11

Warm-Up ReviewWarm-Up Review

Time Value of MoneyTime Value of Money Calculation of Future ValueCalculation of Future Value Calculation of Current ValueCalculation of Current Value Simple interests and Simple interests and

compound interestscompound interests Continuous compoundingContinuous compounding

22

Basics of Time Value of MoneyBasics of Time Value of Money Interest rateInterest rate

reward for use of capitalreward for use of capital$$ usually expressed in % per yearusually expressed in % per year

Simple Interest Simple Interest (self-study)(self-study) Only the principal earns interestOnly the principal earns interest Interest amount =P • i • nInterest amount =P • i • n Future value = P + P • i • n = P (1 + i • n)Future value = P + P • i • n = P (1 + i • n)

33

Basics of Time Value of MoneyBasics of Time Value of Money Compound InterestCompound Interest

Interest on interestInterest on interest• dependant on compounding period dependant on compounding period

(yearly, semi-annually, monthly)(yearly, semi-annually, monthly) For 2 years:For 2 years:

• Future value = P ( 1+i) + i • P (1+i) = P (1+ i)Future value = P ( 1+i) + i • P (1+i) = P (1+ i)22

For n years:For n years:• Future value = P (1+ i)Future value = P (1+ i)nn

• see column 2 of interest tablessee column 2 of interest tables

44

What is What is Time ValueTime Value?? We say that money has a time We say that money has a time

value because that money can be value because that money can be invested with the expectation of invested with the expectation of earning a positive rate of returnearning a positive rate of return

In other words, “In other words, “a dollar received a dollar received today is worth more than a dollar today is worth more than a dollar to be received tomorrowto be received tomorrow””

That is because today’s dollar That is because today’s dollar can be invested so that we have can be invested so that we have more than one dollar tomorrowmore than one dollar tomorrow

55

Compound InterestCompound Interest Note from the example that the future value is Note from the example that the future value is

increasing at an increasing rateincreasing at an increasing rate In other words, In other words, the amount of the amount of interestinterest earned each earned each

year is increasingyear is increasing Year 1: $10Year 1: $10 Year 2: $11Year 2: $11 Year 3: $12.10Year 3: $12.10

The reason for the increase is that each year you are The reason for the increase is that each year you are earning interest on the interest that was earned in earning interest on the interest that was earned in previous years in addition to the interest on the previous years in addition to the interest on the original original principle amountprinciple amountchangechange

66

Interest FormulationInterest Formulation

Simple Interest

iN)P(1IPF(iP)NI

Compound Interest

2i)P(1

i)i)(1P(1

i)]i[P(1i)P(1

After N periods, the total accumulated value F will grow to

Ni)P(1F Ni

FP

)1(

77

Continuous CompoundingContinuous Compounding

There is no reason why we need to stop increasing the There is no reason why we need to stop increasing the compounding frequency at dailycompounding frequency at daily

We could compound every hour, minute, or second We could compound every hour, minute, or second We can also compound every instant (i.e., We can also compound every instant (i.e.,

continuously):continuously):

Here, F is the future value, P is the present value, r is the annual rate of interest, t is the total number of years, and e is a constant

88

Topics TodayTopics Today

Cash Flow Diagrams Cash Flow Diagrams

Equivalent IssuesEquivalent Issues

Engineer DecisionEngineer Decision

99

Cash Flow- Cash Flow- expenses and receiptsexpenses and receipts Engineering projects generally have economic Engineering projects generally have economic

consequences that occur over an extended period of consequences that occur over an extended period of timetime For example, if an expensive piece of machinery is For example, if an expensive piece of machinery is

installed in a plant were brought on credit, the installed in a plant were brought on credit, the simple process of paying for it may take several simple process of paying for it may take several yearsyears

Each project is described as cash receipts or expenses Each project is described as cash receipts or expenses at different points in time at different points in time

1010

Categories of Categories of Cash FlowsCash Flows The The expenses and receiptsexpenses and receipts due to engineering due to engineering

projects usually fall into one of the following projects usually fall into one of the following categories:categories: First costFirst cost: expense to build or to buy and install: expense to build or to buy and install Operations and maintenance (Operations and maintenance (O&MO&M): annual expense, ): annual expense,

such as electricity, labor, and minor repairssuch as electricity, labor, and minor repairs Salvage valueSalvage value: receipt at project termination for sale or : receipt at project termination for sale or

transfer of the equipment (can be a salvage cost)transfer of the equipment (can be a salvage cost) RevenuesRevenues: annual receipts due to sale of products or : annual receipts due to sale of products or

servicesservices OverhaulOverhaul: major capital expenditure that occurs during : major capital expenditure that occurs during

the asset’s lifethe asset’s life

1111

Examples of Cash Inflows & OutflowsExamples of Cash Inflows & Outflows

Receipts from customers--operating Receipts from customers--operating activityactivity

