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Contemporary Engineering Economics, 4th

edition 2007

Time Value of Money

Chapter 3-1

Contemporary Engineering EconomicsCopyright 2006

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Chapter Opening StoryTake a Lump Sum or

Annual Installments

Mrs. Louise Outing won alottery worth $5.6 million.

Before playing the lottery,she was offered to choosebetween a single lump sum

$2.912 million, or $5.6million paid out over 20years (or $280,000 peryear).

She ended up taking theannual installment option,

as she forgot to mark theCash Value box, bydefault.

What basis do we comparethese two options?

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Year Option A

(Lump Sum)

Option B

(Installment Plan)

01

2

3

19

$2.912M $283,770$280,000

$280,000

$280,000

$280,000

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What Do We Need to Know?

To make such comparisons (the lotterydecision problem), we must be able tocompare the value of money at different point

in time. To do this, we need to develop a method for

reducing a sequence of benefits and costs to

a single point in time. Then, we will make ourcomparisons on that basis.

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Time Value of Money

Money has a time value

because it can earn moremoney over time (earningpower).

Money has a time valuebecause its purchasing

power changes over time(inflation).

Time value of money ismeasured in terms ofinterest rate.

Interest is the cost ofmoneya costto theborrower and an earningtothe lender

This a two-edged sword whereby earninggrows, but purchasing power decreases(due to inflation), as time goes by.

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The Interest Rate

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Cash Flow Transactions for Two Types of Loan

Repayment

End of Year Receipts PaymentsPlan 1 Plan 2

Year 0 $20,000.00 $200.00 $200.00

Year 1 5,141.85 0Year 2 5,141.85 0

Year 3 5,141.85 0

Year 4 5,141.85 0

Year 5 5,141.85 30,772.48The amount of loan = $20,000, origination fee = $200, interest rate = 9% APR(annual percentage rate)

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Cash Flow Diagram for Plan 2

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End-of-Period Convention

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Methods of Calculating Interest

Simple interest: the practice of charging aninterest rate only to an initial sum (principal

amount). Compound interest: the practice of

charging an interest rate to an initial sum

and to any previously accumulated interestthat has not been withdrawn.

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Simple Interest

P= Principal amount

i = Interest rate

N= Number ofinterest periods

Example:

P= $1,000

i= 10% N= 3 years

End ofYear

Beginning

Balance

Interestearned

EndingBalance

0 $1,000

1 $1,000 $100 $1,100

2 $1,100 $100 $1,200

3 $1,200 $100 $1,300

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Simple Interest Formula

( )

where

= Principal amount

= simple interest rate

= number of interest periods

= total amount accumulated at the end of period

F P iP N

P

i

N

F N

$1,000 (0.10)($1,000)(3)

$1,300

F

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Compound Interest

P= Principal amount

i = Interest rate

N= Number ofinterest periods

Example:

P= $1,000

i = 10% N= 3 years

Endof

Year

BeginningBalance

Interestearned

EndingBalance

0 $1,000

1 $1,000 $100 $1,100

2 $1,100 $110 $1,210

3 $1,210 $121 $1,331

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Compounding Process

$1,000

$1,100

$1,100

$1,210

$1,210

$1,33101

2

3

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0

$1,000

$1,331

12

3

3$1,000(1 0.10)

$1,331

F

Cash Flow Diagram

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Compound Interest Formula

1

2

2 1

0 :

1: (1 )

2 : (1 ) (1 )

: (1 )N

n P

n F P i

n F F i P i

n N F P i

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Some Fundamental Laws

2

F m a

V i R

E m c

The Fundamental Law of Engineering Economy

(1 )NF P i

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Compound Interest

The greatest mathematical discovery of

all time,Albert Einstein

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Market Value

Assume that the companys stock will continue to

appreciate at an annual rate of 23.15% for thenext 24 years.

24$115.802 (1 0.2315)

$17.145 trillions

F M

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EXCEL Template

In 1626 the Indians sold Manhattan Island to Peter Minuitof the Dutch West Company for $24.

If they saved just $1 from the proceeds in a bank account

that paid 8% interest, how much would their descendents

have now?

As of Year 2006, the total US population would be close to

300 millions. If the total sum would be distributed equally

among the population, how much would each person receive?

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Excel Solution

380

$18%

380 years

$1(1 0.08) $5,023,739,194,020

P

i

N

F

=FV(8%,380,0,1)

= $5,023,739,194,020

$5,023,739,194,020Amount per person

300,000,000

$16,746

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Practice Problem

Problem Statement

If you deposit $100 now (n= 0) and $200 twoyears from now (n= 2) in a savings account

that pays 10% interest, how much would youhave at the end of year 10?

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Contemporary Engineering Economics, 4

th

edition 2007

Solution

0 1 2 3 4 5 6 7 8 9 10

$100$200

F

10

8

$100(1 0.10) $100(2.59) $259

$200(1 0.10) $200(2.14) $429$259 $429 $688F

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th

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Practice problem

Problem StatementConsider the following sequence of depositsand withdrawals over a period of 4 years. Ifyou earn a 10% interest, what would be thebalance at the end of 4 years?

$1,000 $1,500

$1,210

0 1

2 3

4

?$1,000

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$1,000$1,500

$1,210

0 1

2

3

4

?

$1,000

$1,100

$2,100 $2,310

-$1,210

$1,100

$1,210

+ $1,500

$2,710

$2,981

$1,000

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s07 chap 3-1