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.

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.

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.

To find the present value of a Survivorship Annuity of £1 on the life of A to be entered on at the death of B, supposing the purchaser to effect an increasing insurance on A's life so that the sum insured may correspond with the amount of his outlay (of purchase money, premiums, and interest) at the end of each year up to the average duration of the joint lives of A and B, and to be at the end of that period of the exact amount which the annuity purchased would cover the annual interest thereon, beside providing for the premiums to keep it insured during the remainder of A's life.

When we consider in a table of mortality the consecutive terms which express the number living at each age, we remark that the differences between those numbers are almost constant in the middle ages of life; they change afterwards, remaining however uniform for another series of terms; so that the table can be considered as a series of decreasing arithmetical progressions. Let c be the difference between lx and lx+1; and let us suppose this difference constant for n years. We shall then have for the value of the life annuity during that time.

Life Assurance provides for the family of the deceased in case of premature death; deferred Annuities provide for old age; but both institutions leave uncovered the risk of premature inability to work. Invalidity Assurance, including the benefits of a deferred Annuity, would be the real complement to Life Assurance. This truth is so deeply felt in Germany, that a good many institutions, employing a large number of officers, workmen, and labourers; many mills, and particularly the Railway Companies, long since directed their attention to the providing for their officers in case of their being invalided. How were they to calculate the annual contribution, how to make the valuation of their liabilities?