It is not my intention in this paper to travel over the same ground as that followed by Mr. Sprague in his paper in the 7th vol. of the Journal, p. 61, “On Certain Methods of Dividing the Surplus among the Assured in a Life Assurance Company; and on the Rates of Premium that should be Charged to Render them Equitable”, and that by Mr. Tucker, in the 9th vol., p. 245, “On the Rates of Premiums Required to Provide certain Periodical Returns to the Assured”, although the changes which have been introduced into the methods described by both writers, and the increased knowledge we possess of life assurance mortality statistics, would fully justify a revision and extension of their labours. Such a work, if undertaken, would fully repay the time spent upon it, and would prove a valuable addition to the information we already possess upon a very important subject.

Both the above-mentioned gentlemen enlarged upon what “H. A. S.”, in the 8th vol. of the Journal, p. 167, describes as “The incongruity existing between the rates of premium charged at certain ages and the benefits accruing therefrom”; and they have shown that when the net cost of the assurances and bonuses is calculated upon a certain basis, the figures differ, often very materially, from those of the premiums charged.

We will now state, with the necessary technical detail, our reasons for the opinion we have already expressed, that it is not permissible to add to the assets, or otherwise bring into the valuation as a credit, the present value of the extra charge made upon half-yearly and quarterly premiums in excess of the aliquot part of an annual premium. This extra charge is made for three different purposes, (1) to cover the cost and trouble of collecting premiums at frequent intervals; (2) to make good the loss of interest on the portion of the premium which is not paid at the beginning of the year; and (3) when the half-yearly or quarterly payments are really premiums, and not instalments of yearly premiums, to compensate for the loss of premium which will occur in some cases through the life assured dying before the full year's premium for the current policy-year has been paid. The proper amount of the extra to be charged for (2) and (3) is easily determined. Since the value of an annuity payable in advance is 1 + a, ¾ + a, or ⅝ + a, according as it is payable yearly, half-yearly, or quarterly, the total yearly net premium for the age x is

according as it is payable yearly, half-yearly, or quarterly.

The problem of finding the rate of interest corresponding to a given annuity-value, has engaged considerable attention at various times; and many formulas have been proposed for obtaining approximate solutions by the aid of existing tables of annuities. Most of these will be found in Mr. Sutton's paper on the Rate of Interest yielded by Foreign Loans (J.I.A., xix, 77). Of the 5 formulas there given, (2) is probably the most generally useful, (3), (4), and especially (5) (Prof. De Morgan's formula), involving a good deal of calculation. The object of the present short paper is to suggest 2 new formulas, which give very accurate results, and at the same time require but little work in their practical application.

Let a be the value of an annuity for n years certain, at a rate of interest i, (which it is required to find), and let a1, a2, a3, be the tabulated annuity-values for the same term at the rates i1, i1 + h, i1 + 2h, where a lies between a1 and a2, and consequently i between i1 and i1 + h.

The ordinary method of approximating to three-life annuities is as follows:—Let the lives be aged respectively x, y, z, (x being the youngest). Find an age w such that at the given rate of interest aw=ayz, then axyz=axw nearly. Baily diminishes the value thus found by .05, but I cannot say if this correction is usually made. It will generally be found to be too small if the lives are very young, but much too large at the older ages.

It has been shown that the above method (neglecting now the correction referred to) is strictly accurate in any mortality table where the mortality conforms to Mr. Gompertz's hypothesis, and where consequently the force of mortality can be represented throughout by the expression

(B and q being constants). The reason for this is, that the force of mortality for two joint lives, or, x, x + t, may be represented by a similar expression

(where qt′ = qt + 1).

It follows from this that if we have

we have also

and, the force of mortality in the joint status being equal to that for the single life, throughout the table, the two annuities are identical, not only as to their total value, but also as to the value of each successive payment.

Upon the selection of lives depends to a large extent the financial success of a life assurance company, as the acceptance of under-average risks at ordinary rates may be as prejudicial to the well-being of an office as an imprudent investment of its funds. The opposition of interests also suggests the necessity for caution, and calls for the constant exercise of sound judgment. The care taken by life offices to exclude inferior risks, invariably has the effect of reducing the number of deaths in the early years of assurance considerably below those indicated in the tables upon which their calculations are based. The extent and duration of this favourable influence have been repeatedly investigated, and the subject is now well understood. In this paper it is proposed to extend the enquiry to the partial death-rates from various classes of diseases, and thus to ascertain the strength of this self-protecting power in different directions. A somewhat wider meaning will be attached to the term “selection” in the following pages than has usually been associated with it by actuarial writers. It will be regarded as including not only the effects of the medical examination, but also those other selecting causes which impart to the mortality of assured lives the characteristics it is seen to possess when compared with that of the general population.