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CBACBA

±± Time Value of Money Time Value of Money

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Key Concepts and SkillsKey Concepts and Skills

� Be able to compute the future value of aninvestment made today

� Be able to compute the present value of cash to be received at some future date

� Be able to compute the return on aninvestment

� Be able to compute the number of periods

that equates a present value and a futurevalue given an interest rate

� Be able to use a financial calculatorand/or a spreadsheet to solve time value

of money problems

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� It deals with the concepts and techniques of analysis useful

in evaluating the worth of systems, products, and services inrelation to their costs

� It is used to answer many different questionsWhich engineering projects are worthwhile?

Has the mining or petroleum engineer

shown that the mineral or oil deposits isworth developing?

Which engineering projects should have ahigher priority?

Has the industrial engineer shown which

factory improvement projects should befunded with the available dollars?

How should the engineering project bedesigned?

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what Is Time Value?what Is Time Value?� the time value of money demonstrates that, all things

being equal, it is better to have money now rather thanlater.

� Money received sooner rather than later allows one to use

the funds for investment or consumption purposes. Thisconcept is referred to as the TIME VALUE OF MONEY TIME VALUE OF MONEY !!

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TIMETIME allows one the opportunity to postponeconsumption and earn INTERESTINTEREST.

NOT having the opportunity to earn interest on

money is calledOPPOR

TUN

ITY COS

T.

Why TIME?Why TIME?Why TIME?Why TIME?

� All Projects involve a cash stream of somesort.

± It is usually a combination of both income and

expenses.± If the net is positive the project ³made money´;

otherwise, it ³lost money.´

� One way to sort out project alternatives isthrough engineering economy.

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Cash FlowCash Flow� Engineering projects generally have

economic consequences that occur overan extended period of time

± For example, if an expensive piece of machinery is installed in a plant were broughton credit, the simple process of paying for itmay take several years

± The resulting favorable consequences may lastas long as the equipment performs its useful

function� Each project is described as cash receipts

or disbursements (expenses) at differentpoints in time

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Categories of Cash FlowsCategories of Cash Flows

� The expenses and receipts due toengineering projects usually fall intoone of the following categories:± First cost: expense to build or to buy and

install

± Operations and maintenance (O&M): annualexpense, such as electricity, labor, andminor repairs

± Salvage value: receipt at projecttermination for sale or transfer of the

equipment (can be a salvage cost)± Revenues: annual receipts due to sale of

products or services

± Overhaul: major capital expenditure thatoccurs during the asset¶s life

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Simple Interest R ateSimple Interest R ate

� Interest formulae play a centralrole in the economic evaluation of

engineering alternatives.� I = Pni

Where:P = principal amount

i = interest rate

n = period( number of years)

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Compound InterestCompound InterestCompound InterestCompound Interest

When interest is paid on not only the principalamount invested, but also on any previousinterest earned, this is called compoundinterest.

FV = Principal + (Principal x Interest)

= 2000 + (2000 x .06)

= 2000 (1 + i)= PV (1 + i)

Note: PV refers to Present Value or Principal

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If you invested $2,000 today in an account$2,000 today in an account

that pays 6that pays 6% interest, with interestcompounded annually, how much will be inthe account at the end of two years if there

are no withdrawals?

Future ValueFuture ValueFuture ValueFuture Value

0 1 2

$2,000$2,000

FVFV

6%

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Finding FVs (moving to the right on a time line) is called

compounding.

� Compounding involves earning interest on interest for

investments of more than one period.

What¶s the FV of an initial $100 after 3What¶s the FV of an initial $100 after 3years if i = 10%? years if i = 10%?

FV = ?100

0 1 2 3

10%

Future Values

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$1,000 now is$1,000 now is equivalentequivalent to $8,916to $8,9161212--years in the future at 20% years in the future at 20%

interestinterest..

Present Money Grows

$0

$2,000

$4,000

$6,000

$8,000

$10,000

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

Years

F u t u r e V a l u e o f $ 1 , 0

eriod 20%

0 $1,000

1 $1,200

2 $1,4403 $1,728

4 $2,074

5 $2,488

6 $2,986

7 $3,583

8 $4,300

9 $5,160

10 $6,192

11 $7,430

12 $8,916Maxwell¶s 1-st Law:Get the Money Up-Front

$8,916 =F =P(1+%)n =$1,000(1.2)12

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Present ValuePresent ValuePresent ValuePresent Value

� Since FV = PV(1 + i)n.

PVPV = FVFV / (1+i)n.

PVPV = FVFV (1+i)-n

� Discounting is the process of translating a future value or a set of future cash flows into a presentvalue.

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Assume that you need to have exactly$4,000$4,000 saved 10 years from now.years from now. Howmuch must you deposit today in anaccount that pays 6% interest,compounded annually, so that you reach

your goal of $4,000?

