Why Engineering Economy

Why Engineering Why Engineering EconomyEconomy

Chapter 1Chapter 1

WHAT IS ECONOMICS ?WHAT IS ECONOMICS ?

The study of how limited resources are The study of how limited resources are used to satisfy unlimited human wants.used to satisfy unlimited human wants.

The study of how individuals and societies The study of how individuals and societies choosechoose to use scarce resources that to use scarce resources that nature and previous generations have nature and previous generations have provided.provided.

• Engineers must work within the realm of economics and justification of engineering projects

• Work with limited funds (capital)

• Capital is not unlimited – rationed

• Capital does not belong to the firm

• Belongs to the Owners of the firm

• Capital is not “free”…it has a “cost”

Section 1.1 DefinitionSection 1.1 Definition

ENGINEERING ECONOMY IS INVOLVED WITH THE APPLICATION

OF DEFINED MATHEMATICAL RELATIONSHIPS THAT AID IN THE

COMPARISON OF ECONOMIC ALTERNATIVES

Problem Solving ApproachProblem Solving Approach

Understand the ProblemUnderstand the Problem Collect all relevant data/informationCollect all relevant data/information Identify the criteria for decision makingIdentify the criteria for decision making Define the feasible alternativesDefine the feasible alternatives Evaluate each alternative using a Evaluate each alternative using a

common perspective.common perspective. Select the “best” alternativeSelect the “best” alternative Implement and monitorImplement and monitor

Economic Issues to be Answered Economic Issues to be Answered before deciding on an alternativebefore deciding on an alternative

How much does the option costHow much does the option cost How much will the option saveHow much will the option save How do we get the money to pay for itHow do we get the money to pay for it What are the tax effectsWhat are the tax effects What is the criteria to be used to decide on What is the criteria to be used to decide on

the optionthe option What are the assumptions used in the What are the assumptions used in the

estimatesestimates How dependent is a decision on the How dependent is a decision on the

assumptions-sensitivity analysisassumptions-sensitivity analysis

Engineering Economy Study Engineering Economy Study ApproachApproach

The parameters associated with an The parameters associated with an alternative include:alternative include:

1.1. First Cost (Initial outlay)First Cost (Initial outlay)2.2. Estimated Useful LifeEstimated Useful Life3.3. Estimated Annual Income or RevenueEstimated Annual Income or Revenue4.4. Estimated Annual Expenses or CostsEstimated Annual Expenses or Costs5.5. Salvage ValueSalvage Value6.6. Interest RateInterest Rate7.7. Tax EffectsTax Effects

Time Value of MoneyTime Value of Money

The time value of money is the change The time value of money is the change in the amount of money over a given in the amount of money over a given time period. time period.

This is the most important concept in This is the most important concept in Engineering Economy.Engineering Economy.

Cash FlowsCash Flows

The parameters listed make up the The parameters listed make up the cash flows associated with an cash flows associated with an alternative.alternative.

Cash flows are said to be positive Cash flows are said to be positive when they flow into the firm (i.e. when they flow into the firm (i.e. revenues)revenues)

Cash flows are said to be negative Cash flows are said to be negative when they flow out of the firm (i.e. when they flow out of the firm (i.e. expenses)expenses)

Alternatives Alternatives

In addition to list of generated In addition to list of generated alternatives, there is the alternatives, there is the do nothingdo nothing alternative. (status quo)alternative. (status quo)

This is the alternative to choose when This is the alternative to choose when none of the generated alternatives none of the generated alternatives achieve the chosen decision criteria.achieve the chosen decision criteria.

Alternatives can be either independent Alternatives can be either independent or mutually exclusive or mutually exclusive

InterestInterest

Interest is a rental for the use of money.Interest is a rental for the use of money. It is what establishes equivalent values It is what establishes equivalent values

for different periods of timefor different periods of time It is the difference between a beginning It is the difference between a beginning

amount and an ending amountamount and an ending amount

Two Interest PerspectivesTwo Interest Perspectives Interest Earned – this is the perspective of a person Interest Earned – this is the perspective of a person

who either saves, who either saves, invests,invests, or loans a sum of money or loans a sum of money out, and at a later time receives a larger sum.out, and at a later time receives a larger sum.

Interest = total amount now – original amount Interest = total amount now – original amount

Interest paid – this is the perspective of a person Interest paid – this is the perspective of a person who who borrowsborrows a sum of money, and at a later time a sum of money, and at a later time repays a larger sum.repays a larger sum.

