CE 533 - ECONOMIC DECISION
ANALYSIS IN CONSTRUCTION
Chapter III- Nominal and Effective
Interest Rates
By
Assoc. Prof. Dr. Ahmet ÖZTAŞ
GAZİANTEP University
Department of Civil Engineering
CHP 3- Nominal and Effective Interest Rates
Contents
Nominal and Effective interest rate staements
Effective interest rate formulation
Compounding and Payment Periods
Equivalence Calculations

- Single Amounts

- Series: PP >= CP

- Series: PP < CP
Using spreatsheets
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3.1 Nominal & Effective Interest Rates
In this chapter, we discuss nominal and effective
interest rates, which have the same basic relationship.
The difference here is that the concepts of nominal
and effective are used when interest is compounded
more than once each year.
For example, if an interest rate is expressed as 1%
per month, the terms nominal and effective interest
rates must be considered.
Every nominal interest rate must be converted into an
effective rate before it can be used in formulas, factor
tables, or spreadsheet functions because they are all
derived using effective rates.
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3.1 Nominal & Effective Interest Rates
Before discussing the conversion from nominal to
effective rates, it is important to identify a stated
rate as either nominal or effective.
There are 3 general ways of expressing interest
rates (See Table 3.1).
Example:

Interest is 12% per year

Interest is 8% per year, compounded monthly

Effctive Interest is 10% per year, compounded
monthly
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3.1 Nominal & Effective Interest Rates
These 3 statements in the top third of the table show that an
interest rate can be stated over some designated time period
without specifying the compounding period.
Such interest rates are assumed to be effective rates with the
compounding period (CP) same as that of the stated interest rate.
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3.1 Nominal & Effective Interest Rates
The above interest statementsd prevail three conditions:
(1) Compounding period is identified, (2) This compounding period
is shorter than the time period over which the interest is stated,
and (3) The interest rate is designated neither as nominal nor as
effective. In such cases, the interest rate is assumed to be
nominal and compounding period is equal to that which is stated.
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3.1 Nominal & Effective Interest Rates
In above statements in Table 3.1, the word effective
precedes or follows the specified, and the compounding
period is also given. These interest rates are obviously
effective rates over the respective time periods stated.
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3.1 Two Common Forms of Quotation
Two types of interest quotation

1. Quotation using a Nominal Interest Rate

2. Quoting an Effective Periodic Interest Rate
Nominal and Effective Interest rates are
common in business, finance, and
engineering economy
Each type must be understood in order
to solve various problems where
interest is stated in various ways.
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3.2 Effective Interest Rate Formulation
Understanding effective Interest rates requires a definition of a nominal
interest rate r as the interest rate per period times the number of periods.
A Nominal Interest Rate, r.
Definition:
A Nominal Interest Rate, r,
is an interest Rate that does
not include
any consideration
of compounding
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3.2 Effective Interest Rate Formulation
The term “nominal”
Nominal means, “in name only”,
not the real rate in
this case.
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3.2 Effective Interest Rate Formulation
Mathematically we have the following
definition:
r=
(interest rate per period)(No. of Periods)
(3.1)
Examples:
1) 1.5% per month for 24 months
Same as: (1.5%)(24) = 36% per 24 months
2) 1.5% per month for 12 months
Same as (1.5%)(12 months) = 18%/year
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3.2 Effective Interest Rate Formulation
Equation for converting a nominal Interest
rate into an effective Interest rate is:
i per period = (1 + r/m)m – 1
(2)
r = interest rate per period x number of periods,
m = number of times interest is comounded
İ = effective interst rate
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3.2 Example 1:
Given:
interest is 8% per year compounded
quarterly”.
What is the true annual interest rate?
Calculate:
i = (1 + 0.08/4)4 – 1
i = (1.02)4 – 1 = 0.0824 = 8.24%/year
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3.2 Example 2:
Given: “18%/year, comp. monthly”
What is the true, effective annual
interest rate?
r = 0.18/12 = 0.015 = 1.5% per month.
1.5% per month is an effective monthly
rate.
The effective annual rate is:
(1 + 0.18/12)12 – 1 = 0.1956 = 19.56%/year
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3.2 Effective Interest Rate Formulation
if we allow compounding to occur more and more
frequently, the compounding period becomes shorter
and shorter. Then m, the number of compounding
periods increases. This situation occurs in businesses
that have a very large number of CF every day.
i = er – 1
Where “r” is the nominal rate of interest
compounded continuously.
This is the max. interest rate for any value of
“r” compounded continuously.
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3.2 Effective Interest Rate Formulation
Example:
What is the true, effective annual interest
rate if the nominal rate is given as:

