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

PLANT DESIGN AND ECONOMICS FOR CHEMICAL ENGINEERS

FIGURE 7-1

Comparison among total amounts accumulated with simple

interest, discrete compound interest, and continuous compound

nominal interest.

and the interest is compounded at half-year periods. A rate of this type would be referred to as “6 percent compounded semiannually.” Interest rates stated in this form are known as nominal interest rates. The actual annual return on the principal would not be exactly 6 percent but would be somewhat larger because of the compounding effect at the end of the semiannual period. It is desirable to express the exact interest rate based on the original principal and the convenient time unit of 1 year. A rate of this type is known as the effective interest rate. In common engineering practice, it is usually preferable to deal with effective interest rates rather than with nominal interest rates. The only time that nominal and effective interest rates are equal is when the interest is compounded annually.

Nominal interest rates should always include a qualifying statement indicating the compounding period. For example, using the common annual basis, $100 invested at a nominal interest rate of 20 percent compounded annually

would amount to $120.00 after 1 year; if compounded semiannually, the amount would be $121.00; and, if compounded continuously, the amount would be $122.14. The corresponding effective interest rates are 20.00 percent, 21.00 percent, and 22.14 percent, respectively. If nominal interest rates are quoted, it is possible to determine the effective interest rate by proceeding from Eq. (5).

s = P(1 + i)n (5)

In this equation, S represents the total amount of principal plus interest due after n periods at the periodic interest rate i. Let r be the nominal interest rate under conditions where there are m conversions or interest periods per year

INTEREST AND INVESTMENT COSTS 221

Then the interest rate based on the length of one interest period is r/m, and the amount S after 1 year is

Designating the effective interest rate as ieff, the amount S after 1 year can be expressed in an alternate form as

By equating Eqs. (6) and (71, the following equation can be obtained for the effective interest rate in terms of the nominal interest rate and the number of periods per year:

Example 1 Applications of different types of interest. It is desired to borrow

$1000 to meet a financial obligation. This money can be borrowed from a loan

agency at a monthly interest rate of 2 percent. Determine the following:

(a) The total amount of principal plus simple interest due after 2 years if no

intermediate payments are made.

(b) The total amount of principal plus compounded interest due after 2 years if no

intermediate payments are made.

(c) The nominal interest rate when the interest is compounded monthly.

(d) The effective interest rate when the interest is compounded monthly.

solwiotl

(a) Length of one interest period = 1 month

Number of interest periods in 2 years = 24

For simple interest, the total amount due after n periods at periodic interest

rate i is

S = P(l + in) (2)

P = initial principal = $1000

i = 0.02 on a monthly basis

n = 24 interest periods in 2 years

S = $lOOO(l + 0.02 x 24) = $1480

(b) For compound interest, the total amount due after n periods at periodic

interest rate i is

S = P(l + i)” (5)

s = $1000(1 + o.02)24 = $1608

(c) Nominal interest rate = 2 X 12 = 24% per year compounded monthly