Introduction to Finance

LEARNING ACTIVITY
Go to the Citibank website, http://www.citibank.com, and identify the types of credit cards 
available to individuals, and the prevailing interest rates on the credit cards.
9.9
More Frequent Time Intervals 
and The Cost of Consumer Credit
More Frequent Than Annual Compounding or Discounting
In many situations, compounding or discounting may occur more often than annually. For 
example, recall from the beginning of this chapter the $1,000 that could be invested at one 
bank at an 8 percent annual interest rate for two years. Remember that the future value at the 
end of two years was as follows:
FV
2
= $1,000(1.08)
2
= 
$1,000(1.166)
= 
$1,166.40
Now let’s assume that another bank off ers the same 8 percent interest rate but with 
semi-annual (twice a year) compounding. We can fi nd the future value of this investment by 
modifying equation 5.1 as follows:
FV
n
= PV(1 + 
r
÷ 
m
)
nxm
(9.9)

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242
C H A PT E R 9 Time Value of Money
where 
m
is the number of compounding periods per year. For this problem,
FV
2
= $1,000(1 + 0.08 ÷ 2)
2 × 2
= 
$1,000(1.04)
4
= 
$1,000(1.1699)
= 
$1,169.90
Thus, by compounding semi-annually the future value would increase by $3.50.
The process for more frequent than annual compounding can be described, operationally, 
as follows. First, divide the annual interest rate of 8 percent by the number of times compound-
ing is to take place during the year (0.08 ÷ 2 = 0.04). We also need to increase the total number 
of periods to refl ect semi-annual compounding. To do this, multiply the number of years for 
the loan times the frequency of compounding within a year (two years × 2 = 4 periods).
Previously in this chapter, it was shown that a $1,000 investment at an 8 percent interest 
rate would grow to $2,158.92 or $2,159 (rounded) at the end of ten years. However, if semi-
annual compounding had been available, the future value of the $1,000 investment would 
have been,
FV
20
= $1,000(1.04)
20
= 
$1,000(2.1911)
= 
$2,191.10
The following subsection shows how the fi nancial calculator solution would be found.