0 55 10

$4,000$4,000

6%

PVPV00

Present ValuePresent ValuePresent ValuePresent Value

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Ali needs to know how large of a depositto make today so that the money willgrow to $2,500$2,500 in 5 years. Assume5 years. Assumetoday¶s deposit will grow at a compoundtoday¶s deposit will grow at a compoundrate of rate of 4% annually.

Present Value ExamplePresent Value ExamplePresent Value ExamplePresent Value Example

0 1 2 3 4 55

$2,500$2,500

PVPV00

4%

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� Calculation based on general formula:

PVPV00 = FVFVnn / (1+i)n

PVPV00 = $2,500/(1.04)$2,500/(1.04)55

= $2,054.81

Present Value SolutionPresent Value SolutionPresent Value SolutionPresent Value Solution

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$10,000 12$10,000 12--years in theyears in thefuture at 20% is equivalentfuture at 20% is equivalent

to $1,122 now.to $1,122 now.

Future Money Shrinks

$0

$2,000

$4,000

$6,000

$8,000

$10,000

$12,000

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

Years

P r e s e n t V a l u e

$ 1 0 , 0

0 0

Per od 20%

12 $10,000

11 $8,333

10 $6,9449 $5,787

8 $4,823

7 $4,019

6 $3,349

5 $2,791

4 $2,326

3 $1,938

2 $1,615

1 $1,346

0 $1,122Maxwell¶s Other Law:Take the Money and Run!

$1,122 =P =F(1+%)-n =$10,000(1.2)-12

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Discount RateDiscount Rate

� Often we will want to know what theimplied interest rate is in aninvestment

� Rearrange the basic PV equation andsolve for r

±FV = PV(1 + r)t

±r = (FV / PV)1/t ± 1

� If you are using formulas, you willwant to make use of both the yx andthe 1/x keys

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General Formula:

FVn = PVPV00(1 + [i /m])mn

n: Number of Years

m: Compounding Periods per Year

i: Annual Interest Rate

FVn,m: FV at the end of Year n

PVPV00: PV of the Cash Flow today

Frequency of Frequency of

CompoundingCompounding

Frequency of Frequency of

CompoundingCompounding

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Frequency of Frequency of Compounding ExampleCompounding Example

� Suppose you deposit $1,000 in anaccount that pays 12% interest,

compounded quarterly. How much willbe in the account after eight years if there are no withdrawals?

PV = $1,000

i = 12% /4 = 3% per quarter

n = 8 x 4 = 32 quarters

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Solution based on formula:Solution based on formula:

FV= PV (1 + i)n

= 1,000(1.03)32

= 2,575.10

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Comparing PV to FVComparing PV to FV

� Remember, both quantities must bepresent value amounts or bothquantities must be future valueamounts in order to be compared.

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How to solve a time value of How to solve a time value of money problem.money problem.

� The ³value four years from today´ isa future value amount.

� The ³expected cash flows of $100per year for four years´ refers to anannuity of $100.

� Since it is a future value problem andthere is an annuity, you need tosolve for a FUTURE VALUE OF AN ANNUITY.

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S ingle Sum S ingle Sum - - F u t u re & Present Val u eF u t u re & Present Val u e

� Assume can invest PV at interest rate i to receive future sum, FV

� Similar reasoning leads to Present Value of a Future sum today.

1 2 30

FV1 = (1+i)PV

FV3 = (1+i)3PV

PV

FV2

= (1+i)2PV

1 2 30

PV = FV1/(1+i)

FV1

PV = FV2/(1+i)2

FV2

PV = FV3/(1+i)3

FV3

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PV = FV

1+ i= FV 1

1+ in

n n

n¨ª© ¸

º¹

FVn = PV(1 + i )n for given PV

$100 = 0.7513 = $75.13.1.10

PV = $1001¨

ª©

¸

º¹3

PVCalculation for $100 received in 3 years

if interest rate is 10%

Single Sum ± FV & PV Formulas

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Present ValuePresent Value ±± ImportantImportant

RelationshipII

RelationshipII

� For a given time period ± the higherthe interest rate, the smaller the

present value±What is the present value of $500

received in 5 years if the interest rate is10%? 15%?

� Rate = 10%: PV = 500 / (1.1)5 = 310.46

� Rate = 15%; PV = 500 / (1.15)5 = 248.58

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Discount RateDiscount Rate ±± Example 1Example 1

� You are looking at an investmentthat will pay $1200 in 5 years if youinvest $1000 today. What is theimplied rate of interest?

±r = (1200 / 1000)1/5 ± 1 = .03714 =3.714%

±Calculator ± the sign convention

matters!!!�N = 5

� PV = -1000 (you pay 1000 today)

� FV = 1200 (you receive 1200 in 5 years)

� CPT I /Y = 3.714%

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lec 12 TVM