vity
since the table was constructed.

Sec. 12. #The single premium for any term.# It is evident that the
natural assessment premium payable at the beginning of the year for
$1000 of insurance for that year is expressed by the death rate, e.g.,
at age 35, the payment of $8.95 by each of the 81,822 living at the
beginning of the year will provide the $732,000 needed to pay the
losses.[5]

In the same manner would be determined the natural assessment premium
for each year of insurance. Now, when it is possible to invest the
premiums so as to yield a minimum rate of income it is a simple matter
to determine the amount of a single premium, at any age, that is
adequate to pay for insurance covering any selected number of years
(term insurance) up to the entire period of each insured person's
life (full life). It is necessary only to apply the formula of present
worth and that of compound interest on investments.[6] Thus the
expected losses of any year according to the table of mortality,
divided by 1 + rate of yield on investments raised to the power 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