Engineering Economic Analysis

Nominal and Effective Interest 109

Nominal and Effective Interest

Consider the situation of a person depositing $100 into a bank that pays 5% interest, compounded

semiannually.How much would be in the savings account at the end of one year?

.SOlU~tO~n

Five percent interest, compoundedsemiannually,means that the bank pays 21/2%every 6 months.

Thus, the initial amountP=$100 would be credited with 0.025(100)=$2.50 interest at the end

of 6 months, or

P--*P+Pi= 100 + 100(0.025) = 100 + 2.50 = $102.50

The $102.50 is left in the savings account; at the end of the second 6-month period, the interest

earned is 0.025(102.50)=$2.56, for a total in the account at the end of one year of 102.50 +

2.56 =$105.06, or

(P+Pi)--*(P+ Pi)+i(P+Pi) =P(1+i)2=100(1+ 0.025)2

=$105.06

Nominal interest rate per year,r,is the annual interest rate without considering the

effect of any compounding.

In the example, the bank pays 21/2%interest every 6 months. The nominal interest rate per

year,r,therefore, is 2 x21h%= 5%.

Effective interest rate per year,ia,is the annual interest rate taking into account the

effect of any compounding during the year.

In Example 4-13 we saw that $100 left in the savings account for one year increased to

$105.06, so the interest paid was $5.06. The effective interest rate per year,ia,is $5.06/

$100.00 =0.0506 =5.06%.

r=Nominal in~erestrate per interest period (usually one year)

i=Effective interest rate per interest period

ia=Effective interest rate per year

m=Number of compounding subperiods per time period

Using the method presented in Example 4-13, we can derive the equation for the effective

interest rate. If a $1 deposit were made to an account that compounded interestmtimes