of reversions and annuities, deduced by an inverse process from the premiums actually charged by the office. In that paper I compared Mr. Tucker's method with the net-premium method of valuation, and gave my reasons for believing the latter to be not only greatly superior to the former, but the method that should be generally employed. Subsequent consideration has however satisfied me that the net-premium method of valuation is also open to very serious objections, and that it is not applicable in all cases; and I propose on the present occasion, first to point out some of these objections, and then to consider briefly what method of valuation should be followed when the net-premium method is inapplicable. The subject however is a very wide one; and the present paper must be considered rather as a contribution to the discussion of a question that is all-important to those who have the responsibility of advising as to the liabilities of Life Insurance Companies, than as claiming to be itself a complete essay on the subject.

To say that the subject of the rate of mortality among assured lives and annuitants is at present very imperfectly understood, and that the process usually adopted to deduce the rate of mortality from observations upon these classes, is not a proper one for the purpose, and leads to erroneous and misleading results, may seem at first sight to be very rash assertions; but I believe that upon consideration they will be found strictly correct.

At the present time the subject of legislation in reference to Insurance Companies is attracting much attention in Germany as well as England. Until quite recently the regulation of insurance matters in Germany was under the sole control of the several States, as is still the case in the United States of America: the former German Confederation possessing neither the organization nor the authority to interfere in the matter. As a consequence of this state of affairs, we are informed that a company organized in one of the German States was regarded in the others as a foreign company, and all companies, whether German or foreign, which desired to extend their business over the whole of Germany, were compelled to put themselves in communication with more than thirty different Governments, each of which had its own special regulations. Some of the larger States had passed laws on the subject, but the provisions of them differed very widely from each other. In the remaining States, each Government imposed on the companies such conditions as it thought proper. This state of things is evidently very unfavourable for Insurance Companies, which require—more than any other undertaking—a wide field for their successful development. Under the old political organization of Germany, a uniform system of legislation, though long and ardently desired, was quite unattainable. But after the events of the year 1866 had led to the formation of the present North German Confederation, which gives a common legislation to at least three-fourths of Germany, the hopes of reformers revived. The 4th article of the Constitution of the Confederation expressly provides that legislation upon insurance shall belong to the province of the Confederation.

our predecessors adopted a ready and (all things considered) a sufficiently effectual method of regulating the charges to be made for granting assurances on lives. They satisfied themselves on the one hand that the mortality table adopted allowed for a much higher mortality than would probably be experienced, and on the other they assumed in their calculations a rate of interest below that which they knew would be realized on their investments. With these precautions they could rely pretty safely upon realizing a considerable profit, and this consideration not unnaturally proved sufficient for the scientific requirements of the time.

The annexed Table contains the values of the function for all values of x differing by ·01, from 0 to 10·4.

To find the value of the function for any intermediate value of x, enter the table with the value of x nearest to the given value, whether greater or less, and take out the quantities Δ and Δ′ opposite the interval to which the given value belongs.

The theory of Probabilities has been characterized by Laplace, one of those who have contributed most largely to its advance,—as “good sense reduced to a system of calculation;” and such, no doubt, it is. But it must be especially noticed that there is hardly any subject to which thought can be applied, which calls for so continuous an application of that excellent quality, or in which it i s easier to make mistakes from simple want of circumspection. And, moreover, that its reduction to calculation is attended with difficulties of a very peculiar nature, such as occur in no other application of mathematical analysis to practical subjects, arising out of the great magnitudes of the numbers concerned, which defeat the ordinary processes of arithmetical and logarithmic calculation, by exhausting the patience of the computer, and require special methods of approximate evaluation to bring them within the compass of human industry. These methods form a conspicuous feature of the general subject, and have furnished scope for very extraordinary displays of mathematical talent and invention. That very large numbers will inevitably be concerned in questions where numerous and independent contingencies may take place, and in any order or mode of combination, will be apparent to any one who considers the astonishing fecundity of such combinations numerically estimated, when the combining elements are many. For example, the number of possible “hands” at whist (regard being had to the trump) is 1,270,027,119,200.

The ordinary formula for the value of a reversionary annuity, ax–axy, is not strictly applicable to the cases which occur in practice. For it assumes that the first payment of the annuity is made at the end of the year in which y dies, and the last payment at the end of the year before that in which x dies. But the conditions which prevail in practice are different; for whenever a reversionary annuity is granted, it will run from the death of y, and the first payment will be made twelve months after that death, i.e., on the average, six months later than is assumed in the above formula. On the other hand, the reversionary annuity will in practice be payable up to the day of the death of x; or, on the average, a full year's payment will be made six months later than the last payment is made according to the above formula.