Page 247 - Plant design and economics for chemical engineers

218 PLANT DESIGN AND ECONOMICS FOR CHEMICAL ENGINEERS

stipulating that interest is due regularly at the end of each interest period. If
payment is not made, the amount due is added to the principal, and interest is
charged on this converted principal during the following time unit. Thus, an
initial loan of $1000 at an annual interest rate of 10 percent would require
payment of $100 as interest at the end of the first year. If this payment were not
made, the interest for the second year would be ($1000 + $100X0.10) = $110,
and the total compound amount due after 2 years would be
$1000 + $100 + $110 = $1210

The compound amount due after any discrete number of interest periods
can be determined as follows:

Interest earned dur-
Principal ing period (i = in-
at start terest rate based on Compound amount S
Period of period length of one period) at end of period
1 P Pi P + Pi = P(l + i)
2 PC1 + i) PC1 + i)(i) P(l + i) + P(1 + i)(i) = P(l + i)*
3 P(1 + $2 P(l + i)*(i) P(l + i)* + P(l + i)*(i) = P(l + i)’
n P(l + ip--l P(1 + +-l(i) P(1 + i)”

Therefore, the total amount of principal plus compounded interest due after n
interest periods and designated as S isi’

s = P(l + i)” (5)

The term (1 + i)” is commonly referred to as the discrete single-payment
compound-amount factor. Values for this factor at various interest rates and
numbers of interest periods are given in Table 1.
Figure 7-1 shows a comparison among the total amounts due at different
times for the cases where simple interest, discrete compound interest, and
continuous interest are used.

NOMINAL AND EFFECTIVE INTEREST RATES
d
In common industrial practice, the length of the discrete interest period is
assumed to be 1 year and the fixed interest rate i is based on 1 year. However,
there are cases where other time units are employed. Even though the actual
interest period is not 1 year, the interest rate is often expressed on an annual
basis. Consider an example in which the interest rate is 3 percent per period

tFor the analogous equation for continuous interest compounding, see Q. (12).