Blank L., Tarquin A. Engineering Economy (McGraw-Hill Series in Industrial Engineering and Management)
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SECTION 4.2
Effective Annual Interest Rates
Solution
(a) The annual interest rate
is
stated, but the compounding period or frequency is not.
Substitute different
m values into Equation [4.5] to obtain the corresponding
i"
value. (Use Equation [4.12] for continuous compounding.) Compare
it
with the
listed
APY rate
in
the Web
ad
(Figure
4-2).
From the results below and rounding to
two decimal places for the estimated
APY rates, it appears that the bank applies
monthly compounding
to
its stated current interest rates.
If
the
The Effective
The
Most
Likely
For an
Stated
Compounding
Annual Rate i
a
,
Compounding
Investm
en
t
In
terest
Frequency m
Is
or
Estimated
Period Used
Term of: Rate
is:
Assumed
to
be:
APY,
is:
by
the
Bank
is:
3 years 3.40% 4 quarters 3.444
12 months 3.453 Monthly
52 weeks 3.457
5 years 4.36% 4 quarters 4.432
12
months 4.448 Monthly
52 weeks 4.455
Continuous 4.456
(b)
Option J: 5-year
CD.
Use the APY rate
of
4.45% (from Figure
4-2)
in
the F / P factor
or the Excel FV function.
F = 10,000(F / P,4.45%,5) = 10,000(1.2432) = $12,432
Option 2: 2 years in stocks, then a 3-year
CD.
This
is
a higher-risk option since the
return
on
stocks
is
not certain. Use 10% per year for the stocks, which
is
the esti-
mated effective annual rate, followed
by
3 years at the 36-month CD effective
annual rate
of
3.45%.
(It
is unlikely that the CD rate will remain at this level for
2 more years, but this
is
the best estimate available now.)
F = 1
O,OOO(F
/
P,
1 0%,2)(F / P,3.45%,3)
= 10,000(1.21)(1.1071) = $13,396
The second option is estimated
to
earn $964 more over the 5 years.
Comment
The interest rates and compounding periods used in this example are only representative;
they change frequently and vary from one institution to another. Check the website
of
any
financial institution offering Internet banking services
to
learn about the current rates.
All the economic situations discussed in this section involve annual nominal
and effective rates and annual cash flows. When cash flows are non-annual, it is
necessary
to
remove the year assumption in the interest rate statement "r% per
year, compounded
m-ly." This is the topic
of
the next section.
135
136
Figure
4-3
One-year cash
flow
dia-
gram for a monthly pay-
ment period (PP) and
semiannual compounding
period
(C
P).
CHAPTER 4 Nominal and Effective Interest Rates
r = nominal 14% per year, compounded semiannually
6
m~~t-h-
S
---'Iom~~ths
'I
2 3 4 5 6 7 8 9 to
11
12
o
I
iii
t t t i
itt
t i
~I
PP
I-
I month
Months
4.3
EFFECTIVE INTEREST
RATES
FOR
ANY
TIME
PERIOD
The concepts
of
nominal and effective annual interest rates have been intro-
duced. Now,
in
addition
to
the compounding period (CP), it
is
necessary to
consider the frequency
of
the payments or receipts, that is, the cash flow transac-
tion period. For simplicity, this is called the
payment
period
(PP). It is important
to
distinguish between the compounding period and the payment period because
in
many instances the two do not coincide. For example, if a company deposits
money each month into
an
account that pays a nominal interest rate
of
14
% per
year, compounded semiannually, the payment period is 1 month while the com-
pounding period is 6 months (Figure
4- 3). Similarly,
if
a person deposits money
each year into a savings account which compounds interest quarterly, the pay-
ment period is 1 year, while the compounding period is 3 months.
To
evaluate cash flows that occur more frequently than annually, that is, PP < ]
year, the effective interest rate over the PP must be used in the engineering econ-
omy relations. The effective annual interest rate formula is easily generalized to
any nominal rate
by
substituting r / m for the period interest rate in Equation [4.5].