Interest = Amount owed now – original amount Interest = Amount owed now – original amount

Interest RateInterest Rate The interest rate is the amount of interest accrued for The interest rate is the amount of interest accrued for

a period of time divided by the original amounta period of time divided by the original amount

The time unit for interest payments is called the interest The time unit for interest payments is called the interest period. Often the interest period is a year. The above period. Often the interest period is a year. The above expression is for a single period. expression is for a single period.

100% X amount original

unit per time accruedinterest % rate Interest

Single Period Interest PaidSingle Period Interest Paid

Example 1.3Example 1.3 You borrow $10,000 for one full yearYou borrow $10,000 for one full year Must pay back $10,700 at the end of one year. Must pay back $10,700 at the end of one year.

Interest Amount? Interest Rate?Interest Amount? Interest Rate? Interest Amount (I) = $10,700 - $10,000Interest Amount (I) = $10,700 - $10,000 Interest Amount = $700 for the year. The $700 Interest Amount = $700 for the year. The $700

represents the return to the lender for this use represents the return to the lender for this use of his/her funds for one yearof his/her funds for one year

Interest rate (i) = 700/$10,000 = 7%/Yr. 7% is Interest rate (i) = 700/$10,000 = 7%/Yr. 7% is the return earned by the lenderthe return earned by the lender

Single Period Interest PaidSingle Period Interest Paid Example 1.4Example 1.4

Borrow $20,000 for 1 year at 9% interest per Borrow $20,000 for 1 year at 9% interest per yearyear

Interest? Total Due?Interest? Total Due? i = 0.09 per year and N = 1 Yeari = 0.09 per year and N = 1 Year Interest (I) = (0.09)($20,000) = $1,800Interest (I) = (0.09)($20,000) = $1,800 Pay $20,000 + (0.09)($20,000) at end of 1 Pay $20,000 + (0.09)($20,000) at end of 1

yearyear Total amt. paid one year henceTotal amt. paid one year hence

$20,000 + $1,800 = $20,000 + $1,800 = $21,800$21,800

Example 1.4Example 1.4

Note the followingNote the following Total Amount Due one year hence isTotal Amount Due one year hence is

($20,000) + 0.09($20,000)($20,000) + 0.09($20,000)

=$20,000(1+.09) = $21,800=$20,000(1+.09) = $21,800

The (1.09) factor accounts for the The (1.09) factor accounts for the repayment of the $20,000 and the interest repayment of the $20,000 and the interest amountamount

This (1+i) factor will be one of the important This (1+i) factor will be one of the important interest factors to be seen laterinterest factors to be seen later

Interest Earned and Rate of Return (Ex 1.4 data)Interest Earned and Rate of Return (Ex 1.4 data)

Assume you invest $20,000 for one year Assume you invest $20,000 for one year in a venture that will return to you, 9% in a venture that will return to you, 9% per year.per year. Original $20,000 backOriginal $20,000 back Plus……..Plus…….. The 9% return on $20,000 = $1,800The 9% return on $20,000 = $1,800

We say that you earned 9%/year on the We say that you earned 9%/year on the investment! This is your RATE of RETURN investment! This is your RATE of RETURN (ROR) on the investmen(ROR) on the investmen

100% X amount original

unit time per accrued interest Return of Rate (%)

Interest rateInterest rate

ROI or Return on Investment is another ROI or Return on Investment is another term for rate of return used in settings term for rate of return used in settings where the original amount invested is where the original amount invested is provided by capital funds provided by capital funds

The calculation for determining interest The calculation for determining interest rate, rate of return, and return on rate, rate of return, and return on investment are identical. investment are identical.

The time unit for which the amount of The time unit for which the amount of interest accrues is the interest period. interest accrues is the interest period.

