By Assoc. Prof. Dr. Ahmet ÖZTAŞ GAZİANTEP University Department of Civil Engineering

By

Assoc. Prof. Dr. Ahmet ÖZTAŞ

GAZİANTEP University Department of Civil Engineering

CE 533 - ECONOMIC DECISION ANALYSIS IN CONSTRUCTION

Chapter III- Nominal and Effective Interest Rates

CE 533 Economic Decision Analysis 2 / 54

CHP 3- Nominal and Effective Interest Rates

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

Contents


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.


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



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.



The above interest statementsd prevail three conditions:(1) Compounding period is identified, (2) This compounding

periodis shorter than the time period over which the interest is

stated,and (3) The interest rate is designated neither as nominal

nor aseffective. In such cases, the interest rate is assumed to benominal and compounding period is equal to that which is

stated.



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.



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.


3.2 Effective Interest Rate Formulation

A Nominal Interest Rate, r.Definition:

A Nominal Interest Rate, r, is an interest Rate that does

not includeany consideration of compounding

Understanding effective Interest rates requires a definition of a nominal interest rate r as the interest rate per period times the number of periods.


The term “nominal”

Nominal means, “in name only”,not the real rate in

this case.



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 months2) 1.5% per month for 12 months

Same as (1.5%)(12 months) = 18%/year



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


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


3.2 Example 2:

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

Given: “18%/year, comp. monthly”


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.



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!



To find the equivalent nominal rate given the i when interest is compounded continuously, apply:

ln(1 )r i



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.



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” ………



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.



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 orreceipts 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.



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.



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.


3.4 Equivalence Calculations of Single Amount Factors

There are many correct combinations of i and n that can be used when only single amountfactors (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 ofthe i and corresponding n values shown in Table 3.4 could be used (as well asmany others not shown) in the factorso For example, if an effective quarterly



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.



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).



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;


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



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.



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


3.4 Example 2.

Consider

0 1 2 3 4 5 6 7 8 9 10

$1,000

$3,000

$1,500

F 10 = ?

r = 12%/yr, c.s.a.

Suggest you work this in 6- month time frames

Count “n” in terms of “6-month” intervals


3.4 Example 2.

Renumber the time line

0 2 4 6 8 10 12 14 16 18 20

$1,000

$3,000

$1,500

F 10 = ?

r = 12%/yr, c.s.a.

i/6 months = 0.12/2 = 6%/6 months; n counts 6-month time periods


3.4 Example 2.

Compound Forward

0 2 4 6 8 10 12 14 16 18 20

$1,000

$3,000

$1,500

F 20 = ?

r = 12%/yr, c.s.a.

F20 = $1,000(F/P,6%,20) + $3,000(F/P,6%,12) +

$1,500(F/P,6%,8) = $11,634


3.4 Example 2. Let n count years….

Compound Forward

0 1 2 3 4 5 6 7 8 9 10

$1,000

$3,000

$1,500

F 10 = ?

r = 12%/yr, c.s.a.

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%!


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 = CPType 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.

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.

3.5 Equivalence Calculations Involving Series With PP >= CP


3.5 Series Example

Consider:

0 1 2 3 4 5 6 7

A = $500 every 6 months

F7 = ??

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.


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


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!


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.


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.

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.”

0 1 2 3 4 5 6 7 8 9 10 11 12

$90

$120

$45

$150

$200

$75 $100$50



CP-2CP-1

r =12%/yr. c.q.

0 1 2 3 4 5 6 7 8 9 10 11 12

$90

$120

$45

$150

$200

$75 $100$50

CP-3 CP-4

Note where some of the cash flow amounts fall with respect to the compounding periods!



CP-1

0 1 2 3 4 5 6 7 8 9 10 11 12

$90

$120

$45

$150

$200

$75 $100$50

Will any interest be earned/owed on the $200 since interest is compounded at the end of each quarter?

The $200 is at the end of month 2 and will it earn interest for one month to go to the end of the first compounding period?

The last month of the first compounding period. Is this an interest-earning period?



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…..



0 1 2 3 4 5 6 7 8 9 10 11 12

Revised CF Diagram$90

$165

$45

$150

$200

$75 $100$50

$200$175

$90

$50

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.



Revised CF Diagram

0 1 2 3 4 5 6 7 8 9 10 11 12

$165

$150

$200$175

$90

$50

Now, determine the future worth of this revised series using the F/P factor on each cash flow.



With the revised CF compute the future worth.

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

“r” = 12%/year, compounded quarterly

“i” = 0.12/4 = 0.03 = 3% per quarter



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.


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.


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.


3.7 Using Excel for i calculations

Study Example 3.7: “Use EXCEL to find the semiannual cash flow requested in Example 3.5.”


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.

Chapter III

End of the Chapter III

By Assoc. Prof. Dr. Ahmet ÖZTAŞ GAZİANTEP University Department of Civil Engineering

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