r = 18%, compounded continuously

Or, r = 18% c.c.
Solve e0.18 – 1 = 1.1972 – 1 = 19.72%/year
The 19.72% represents the MAXIMUM i for 18%
compounded anyway you choose!
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3.2 Effective Interest Rate Formulation
To find the equivalent nominal rate given the
i when interest is compounded continuously,
apply:
r  ln(1  i )
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3.2 Effective Interest Rate Formulation
Example
Given r = 18% per year, cc, find:

A. the effective monthly rate

B. the effective annual rate
a. r/month = 0.18/12 = 1.5%/month
Effective monthly rate is e0.015 – 1 = 1.511%
b. The effective annual interest rate is e0.18 – 1 = 19.72%
per year.
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3.2 Effective Interest Rate Formulation
Example
An investor requires an effective return of at
least 15% per year.
What is the minimum annual nominal rate
that is acceptable if interest on his investment
is compounded continuously?
To start: er – 1 = 0.15
Solve for “r” ………
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3.2 Effective Interest Rate Formulation
Example - Solution
er – 1 = 0.15
er = 1.15
ln(er) = ln(1.15)
r = ln(1.15) = 0.1398 = 13.98%
A rate of 13.98% per year, cc. generates the same
as 15% true effective annual rate.
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3.3 Reconciling Compounding periods &
Payment Periods (PP)
The concepts of nominal and effective Interest
rates are introduced, considering the compounding
period.
Now, let’s consider the frequency of the payments
of receipts within the cash-flow time interval.
For simplicity, the frequency of the payments or
receipts is known as the payment period (PP).
It is important to distinguish between the
compounding period (CP) and the payment period
because in many instances the two do not coincide.
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3.3 Reconciling Compounding periods &
Payment Periods (PP)
For example, if a company deposited money each
month into an account that pays a nominal interest
rate of 6% per year compounded semiannually, the
payment period would be 1 month while the CP
would be 6 months as shown in below Figure.
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3.3 Reconciling Compounding periods &
Payment Periods (PP)
So, to solve problems first step is to determine the
relationship between the compounding period and
the payment period.
The next three sections deseribe procedures for
determining the correct i and n values for use in
formulas, factor tables, and spreadsheet functions.
In general, there are three steps:
1. Compare the lengths of pp and CP.
2. Identify the CF series as involving only single
amounts (P and F) or series amounts (A, G, or g).
3. Select the proper i and n values.
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3.4 Equivalence Calculations of Single
Amount Factors
There are many correct combinations of i and n
that can be used when only single amount
factors (F/P and P/F) are involved. This is because
there are only two requirements:
(1) An effective rate must be used for i, and
(2) Time unit on n must be the same as that on i.
In standard factor notation, the single-payment
equations can be generalized.
P= F(P/F, effective i per period, number of periods)
F= P(F/P, effective i per period, number of periods)
Thus, for a nominal interest rate of 12% per year
compounded monthly, any of
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3.4 Equivalence Calculations of Single
Amount Factors
Thus, for a nominal interest rate of 12% per year
compounded monthly, any of the i and corresponding n
values shown in Table 3.4 could be used in the factors.
Example: if an effective quarterly interest rate is used for i,
that is, (1.01)3 - 1 = 3.03%, then the n time unit is 4
quarters.
Alternatively, it is always
correct to determine the effective i
per payment period using
Equation [3.2] and to use
standard factor equations to
calculate P, F, or A.
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3.4 Equivalence Calculations of Single
Amount Factors
Example: Sherry expects to deposit $1000 now, $3000 4
years from now, and $1500 6 years from now and eaen at a
rate of 12% per year compounded semiannually through a
company-sponsored savings plan.
What amount can she withdraw 10 years from now?
Solution:
Only single-amount P and F values are involved (See Figure
below).
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3.4 Equivalence Calculations of Single
Amount Factors
Since only effective rates can be present in the factors, use
an effective rate of 6% per semiannual compounding period
and semiannual payment periods.
The future worth is calculated as;
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3.4 Single Amounts: PP >= CP
Example:
“r” = 15%, c.m.
(compounded monthly)
Let P = $1500.00
Find F at t = 2 years.
15% c.m. = 0.15/12 = 0.0125 =
1.25%/month.
n = 2 years OR 24 months
Work in months or in years
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3.4 Single Amounts: PP >= CP
Approach 1. (n relates to months)
State:

F24 = $1,500(F/P,0.15/12,24);

i/month = 0.15/12 = 0.0125 (1.25%);

F24 = $1,500(F/P,1.25%,24);

F24 = $1,500(1.0125)24 = $1,500(1.3474);

F24 = $2,021.03.
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3.4 Single Amounts: PP >= CP
Approach 2. (n relates to years)
State:


F24 = $1,500(F/P,i%,2);
Assume n = 2 (years) we need to apply an annual
effective interest rate.

i/month =0.0125

Effective I = (1.0125)12 – 1 = 0.1608 (16.08%)

F2 = $1,500(F/P,16.08%,2)

F2 = $1,500(1.1608)2 = $2,021.19

Slight roundoff compared to approach 1
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3.4 Example 2.
F 10 = ?
Consider
r = 12%/yr, c.s.a.
0
1
2
3
4
$1,000
5
6
7
8
9
10
$1,500
$3,000
Suggest you work this in 6- month time frames
Count “n” in terms of “6-month” intervals
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3.4 Example 2.
F 10 = ?
Renumber the time line
r = 12%/yr, c.s.a.
0
2
4
6
8
$1,000
10
12
14
16
18
20
$1,500
$3,000
i/6 months = 0.12/2 = 6%/6 months; n counts 6month time periods
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3.4 Example 2.
F 20 = ?
Compound Forward
r = 12%/yr, c.s.a.
0
2
4
6
8
$1,000
10
12
14
16
18
20
$1,500
$3,000
F20 = $1,000(F/P,6%,20) + $3,000(F/P,6%,12) +
$1,500(F/P,6%,8) = $11,634
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3.4 Example 2. Let n count years….
F 10 = ?
Compound Forward
r = 12%/yr, c.s.a.
0
1
2
3
4
$1,000
5
6
7
8
9
10
$1,500
$3,000
IF n counts years, interest must be an annual rate.
Eff. A = (1.06)2 - 1 = 12.36%
Compute the FV where n is years and i = 12.36%!
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3.5 Equivalence Calculations Involving
Series With PP >= CP
When CF of the problem dictates the use of one or more of
the uniform series or gradient factors, the relationship
between CP and PP must be determined.
The relationship will be one of the following three cases:
Type 1. Payment period equals compounding period,
PP = CP
Type 2. Payment period is longer than compounding period,
PP > CP.
Type 3. Payment period is shorter than compounding
period, PP < CP.
The procedure for the first two CF types is the same.
Type 3 problems are discussed in the following section.
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3.5 Equivalence Calculations Involving
Series With PP >= CP
When PP = CP or PP > CP, the following procedure
always applies:
Step 1. Count the number of payments and use
that number as n.
For example, if payments are made quarterly for 5
years, n is 20.
Step 2. Find the effective interest rate over the
same time period as n in step 1.
For example, if n is expressed in quarters, then the
effective interest rate per quarter must be used.
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3.5 Series Example
F7 = ??
Consider:
0
1
2
3
4
5
6
7
A = $500 every 6 months
Find F7 if “r” = 20%/yr, c.q. (PP > CP)
We need i per 6-months – effective.
i6-months = adjusting the nominal rate to fit.
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3.5 Series Example
Adjusting the interest
r = 20%, c.q.
i/qtr. = 0.20/4 = 0.05 = 5%/qtr.
2-qtrs in a 6-month period.
i6-months = (1.05)2 – 1 = 10.25%/6-months.
Now, the interest matches the payments.
Fyear 7 = Fperiod 14 = $500(F/A,10.25%,14)
F = $500(28.4891) = $14,244.50
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3.5 This Example: Observations
Interest rate must match the frequency of the
payments.
In this example – we need effective interest
per 6-months: Payments are every 6-months.
The effective 6-month rate computed to equal
10.25% - un-tabulated rate.
Calculate the F/A factor or interpolate.
Or, use a spreadsheet that can quickly
determine the correct factor!
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3.5 This Example: Observations
Do not attempt to adjust the payments to fit
the interest rate!
This is Wrong!
At best a gross approximation – do not do it!
This type of problem almost always results in
an un-tabulated interest rate