Economic EquivalenceEconomic Equivalence

Economic EquivalenceEconomic Equivalence Two sums of money at two different points in Two sums of money at two different points in

time can be economically equivalent if:time can be economically equivalent if: We consider an interest rate and,We consider an interest rate and, No. of time periods between the two sumsNo. of time periods between the two sums This illustrates the time value of money This illustrates the time value of money

conceptconcept

Equivalence ExampleEquivalence Example Return to Example 1.4 i = 9%Return to Example 1.4 i = 9% Diagram the loan (Cash Flow Diagram)Diagram the loan (Cash Flow Diagram) The company’s perspective is shownThe company’s perspective is shown

T=0 t = 1 Yr

$20,000 is received here

$21,800 paid back here

+

_

Equivalence ExampleEquivalence Example

$20,000 now is economically equivalent to $20,000 now is economically equivalent to $21,800 one year from now IF the interest rate is $21,800 one year from now IF the interest rate is set to equal 9%/yearset to equal 9%/year

T=0 t = 1 Yr

$20,000 is received here

$21,800 paid back here

The Cash Flow Diagram: The Cash Flow Diagram: Extremely valuable analysis toolExtremely valuable analysis tool First step in the solution processFirst step in the solution process Graphical Representation on a time Graphical Representation on a time

scalescale Does not have to be drawn “to exact Does not have to be drawn “to exact

scale”scale” But, should be neat and properly labeledBut, should be neat and properly labeled Will be helpful on most in-class examsWill be helpful on most in-class exams

Important TERMSImportant TERMS

CASH INFLOWSCASH INFLOWS Money flowing INTO the firm from outsideMoney flowing INTO the firm from outside Revenues, Savings, Salvage Values, etc.Revenues, Savings, Salvage Values, etc.

CASH OUTFLOWSCASH OUTFLOWS DisbursementsDisbursements First costs of assets, labor, salaries, taxes First costs of assets, labor, salaries, taxes

paid, utilities, rents, interest, etc.paid, utilities, rents, interest, etc.

Net Cash FlowsNet Cash Flows

A A NET CASH FLOWNET CASH FLOW is is Cash Inflows – Cash Outflows Cash Inflows – Cash Outflows

• (for a given time period)(for a given time period)

We normally assume that all cash We normally assume that all cash flows occur:flows occur: At the At the ENDEND of a given time period. This of a given time period. This

is the end-of-period conventionis the end-of-period convention

Cash Flow diagrams - timelineCash Flow diagrams - timeline

Assume a 5-year problemAssume a 5-year problem The basic time line is shown belowThe basic time line is shown below

The present is time 0, generally.

Displaying Cash FlowsDisplaying Cash Flows A sign convention is appliedA sign convention is applied

Positive cash flows (Inflows) are normally Positive cash flows (Inflows) are normally drawn upward from the time linedrawn upward from the time line

Negative cash flows (Outflows) are normally Negative cash flows (Outflows) are normally drawn downward from the time linedrawn downward from the time line

Sample CF DiagramSample CF DiagramPositive CF at t = 1

Negative CF’s at t = 2 & 3


A father wants to deposit anA father wants to deposit an unknown unknown lump‑sum amount into an investment lump‑sum amount into an investment opportunity opportunity 2 years from now2 years from now that is that is large enough to withdraw $4,000 per large enough to withdraw $4,000 per year for state university tuition for 5 year for state university tuition for 5 years years starting 3 years from now. starting 3 years from now.

Construct the cash flow diagram, Construct the cash flow diagram, assume the rate of return is estimated assume the rate of return is estimated to be 15.5% per year.to be 15.5% per year.

Example 1.17 CF DiagramExample 1.17 CF Diagram

Using PV(.155,5,4000), P = $13,251.40

Multi-Period Simple and Multi-Period Simple and Compound InterestCompound Interest

Prior discussion on interest and interest rate Prior discussion on interest and interest rate were for one interest periodwere for one interest period

For more than one interest period there are For more than one interest period there are two “types” of interest calculationstwo “types” of interest calculations Simple InterestSimple Interest Compound InterestCompound Interest

Compound Interest is more common Compound Interest is more common worldwide and applies to most analysis worldwide and applies to most analysis situationssituations

Simple Interest Over TimeSimple Interest Over Time

Simple InterestSimple Interest Calculated on the principal amount Calculated on the principal amount

onlyonly Easy (simple) to calculateEasy (simple) to calculate Simple Interest is:Simple Interest is:

(principal)*(interest rate)*(number of (principal)*(interest rate)*(number of periods)periods)

$I = (P)*(i)*(n) i = interest rate I = interest$I = (P)*(i)*(n) i = interest rate I = interest


Ex 1.7Ex 1.7 An engineer borrows $1,000 for 3 years An engineer borrows $1,000 for 3 years

at 5% per year, simple interestat 5% per year, simple interest Let “P” = the principal sum ($1,000)Let “P” = the principal sum ($1,000) i = the interest rate (5%/year)i = the interest rate (5%/year) Let n = number of years (3)Let n = number of years (3)


