CE533-Chp3-Nominal rate.ppt

7/28/2019 CE533-Chp3-Nominal rate.ppt

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By

Assoc. Prof. Dr. Ahmet ZTA

GAZANTEP UniversityDepartment of Civil Engineering

CE 533 - ECONOMIC DECISIONANALYSIS IN CONSTRUCTION

Chapter III- Nominal and EffectiveInterest Rates


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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 PeriodsEquivalence Calculations

- Single Amounts

- Series: PP >= CP

- Series: PP < CP

Using spreatsheets

Contents


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3.1 Nominal & Effective Interest Rates

In this chapter, we discuss nominal and effectiveinterest rates, which have the same basic relationship.

The difference here is that the concepts of nominaland effective are used when interest is compoundedmore than once each year.

For example, if an interest rate is expressed as 1%per month, the terms nominal and effective interestrates must be considered.

Every nominal interest rate must be converted into aneffective rate before it can be used in formulas, factortables, or spreadsheet functions because they are allderived using effective rates.


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Before discussing the conversion from nominal toeffective rates, it is important to identifya statedrate 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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These 3 statements in the top third of the table show that aninterest rate can be stated over some designated time periodwithout specifying the compounding period.Such interest rates are assumed to be effective rates with thecompounding period (CP)same as that of the stated interest rate.



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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 benominaland compounding period is equal to that which is stated.



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In above statements in Table 3.1, the word effectiveprecedes or follows the specified, and the compoundingperiod is also given. These interest rates are obviouslyeffective 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, andengineering economy

Each type must be understood in order

to solve various problems whereinterest is stated in various ways.


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3.2 Effective Interest Rate Formulation

A NominalInterest Rate, r.

Definition:

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

not includeany considerationof compounding

Understanding effective Interest rates requires a definition of a nominal

interest rate ras the interest rate per period times the number of periods.


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The term nominal

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

this case.



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Mathematically we have the following

definition:

r =

(interest rate per period)(No. of Periods) (3.1)

Examples:

1) 1.5% per month for 24 monthsSame as: (1.5%)(24) = 36% per 24 months

2) 1.5% per month for 12 monthsSame as (1.5%)(12 months) = 18%/year



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Equation for converting a nominal Interestrate 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

= 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:

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 monthlyrate.

The effective annual rate is:

(1 + 0.18/12)12 1 = 0.1956 = 19.56%/year

Given:18%/year, comp. monthly


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if we allow compounding to occur more and more

frequently, the compounding period becomes shorterand shorter. Then m,the number of compoundingperiods increases. This situation occurs in businessesthat 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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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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To find the equivalent nominal rate given thei when interest is compounded continuously,

apply:

ln(1 )r i



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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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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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Example - Solution

er 1 = 0.15

er

= 1.15ln(er) = ln(1.15)

r = ln(1.15) = 0.1398 = 13.98%

A rate of 13.98% per year, cc. generates the sameas 15% true effective annual rate.



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3.3 Reconciling Compounding periods &Payment Periods (PP)

The concepts of nominal and effective Interestrates are introduced, considering the compoundingperiod.

Now, lets consider the frequency of the paymentsof 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 thecompounding period (CP) and the payment periodbecause in many instances the two do not coincide.


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For example, if a company deposited money eachmonth into an account that pays a nominal interestrate of 6% per year compounded semiannually, thepayment period would be 1 month while the CP

would be 6 months as shown in below Figure.


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So, to solve problems first step is to determine therelationship between the compounding period andthe payment period.

The next three sections deseribe procedures for

determining the correct iand nvalues for use informulas, 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 singleamounts (Pand F)or series amounts (A, G, or g).

3. Select the proper iand nvalues.


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3.4 Equivalence Calculations of SingleAmount Factors

There are many correct combinations of i and nthat can be used when only single amount

factors (F/Pand P/F)are involved. This is becausethere are only two requirements:

(1) An effective rate must be used for i, and(2) Time unit on nmust be the same as that on i.

In standard factor notation, the single-paymentequations 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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Thus, for a nominal interest rate of 12% per yearcompounded monthly, any of the i and corresponding nvalues 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 ntime unit is 4

quarters.

Alternatively, it is alwayscorrect to determine the effective iper payment period usingEquation [3.2] and to usestandard factor equations tocalculate P, F, orA.


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Example: Sherry expects to deposit $1000 now, $3000 4years from now, and $1500 6 years from now and eaen at arate of 12% per year compounded semiannually through acompany-sponsored savings plan.

What amount can she withdraw 10 years from now?

Solution:

Only single-amount Pand Fvalues are involved (See Figurebelow).


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Since only effective rates can be present in the factors, usean effective rate of 6% per semiannual compounding periodand 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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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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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.

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


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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


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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


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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%!

3 5 E i l C l l ti I l i


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When CF of the problem dictates the use of one or more ofthe uniform series or gradient factors, the relationshipbetween 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 compoundingperiod, 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 InvolvingSeries With PP >= CP

3 5 E i l C l l ti I l i


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When PP = CP or PP > CP, the following procedurealwaysapplies:

Step 1. Count the number of payments and use

that number as n.For example, if payments are made quarterly for 5years, nis 20.

Step 2. Find the effectiveinterest rate over the

same time periodas nin step 1.

For example, ifnis expressed in quarters, then theeffective interest rate per quarter mustbe used.

3.5 Equivalence Calculations InvolvingSeries With PP >= CP


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3.5 Series Example

Consider:

0 1 2 3 4 5 6 7

A = $500 every 6 months

F7 = ??

Find F7if 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 computethe factor or a spreadsheet model to achieve

exact result.

3 6 Equivalence Calculations Involving


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This situation is different than the last.

Here, PP is less than the compounding period (CP).

Raises questions?

Issue ofinterperiod compoundingAn example follows.

3.6 Equivalence Calculations InvolvingSeries With PP < CP



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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




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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 withrespect to the compounding periods!




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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 ofmonth 2 and will it earninterest for one month to goto the end of the first

compounding period?The last month of the first compounding period.

Is this an interest-earning period?




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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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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 CFs move to the end of their respectivequarters and all positive CFs move to the beginningof their respective quarters.




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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 seriesusing the F/P factor on each cash flow.




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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



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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 rateis rand must be expressed overthe same time period as that of the effective raterequested.

The compounding frequencyis m,which must equalthe number of times interest is compounded for theperiod 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, enterthe nominal rate per quarter (3.75%) to get aneffective rate per quarter, and enter m= 3, sincemonthly compounding occurs 3 times in a quarter.


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3.7 Using Excel for i calculations

Find nominal:

NOMINAL(effective rate, compounding frequencyper year)

This function always displays the annualnominalrate. Accordingly, the mentered must equal thenumber of times interest is compounded annually. ifthe 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 inExample 4 of Figure 3.6 is divided by 2.

i l f i l l i


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3.7 Using Excel for i calculations

Study Example 3.7:Use EXCEL to find the semiannualcash flow requested in Example 3.5.

Ch S


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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 iin order to compare.


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Chapter III

End of the ChapterIII

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CE533-Chp3-Nominal rate.ppt