2 Let P he the present worth of all the policies for a group of the same age, p the present worth of one policy, X the total insured at the beginning of the period. f the natural assessment premium this year, or the natural premium required for any year. Then 2 P (1 r) (1 (1 (14-r)n P definite term provides a reserve fund sufficient, on the as sumptions made, to carry all the insurance without further payments. Each year there is added to the fund the income earned on investments, and there is subtracted the amount of the losses for the year, until the death of the last member of the insured group. If the deaths in the earlier years are fewer than were expected in the mortality table, this will be offset eventually by more deaths at the advanced years: but in the meantime a reserve larger than was expected is yield ing income, thus providing a larger sum than is needed to pay all the policies at maturity. This surplus might be dis tributed as so-called "dividends" from time to time to those surviving, or be added pro-rata, at intervals, to the amount of the policies as accumulated dividends.
§ 4. Level annual term premiums and reserves. It is a matter of no very abstruse mathematics (in principle) to find the equivalent of this single premium in any one of many other forms of premium payment. The processes are but variations of present worth and compound interest calcula tions. Such calculations, however, lead into many complexi ties of practical detail difficult to explain in brief compass, and are the special task of the actuary (the mathematical expert dealing with such problems in the insurance business). The most• useful actuarial equivalent of the single premium is the level annual premium for any period (term or life). Almost all policies now written have the level annual premium as a feature. The amount of the level annual premiums at first is greater than the losses: this causes for a time the steady accumulation of a reserve that yields income. Then. as the losses grow, they overtake and finally surpass the amount of the annual premiums. Therefore, the total re serve for any group of insured, within the definite term fdr which insured, increases year by year to a maximum and then declines until it reaches zero with the payment of the last claim. The individual reserve for each policy not yet matured increases steadily the longer it is in force, what ever be the term. The total reserve is essential to the sol vency of the company and the payment of all the policies as they fall due.
The companies that issue policies on the level premium plan or reserve plan are known as ."old line" companies, or as "legal reserve" companies, because the state laws require every company of this type to maintain the reserves cal culated on the basis of a certain rate of yield. The growth
of the legal reserve companies in recent times constitutes one of the financial marvels of the age. They had in 1919 more than 58,000,000 policies in force, for a total of nearly $36,000,000,000 of indemnity (insurance in force) ; their total income was nearly $1,600,000,000 (about one fortieth being from investments, the remainder from premiums), and their total assets $6,700,000,000. These figures grow so rapidly that any statistics are soon out of date. The up ward curve may be seen in the following data: Number of Amount of Total Total policies in force insurance in force income of year assets Reserve insurance is carried on by both mutual and stock companies; of late some large stock companies, such as the Equitable and the Prudential, have been transformed into mutual companies. The mutual company legally belongs to the policyholders, though its control is actually in the hands of a self-perpetuating group of trustees and officers, more or less supervised by state officials. The gross premiums in reserve insurance are, for the purpose of safety, fixed at a figure larger than the expected cost of the insurance, and normally the earnings from interest are higher, the mortality is lower, and expenses are less than those on which the cal culation of rates is based. From the excess of income result ing, the company sets aside a surplus and then divides the rest among the policyholders. These returns, virtually but the refund of excess premiums, are called "dividends" (a somewhat misleading term, not to be confused with dividends on corporate stock). The policies that receive dividends are called "participating" and are said to participate in the earnings. Formerly the majority of policies paid "deferred" dividends after five, ten, or twenty years, according to vari ous tontine and semi-tontine plans, the survivors to these periods receiving their dividends plus those of the other pol icyholders who had died or had withdrawn from the com pany. This form of policy was objectionable in that it in volved a lottery element, the survivors winning the "divi dends" that should have been paid to the deceased; it was made illegal in New York and other states, and in most cases dividends are now paid annually. The stock company, or ganized for profit, frequently charges lower premiums for "non-participating" policies, and then retains such profits as may result from keeping expenses below receipts.