of years distant, equals the present worth of insuring the entire group for that year.  The sum of the discounted cost of insurance for all the years of the term divided by the number living at the beginning of the period, gives the single premium for each of the insured.  Let P be 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

f           f1          f2          fn
P = __________ + _________ + ________ + _________
(l + r)      (l + r)^2   (l + r)^3   (l + r)^n

P
p = _________
X

The payment in advance of the single premium for any selected period provides a reserve fund sufficient, on the assumptions 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 yielding income, thus providing a larger sum than is needed to pay all the policies at maturity.  This surplus might be distributed 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.

Sec. 13. #Level annual 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 mainly but variations of present worth and compound interest calculations.  Such calculations, however, lead into many complexities 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 which yields income.  Then, as the losses grow, they overtake and finally surpass the amount of the annual premiums.  Therefore, the total reserve for any group of 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.  The total reserve is essential to the solvency of the company and the payment of all the policies as they fall due.  The companies which 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 calculated 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.