Simple Interest DefinitionSimple Interest Definition I = P(i)(N)I = P(i)(N) For Ex. 1.7:For Ex. 1.7:

I = $1,000(0.05)(3) = I = $1,000(0.05)(3) = $150.00$150.00 Total Interest over 3 YearsTotal Interest over 3 Years

Simple Interest Over TimeSimple Interest Over Time Year-by-Year Analysis: Simple InterestYear-by-Year Analysis: Simple Interest Year 1Year 1

II11 = $1,000(0.05) = $50.00 accrues = $1,000(0.05) = $50.00 accrues

1 2 3

P=$1,000

I1=$50.00

$50 interest accrues but is not paid

0


Year 2Year 2 II22 = $1,000(0.05) = $50.00 accrues = $1,000(0.05) = $50.00 accrues

1 2 3

I1=$50.00

P=$1,000

I2=$50.00

Another $50.00 interest accrues but is not paid

0


Year 3Year 3 II33 = $1,000(0.05) = $50.00 accrues = $1,000(0.05) = $50.00 accrues

I2=$50.00

1 2 3

I1=$50.00

P=$1,000

I3=$50.00

Pay back $1,000 + $150 of interest

The unpaid interest did not earn interest over the

3-year period

0

Simple Interest SummarySimple Interest Summary

In a multi-period situation with simple In a multi-period situation with simple interest:interest: The accrued interest does not earn interest The accrued interest does not earn interest

during the succeeding time period.during the succeeding time period. Normally, the total sum borrowed (lent) is Normally, the total sum borrowed (lent) is

paid back paid back at the end of the agreed time at the end of the agreed time periodperiod PLUS the accrued (owed but not paid) PLUS the accrued (owed but not paid) interest.interest.

Compound Interest Over TimeCompound Interest Over Time Compound Interest is differentCompound Interest is different For compound interest the interest accrued for For compound interest the interest accrued for

each interest period is calculated on the each interest period is calculated on the principal principal plus the total amount of interest plus the total amount of interest accumulated in all prior periods. accumulated in all prior periods. This accrued This accrued interest is then added to the prior balance to interest is then added to the prior balance to form a new principal balance.form a new principal balance.

Interest then “earns interest”Interest then “earns interest” Compound interest is the interest type used to Compound interest is the interest type used to

calculate the time value of money for calculate the time value of money for Engineering Economy AnalysisEngineering Economy Analysis

Compound Interest Over TimeCompound Interest Over Time Unlike simple interest, compound interest Unlike simple interest, compound interest

does not have a formula for calculating the does not have a formula for calculating the total amount of interest over several total amount of interest over several interest periods.interest periods.

Compound interest has to be calculated Compound interest has to be calculated each period to determine the principal plus each period to determine the principal plus accumulated interest that the interest rate accumulated interest that the interest rate is applied to in the next period.is applied to in the next period.

Interest = (interest rate) X [principal + Interest = (interest rate) X [principal + accrued interest]accrued interest]

Compound Interest Example 1.8Compound Interest Example 1.8

Here an engineer borrows $1000 @ 5% Here an engineer borrows $1000 @ 5% per year compound interest. How much is per year compound interest. How much is due after 3 years.due after 3 years. P = $1,000P = $1,000 i = 5% per year compounded annually i = 5% per year compounded annually

(C.A.)(C.A.) N = 3 yearsN = 3 years

Example 1.8 Cash FlowExample 1.8 Cash Flow

I2=$52.50

1 2 3

P=$1,000

I3=$55.13

I1=$50.00

Owe at t = 3 years:

$1,000 + 50.00 + 52.50 + 55.13

= $1,157.63

Owe at t = 3 years:

$1,000 + 50.00 + 52.50 + 55.13

= $1,157.63

$1157.63

For 3 yrs of compound interest, accrued not paid:

i = 5%

0

+

-

Compound interestCompound interest

For the example:For the example: PP00 = +$1,000 = +$1,000

II11 = $1,000(0.05) = $50.00 = $1,000(0.05) = $50.00

Owe POwe P11 = $1,000 + 50 = $1,050 (but we = $1,000 + 50 = $1,050 (but we

don’t pay yet!)don’t pay yet!) New Principal sum at end of t = 1: = New Principal sum at end of t = 1: =

$1,050.00$1,050.00

Compound Interest: t = 2Compound Interest: t = 2

Principal at end of year 1: $1,050.00Principal at end of year 1: $1,050.00 II22 = $1,050(0.05) = $52.50 (owed but not = $1,050(0.05) = $52.50 (owed but not

paid)paid) Add to the current unpaid balance yields:Add to the current unpaid balance yields:

$1,050 + 52.50 = $1,102.50$1,050 + 52.50 = $1,102.50 New unpaid balance or New Principal New unpaid balance or New Principal

AmountAmount Now, go to year 3…….Now, go to year 3…….

