F301_CH6-1
We will deal with 3 different
rates:
iNom = nominal, or stated, or
quoted, rate per year.
iPer = periodic rate.
effective annual
EAR = EFF% =
.
rate
F301_CH6-2
 iNom is stated in contracts. Periods per year (m)
must also be given. This is also frequently
referred to as an Annual Percentage Rate (APR).
 Examples:

8%; Quarterly

8%, Daily interest (365 days)
F301_CH6-3
 Periodic rate = iPer = iNom/m, where m is number
of compounding periods per year. m = 4 for
quarterly, 12 for monthly, and 360 or 365 for
daily compounding.
 Examples:
8% quarterly: iPer = 8%/4 = 2%.
8% daily (365): iPer = 8%/365 = 0.021918%.
F301_CH6-4
 Effective Annual Rate (EAR = EFF%):
Our bank offers a Certificate of Deposit (CD)
with a 10% nominal rate, compounded semiannually. What is the true rate of interest that
they pay annually (the effective annual rate)?
0
6mo
5%
$1
12mo
5%
F301_CH6-5
How do we find EFF% for a nominal rate
of 10%, compounded semiannually?
iNom
EFF% = 1 +
m
m
(
) -1
= (1 + 0.10) - 1.0
2
2
= (1.05)2 - 1.0
= 0.1025 = 10.25%.
Any PV would grow to the same FV at 10.25% annually or 10% semiannually. In both
instances, our money earns 10.25 percent each year.
F301_CH6-6
For a given rate, more frequent compounding
increases returns
12 percent compounded annually
EARA
= 12%.
12 percent compounded semi-annually
EARs
= (1 + 0.12/2)2 - 1
= 12.36%.
12 percent compounded monthly
EARM
= (1 + 0.12/12)12 - 1
= 12.68%.
12 percent compounded daily
EARD(365) = (1 + 0.12/365)365 - 1 = 12.75%.
F301_CH6-7
What is the FV of $100, if you earn 10
percent compounded semi-annually?
When using semi-annual periods we use the periodic rate:
INPUTS
6
N
5
I/YR
100
PV
0
PMT
FV
OUTPUT
Cpt = 134.01
When using annual periods we use the EAR:
10.25
100
0
3
INPUTS
N
OUTPUT
I/YR
PV
PMT
FV
Cpt = 134.01
The nominal rate is never used in calculations!
F301_CH6-8
Can the effective rate ever be equal to
the nominal rate?
 Yes, but only if annual compounding is
used, i.e., if m = 1.
 If m > 1, EFF% will always be greater than
the nominal rate.
F301_CH6-9
When is each rate used?
The use of the periodic rate and the effective annual rate is
determined by the choice of N. If N is the number of
years, the EAR is used. If N is the number of periods (shorter
than 1 year), the periodic rate is used. The nominal rate is not
used in calculations. As shown later, the choice of N will be
determined by the number of payments (PMT).
iNom:
Written into contracts, quoted
by banks and brokers. Not
used in calculations or shown
on time lines.
F301_CH6-10
Nominal, Periodic, or Effective Annual Rate?
 We never use the nominal rate calculating future or
present values. We always use either the effective
annual rate or the periodic rate of return.
 If PMT is 0, either the periodic rate or the effective rate
may be used.
 Ex: Find the present value of $100 to be received in 5 years.
The nominal rate is 6 percent, compounded monthly.
6/12 compute PV= $74.14
Periodic: FV = 100, N = 5*12, I = _______;
6.1678
E.A.R.: FV = 100, N = 5, I = __________;
compute PV= $74.14
F301_CH6-11
 If PMT has a value, N must equal the number of
payments. Since N is determined by the number of
payments, we must make I (the interest rate)
consistent with how we defined N.
 Ex: Find the present value of $100 per year for 5 years.
The nominal rate is 6 percent, compounded monthly.
6.1678 compute PV = $419.32
PMT = 100, N = 5, I = _________;
 Ex: Find the present value of $100 per month for 5 years.
The nominal rate is 6 percent, compounded monthly.
6/12
PMT = 100, N = 5*12, I = ________;
compute PV = $5,172.56
F301_CH6-12
A few other points to note.
 If the rate is 12% compounded monthly,
your EAR is 12.6825%, this is used if the
N is in number of years.
 If the rate is quoted at 12% compounded
monthly, the periodic rate would be 1%
(12%/12). The 1% is used if the N is in
number of months.
 If the rate is 12% compounded monthly,
you cannot divide the 12% by anything
besides its number of compounding
periods (in this case 12 monthly
periods), or you change the interest rate.
Example
F301_CH6-13
 What is the future value of 4 quarterly payments of $100
each, given a nominal rate of 12 percent, compounded
monthly.
 Note that the quarterly payments and monthly
compounding are different, so we have to transform the
monthly compounded rate to a quarterly compounded
rate. This takes some work.
First, find the EAR of the monthly rate:
EARM = (1 + 0.12/12)12 - 1 = 12.68%.
Second, find the nominal rate with quarterly
compounding that has an effective annual rate of 12.68%
(1 + Inom/4)4 - 1
= 12.68%.
Inom = 12.12%
Third, compute the future value of the payments
N = 4, I = 12.12/4, PV = 0, PMT = 100, cpt FV = $418.55
 See the next slide for the Excel Example
Excel Calculation Example
F301_CH6-14
Input area:
Quarterly Payments
Nominal Rate
# of times compounded
Quarter Compounding Periods
$100
12.00%
12
4
Output area:
Effective Annual Rate
12.68%
=+EFFECT(C5,C6)
Nominal Rate with 4 periods
12.12%
=NOMINAL(C13,C7)
Future Value of Quarterly PMTS
$418.55
=FV(C15/C7,C7,-C4,0)