Loans made to other firms--investing Loans made to other firms--investing activityactivity

Dividend payments--financing activityDividend payments--financing activity Payments to investing activityPayments to investing activity Payments of taxes--operating activityPayments of taxes--operating activity

Slide 14.8

1212

Types of Cash FlowsTypes of Cash Flows

Single cash flowSingle cash flow Uniform seriesUniform series Linear gradient seriesLinear gradient series Geometric gradient seriesGeometric gradient series Irregular seriesIrregular series

1313

Cash Flow DiagramsCash Flow Diagrams

The costs and benefits of engineering The costs and benefits of engineering projects over time are projects over time are summarized on a summarized on a cash flow diagram. cash flow diagram.

Cash flow diagram illustrates the size, Cash flow diagram illustrates the size, sign, and timing of individual cash sign, and timing of individual cash flows, and forms the basis for flows, and forms the basis for engineering economic analysisengineering economic analysis

Tool! To show Tool! To show expenses and receiptsexpenses and receipts

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Cash Flow DiagramsCash Flow Diagrams

Pictorial representation of Pictorial representation of engineering economic problem engineering economic problem incomes and expendituresincomes and expenditures time periodtime period interest rateinterest rate

1515

Cash Flow diagrams--Cash Flow diagrams--HowHow A cash flow diagram is created by first A cash flow diagram is created by first

drawing a segmented drawing a segmented time-based time-based horizontal linehorizontal line, divided into appropriate , divided into appropriate time unit. Each time when there is a time unit. Each time when there is a cash flow, a vertical arrowcash flow, a vertical arrow is added is added pointing down for costs and up for pointing down for costs and up for revenues or benefits. The cost flows are revenues or benefits. The cost flows are drawn to relative scaledrawn to relative scale

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Cash Flow DiagramsCash Flow Diagrams

P-Pattern1 2 3 n

“present”

F-Pattern1 2 3 n

“future”

A-Pattern1 2 3 n

“annual”

G-Pattern1 2 3 n

“gradient”

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Cash Flow DiagramsCash Flow Diagrams

1 2 3 4 5

0Time (# of interest periods)

Positive net Cash flow(receipts)

Negative net Cash Flow(payments)

$15,000

$2000

$13,000 is net positive cash flow

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Single Cash FlowSingle Cash Flow

P

F

Compounding Process

Discounting Process

Ni)P(1F Ni)(1F

P

P=Present equivalent value

F= Future equivalent value

A=Annual equivalent value

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Example: Value and InterestExample: Value and Interest

The “value” of money depends on the The “value” of money depends on the amountamount and and when it is received or spentwhen it is received or spent..

1 2

$1000

$1166

Example: What amount must be paid to settle a current debt of $1000 in two years at an interest rate of 8% ?

Solution: $1000 (1 + 0.08) (1 + 0.08) = $1166

2020

An Example of Cash Flow DiagramAn Example of Cash Flow Diagram

Boney (right) borrowed Boney (right) borrowed $1,000$1,000 from a bank at from a bank at 8%8% interest. interest. Two end-of-year Two end-of-year paymentspayments: at the end of the : at the end of the first year, he will repay half of first year, he will repay half of the the $1000 principal plus the $1000 principal plus the interest that is dueinterest that is due. At the end . At the end of the second year, he will of the second year, he will repay the repay the remaining half plus remaining half plus the interest for the second the interest for the second year.year.

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An Example of Cash Flow DiagramAn Example of Cash Flow Diagram

Cash flow for this problem is:Cash flow for this problem is:End of year Cash flowEnd of year Cash flow 0 +$10000 +$1000 1 -$580 (-$500 - $80)1 -$580 (-$500 - $80) 2 -$540 (-$500 - $40)2 -$540 (-$500 - $40)

2222

Cash Flow DiagramCash Flow Diagram

$1,000

0

1 2

$580$540

Time (# of interest periods)

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Uneven Payment SeriesUneven Payment SeriesFind the present worth of any uneven stream of payments by calculating the present value of eachindividual payment and summing the results

Future worth can then be calculated by using the interest formula

P0

P1 P2P3

P4

P5 P6

Years

Ni)(1F

P

0

2424

Equal Payment SeriesEqual Payment Series

0 1 2 3 N-1N

F

A A A A A A

Ai)......A(12N

i)A(11N

i)A(1F

1Ni)A(1.....2i)A(1i)A(1AF

Ni)A(1....2i)A(1i)A(1i)F(1

Ni)A(1A- Fi)F(1

i

1Ni)(1AFSubtracting two above equations from each other yields:

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Linear Gradient SeriesLinear Gradient Series

0 1 2 N-1 N0

G

(N-1)G

2G

N2

N

i)(1i1iNi)(1

GP

Composite Series: uniform series + linear gradient

Find P, given A1, G, I, N

A1

Uniform Series

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Geometric Gradient SeriesGeometric Gradient Series

•Particularly relevant to construction costs•Cash flows increase by a constant %(g); compound growth•Example: price changes due to inflation

0

1 2 3 NN-1

A1

A1(1+g)

A1(1+g)N-1

g > 0P

n

1nN

1n1

n1n1

nnn

i)(1

g)(1AP

i)(1g)(1Ai)(1AP

Present Worth, Pn, of any Cash Flow An

g....igi

Ni)(1Ng)(111AP

If i=g, then P=?