Compound Interest: t = 3Compound Interest: t = 3 New Principal sum: $1,102.50New Principal sum: $1,102.50 II33 = $1102.50(0.05) = $55.125 = $55.13 = $1102.50(0.05) = $55.125 = $55.13

Add to the beginning of year principal Add to the beginning of year principal yields:yields: $1102.50 + 55.13 = $1157.63$1102.50 + 55.13 = $1157.63 This is the loan payoff at the end of 3 yearsThis is the loan payoff at the end of 3 years

Note how the interest amounts were Note how the interest amounts were added to form a new principal sum with added to form a new principal sum with interest calculated on that new amountinterest calculated on that new amount

Comparison of simple and Comparison of simple and compound interest, Ex 1.7 and 1.8compound interest, Ex 1.7 and 1.8

Comparison of simple and compound interest, Ex 1.7 and 1.8

SS

S

CC

C

900

1000

1100

1200

End of Year

To

tal

Ow

ed

S

C

S 1050 1100 1150

C 1050 1102.5 1157.63

1 2 3

Example 1.9Example 1.9 Five plans are shown that will pay off a loan of Five plans are shown that will pay off a loan of

$5,000 over 5 years with interest at 8% per year. $5,000 over 5 years with interest at 8% per year. We illustrate differences in equivalence We illustrate differences in equivalence depending upon interest type, interest timing, depending upon interest type, interest timing, and method for repaying principal.and method for repaying principal.

Plan 1. Simple Interest, pay all at the endPlan 1. Simple Interest, pay all at the end Plan 2. Compound Interest, pay all at the endPlan 2. Compound Interest, pay all at the end Plan 3. Simple interest, pay interest at end of Plan 3. Simple interest, pay interest at end of

each year. Pay the principal at the end of N = 5each year. Pay the principal at the end of N = 5 Plan 4. Compound Interest, pay interest and Plan 4. Compound Interest, pay interest and

part of the principal each year (pay 20% of the part of the principal each year (pay 20% of the Prin. Amt.)Prin. Amt.)

Example 1.9Example 1.9• Plan 5. Compound Interest, make equal Plan 5. Compound Interest, make equal

payments of the compound interest and principal payments of the compound interest and principal reduction over 5 years with end-of-year reduction over 5 years with end-of-year payments.payments.

Note: The following tables will show the five Note: The following tables will show the five approaches. For now, do not try to understand approaches. For now, do not try to understand how all of the numbers are determined (that will how all of the numbers are determined (that will come later!). Focus on the plans and how these come later!). Focus on the plans and how these tables illustrate economic equivalence.tables illustrate economic equivalence.

Plan 1: @ 8% Simple InterestPlan 1: @ 8% Simple Interest

Simple Interest: Pay all at end on $5,000 Simple Interest: Pay all at end on $5,000 LoanLoan

Plan 2: Compound Interest @ Plan 2: Compound Interest @ 8%/yr8%/yr

• Compound interest: Pay all at the End Compound interest: Pay all at the End of 5 Yearsof 5 Years

Plan 3: Simple Interest Pd. Plan 3: Simple Interest Pd. AnnuallyAnnually

Principal Paid at the End (balloon Note)Principal Paid at the End (balloon Note)

Plan 4: Compound InterestPlan 4: Compound Interest 20% of Principal and accrued interest 20% of Principal and accrued interest

Paid back annuallyPaid back annually

Plan 5: Equal Repayment Plan with Plan 5: Equal Repayment Plan with Compound InterestCompound Interest

• Equal Annual Payments (Part Principal Equal Annual Payments (Part Principal and Part Interestand Part Interest