You have to use your calculator to compute
the factor or a spreadsheet model to achieve
exact result.
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3.6 Equivalence Calculations Involving
Series With PP < CP
This situation is different than the last.
Here, PP is less than the compounding period (CP).
Raises questions?
Issue of interperiod compounding
An example follows.
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3.6 Equivalence Calculations Involving
Series With PP < CP
Consider a one-year cash flow situation.
Payments are made at end of a given month.
Interest rate is “r = 12%/yr, c.q.”
$120
$90
$45
0
1
2
3
4
$75
$150
5
6
7
$100
8
9
10
11
12
$50
$200
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3.6 Equivalence Calculations Involving
Series With PP < CP
r =12%/yr. c.q.
$120
$90
$45
CP-1
0
1
CP-2
2
3
4
$75
$150
CP-3
5
6
7
$100
8
CP-4
9
10
11
12
$50
$200
Note where some of the cash flow amounts fall with
respect to the compounding periods!
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3.6 Equivalence Calculations Involving
Series With PP < CP
Will any interest be earned/owed on the
$120
$200 since interest is compounded
at the end
$90
of each quarter?
$45
CP-1
0
1
2
$150
$200
3
The $200 is at the end of
4
5
6
7
8
10
11
month
2 and
will 9it earn
$50to go
for one month
$75interest
$100
to the end of the first
compounding period?
12
The last month of the first compounding period.
Is this an interest-earning period?
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3.6 Equivalence Calculations Involving
Series With PP < CP
The $200 occurs 1 month before the end of
compounding period 1.
Will interest be earned or charged on that
$200 for the one month?
If not then the revised cash flow diagram for
all of the cash flows should look like…..
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3.6 Equivalence Calculations Involving
Series With PP < CP
$165
Revised CF Diagram
$90
$90
$45
0
1
2
3
4
5
$75
$150
$200 $200
6
7
$100
8
9
10
11
12
$50
$50
$175
All negative CF’s move to the end of their respective
quarters and all positive CF’s move to the beginning
of their respective quarters.
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3.6 Equivalence Calculations Involving
Series With PP < CP
$165
Revised CF Diagram
$90
0
1
2
3
4
5
6
7
8
9
10
11
12
$50
$150
$200
$175
Now, determine the future worth of this revised series
using the F/P factor on each cash flow.
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3.6 Equivalence Calculations Involving
Series With PP < CP
With the revised CF compute the future
worth.
“r” = 12%/year, compounded quarterly
“i” = 0.12/4 = 0.03 = 3% per quarter
F12 = [-150(F/P,3%,4) – 200(F/P,3%,3) + (-175
+90)(F/P,3%,2) + 165(F/P,3%,1) – 50]
= $-357.59
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3.7 Using Excel for i Computations
In Excel, two functions are used to convert between
nominal and effective interest rates:
the EFFECT or NOMINAL functions.
Find effective rate:
EFFECT(nominal-rate, compounding frequency)
The nominal rate is r and must be expressed over
the same time period as that of the effective rate
requested.
The compounding frequency is m, which must equal
the number of times interest is compounded for the
period of time used in the effective rate.
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3.7 Using Excel for i Computations
Therefore, in the second example of Figure 3.6
where effective quarterly rate is requested, enter
the nominal rate per quarter (3.75%) to get an
effective rate per quarter, and enter m = 3, since
monthly compounding occurs 3 times in a quarter.
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3.7 Using Excel for i calculations
Find nominal:
NOMINAL(effective rate, compounding frequency
per year)
This function always displays the annual nominal
rate. Accordingly, the m entered must equal the
number of times interest is compounded annually. if
the nominal rate is needed for other than annually,
use Equation [3.1] below to calculate it.
r = (interest rate per period)(No. of Periods)
This is why the result of the NOMINAL function in
Example 4 of Figure 3.6 is divided by 2.
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3.7 Using Excel for i calculations
Study Example 3.7:
“Use EXCEL to find the semiannual
cash flow requested in Example 3.5.”
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Chapter Summary
Many applications use and apply nominal and
effective compounding
Given a nominal rate – must get the interest
rate to match the frequency of the payments.
Apply the effective interest rate per payment
period.
When comparing varying interest rates, must
calculate the Effective “i” in order to compare.
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Chapter III
End of the Chapter III