Effective
i =
(1
+ r/
m)m
- 1
[4.8]
where:
.7
r = nominal interest rate per payment period (PP)
m = number
of
compounding periods per payment period (CP per PP)
Instead
of
i(f,
this general expression uses i
as
the symbol for effective interest.
This conforms to all other uses
of
i for the remainder
of
the text. With Equation
[4.8], it is possible to take a nominal rate
(r%
per year or any other time period)
and convert it
to
an effective rate i for any time basis, the most common
of
which
will be the
PP time period. The next two examples illustrate how
to
do this.
EXAMPLE
4.4
~,'~'
Visteon, a spin-off company
of
Ford Motor Company, supplies major automobile compo-
nents
to
auto manufacturers worldwide and is Ford's largest supplier.
An
engineer
is
on a
Visteon committee to evaluate bids for new-generation coordinate-measuring machinery
to
be directly linked to the automated manufacturing
of
high-precision components.
TIu-ee
vendor bids include the interest rates
011
the next page. Visteon will make payments
SECTION 4.3 Effective Interest Rates for Any Time Period
on a
semiaJU1Ual
basis only. The eng
in
eer is confused about the effective interest rat
es-
what they are annually and over the payment period
of
6-months.
Bid #1: 9% per year, compo
un
ded quarterly
Bid
#2: 3% per quarter, compounded quarterly
Bid #3: 8.8% per year, compounded monthly
(a) Determ
in
e the effective rate for each bid
on
tbe basis
of
semiannual payme
nt
s,
an
d
construct cash
flow
diagrams similar
to
Figure
4-3
for each bid rate.
(b) What are the effective annual rates? These are
to
be a part
of
the
final
bid selec
ti
on.
(c) W
hi
ch bid h
as
the lowest effective annual rate?
Solution
(a) Set the payment peri
od
(PP) at 6-mo
nth
s, convert the nominal rate r%
to
a semi-
annual basis, then determine
m. Fina
ll
y,
use Equation [4.8]
to
calculate
the
effective
semia
nn
ua
l
in
terest rate i. For Bid #1, t
he
fo
ll
owing are correct:
PP
= 6
mo
nths
r = 9% per year = 4.5% per 6-months
m = 2 quarters per 6-months
Effective i% per 6-months
=
(1
+
0.~45
r -1 = 1.0455 - 1 = 4.55%
Table 4- 4 (left section) summarizes the effective semiannual rates for all three bids.
Figure
4- 4a
is
the cash
flow
diagram for bids
#1
and #2, semiannual payments
(PP
= 6 months) and quarterly compounding (CP = I quarter). Figure
4-4b
is
the
same for monthly compounding (bid #3).
(b) For the effective annual rate, the time basis in Equation
[4
.
8]
is
1 year. T
hi
s
is
the
same
as
PP = 1 year. For bid
#1
,
r = 9% per year
m = 4 quarters per year
Effective i% per year
=
(1
+
0~9
r -1 = 1.0931 - 1 = 9.31 %
The right secti
on
of
Table
4-4
includes a summary
of
the effective annual rates.
(c) Bid #3 includes the lowest effecti
ve
annual rate
of
9.16%, which
is
equivalent
to
an
effecti
ve
semiannual rate
of
4.48%.
TABLE
4-4
Effec
ti
ve Se
mi
annual and Annual Interest Rates
fo
r Three Bid Rates,
Example 4.4
Semiannual Rates
Annual Rates
Nominal
per
CP
per
Equation
[4.8], Nominal
per
CP
per
Equation
[4
.
8]
,
Bid 6
Months
, r
PP,
m
Effective;
Year, r
Year, m
Effecti
ve
;
#1
4.5% 2 4.55%
9% 4 9.31%
#2
6.0% 2 6.09% 12%
4
12
.
