Page 260 - Plant design and economics for chemical engineers

230 PLANT DESIGN AND ECONOMICS FOR CHEMICAL ENGINEERS

Special Types of Annuities

One special form of an annuity requires that payments be made at the
beginning of each period instead of at the end of each period. This is known as
an annuity due. An annuity in which the first payment is due after a definite
number of years is called a deferred annuity. Determination of the periodic
payments, amount of annuity, or present value for these two types of annuities
can be accomplished by methods analogous to those used in the case of ordinary
annuities.

PERPETUITIES AND CAPITALIZED COSTS
A perpetuity is an annuity in which the periodic payments continue indefinitely.
This type of annuity is of particular interest to engineers, for in some cases they
may desire to determine a total cost for a piece of equipment or other asset
under conditions which permit the asset to be replaced perpetually without
considering inflation or deflation.
Consider the example in which the original cost of a certain piece of
equipment is $12,000. The useful-life period is 10 years, and the scrap value at
the end of the useful life is $2000. The engineer reasons that this
piece of equipment, or its replacement, will be in use for an indefinitely long
period of time, and it will be necessary to supply $10,000 every 10 years in order
to replace the equipment. He therefore wishes to provide a fund of sufficient
size so that it will earn enough interest to pay for the periodic replacement. If
the discrete annual interest rate is 6 percent, this fund would need to be
$12,650. At 6 percent interest compounded annually, the fund would amount to
($12,650)(1 + 0.06)” = $22,650 after 10 years. Thus, at the end of 10 years, the
equipment can be replaced for $10,000 and $12,650 will remain in the fund.
This cycle could now be repeated indefinitely. If the equipment is to perpetuate
itself, the theoretical amount of total capital necessary at the start would be
$12,000 for the equipment plus $12,650 for the replacement fund. The total
capital determined in this manner is called the capitalized cost. Engineers use
capitalized costs principally for comparing alternative choices.?
In a perpetuity, such as in the preceding example, the amount required for
the replacement must be earned as compounded interest over a given length of
time. Let P be the amount of present principal (i.e., the present worth) which
can accumulate to an amount of S during n interest periods at periodic interest
rate i. Then, by Eq. (5),
S = P(l + i)” (5)

tFor further discussion of capitalized costs used in engineering, see Chap. 10 and F. C. Jelen and
M. S. Cole. Methods for Economic Analysis, Part I, Hydrocarbon Proc., 53(7):133 (1974); Part II,
Hydrocarbon Proc., 53(9):227 (1974).