ComparisonComparisonPlanPlan How PaidHow Paid Total PaidTotal Paid

11 Simple interestSimple interestPaid at endPaid at end

$7,000$7,000

22 Compound interest Compound interest Paid at endPaid at end

$7,346.64$7,346.64

33 Interest only paid Interest only paid annuallyannually

$7,000$7,000

44 Interest and 20% Interest and 20% Prin. Paid annuallyPrin. Paid annually

$6,200$6,200

55 Equal annual Equal annual paymentspayments

$6,261.41$6,261.41

AnalysisAnalysis Note that the amounts of the annual Note that the amounts of the annual

payments are different for each payments are different for each repayment schedule and that the total repayment schedule and that the total amounts repaid for most plans are amounts repaid for most plans are different, even though each repayment different, even though each repayment plan requires exactly 5 years. plan requires exactly 5 years.

The difference in the total amounts The difference in the total amounts repaid can be explained (1) by the time repaid can be explained (1) by the time value of money, (2) by simple or value of money, (2) by simple or compound interest, and (3) by the partial compound interest, and (3) by the partial repayment of principal prior to year 5.repayment of principal prior to year 5.

Terminology and SymbolsTerminology and Symbols Specific symbols and their respective Specific symbols and their respective

definitions have been developed for use definitions have been developed for use in engineering economy.in engineering economy.

Symbols tend to be standard in most Symbols tend to be standard in most engineering economy texts world-wide.engineering economy texts world-wide.

Mastery of the symbols and their Mastery of the symbols and their respective meanings is most important respective meanings is most important in understanding the subsequent in understanding the subsequent material! material!

Terminology and SymbolsTerminology and Symbols

PP = value or amount of money at a time = value or amount of money at a time designated as the present or time 0. A designated as the present or time 0. A single dollar amount.single dollar amount.

In an equivalence context, P is referred In an equivalence context, P is referred to as present worth (PW), present value to as present worth (PW), present value (PV), net present value (NPV), (PV), net present value (NPV), discounted cash flow (DCF), and discounted cash flow (DCF), and capitalized cost (CC); dollars. capitalized cost (CC); dollars.


FF = value or amount of money at some = value or amount of money at some future time. A single dollar amount.future time. A single dollar amount.

Also, F is called future worth (FW) and Also, F is called future worth (FW) and future value (FV); dollarsfuture value (FV); dollars

Terminology and SymbolsTerminology and Symbols• AA = series of consecutive, equal, = series of consecutive, equal,

end‑of‑period amounts of money. end‑of‑period amounts of money.

• Also, A is called the annual worth Also, A is called the annual worth (AW), annuity, equivalent uniform (AW), annuity, equivalent uniform annual worth (EUAW); dollars per annual worth (EUAW); dollars per year, dollars per month year, dollars per month

• nn = number of relevant interest = number of relevant interest periods; years, months, days,periods; years, months, days,


ii = interest rate or rate of return per time = interest rate or rate of return per time period; percent per year, percent per period; percent per year, percent per month month

t = time, stated in periods; years, t = time, stated in periods; years, months, days, etcmonths, days, etc

P and FP and F The symbols P and F represent one-The symbols P and F represent one-

time occurrences: time occurrences: Specifically:Specifically:

$P

$F

t = n0 1 2 … … n-1 n

+

-

P and F:P and F:• It should be clear that a present value P It should be clear that a present value P

represents a single sum of money at represents a single sum of money at some time prior to a future value F some time prior to a future value F

Annual AmountsAnnual Amounts• It is important to note that the symbol A always It is important to note that the symbol A always

represents a uniform amount (i.e., the same represents a uniform amount (i.e., the same amount each period) that extends through amount each period) that extends through consecutive interest periods. consecutive interest periods.

• Cash Flow diagram for annual amounts might Cash Flow diagram for annual amounts might look like the following:look like the following:

0 1 2 3 .. N-1 n

$A

$A

$A

$A

$A

A = equal, end of period cash flow amounts

+

-

Interest Rate – i% per periodInterest Rate – i% per period

The interest rate i is assumed to be a The interest rate i is assumed to be a compound rate, unless specifically statedcompound rate, unless specifically stated

as “simple interest”as “simple interest”

The effective rate i is expressed in The effective rate i is expressed in percent per interest period; for example, percent per interest period; for example, 12% per year. 12% per year.

Terminology and SymbolsTerminology and Symbols Most engineering economy Most engineering economy

problems:problems: Involve the dimension of timeInvolve the dimension of time At least 4 of the symbols { P, F, A, i% and At least 4 of the symbols { P, F, A, i% and

n } will be used, andn } will be used, and At least 3 of 4 have a known estimated At least 3 of 4 have a known estimated

value.value.

