Chapter 5 Time Value of Money 147conversions are done as follows, where I is the stated annual rate, M is the number
of compounding periods per year, and N is the number of years:
Periodic rate (IPER)! __Stated annual rate
Number of payments per year
! I/M 5-8
With a stated annual rate of 5% compounded semiannually, the periodic rate is 2.5%:
Periodic rate! 5%/2! 2.5%
The number of compounding periods per year is found with Equation 5-9:
Number of periods! (Number of years)(Periods per year)! NM 5-9
With 10 years and semiannual compounding, there are 20 periods:
Number of periods! 10(2)! 20 periods
Under semiannual compounding, our $100 investment will earn 2.5% every
6 months for 20 semiannual periods, not 5% per year for 10 years. The periodic rate
and number of periods, not the annual rate and number of years, must be shown
on time lines and entered into the calculator or spreadsheet whenever you are
working with nonannual compounding.^9
With this background, we can! nd the value of $100 after 10 years if it is held
in an account that pays a stated annual rate of 5.0% but with semiannual com-
pounding. Here’s the time line and the future value:
Periods 0 1 2 19
Cash "ows
I = 2.5%^20
!$100 PV (1 + I)N = $100(1.025)^20 = FV 20 = $163.86
With a! nancial calculator, we get the same result using the periodic rate and num-
ber of periods:
N I/YR PV PMT FV20 2.5 –100 0163.86The future value under semiannual compounding, $163.86, exceeds the FV under
annual compounding, $162.89, because interest starts accruing sooner; thus, you
earn more interest on interest.
How would things change in our example if interest was compounded quarterly
or monthly or daily? With quarterly compounding, there would be NM! 10(4)!
40 periods and the periodic rate would be I/M! 5%/4! 1.25% per quarter. Using
those values, we would! nd FV! $164.36. If we used monthly compounding, we
would have 10(12)! 120 periods, the monthly rate would be 5%/12! 0.416667%,
and the FV would rise to $164.70. If we went to daily compounding, we would have
10(365)! 3,650 periods, the daily rate would be 5%/365! 0.0136986% per day, and
the FV would be $164.87 (based on a 365-day year).
(^9) With some! nancial calculators, you can enter the annual (nominal) rate and the number of compounding peri-
ods per year rather than make the conversions we recommend. We prefer the conversions because they must be
used on time lines and because it is easy to forget to reset your calculator after you change its settings, which
may lead to an error on your next problem.