ch4_brief

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Slide Sets to accompany Blank & Tarquin, Engineering

Economy, 6th Edition, 2005

2005 by McGraw-Hill, New York, N.Y All Rights Reserved4-1

Developed By:

Dr. Don Smith, P.E.

Department of IndustrialEngineering

Texas A&M University

College Station, Texas

Executive Summary Version

Chapter 4

Nominal and EffectiveInterest

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LEARNING OBJECTIVES

1. Nominal and effectiveinterest rates2. Effective annual interest

rates

3. Effective interest rates4. Compare PP and CP5. Single amounts

with PP CP

6. Series with PP CP7. Single and serieswith PP < CP

8. Continuous

compounding9. Varying interestrates

Notation:

CP = Compounding Period PP = Payment Period

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Sct 4.1 Nominal and EffectiveInterest Rate Statements

Review simple interest and compound interest

definitions (from Chapter 1)

Compound Interest Interest computed on interest

For a given interest period

The time standard for interest computations One Year

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Nominal Rate of Interest

Nominal interest rate definition:

An interest rate that does not include anyconsideration of compounding

For example, 8% per year is a nominal rate

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Effective Interest Rate

Definition:The effective interest rate is the actual rate

that applies for a stated period of time.

The compounding of interest during the timeperiod of the corresponding nominal rate isaccounted for by the effective interest rate.

The effective rate is commonly expressed onan annual basis denoted as ia

All interest formulas, factors, tabulated values, and spreadsheet relations musthave the effective interest rate to properly account for the time value of money.

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Three Time Based Units

Time Period The period over which the interest isexpressed (always stated). Ex: 1% per month

Compounding Period (CP) The shortest time unit

over which interest is charged or earned. Ex: 8% per year, compounded monthly

Compounding Frequency The number of times mthat compounding occurs within time period t.

Ex: 1% per month, compounded monthly has m = 1 Ex: 10% per year, compounded monthly has m = 12

One Year is segmented into:365 days, 52 weeks, 12 monthsOne quarter is: 3 months with 4 quarters/year

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The Effective Rate Per CP

% per time period t r

m compounding periods per t m

r

The Effective rate per compounding period (CP) is:

Ex: r = 9% per year, compounded monthly:

m = 12.(12 months in a year)

i per month = 0.09/12 = 0.0075 or 0.75%/month

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Two Common Forms of Quotation

Two types of interest quotation:

1. Quotation using a Nominal Interest Rate

2. Quoting using an Effective Interest Rate

Nominal and Effective interest rates are common in

business, finance, and engineering economy

Each type must be understood in order to solve

problems where interest is stated in various ways

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Definition of a Nominal InterestRate

Nominal interest rate definition:

An interest rate that does not include any consideration ofcompounding

Means in name only, not the true, effective rate

8% per year, compounded monthly 8% is NOT the true effective rate (per year)

8% represents the nominal rate

Effective rate will consider the monthly compounding

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Examples of Nominal InterestRates

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

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

1.5% per 6-month period for 1 year

Same as: (1.5%)(2 six-month periods) = 3% per year1%per week for 1 yearSame as: (1%)(52 weeks) = 52% per year

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Sct 4.2 Effective Annual InterestRate

r = nominal interest rate peryear

m = number ofcompounding periods peryear

i = effective interest rateper compounding period(CP) = r/m

ia = effective interest rateper year

1/(1 ) 1meff ai i

r/year = eff i / CP ) X (CP / year)=(i)X(m)

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Sct 4.3 Effective Interest Ratesfor Any Time Period

How to calculate true, effective, annual interest rates.

We assume the year is the standard of measure for time.

The year can be comprised of various numbers of compounding

periods (within the year).

Equation [4.8] in the text is the effective interest rate

relation

ri = (1+ ) 1

m

mEffective

C ff

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Example: Calculating the EffectiveRate

Interest is 8% per year, compounded quarterly

What is the effective annual interest rate?

Use Equation [4.8] with r = 0.08, m = 4

Effective i = (1 + 0.08/4)4

1= (1.02)4 1

= 0.0824 or8.24%/year

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Sct 4.4 Equivalence Relations:Lengths of Payment Period (PP)

and Compounding Period (CP)To be considered:

Frequency of cash flows may or may not equal

the frequency of interest compounding

If the frequency of the cash flow equals the

frequency of the interest compounding No

Problem! If not, must make adjustments

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Situations

Situation Text Reference

PP = CP Sections 4.5 and 4.6

PP > CP Sections 4.5 and 4.6

PP < CP Section 4.7

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Sct 4.5 Equivalence Relation:Single Amounts with PP CP

There are only single amount cash flows, that is, P and F values

To determine P or F using P = F(P/F,i,n) or F = P(F/P,i,n), there aretwo equivalent methods to determine i and n in the factors.

