In the month of January, 1859, I had the honour to read before the Institute of Actuaries a paper upon the determination of the rates of premium for assuring against issue. Since that period the subject has lost none of its interest; on the contrary, its importance is yearly increasing, and it bids fair to form an important branch of assurance business. It is true that, out of the numerous inquiries made at Assurance Offices respecting the rates of premium for these risks, not 20 per cent, become the subject of policies absolutely granted, still it is desirable that there should be some sort of consistency in the premiums quoted. It would appear that considerable differences of opinion exist among actuaries upon the subject; and I feel, therefore, that I owe no apology to the Institute for departing from our usual routine subjects of rates of interest and mortality, and introducing once more the question of issue policies and the statistics of marriage.

I Hope, from time to time, to be able to give some further tables bearing on the subject, and to state how far I think subsequent experience and new information confirm or modify the interim conclusions which I ventured to submit in the present state of our knowledge; and I shall be very glad that my recent observations may receive attention, and correction where required, from critics abler than their author.

In the third of my papers entitled “Contributions to the History of Insurance and of the Theory of Life Contingencies” after treating upon the earliest examples which, after a somewhat tedious search, I could find of tontine associations practically carried out—examples confined, as it will be recollected, to some comparatively small projects of the Dutch municipalities of Kampen, Middelburg, and Groningen, employed for raising capital sums, by the grant of tontine life annuities, in 1670 and 1671—the circumstance was mentioned, that the credit of the invention was properly accorded to the Italians, or more particularly to the Italian Tonti, whose name has been so effectually incorporated into several languages through the word “tontine.”

We have now to give some examples of compound benefits, which are those consisting of two or more simple benefits; but the combinations which may be formed of these being obviously very numerous, it would be beside our present purpose to attempt giving a complete list of them. Our object will be, in selecting a few of them for illustration, to indicate the method of dealing with the more complicated cases, and also to prepare the way for the most general application of the Commutation Tables, which application will form the subject of the concluding portion of this paper. A very complete list of the formulae for the more elementary of these benefits is contained in Professor De Morgan's first paper on the subject; and as it is hoped that little difficulty will be experienced with these, after the illustrations to which our space limits us, we shall not scruple, as we have occasion, in the solution of any of the problems with which the present paper will be occupied, to refer to any of the learned gentleman's formulæ which we may not have deduced for ourselves. Our references will be made in the following manner, which is rendered necessary in consequence of the formulae not forming one consecutive series. Formula 10, on page 16, for example, will be denoted thus, [16,10]; formula 72, on page 18, thus, [18, 72]; and so on.