or
or
ELEMENTS WHICH REDUCE VALUE
of gradual development but of full growth and mature age, com-
posed of numerous units, the useful life of all the units or parts of
which is # years, it is desired to know what amount is in the
replacement fund at any time, that fund being assumed to re-
ceive such an increment at the end of each year that, during the
life of each unit, this annuity, with interest, will amount to the
original cost of this unit.
Being composed of a large number of elements — each year
having added new ones — the addition to it per year will be
taken for the purpose of this illustration at one-nth of the total
plant as it stands at the end of the nth year.
For each dollar invested on this assumption in the first
year, there will be $1 invested in each succeeding year, and
for each dollar thus invested there will be # dollars of total
investment.
Let a represent the annual contribution to the replacement
fund for each dollar invested.
Assume this contribution to be available at the end of each
year.
Then after n years, na will be the annual contribution to the
amortization fund for each dollar of the annual investment.
Let m represent any number of years greater than 7.
Let ¢ represent the interest rate expressed in hundredths, i.e.,
for 6 per cent, 2 = 0.06.
During the first # years, after beginning the construction of
the plant, there will be no replacements, and the replacement
fund continues to grow. At the end of the nth year the re-
placement requirement, assuming permanency in character and
cost, will be $1 for each dollar of annual investment, and this
replacement requirement will continue at this rate thereafter.
At the end of the nth year the replacement fund will contain:
For each dollar invested the first year:
a(x +2)" +a(xr +4924 «+. + adollars
HC + 2)» — 1] dollars.