Method 1. For the effective interest rate, i, in the factor: Determine i over the CP using i= r/m

For the total number of periods, n, in the factor: Determine the number of CP between

occurrence of P and F values usingn = (m)(number of payment periods)

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Sct 4.5 Equivalence Relation:Single Amounts with PP CPThere are only single amount cash flows, that is, P and F values

To determine P or F using P = F(P/F,i,n) or F = P(F/P,i,n), there aretwo equivalent methods to determine i and n in the factors.

Method 2. Find the effective interest rate for the time period of the nominal

rate using effective i formula, Eq. [4.8]

Set n to the number of periods in the nominal rate statement

r

Effective i = (1+ ) 1m

m

Si l A N i l E l

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Single Amounts: Numerical ExampleUsing Method 1

Find future worth in 5 years if $5000 now earnsinterest at 6% per year, compounded monthly.

Effective i per month is i = 6%/12 = 0.5%

Total number of CP for year and m = 12 times peryear is n = (12)(5) = 60 periods

F = 5000(F/P,0.5%,60) = 5000(1.3489)= $6744

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Sct 4.6 Equivalence Relations:Series with PP CP

When cash flows involve a series (A, G, or g)

the PP is defined by the frequency of the cash

flowsIF PP CP

Calculate the effective iper payment period

Apply the correct n for the total number ofpayments periods.

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Series: Numerical Example

A = $500 every 6 months

F7 = ?

PP > CP since PP = 6-months and CP = quarterCalculate effective i per PP of 6-months

i6-months means adjusting r to fit the PP

r= 20% per year, compounded quarterly

0 1 2 3 4 5 6 7 Years

S i N i l E l

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Series: Numerical Example

Adjusting the interest rate

r = 20% per year, compounded quarterly

i/qtr = 0.20/4 = 0.05 = 5% per quarter

2 quarters in the 6-month payment period

Effective i = (1.05)2 1 = 10.25% per 6-month

Now, the interest matches the payment period

Finding Fyear 7 = Fperiod 14

F = 500(F/A,10.25%,14) = 500(28.4891)

= $14,244.50

S 4 Si l A d

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Sct 4.7 Single Amounts andSeries with PP < CP

This situation is different from the last where PP

CP

Here, PP is less than the compounding period,

CP

Raises questions of how interperiodcompoundingis handled

Study Example 4.10

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Sct 4.8 Effective Interest Rate

for Continuous Compounding Recall that effective i =(1 + r/m)m 1

What happens if the compounding frequency, m,

approaches infinity? This means an infinite number of compounding periods within a

payment period, and

The time between compounding approaches 0

A limiting value of i will be approached for a given value

of r

D i ti f C ti C di

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Derivation of Continuous CompoundingEffective Rate

Rewrite the effective irelation as

(1 ) 1 1 1

rm

rmr r

m m

(1 ) 1mr

im

Now, examine the impactof letting m approach

infinity. This requirestaking the limit of theabove expression as

m Remember the definitionof the number e

1lim 1 2.71828

h

he

h

D i ti f C ti C di

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Derivation of Continuous CompoundingEffective Rate

So that: lim 1 1 1.

rm

rr

m

ri e

m

The effective i when interest is compounded

continuously is then: Effective i = er 1

To find the equivalent nominal rate given i when

interest is compounded continuously, apply:

ln(1 )r i

S t 4 9 I t t R t Th t V

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Sct 4.9 Interest Rates That VaryOver Time

In practice, interest rates do not stay the same

over time unless by contractual obligation.

There can exist variation of interest rates over

time quite normal!

If required, how is this situation handled?

Varying Rates: Finding the

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Varying Rates: Finding thePresent Worth

To find the present worth:

Bring each cash flow amount back to the desired

point in time at the interest rate for each period

according to:

P = F1(P/F,i1,1) + F2(P/F,i1,1)(P/F,i2,1) +

+ Fn(P/F,i1,1)(P/F,i2,1)(P/F,i3,1)(P/F,in,1)

This pro cess can get computat ional ly invo lved!

Varying Rates: Observations

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Varying Rates: Observations

We seldom evaluate problem models withvarying interest rates except in special

cases.

If required, it is best to build a

spreadsheet model

It can be a cumbersome task to perform

Chapter Summary

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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 interest rates over different

payment and compounding periods, must calculate

the effective i to correctly compare P, F or A values

Chapter Summary continued

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Chapter Summary - continued

All time value of money interest factors

require use of an effective (true) periodic

interest rate

The interest rate, i, and the payment or cashflow periods must have the same time unit

One may encounter varying interest rates.

An exact answer requires a sequence of

interest rates for each period

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Chapter 4

End of Slide Set

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