ch 3 combining factors

8/12/2019 Ch 3 Combining Factors

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2012 by McGraw-Hill, New York, N.Y All Rights Reserve!-1

Lecture slides to accompany

Engineering Economy7thedition

Leland Blank

Anthony Tarquin

Chapter 3Cha

pter 3

CombiningCombining

Factors andFactors andSpreadsheetSpreadsheet

FunctionsFunctions


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LEA!"!# $%TC$&ESLEA!"!# $%TC$&ES

1. Shifted uniform series

2. Shifted series andsingle cash flows

3. Shifted gradients


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Shi'ted %ni'orm SeriesShi'ted %ni'orm SeriesA shi'ted uni'orm series starts at a time other than period 1

"he cash #low iagra$ below is a% e&a$'le o# a shi#te series (eries starts i% 'erio 2, %ot 'erio 1

Shi'ted series

usuallyrequire the use

o'

multiple factors

Remember: (hen using )*A or A*) 'actor+ )Ais al,ays one year ahead

o' 'irst A

(hen using F*A or A*F 'actor+ FAis in sameyear as last A

)A* +


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E-ample %sing )*A Factor. Shi'ted %ni'orm SeriesE-ample %sing )*A Factor. Shi'ted %ni'orm Series

0 * +

A / 012+222

0 1 2 ! /

i / 12

The present ,orth o' the cash 'lo, sho,n belo, at i / 12 is.

4a5 06+328 4b5 069+:6 4c5 038+8:6 4d5 037+92;

Solution 1 3se 4A #actor with % * #or arrows to get 1in year 1

2 3se 4) #actor with % * 1 to $ove 1back #or 0in year 2

)2

/ )1

4)*F+12+15 / A4)*A+12+54)*F+12+15 / 12+22243


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=o, much money ,ould be a>ailable in year 12 i' 0;222 is deposited eachyear in years 3 through 12 at an interest rate o' 12 per year?

0 1 2 ! / 6 7 8 10

FA/ ?

A / 0;222

i / 12

Solution Re-%5$ber iagra$ to eter$i%e % * 7 %5$ber o# arrows

0 1 2 ! / 6 7

Cash flow diagram is:

)A* 7000)4A,109,7

* 700011.!8

*091+8;7

E-ample %sing F*A Factor. Shi'ted %ni'orm SeriesE-ample %sing F*A Factor. Shi'ted %ni'orm Series

Act5al year

(eries year


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Shi'ted Series and andom Single AmountsShi'ted Series and andom Single Amounts

)or cash #lows that i%cl5e uniform series andrandomly placed single amounts

Uniform series procedures are applied to the series amounts

Single amount formulasare applied to the one-time cash flows

The resulting >alues are then combinedper the problem statement

"he #ollowi%g slies ill5strate the 'roce5re


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E-ample. Series and andom Single AmountsE-ample. Series and andom Single Amounts)i% the 'rese%t worth i% year 0 #or the cash #lows

show% 5si%g a% i%terest rate o# 109 'er year.

0 1 2 ! / 6 7 8

10

)T/ ?

A / 0222

i / 12

)irst, re-%5$ber cash #low iagra$ to get % #or 5%i#or$ series % * 7

06222

0 1 2 ! / 6 7 8

10

)T/ ?

A / 0222

i / 12

06222

0 1 2 ! / 6 7

Solution.

Act5al year

(eries year


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E-ample (orked a @i''erent (ayE-ample (orked a @i''erent (ay4%sing F*A instead o' )*A 'or uni'orm series5

0 1 2 ! / 6 7 810

)T/ ?

A / 0222

i / 12

06222

0 1 2 ! / 6 7

"he sa$e re-%5$bere iagra$ #ro$ the 'revio5s slie is 5se

Solution. 3se )4A to get )Ai% act5al year 10 )A* 000)4A,109,7 * 00011.!8 * :6,170

Move :2000 si%gle a$o5%t back to year 0 2000* 20004),109,7 * 20000.// * :8!!Now, a two val5es to get " "* 22,0! ; 8!! * :22,86/ Same as be'ore

Move )Aback to year 0 5si%g 4) 0* 6,1704),109,10 * 6,1700.!7 * :22,0!