55
%
#3 4.4%
6
4.48%
8.8% 12 9.16%
137
138
CHAPTER 4 Nominal and Effective Interest Rates
~
C
P-~
CP
~CP~CP=1
I I quarter I 1 quarter I I quarter I 1 quarter I
I
mTTf"ihTTril
I 2 3 4 2 3 4 5 6 7 8 9
LO
II 12
1-,----
pp----
.
f+-
----pp---
--+J
6 months 6 months
I
I
-,----PP---~~
~
--
--p
p---~~
,
6 months 6 months
(a) Quarte
rl
y compounding (b) Monthly compounding
Figure
4-4
Cash
fl
ow
diagram
s
howin
g
CP
and
PP
for
(a) bids I
and
2 and (b) bid
3,
Example
4.4.
Comment
The effective rates for bid #2 only may be found direct
ly
in
Table 4
-3
. For the effective
semia
nnu
al rate, look at the nominal
60/0
line under m = 2, which
is
the number
of
quarters
per 6-months. The effective semiannual rate
is
6.09
0/0
. Similarly, for
th
e nominal
120/0
rate,
there are m
= 4 quarters per year, so effective annual i = 12.551
0/0
. Although Table
4-
3
was orig
in
a
ll
y designed for nominal annual rates,
it
is correct for other nominal rate peri-
ods, provided the appropriate
m va
lu
e is included
in
the column headings.
A dot-com company
pl
a
ns
to place money
in
a new venture capital fund that cun·ently
returns
IS0/0
per year, compounded daily. What effective rate
is
th
is (a) yearly and
(b) semiannually?
Solution
(a) Use Equation [4.8], with r = O.IS and m = 365.
Effective
i
O/O
per year = (1 +
~.~;
t
5
- 1 = 19.716
0/0
(b) Here r = 0.09 per 6-months a
nd
m =
IS
2 days.
Effective
i
O/O
per 6-months = (1 +
~.~~
)
182
- 1 =
9.41
5
0/0
4.4
EQUIVALENCE RELATIONS: COMPARING PAYMENT
PERIOD
AND
COMPOUNDING
PERIOD
LENGTHS
(PP
VERSUS
CP)
In
a large
percentage
of
equivale
n
cy
com
put
ations,
the
frequency
of
cash
flows
does not equal the frequency
of
interest
compo
unding.
For
example,
cash
flows
may
occ
ur
monthly
, and
compounding
occur
s annually,
quart
erly,
or
more often.
SECTION 4.5 Equivalence Relation
s:
Single Amounts w
ith
PP ~ CP
TABLE
4-5
Section References for Equivalence Calculations Based
on
Payment Period and Compounding Period Comparison
Involves Involves Uniform
Series
Length
Single
Amounts
or
Gradient
Series
of
Time
(P
and
F Only) (A, G,
or
g)
PP = CP Section 4.5 Section 4.6
PP
> CP Section 4.5
Section 4.6
PP < CP
Section 4.7 Section 4.7
Consider deposits made to a savings account each month, while the earning rate
is
compounded quarterly.
The
length
of
the
CP
is a quarter, while the
PP
is a
month. To correctly perform any equivalence computation, it
is
essential that the
compounding period and payment period be placed on the same time basis, and
that the interest rate be adjusted accordingly.
The next three sections describe procedures to determine correct
i and n val-
ues for engineering economy factors and spreadsheet solutions. First, compare
the length
of
PP
and
CP,
then identify the cash flow series
as
only single amounts
(P and
F)
or as a selies (A, G, or g). Table
4-5
provides the section reference.
When only sing
le
amounts are
in
volved, there is no payment period
PP
per se
defined by the cash flows.
The
length
of
PP
is, therefore, defined
by
the time
period
t
of
the interest rate statement.