Computer SolutionsComputer Solutions Use of a spreadsheet similar to Use of a spreadsheet similar to

Microsoft’s Excel is fundamental to the Microsoft’s Excel is fundamental to the analysis of engineering economy analysis of engineering economy problems.problems.

Appendix A of the text presents a Appendix A of the text presents a primer on spreadsheet use.primer on spreadsheet use.

All engineers are expected by training to All engineers are expected by training to know how to manipulate data, macros, know how to manipulate data, macros, and the various built-in functions and the various built-in functions common to spreadsheets.common to spreadsheets.

SpreadsheetsSpreadsheets

Excel supports (among many others) Excel supports (among many others) six built-in functions to assist in time six built-in functions to assist in time value of money analysisvalue of money analysis

Master each on your own and set up a Master each on your own and set up a variety of the homework problems (on variety of the homework problems (on your own)your own)

Excel’s Financial FunctionsExcel’s Financial Functions To find the present value P, when there is either a To find the present value P, when there is either a

single future payment and/or an annuity: single future payment and/or an annuity: PV (i%, n, A, F) PV (i%, n, A, F)

To find the present value P of any series: To find the present value P of any series: NPV (i%, second_cell:last_cell) + first_cellNPV (i%, second_cell:last_cell) + first_cell To find the equal, periodic value A, when there is a To find the equal, periodic value A, when there is a

single P and /or F payment : PMT(i%, n, P, F) single P and /or F payment : PMT(i%, n, P, F) To find the future value F, when there is either a To find the future value F, when there is either a

single P or and/or an annuity: FV(i%, n, A, P) single P or and/or an annuity: FV(i%, n, A, P)

Nesting of 1 function for the argument of another is Nesting of 1 function for the argument of another is allowed.allowed.

Financial FunctionsFinancial Functions To find the number of periods n: To find the number of periods n:

NPER (i%, A, NPER (i%, A, P, F) P, F)

To find the compound interest rate i: To find the compound interest rate i: RATE (n, A, RATE (n, A,

P, F) P, F) To find the compound interest rate i: To find the compound interest rate i:

IRR (first_ cell:last_ cell) IRR (first_ cell:last_ cell)

MMinimum inimum AAttractive ttractive RRate of ate of RReturneturn

An investment is a commitment of funds and An investment is a commitment of funds and resources in a project with the expectation of resources in a project with the expectation of earning a return over and above the worth of earning a return over and above the worth of the resources that were committed.the resources that were committed.

A firm’s financial managers set a minimum A firm’s financial managers set a minimum interest rate that that all accepted projects interest rate that that all accepted projects must meet or exceed.must meet or exceed.

The rate, once established by the firm is The rate, once established by the firm is termed the Minimum Attractive Rate of termed the Minimum Attractive Rate of Return (MARR).Return (MARR).

The The MARRMARR The MARR is expressed as a percent per The MARR is expressed as a percent per

year.year. Numerous models exist to aid a firm’s Numerous models exist to aid a firm’s

financial managers in estimating what this financial managers in estimating what this rate should be in a given time period.rate should be in a given time period.

In some circles, the MARR is termed the In some circles, the MARR is termed the Hurdle Rate.Hurdle Rate.

Capital (investment funds) is not free.Capital (investment funds) is not free. It costs the firm money to raise capital or to It costs the firm money to raise capital or to

use the owners of the firm’s capital.use the owners of the firm’s capital. This cost is often expresses as a % per year.This cost is often expresses as a % per year.

Costs of CapitalCosts of CapitalFirms raise capital from the following Firms raise capital from the following sources:sources:

EQUITY – using the owner’s funds EQUITY – using the owner’s funds (retained earnings, cash on hand–(retained earnings, cash on hand–belongs to the owners, or new stock)belongs to the owners, or new stock) Owners expect a return on their money Owners expect a return on their money

and hence, there is a cost to the firmand hence, there is a cost to the firm DEBT – the firm borrows from outside DEBT – the firm borrows from outside

the firm and pays an interest rate on the the firm and pays an interest rate on the borrowed fundsborrowed funds

Costing CapitalCosting Capital

Financial models exist that will Financial models exist that will approximate the costs of the components approximate the costs of the components making up the weighted average cost of making up the weighted average cost of capital for a given time period.capital for a given time period.

