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CHAPTER 3
COMBINING FACTORS
Mc
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ENGINEERING ECONOMY, Sixth Edition by Blank and
Tarquin
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Learning Objectives
Shifted SeriesShifted Series and Single AmountsShifted GradientsDecreasing GradientsSpreadsheets
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Section 3.1
Calculations For Uniform Series That
Are Shifted
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1. Uniform Series that are SHIFTED
A shifted series is one whose present worth point in time is NOT t = 0.Shifted either to the left of “0” or to the right of t = “0”.Dealing with a uniform series: The PW point is always one period to the
left of the first series value No matter where the series falls on the time
line.
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Shifted Series
0 1 2 3 4 5 6 7 8
A = -$500/year
Consider:
P of this series is at t = 2 (P3 or F3)
P3 = $500(P/A,i%,4) or, could refer to as F3
P0 = P3(P/F,i%,2) or, F3(P/F,i%,2)
P3P0
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Shifted Series: P and F
A = -$500/year
Consider:
F for this series is at t = 6
F6 = A(F/A,i%,4)
0 1 2 3 4 5 6 7 8
P3P0
F at t = 6
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Figure 3.3, Suggested Steps
Draw and correctly label the cash flow diagram that defines the problemLocate the present and future worth points for each seriesWrite the time value of money equivalence relationshipsSubstitute the correct factor values and solve
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Section 3.2
Uniform Series and Randomly Placed Single Amounts
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3.2 Series with other single cash flows
It is common to find cash flows that are combinations of series and other single cash flows.Solve for the series present worth values then move to t = 0Solve for the PW at t = 0 for the single cash flowsAdd the equivalent PW’s at t = 0
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3.2 Series with Other cash flows
Consider:
0 1 2 3 4 5 6 7 8
A = $500
F5 = -$400
F4 = $300
•Find the PW at t = 0 and FW at t = 8 for this cash flow
i = 10%
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3.2 The PW Points are:
F5 = -$400
F4 = $300
A = $500
0 1 2 3 4 5 6 7 8
i = 10%
t = 1 is the PW point for the $500 annuity;“n” = 3
1 2 3
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3.2 The PW Points are:
F5 = -$400
F4 = $300
A = $500
0 1 2 3 4 5 6 7 8
i = 10%
t = 1 is the PW point for the two other single cash flows
1 2 3
Back 4 periods
Back 5 Periods
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3.2 Write the Equivalence Statement
P = $500(P/A,10%,3)(P/F,10%,2)+ $300(P/F,10%,4)-
400(P/F,10%,5)
Substituting the factor values into the equivalence expression and solving….
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3.2 Substitute the factors and solve
P = $500( 2.4869 )( 0.8264 )+ $300( 0.6830 )-
400( 0.6209 )=
$831.06
$1,027.58
$204.90
$248.36
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Section 3.3
Calculations for Shifted Gradients
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3.3 The Linear Gradient - revisited
The Present Worth of an arithmetic gradient (linear gradient) is always located: One period to the left of the first cash
flow in the series ( “0” gradient cash flow) or,
Two periods to the left of the “1G” cash flow
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3.3 Shifted Gradient
A Shifted Gradient is one whose present value point is removed from time t = 0.A Conventional Gradient is one whose present worth point is t = 0.
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3.3 Example of a Conventional Gradient
Consider:
……..Base Annuity ……..
Gradient Series
0 1 2 … n-1 n
This Represents a Conventional Gradient.
The present worth point is t = 0.
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3.3 Example of a Shifted Gradient
Consider:
……..Base Annuity ……..
Gradient Series
0 1 2 … n-1 n
This Represents a Shifted Gradient.
The Present Worth Point for the Base Annuity and the Gradient
would be here!
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3.3 Shifted Gradient: ExampleGiven:
Base Annuity = $100
G = +$100
0 1 2 3 4 ……….. ……….. 9 10
Let C.F start at t = 3:
$500/ yr increasing by $100/year through year 10; i = 10%; Find the PW at t = 0
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3.3 Shifted Gradient: ExamplePW of the Base Annuity
Base Annuity = $100
0 1 2 3 4 ……….. ……….. 9 10
P2 = $100( P/A,10%,8 ) = $100( 5.3349 ) = $533.49
Nannuity = 8 time periods
P0 = $533.49( P/F,10%,2 ) = $533.49( 0.8264 )
= $440.88
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3.3 Gradient Present Worth
For the gradient component
0 1 2 3 4 ……….. ……….. 9 10
G = +$100
• PW of gradient is at t = 2:
•P2 = $100( P/G,10%,8 ) = $100( 16.0287 ) = $1,602.87
•P0 = $1,602.87( P/F,10%,2 ) = $1,602.87( 0.8264 )
• = $1,324.61
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3.3 Example: Final Solution
For the Base Annuity P0 = $440.88
For the Linear Gradient P0 = $1,324.61
Total Present Worth: $440.88 + $1,324.61 = $1,765.49
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3.3 Shifted Geometric Gradient
Conventional Geometric Gradient
0 1 2 3 … … … n
A1
Present worth point is at t = 0 for a conventional geometric gradient!