If
the rate is 8% per 6-months, com-
pounded quarterly, the
PP
is 6-months, the
CP
is
3 months, and PP >
CPo
Note that the section references in Table
4-5
are the same when PP = CP and
PP >
CPo
The
equations to determine i and n are the same. Additionally, the tech-
nique to account for the time value
of
money
is
the same because it
is
only when
cash flows occur that the effect
of
the interest rate is determined. For example,
assume that cash flows occur every 6 months
(PP is semiannual), and that inter-
est is compou
nd
ed each 3 months (CP is a quarter). After 3 months there is no
cash flow and no need to determine the effect
of
quarterly compounding. How-
ever, at the 6-month time point, it is necessary to consider the interest accrued
during the previous two quarterly compounding periods.
4.5 EQUIVALENCE RELATIONS: SINGLE
AMOUNTS
WITH
PP
2:
CP
When only single-amount cash flows are involved, there are two equally correct
ways
to
determine i and n for P / F and F / P factors. Method 1 is easier to apply,
because the interest tables
in
the back
of
the text can usually provide the factor
value. Method 2 likely requires a factor formula calculation, because the result-
ing effective interest rate is not an integer. For spreadsheets, either method is ac-
ceptable; however, method 1 is usually easier.
Method
1:
Determine the effective interest rate over the compounding period
CP, and set n equal to the number
of
compounding periods between P and F.
The
139
140
"
CHAPTER
4 Nominal and Effective Interest Rates
relations to calculate P and
Fare
P =
F(P
/
F,
effective i%
per
CP,
total
number
of
periods
n)
F =
P(F
/P, effective i%
per
CP
,
total
number
of
periods
n)
[4.9]
[4.10]
For
example, assume that a nominal 15% per year, compounded monthly, is the
stated credit card rate. Here
CP
is
a month.
To
find P or F over a 2-year span, cal-
culate the effective monthly rate
of
15%/12
= 1.25% and the total months
of
2(12) = 24. Then 1.25% and 24 are used in the P / F and F / P factors.
Any time period can be used to determine the effective interest rate; however,
CP
is
the best basis. The
CP
is
best because only over the CP can the effective
rate have the same numerical value as the nominal rate over the same time period
as the
CP.
This was discussed in Section 4.1 and Table
4-1.
This means that the
effective rate over
CP
is usually a whole number. Therefore, the factor tables
in
the back
of
the text can be used.
Method
2: Determine the effective interest rate for the time period t
of
the
nominal rate, and set n equal to the total number
of
periods using this same time
period. The
P and F relations are the same as in Equations [4.9] and [4.10] with
the term
effective i%
per
t substituted for the interest rate.
For a credit card rate
of
15
% per year, compounded monthly, the time period
t is I year.
The
effective rate over 1 year and n values are
Effective
i% per year = (1 +
0~~5)1
2
- 1 = 16.076%
n = 2 years
The
P/F
factor is the same by both methods:
(P/F,1.25%,24)
= 0.7422 using
Table 5; and
(P
/ F, 16.076%,2) = 0.7422 using the P / F factor formula.
EXAMPLE
4.6
..
An engineer working as a pl;vate consultant
made
deposits into a special account to
cover
unreimbursed travel expenses. Figure
4-5
is
the cash flow diagram. Find the
amount
in the account after
10
years at an interest rate
of
12% per year, compounded
semiannually.
Solution .
Only P and F values are involved. Both methods are illustrated to find
Fin
year It.
Method 1: Use the semiannual CP to express the effective semiannual rate
of
6%
per
6-month period.
There
are n = (2)(number
of
years) semiannual periods for each cash
flow.
Using the factor values from Table 11,
we
see that the future worth by Equation
[4.10]
is
F = 1000(F/P,6%,20) + 3000(F/ P,6
%,
12)
+ 1500(F/P,6
%,
8)
=
LOOO(3.2071)
+ 3000(2.0122) + 1500(1.5938)
=
$11
,634
SECTION 4.5 Equivalence Relations: Single Amounts with PP 2':
CP
F=
?
0
2 3
4
5 6
7
8 9
10
J
!
I I
I
I
I
I I I
$
1000
$1500
$
3000
Figure
4-5
Ca
sh
flow
diagram,
Exampl
e 4.
