The object of the actuary, in applying the science of vital statistics to pecuniary transactions, being to determine the values of payments dependent upon conditions of human survivorship, and therefore necessarily deferred for periods of longer or shorter duration, it is obvious that the rate of interest at which money may be increased is an element of his calculations, nearly if not quite as important as the probable risk of mortality that may affect the lives involved. The labours of actuaries have been generally directed to determining with precision the laws of mortality, rather than towards the present more humble branch of inquiry–the usefulness of which, however, must always become apparent with the earliest professional experience. Believing that an extensive and well digested collection of facts relating to the rate of interest would lead to some valuable and important conclusions as to future fluctuations, I trust that the present attempt to contribute to such a result will not be without interest for the members of the Institute, although I make no pretension to give a complete view of the whole subject, still less to discuss the important and intricate questions arising out of it; my object being merely to offer a general historical sketch, leaving the full prosecution of the inquiry to others, possessing more leisure and better qualifications for the task. Apart from its professional, the subject is one of historical interest. Men's pecuniary affairs afford unfailing indications of their characters, habits, and inclinations; and it has been truly remarked by a philosophical writer, that “a right measure and manner in getting, saving, spending, giving, taking, lending, borrowing and bequeathing, would almost argue a perfect man.” What is true in this respect of individuals is true to nearly the same extent of communities; and hence the conviction, becoming every day more general, of the importance of financial, for the illustration of general history.

The efforts made by Fire Insurance Companies to collect accurate particulars of the state of their business, and the profit or loss on different classes of risks, evince at once their appreciation of correct statistical information and the difficulty of obtaining it. Most Offices of any standing keep registers in which risks are divided into distinct classes—such as cotton mills, corn mills, farm stock, dwelling-houses, carpenters' shops, &c. &c.; and in one set enter under their proper heads the sums insured, premiums received, and number of risks, so that they can tell from inspection how many corn or cotton mills, or other risks, they insure each year, the total sums insured on them, and the amounts of premiums received: while in another set they enter under their proper heads the number of corn or cotton mills, or other risks, in which loss has occurred, and the losses paid—so that by inspection they can see what losses have been sustained on each class. The difference between the premiums received and the losses paid, leaving expenses of management out of view, constitutes a surplus or deficiency from which they judge of the profit or loss on each class of their insurances.

In the present paper I propose to give some numerical illustrations of the working of the method of which the formulæ were given in the last Number of this Magazine, and to add some further results which have been subsequently arrived at.

In order to apply the formulæ to numerical cases, it is necessary to assume some law according to which the successive bonuses proceed. It is obvious, that whatever law is assumed must be to a great extent arbitrary, as it is of course quite impossible to say what will be the state of a particular Office after the lapse of a number of years. But it seems probable that, on an average, an Office may return to the assured in the form of bonus 20 per cent. of the premiums paid, and it is accordingly assumed in the calculations that each quinquennial cash bonus declared on a policy is equal in amount to one year's premium. This will probably not be far from the truth where the method of division of profits adopted in the Office is one that leads to nearly constant cash bonuses to each policy: such as the method proposed by Mr. Jellicoe, which divides the profits in proportion to the loading of the premium, accumulated at compound interest for the interval between two successive bonuses. But in cases where one of the more common methods of division is adopted, which lead to a series of increasing cash bonuses on each policy, the above illustration will not be applicable, and the conditions of the problem will be greatly changed. The calculations are made by means of the “Experience” mortality, assuming interest at 4 per cent.

The investigation of the laws by which the general mortality of the human species appears to be governed is of equal importance to the statesman, the physician, the natural philosopher, and the mathematician; and as you have had occasion to pay particular attention to the subject, I trust that it will not be disagreeable to you to receive the results of an inquiry, into which I have entered, for the purpose of appreciating, if not of reconciling, the many discordant opinions that have been advanced respecting the comparative mortality of mankind, at different times and under different circumstances.