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3.3 Shifted Geometric Gradient
Conventional Geometric Gradient
0 1 2 3 … … … n
A1
Present worth point is at t = 2 for this example
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3.3 Geometric Gradient Example
0 1 2 3 4 5 6 7 8
A = $700/yr
12% Increase/yr
i = 10%/year
A1 = $400 @ t = 5
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3.3 Geometric Gradient Example
0 1 2 3 4 5 6 7 8
A = $700/yr
12% Increase/yr
i = 10%/year
PW point for the gradient
PW point for the annuity
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3.3 The Gradient Amounts
t Base Amt
1 $400.00
2 $448.00
3 $501.76
4 $561.97
5
6
7
8
Present Worth of the Gradient at t = 4
P4 = $400{ P/A1,12%,10%,4 } = 1,494.70$
3.73674
P0 = $1,494.70( P/F,10%,4) = $1,494.70( 0. 6830 )
P0 = $1,020,88
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3.3 The Annuity Present Worth
PW of the Annuity
0 1 2 3 4 5 6 7 8
i = 10%/year
P0 = $700(P/A,10%,4)
= $700( 3.1699 ) = $2,218.94
A = $700/yr
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3.3 Total Present Worth
Geometric Gradient @ t = P0 = $1,020,88
Annuity P0 = $2,218.94
Total Present Worth”
$1,020.88 + $2,218.94
= $3,239.82
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Section 3.4
Shifted Decreasing Arithmetic Gradients
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3.4 Shifted Decreasing Linear Gradients
Given the following shifted, decreasing gradient:
0 1 2 3 4 5 6 7 8
F3 = $1,000; G=-$100
i = 10%/year
Find the Present Worth @ t = 0
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3.4 Shifted Decreasing Linear Gradients
Given the following shifted, decreasing gradient:
0 1 2 3 4 5 6 7 8
F3 = $1,000; G=-$100 i = 10%/year
PW point @ t = 2
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3.4 Shifted Decreasing Linear Gradients
0 1 2 3 4 5 6 7 8
F3 = $1,000; G=-$100 i = 10%/year
P2 or, F2: Take back to t = 0
P0 here
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3.4 Shifted Decreasing Linear Gradients
0 1 2 3 4 5 6 7 8
F3 = $1,000; G=-$100
i = 10%/year
P2 or, F2: Take back to t = 0
P0 here
Base Annuity = $1,000
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3.4 Time Periods Involved
0 1 2 3 4 5 6 7 8
F3 = $1,000; G=-$100
i = 10%/year
P2 or, F2: Take back to t = 0
P0 here
1 2 3 4 5
Dealing with n = 5.
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3.4 Time Periods Involved
0 1 2 3 4 5 6 7 8
F3 = $1,000; G=-$100
i = 10%/year
1 2 3 4 5
P2 = $1,000( P/A,10%,5 ) – 100( P/G,10%.5 )
$1,000 G = -$100/yr
P2= $1,000( 3.7908 ) - $100( 6.8618 ) = $3,104.62
P0 = $3,104.62( P/F,10%,2 ) = $3104.62( 0 .8264 ) = $2,565.65
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Section 3.5
Spreadsheet Applications
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3.5 Spreadsheet Applications
Assume Excel is the spreadsheet of choice Instructors may vary on the degree of emphasis placed on spreadsheet useStudent’s Goal: Learn the Excel Financial Functions Create your own spreadsheets to solve
a variety of problems
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3.5 NPV Function in Excel
NPV function is basicRequires that all cell in the range so defined have an entry.The entry can be $0…but not blank!Incorrect results can be generated if one or more cells in the defined range is left blank . A “0” value must be entered.
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3.5 Spreadsheets
It is assumed if an instructor desires to apply spreadsheets, he or she will provide examples and go over each example and the associated cell formulas.See Appendix A for further details on Excel applications
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End of Slide Set
Mc
GrawHill
ENGINEERING ECONOMY, Sixth Edition
Blank and Tarquin