Find P, given A1, g, i, N

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Principal Uses of A Statement of Cash FlowsPrincipal Uses of A Statement of Cash Flows

EvaluateEvaluate a business’s ability to produce positive a business’s ability to produce positive cash flows in the future.cash flows in the future.

DetermineDetermine whether a company can satisfy its whether a company can satisfy its financial obligations.financial obligations.

IdentifyIdentify sources of differences between a sources of differences between a business’s net income and its related (net) cash business’s net income and its related (net) cash flow from revenue and expense transactions.flow from revenue and expense transactions.

AnalyzeAnalyze the impact on a business’s financial the impact on a business’s financial condition of its major investing and financing condition of its major investing and financing transactions.transactions.

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Cash-Flow Data Can Be Used to AddressCash-Flow Data Can Be Used to Address

Will a company generate Will a company generate sufficient cash to retire a long-sufficient cash to retire a long-term debt that matures soon?term debt that matures soon?

Why doesn’t a company’s Why doesn’t a company’s record profits translate into record profits translate into positive cash flows?positive cash flows?

Is a company likely to suspend Is a company likely to suspend (or increase) its dividend (or increase) its dividend payments?payments?

How does the composition of a How does the composition of a company’s cash flows compare company’s cash flows compare to that of its competitors?to that of its competitors?

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Three Major Types of Business ActivitiesThree Major Types of Business Activities

Operating activitiesOperating activities: those : those

transactions and events transactions and events related to the production related to the production and delivery of goods and delivery of goods and services.and services.

Investing activitiesInvesting activities: include : include

the making and collecting of the making and collecting of loans, the acquisition and disposal of PP&E loans, the acquisition and disposal of PP&E assets, and the purchase and sale of assets, and the purchase and sale of securities other than trading securities and securities other than trading securities and cash equivalents.cash equivalents.

Financing activitiesFinancing activities: :

involve obtaining cash from lenders and involve obtaining cash from lenders and repaying those amounts and obtaining cash repaying those amounts and obtaining cash from investors and providing them with a from investors and providing them with a return of and a return on their investments.return of and a return on their investments.

Slide 14.7

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Economic EquivalenceEconomic EquivalenceWhich one would you prefer?

•$20,000 today•$50,000 ten years from now•$ 8,000 each year for the next ten years

We need to compare their economic worth!

Economic equivalence exists between cash flows ifthey have the same economic effect.

Convert cash flows into an equivalent cash flow atany point in time

3131

Equivalence PrinciplesEquivalence Principles

1 Use a common time basisUse a common time basis Equivalent cash flows are equivalent at any Equivalent cash flows are equivalent at any

common point in timecommon point in time Use the present time = Use the present time = present worthpresent worth Use some future point in time = Use some future point in time = future worthfuture worth

2 Equivalence depends on interest rateEquivalence depends on interest rate Changing the interest rate destroys Changing the interest rate destroys

equivalenceequivalence

3232

Equivalence PrinciplesEquivalence Principles

3 Requires conversion Requires conversion of multiple payment of multiple payment cash flows to a cash flows to a single cash flowsingle cash flow

4 Equivalence is Equivalence is maintained maintained regardless of the regardless of the point of viewpoint of view

3333

The Decision Making ProcessThe Decision Making Process

Define problemDefine problem Choose objectivesChoose objectives Identify alternativesIdentify alternatives Evaluate consequencesEvaluate consequences Select the bestSelect the best ImplementImplement Audit resultsAudit results

3434

Making DecisionsMaking Decisions

Preferences

Politics People

Facts

Cost

s

Mark

et

rese

arc

h

Expert

op

inio

n

…

3535

Example: buying a carExample: buying a car

57 Chevy57 Chevy 97 Neon97 Neon 93 Mercedes93 Mercedes

PurchasePurchase $12,000$12,000 $7,000$7,000 $20,000$20,000

OperationOperation 200/mth200/mth 50/mth50/mth 150/mth150/mth

ResaleResale $13,000$13,000 $5,000$5,000 $20,000$20,000

3636

Modeling Modeling

Real World

The ModelAnalysis Information for decision making