The Mathematics of Money

Copyright © 2008, The McGraw-Hill Companies, Inc.

illustrate this by finding a future value with a number of commonly used compounding

frequencies.

Example 3.2.5 Find the future value of $5,000 in 5 years at 8% interest

compounded annually, semiannually, quarterly, monthly, biweekly, weekly, and daily.

The results are displayed in the table below. (You may want to work through these calcula-

tions yourself to get some more practice in using your calculator.)

Frequency Times/Yr i n

Formula

FV 
 PV(1  i)n

Future

Va l u e

Annual 1 0.08 5 FV 
 $5,000(1  0.08)^5 $7,346.64

Semiannual 2 0.08/2 5(2)
10 FV
$5,000  1 + 0.08_____ 2 

10

$7,401.22

Quarterly 4 0.08/4 5(4)
20 FV
$5,000  1  0.08_____ 4 

20

$7,429.74

Monthly 12 0.08/12 5(12)
60 FV
$5,000  1  0.08_____ 12 

60

$7,449.23

Biweekly

(Fortnightly)^26 0.08/26 5(26) 
^130 FV 
 $5,000 ^1 ^

0.08_____

26 ^

130

$7,454.54

Weekly 52 0.08/52 5(52) 
 260 FV 
 $5,000  1  0.08_____

52 ^

260

$7,456.83

Daily

(bankers’ rule)

360 0.08/360 5(360)
1,800 FV
$5,000  1  0.08_____ 360 

1,800

$7,458.79

Daily (exact

method)^365 0.08/365 5(365) 
 1,825 FV 
 $5,000 ^ 1+

_____0.08

365 ^

1,825

$7,458.80

As we would have expected, the table shows that more frequent compounding does indeed

result in more interest. This is true in general, though it may be a bit disappointing that the

gain in interest as the compounding frequency increases is not all that impressive beyond

a certain point. The gain in interest between annual and, say, monthly is far greater than

the gain between monthly and daily. This effect is due to the fact that, while a small a

time interval means plenty of compoundings, it also means that the interest rate is divided

among so many compoundings that each time interval brings only a miniscule amount of

interest. This is illustrated by the following very silly example.

Example 3.2.6 Find the future value of $5,000 at 8% interest for 5 years, assuming

that the interest compounds every minute.

This problem is basically the same as the previous example, except that we fi rst need to

determine how many minutes there are in a year. Since each day has 24 hours, and each

hour has 60 minutes:

(365 days/year)(24 hours/day)(60 minutes/hour) 
 525,600 minutes/year And so we add a

new line to the table from Example 3.2.3

Frequency Times/Year i n

Formula

FV 
 PV(1  i)n

Future

Value

Every minute 525,600 0.08/525,600 5(525,600)


2,628,000

FV 
 $5,000  1  ________0.08

525,600

(^) 
2,62
$7,459.12
While more than 2^1 ⁄ 2 million compoundings sounds astounding, in fact each compounding
contributes such a miniscule amount of interest that the end result produces just a whop-
ping 32 cents more than plodding along with daily compounding! Ridiculously frequent
compounding actually produces rather dull results, and thus (aside from the occasional
3.2 Compounding Frequencies 105