.2. Life Assurance, in its widest sense, is a contract entered into by the assurer to pay a certain contingent upon the duration of one or more lives. The "present value" or single premium corresponding to an assurance of £1. payable at the end of the year of death of an individual, is deduced from the value of an annuity on the same life (see ANNUITY), and is expressed by the formula v — (1 — v) where v is the sum' which 1 will amount to £1 in one year (therefore equal to -----r being the interest of £1 for a 1 + /.' year), and A. is the value of an annuity of £1 per annum on the life aged z.
The more common form in which a life assurance is carried out is, however, by the payment of an annual premium to the company assuring, and this is determined (using 1 the same symbols as above) by the formula 1+ (1 v). The truth of which is thus demonstrated in a popular form by Mr. Gray. The present value of an " immediate" annuity on a life aged of an annuity of which the first payment falls to be made at the commencement of the transaction—being 1+A., it is easily deduced by propor 1 tion that £1 will purchase an immediate annuity of 1 + the reciprocal of the first value; and this would be the proper premium for the benefit if the latter were paid to the assured at the beginning, of the first and not at the end of the last year of the duration of the policy; but inasmuch as the benefit is not paid until the close of the stipulated period, the difference between its immediate value and its value if due a year hence (1 — v) has to be deducted from each year's premium, and the formula is the result.
The three important elements that have to he taken into account in the calculation of office premiums are—the rate of interest which is to accrue from their investment, the mortality returns with which the future experience of the insured is expected to agree, and the proportion or " loading" to be added to the net rates to meet expenses of management, and afford a profit to the insurer. The rate of 3 per cent has, with a very few exceptions, been adopted as a basis for such calculations, as the nearest to what can be expected to be realized on good security for transactions extending over many years. The mortality table most generally in use is that originally published by Mr. Milne, derived from the observations of Dr. Heysham on the rate of mortality in Carlisle during the nine years 1779 to 1787 inclusive, and hence known as the Carlisle table. This, how
ever, is now being superseded by the mortality experience of life assurance companies, collected by the institute of actuaries, and published in May, 1869, exhibiting certainly the most correct standard of assured life in this country, and possessing, by reason of the great skill with which it has been graduated, a complete adaptation for all practical purposes.
The following are examples of net premiums calculated on the institute data: The question of the addition to be made to such (net) premiums is influenced by differ ent considerations having regard to the practice of the office using the table.
Assurance companies are divided into three classes: 1. Proprietary companies, being those offices possessing a capital the property of the partners, and which, in addition to the accumulated premiums, is pledged to the policyholders as a guarantee for the fulfillment of their claims. As the liability in such companies is limited to the net sums assured, the addition made to the premiums requires to be only such a proportion as will cover the actual outlay for management, and remunerate the shareholders for the risk of loss which they run by fluctuation in the mortality, or from bad investments. A comparison of the above premiums with the "non-participation" rates usually adver tised, will show that the prevailing competition has induced the construction of tables very favorable to the public. 2. Hatual offices, where the members themselves constitute the company, being liable to each other for all claims. Here, in the absence of a capital, it is usual to adopt a scale of premiums known to be in excess of what is required to meet the sums insured. The profit arising therefrom is periodically ascer tained, and allotted to the assured, most frequently in the form of " bonuses" or additions to the claims payable under the policies. Some companies doing a large business are of this class, and in point of stability and irreproachable management bear the highest character. 3. Mixed companies are proprietary companies charging such increased rates as will yield a bonus, but which, in return for the expenses of management and guar antee of their capital, reserve for their proprietors a stipulated proportion of the profits.