Page 253 - Plant design and economics for chemical engineers

INTEREST AND INVESTMENT COSTS 223

Thus, with continuous interest compounding at a nominal annual interest rate
of r, the amount S an ’ ‘tial principal P will compound to in n years is?+
7 S = Pe’” (12)
Similarly, from Eq. (8), the effective annual interest rate ieff, which is the
conventional interest rate that most executives comprehend, is expressed in
terms of the nominal interest rate r compounded continuously as

’ =e’-1 (13)
leff
r = In(i,, + 1) (14)
Therefore,
e ‘n = (1 + ieff)” (15)
and

S = Pe’” = P(l + i,ff)n (16)
As is illustrated in the following example, a conventional interest rate (i.e.,
effective annual interest rate) of 22.14 percent is equivalent to a 20.00 percent
nominal interest rate compounded continuously. Note, also, that a nominal
interest rate compounded daily gives results very close to those obtained with
continuous compounding.

tThe same result can be obtained from calculus by noting that, for the case of continuous
compounding, the differential change of S with time must equal the nominal continuous interest
rate times S, or dS/dn = KS. This expression can be integrated as follows to give Eq. (12):

,,5; = rpn

In f = m or S = Pern (12)

$A generalized way to express both Eq. (12) and Eq. (9, with direct relationship to the other
interest equations in this chapter, is as follows:
Future worth = present worth x compound interest factor
s = PC
or
Future worth x discount factor = present worth
SF = P
1 1
Discount factor = F = c-...
compound interest factor C
Although the various factors for different forms of interest expressions are derived in terms
of interest rate in this chapter, the overall concept of interest evaluations is simplified by the use of
the less-complicated nomenclature where designated factors are applied. Thus, expressing both Eqs.
(12) and (5) as SF = P would mean that F is e-‘” for the continuous interest case of Eq. (12) and
(1 + i)-” for the discrete interest case of Eq. (5). See Table 4 for further information on this
subject.