Page 252 - Plant design and economics for chemical engineers

222 PLANT DESIGN AND ECONOMICS FOR CHEMICAL ENGINEERS

(d) Number of interest periods per year = m = 12
Nomin
m
- 1 (8)
( 02,) 1 2
Effective interest rate = 1 + 12 - 1 = 0.268 = 26.8%

CONTINUOUS INTEREST
The preceding discussion of types of interest has considered only the common
form of interest in which the payments are charged at periodic and discrete
intervals, where the intervals represent a finite length of time with interest
accumulating in a discrete amount at the end of each interest period. Although
in practice the basic time interval for interest accumulation is usually taken as
one year, shorter time periods can be used as, for example, one month, one day,
one hour, or one second. The extreme case, of course, is when the time interval
becomes infinitesimally small so that the interest is compounded continuously.
The concept of continuous interest is that the cost or income due to
interest flows regularly, and this is just as reasonable an assumption for most
cases as the concept of interest accumulating only at discrete intervals. The
reason why continuous interest has not been used widely is that most industrial
and financial practices are based on methods which executives and the public
are used to and can understand. Because normal interest comprehension is
based on the discrete-interval approach, little attention has been paid to the
concept of continuous interest even though this may represent a more realistic
and idealized situation.

The Basic Equations for Continuous
Interest Compounding
Equations (6), (7), and (8) represent the basic expressions from which continu-
ous-interest relationships can be developed. The symbol r represents the
nominal interest rate with m interest periods per year. If the interest is
compounded continuously, m approaches infinity, and Eq. (6) can be written as
mn r (m/rXm)
S after n years = P lim 1 + L 1 = P lim 1 + - 1 (10)
m-+m
m
m-m
m
(
(
The fundamental definition for the base of the natural system of loga-
rithms (e = 2.71828) is?
= e = 2.71828.. . (11)

tsee any book on advanced calculus. For example, W. Fulks, “Advanced Calculus,” 3d ed., pp.
55-56, John Wiley & Sons, Inc., New York, 1978.