6.
Method 2: Express the effective annual rate, based on semiannual compounding.
Effective i% per year
=
(I
+
0.~2
r -J = 12.36%
The
n value
is
the actual number
of
years. Use the factor formula (F/ P,i,n) = (1.1236)"
a
nd
Equation [4.10] to obtain the same answer as with method
I.
F =
JOOO
(F/ P,12.
36
%,IO) +
3000(F
/ P,12.36%,6) + 1500(F/ P,12.36%,4)
= J 000(3.2071) + 3000(2.0 J 22) + 1500(1.5938)
=
$11
,634
Comment
For single-amount cash flows, any combination
of
i and n derived from the stated nom-
inal rate can be used
in
the factors, provided they are on the same time basis. Using 12%
per year, compounded monthl
y,
Table
4-6
presents various acceptable combinations
of
i and
n.
Other combinations are
con
'ect, such as the effective weekly rate for i and
weeks for
n.
TABLE
4-6
Various i and
Il
Values for Single-
Amount Equations
Using r =
12
%
per
Year, Compounded Monthly
Effective Rate ;
1% per month
3.03% per quarter
6.15% per 6 months
12.68% per year
26.97% per 2 years
Units for n
Months
Quarters
Serniannual periods
Years
2-year periods
141
142
CHAPTER 4
Nominal and Effective Interest Rates
TABLE
4-7
Examples
of
nand
i Values Where PP = CP or PP > CP
Cash Flow
What
to
Find;
Standard
Series
Interest
Rate
What
Is
Given
Notation
$500 se
mi
annually
16% per year, Find
P; given A
P = 500(P/A,8%,
IO)
for 5 years
compounded
semiannually
$75 monthly for
24% per year,
Find
F; given A F =
75(F
/A,2%,36)
3 years compounded
monthly
$180 quarterly for
5% per quarter Find
F;
given A
F = 180
(F
/ A,5
%,
60)
15
years
$25 per month
1 % per month Find
P; given G P = 25(P/ G,1%,48)
increase for
4 years
$5000 per quarter
J%
per month
Find
A;
given P
A =
5000(A/
P
,3
.03%,24)
for 6 years
4.6
EQUIVALENCE RELATIONS:
SERIES
WITH
PP
2':
CP
When uniform or gradient series are included in the cash flow sequence, the pro-
cedure is basically the same
as
method 2 above, except that PP
is
now defined
by
the frequency
of
the cash flows. This also establishes the time unit
of
the effec-
tive interest rate. For example,
if
cash flows occur
Oi1
a quarterly basis, PP is a
quarter and the effective quart
er
ly rate is necessary. The n value is the total num-
ber
of
quarter
s.
If
PP
is
a quarter, S years translates to an n value
of
20 quarter
s.
This
is
a direct application
of
the following general guideline:
When cash flows involve a series (i.e.,
A,
G, g)
and
the
payment
period
equals
or
exceeds the compounding period in length,
• Find the effective i
per
payment
period.
• Determine n as the total
number
of
payment
periods.
In performing equivalence computations for series, only these values
of
i and n
can be used
in
interest tables, factor formulas, and spreadsheet functions. In other
words, there are no other combinations that give the correct answers, as there are
for single amount cash flows.
Table
4-7
shows the correct formulation for several cash flow series and
interest rate
s.
Note that n is always equal to the total number
of
payment periods
and
i
is
an effective rate expressed over the same time period as n.
EXAMPLE
4.7
'
For the past 7 years, a quality manager has paid $500 every 6 months for the software main-
tenance contract
of
a LAN. What is the equivalent amount after the last payment, if these
funds are taken from a pool
th
at
ha
s been returning 20% per year, compounded quarterly?
SECTION 4.6 Equivalence Relations: Series with PP
2:
CP
Solution
The cash flow diagram is shown
in
Figure
4-6.