As show%, there are 5s5ally $5lti'le ways to work e


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E-ample. Series and andom AmountsE-ample. Series and andom Amounts


Con>ert the cash 'lo,s sho,n belo, 4black arro,s5 into

an equi>alent annual ,orth Ain years 1 through ; 4red arro,s5at i / 12 per yearert all cash 'lo,s into ) in year 2 a% 5se A4 with % * 7

6< Find F in year ; a% 5se A4) with % * 7

(olve #or ) F / 32224F*A+12+5 12224F*)+12+15

/ 32224:


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E-ample. Shi'ted Arithmetic #radientE-ample. Shi'ted Arithmetic #radient

Solution.

=%g. Ali Ah$a e&'ects the cost o# a tractor 'art to i%crease by : 'er year begi%%i%g

years #ro$ %ow. ># the cost i% years 1-! is :/0, eter$i%e thepresent worth in year 0

o# the cost thro5gh year 10 at a% i%terest rate o# 129 'er year.

0 1 2 ! 10

/0 /0 :2 /60

8

)T/ ?i / 16

First 'ind )6'or # / 0 andbase amount 40:25 in actual year 6

2 * /04A,129,7 ; 4G,129,7 * :!60.1

!e-t+ mo>e )6back to year 20 * 24),129,2 * :28.28

!e-t+ 'ind )A 'or the 0:2 amounts o' years 1 and 6 A * /04A,129,2 * :101.1

Finally+ add )2 and )Ato get )Tin year 2 )T

/ )2 )

AA / 039:


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Shi'ted #eometric #radientsShi'ted #eometric #radients

Shi'ted gradient begins at a time other than bet,een periods 1 and 6

Equation yields )g'orallcash 'lo,s 4base amount A

1is included5

For negati>e gradient+ change signs on both g >alues

)g/ A 11 D41g5*41i5n*4ig5Equation 4i G g5.

There are no tables 'or geometric gradient 'actors


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E-ample. Shi'ted #eometric #radientE-ample. Shi'ted #eometric #radientNabl5s Natio%al (teel sig%e a -year co%tract to '5rchase water treat$e%t

che$icals #ro$ a local istrib5tor #or :6000 'er year. ?he% the co%tract e%s,the cost o# the che$icals is e&'ecte to i%crease by 129 'er year #or the %e&t 7

years. ># a% i%itial i%vest$e%t i% storage ta%ks is :!,000, eter$i%e the

e


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E-ample. Shi'ted #eometric #radientE-ample. Shi'ted #eometric #radient

Graie%t starts betwee% act5al years a% /@ these are graie%t years 1 a% 2.

gis locate i% graie%t year 0, which is act5al year

Move ga% other cash #lows to year 0 to calc5late ")

T/ 3+222 72224)*A+1+85 89+8214)*F+1+85 / 0;3+636

g * 60001-B1;0.1241;0.1C840.1-0.12D * :8,01


16/18

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!egati>e Shi'ted #radients!egati>e Shi'ted #radients

For negati>earithmetic gradients+ change sign on # term 'rom to

For negati>e geometricgradients+ change signs on both g >alues

All other procedures are the same as 'or positi>e gradients

Ge%eral e


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E-ample. !egati>e Shi'ted Arithmetic #radientE-ample. !egati>e Shi'ted Arithmetic #radient

For the cash 'lo,s sho,n+ 'ind the 'uture ,orth in year 7 at i / 12 per year

F / ?

0 1 2 ! / 6

600/0

00 00

/00

# / 02

Graie%t G #irst occ5rs betwee% act5al years 2 a% !@ these are graie%t years 1 a% 2Solution.

0 1 2 ! /

Actual years

#radient years

Gis locate i% graie%t year 0 act5al year 1@ base a$o5%t o# :600 is i% graie%t years 1-/

)#/ 7224)*A+12+:5 H 24)*#+12+:5 / 72248


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ch 3 combining factors

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