Once this “cost” is approximated, then Once this “cost” is approximated, then new projects up for funding MUST return new projects up for funding MUST return at least the cost of the funds used in the at least the cost of the funds used in the project PLUS some additional percent project PLUS some additional percent return.return.

Costs of capital are expressed as a % Costs of capital are expressed as a % per year just like an interest rate.per year just like an interest rate.

Setting a MARRSetting a MARR If we start with the WACC…If we start with the WACC… Add a buffer percent (?? Varies from firm Add a buffer percent (?? Varies from firm

to firm)to firm) This yields an approximation to a This yields an approximation to a

reasonable MARRreasonable MARR This becomes the Hurdle Rate that all This becomes the Hurdle Rate that all

prospective projects should earn in order prospective projects should earn in order to be considered for funding.to be considered for funding.

Graphical Presentation: MARRGraphical Presentation: MARR

0%

MARR - %

Safe Investment

WACC - %

Acceptable range for new projects

ROR - %

The MARR as an Opportunity The MARR as an Opportunity ForgoneForgone

Assume a firm’s MARR = 12%Assume a firm’s MARR = 12% Two independent projects, A and BTwo independent projects, A and B A costs $400,000 and presents an A costs $400,000 and presents an

estimated 13% Rate of Return per year.estimated 13% Rate of Return per year. B cost $100,000 with an estimated Rate B cost $100,000 with an estimated Rate

of Return of 14.5%of Return of 14.5%

Opportunity ForgoneOpportunity Forgone What if the firm has a budget of, say, What if the firm has a budget of, say,

$150,000?$150,000? A cannot be funded – not sufficient A cannot be funded – not sufficient

funds!funds! B is funded and earns 14.5% return or B is funded and earns 14.5% return or

moremore A is not funded, hence the firm loses A is not funded, hence the firm loses

the OPPORTUNITY to earn 13%the OPPORTUNITY to earn 13% This often happens!This often happens!

Opportunity ForgoneOpportunity Forgone

In the event a MARR is unknown, and In the event a MARR is unknown, and there are insufficient funds for all there are insufficient funds for all projectsprojects, the de facto MARR becomes , the de facto MARR becomes the rate of return of the first unfunded the rate of return of the first unfunded project. project.

Problem PerspectivesProblem Perspectives Before solving, one must decide upon Before solving, one must decide upon

the perspective of the problemthe perspective of the problem Most problems will present two Most problems will present two

perspectivesperspectives Assume a borrowing situation; for Assume a borrowing situation; for

example:example: Perspective 1: From the lender’s viewPerspective 1: From the lender’s view Perspective 2: From the borrower’s viewPerspective 2: From the borrower’s view Impact upon the sign conventionImpact upon the sign convention

Lending – Borrowing ExampleLending – Borrowing Example

Assume $5,000 is borrowed and Assume $5,000 is borrowed and payments are $1,100 per year.payments are $1,100 per year.

Draw the cash flow diagram for thisDraw the cash flow diagram for this First, whose perspective will be used?First, whose perspective will be used? Lender’s or the Borrower’s ? ? ?Lender’s or the Borrower’s ? ? ? Problem will “infer” or you must decide….Problem will “infer” or you must decide….

Lending – BorrowingLending – Borrowing

From the lenders perspective:From the lenders perspective:

0 1 2 3 4 5

-$5,000

A = +$1,100/yr

Lending - BorrowingLending - Borrowing

From the borrower’s perspective:From the borrower’s perspective:

0 1 2 3 4 5

P = +$5,000

A = -$1,100/yr

Spreadsheet ApplicationsSpreadsheet Applications


a – For the series of positive cash flows a – For the series of positive cash flows given in the problem, find the equivalent given in the problem, find the equivalent future value for the end of year 4. Obtain future value for the end of year 4. Obtain answers for both simple interest and answers for both simple interest and compound interest. compound interest.

Spreadsheet ApplicationsSpreadsheet Applications


b – For the revised series of positive cash b – For the revised series of positive cash flows given in the problem, (changing flows given in the problem, (changing $300,000 in years 3 and 4 to $600,000), $300,000 in years 3 and 4 to $600,000), find the equivalent future value for the find the equivalent future value for the end of year 4. Obtain answers for both end of year 4. Obtain answers for both simple interest and compound interest. simple interest and compound interest.

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