The
payment period (6 months) is
longer than the compounding period (quarter); that is,
PP >
CP.
Applying the guideline,
we
need to determine an effective semiannual interest rate. Use Equation [4.8) with r =
0.10 per 6-month period and
/11.
= 2 quarters per semiannual period.
(
0.10)2
Effective i% per 6-months = J + 2 - I = 10.25%
The effect
iv
e semiannual interest rate can also be obtained from Table
4-3
by
using the
r value
of
10
% and
/11.
= 2 to get i =
10
.25%.
The
va
lu
e i = 10.25% see
ms
reasonable, since we expect the effecti ve rate to be
slightly higher than the nominal rate
of
10
% per 6-month period. The total number
of
semiannual payment periods
is
n = 2(7) = J
4.
The relation for F
is
o
I
Figure
4-6
F = A(F/A,10.25%,14)
= 500(28.4891)
= $14,244.50
i = 20% per year. compounded quarterly
2
3
4
5
A = $500
Diagram
of
semiannual deposits used to determine
F,
Example 4.7.
EXAMPLE 4.8
,<.
F=?
6 7
Years
Suppose you plan to purchase a car and carry a loan
of
$12,500
at
9% per year, com-
pounded monthly. Payments will be made monthly for 4 years. Determine the monthly
payment. Compare the computer and hand solution
s.
Solution
A monthly series A
is
sought; the PP and
CP
are both a mon
th
. Use the steps for PP =
CP
when a uniform series is present. The effective interest per month
is
9%/
12
= 0.75%, and
fIJ
the number
of
payments is (4 years)(12 months per year) = 48.
Enter PMT(9
%/
12
,48, - 12500) into any cell to display $3 11.06.
Figure
4-7
shows a complete spreadsheet with the PMT function in cell B5 using ce
ll
Q-
So
lv
reference format. This monthly payment
of
$311.06
is
equivalent to the following solution
by
hand, using standard notation and the factor tables.
A = $12,500(A/ P,0.75%,48) = 12,500(0.02489) =
$31
1.13
143
144
,
CHAPTER 4 No
mi
na
l
an
d Effective Interest Rates
_ 0 x
B
c
Purchase price $12,500
Number of payments
48
I nterest rate 9%
Monthly payments $311 .06
=
PM
T(B31l
2,
B2, - B I)
1
<4
J
<4
~
'
.,
\
Sh
ee
t 1
She~
Sheet5
~
eet
fJ(
Sheet7
.Jl
~
I
J
Re
ady
I
NUM
Fi
gure
4-7
Spreadsheet
fo
r
Ex
ample
4.
8.
Comment
It
is incorrect to use
th
e effective annual rate of i =
9.
3
81
% and n = 4 years to calculate
th
e monthly A value, whether solv
in
g by hand or by
co
mpute
r.
The payment pe
ri
od,
th
e ef-
fective rate, and
th
e number
of
payments must all be on
th
e same time
ba
sis, which is
th
e
month in
thi
s example.
The
Scott a
nd
White Health Plan (SWHP) has purchased a robotized prescription fulfill-
ment system for faster a
nd
more accurate delivery to patients with stable, pill-form med-
ic
ation for chronic health problems, such as diabetes, thyroid, and
hi
gh blood pressur
e.
Assume
thi
s
hi
gh-volume system costs $3 mil lion to
in
sta
ll
and an estimated $200,000 per
year for a
ll
mate
ri
als, operating, personnel, and maintenan
ce
costs. The expected life is
10
years. An SWHP biomedical eng
in
ee
r wants to estimate the total reve
nu
e requirement
for each 6-month pe
ri
od
th
at is necessary
to
recov
er
the investment,
in
terest, a
nd
annual
cost
s.
Find
thi
s semiannual A value bo
th
by hand a
nd
by computer, if capital funds are eval-
uated at 8% per year us
in
g two different compounding periods:
l.
8% per yea
r,
compounded semiannually.
2. 8% per year, compounded monthly.