1. Adopted and modified by Dr. W-.W. Li of UTEP, Fall, 2003
ENGINEERING ECONOMY Fifth Edition Mc
Graw
Hill
Blank and Tarquin
CHAPTER III
COMBINING FACTORS
Adopted and modified by Dr. W-.W. Li of UTEP, Fall, 2003
Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University

2. 3.1 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.
Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University

3. 3.1 Shifted Series Consider: P of this series is at t = 2 (P2 or F2)
A = -$500/year P2 P0
P of this series is at t = 2 (P2 or F2)
P2 = $500(P/A,i%,4) or, could refer to as F2
P0 = P2(P/F,i%,2) or, F2 (P/F,i%,2)
Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University

4. 3.1 Shifted Series: P and F Require F6 F for this series is at t = 6
A = -$500/year P0 P3
F6 = ??
F for this series is at t = 6
F6 = A(F/A,i%,4) where n = 4 time periods forward.
Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University

5. 3.1 Suggested Steps
Draw and correctly label the cash flow diagram that defines the problem
Locate the present and future worth points for each series
Write the time value of money equivalence relationships
Substitute the correct factor values and solve
Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University

6. 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 = 0.
Solve for the PW at t = 0 for the single cash flows.
Add the equivalent PW’s at t = 0.
Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University

7. 3.2 Series with Other cash flows
Consider:
A = $500
F5 = -$400
i = 10%
Find the PW at t = 0 and FW at t = 8 for this cash flow –watch the signs!
Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University

8. t = 1 is the PW point for the $500 annuity;
3.2 The PW Points are:
F4 = $300
A = $500
i = 10% 1 2 3
F5 = -$400
t = 1 is the PW point for the $500 annuity;
“n” = 3
Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University

9. t = 1 is the PW point for the two other single cash flows
3.2 The PW Points are:
Back 4 periods
F4 = $300
A = $500 1 2 3
i = 10%
F5 = -$400
Back 5 Periods
t = 1 is the PW point for the two other single cash flows
Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University

10. 3.2 Write the Equivalence Statement
P = $500(P/A,10%,3)(P/F,10%,1) +
$300(P/F,10%,4) -
400(P/F,10%,5)
Substituting the factor values into the equivalence expression and solving….
Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University

11. 3.2 Substitute the Factors and Solve
P = $500( )( ) +
$300( ) -
400( ) = $
$1,130.30
$204.90
$248.36
Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University

12. 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
Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University

13. 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.
Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University

14. 3.3 Example of a Conventional Gradient
Consider:
Gradient Series
……..Base Annuity …….. …
n n
This Represents a Conventional Gradient.
The present worth point is t = 0.
Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University

15. 3.3 Example of a Shifted Gradient
Consider:
Gradient Series
……..Base Annuity ……..
n n
The Present Worth Point for the Base Annuity and the Gradient would be here!
This Represents a Shifted Gradient.
Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University

16. 3.3 Shifted Gradient: Example
Given:
G = +$100
Base Annuity = $100
……… ………
Let Cash Flow (CF) start at t = 3:
$100/ yr increasing by $100/year through year 10; i = 10%; Find the PW at t = 0.
Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University

17. 3.3 Shifted Gradient: Example
PW of the Base Annuity
P2 = $100(P/A,10%,8)
Base Annuity = $100
……… ………
nannuity = 8 time periods
P2 = $100( P/A,10%,8 ) = $100( ) = $533.49
P0 = $533.49( P/F,10%,2 ) = $533.49( )
= $440.88
Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University

18. 3.3 Gradient Present Worth
For the gradient component
P2 = $100(P/G,10%,8)
G = +$100
……… ………
PW of gradient is at t = 2:
P2 = $100( P/G,10%,8 ) = $100( ) = $1,602.87
P0 = $1,602.87( P/F,10%,2 ) = $1,602.87( )
= $1,324.61
Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University

19. 3.3 Example: Final Solution
For the Base Annuity
P0 = $440.88
For the Linear Gradient
P0 = $1,324.61
Total Present Worth:
$ $1, = $1,765.49
Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University

20. 3.3 Shifted Geometric Gradient
Conventional Geometric Gradient
P of G. Grad. Is at t = “0” A1
… … … n
Present worth point is at t = 0 for a conventional geometric gradient!
Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University

21. 3.3 Shifted Geometric Gradient
Present worth point is at t = 2 for this example
Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University

22. 3.3 Geometric Gradient Example
i = 10%/year
A = $700/yr
A1 = t = 5
12% Increase/yr
Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University

23. 3.3 Geometric Gradient Example
i = 10%/year
A = $700/yr
PW point for the annuity
12% Increase/yr
PW point for the gradient
Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University

24. 3.3 The Gradient Amounts Present Worth of the Gradient at t = 45 6 7 8
Present Worth of the Gradient at t = 4
P4 = $400{ P/A1,12%,10%,4 } =
P0 = $1,494.70( P/F,10%,4) = $1,494.70( )
P0 = $1,020,88
Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University

25. 3.3 The Annuity Present Worth
PW of the Annuity
i = 10%/year
A = $700/yr
P0 = $700(P/A,10%,4)
= $700( ) = $2,218.94
Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University

26. 3.3 Total Present Worth Geometric Gradient @ t = Annuity
$1, $2,218.94
= $3,239.82
Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University

27. 3.4 Shifted Decreasing Linear Gradients
Given the following shifted, decreasing gradient:
F3 = $1,000; G=-$100
i = 10%/year
Find the Present t = 0
Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University

28. 3.4 Shifted Decreasing Linear Gradients
Given the following shifted, decreasing gradient:
F3 = $1,000; G=-$100
i = 10%/year
PW t = 2
Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University

29. 3.4 Shifted Decreasing Linear Gradients
F3 = $1,000; G=-$100
i = 10%/year
Use (P/F,10%,2)
P2 or, F2: Then take back to t = 0
P0 here
Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University

30. 3.4 Shifted Decreasing Linear Gradients
F3 = $1,000; G=-$100
Base Annuity = $1,000
i = 10%/year
(P/F,10%,2)
P2 or, F2: Then take back to t = 0
P0 here
Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University

31. 3.4 Time Periods Involved F3 = $1,000; G=-$100 P0 here
i = 10%/year 1 2 3 4 5
P2 or, F2: Take back to t = 0
P0 here
Dealing with n = 5 periods.
Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University

32. 3.4 Time Periods Involved F3 = $1,000; G=-$100 G = -$100/yr
i = 10%/year 1 2 3 4 5
P2 = $1,000( P/A,10%,5 ) – 100( P/G,10%.5 )
P2= $1,000( ) - $100( ) = $3,104.62
P0 = $3,104.62( P/F,10%,2 ) = $ ( ) = $2,565.65
Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University

33. 3.5 Spreadsheet Applications
Assume Excel is the spreadsheet of choice
Instructors may vary on the degree of emphasis placed on spreadsheet use
Student’s Goal:
Learn the Excel Financial Functions
Create your own spreadsheets to solve a variety of problems
Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University

34. 3.5 NPV Function in Excel NPV function is basic
Requires 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.
Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University

35. 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
Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University