1. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University 1 CHAPTER III COMBINING FACTORS McMc Graw Hill ENGINEERING ECONOMY Fifth Edition Blank and Tarquin

2. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University2 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.

3. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University3 3.1 Shifted Series 0 1 2 3 4 5 6 7 8 A = -$500/year Consider: P of this series is at t = 2 (P 2 or F 2 ) P 2 = $500(P/A,i%,4) or, could refer to as F 2 P 0 = P 2 (P/F,i%,2) or, F 2 (P/F,i%,2) P2P2 P0P0

4. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University4 A = -$500/year 0 1 2 3 4 5 6 7 8 P0P0 3.1 Shifted Series: P and F Require F 6 F for this series is at t = 6 F 6 = A(F/A,i%,4) where n = 4 time periods forward. P3P3 F 6 = ??

5. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University5 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

6. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University6 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.

7. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University7 3.2 Series with Other cash flows Consider: 0 1 2 3 4 5 6 7 8 A = $500 F 5 = -$400F 4 = $300 Find the PW at t = 0 and FW at t = 8 for this cash flow –watch the signs! i = 10%

8. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University8 3.2 The PW Points are: F 5 = -$400 F 4 = $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” = 3 123

9. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University9 3.2 The PW Points are: F 5 = -$400 F 4 = $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 123 Back 4 periods Back 5 Periods

10. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University10 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….

11. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University11 3.2 Substitute the Factors and Solve P = $500( 2.4869 )( 0.9090 ) + $300( 0.6830 ) - 400( 0.6209 ) = $1086.84 $1,130.30 $204.90 $248.36

12. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University12 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

13. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University13 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.

14. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University14 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.

15. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University15 3.3 Example of a Shifted Gradient Consider: ……..Base Annuity …….. Gradient Series 0 1 2 3 4 n-1 n This Represents a Shifted Gradient. The Present Worth Point for the Base Annuity and the Gradient would be here!

16. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University16 3.3 Shifted Gradient: Example Given: Base Annuity = $100 G = +$100 0 1 2 3 4 ……….. ……….. 9 10 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.

17. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University17 3.3 Shifted Gradient: Example PW of the Base Annuity Base Annuity = $100 0 1 2 3 4 ……….. ……….. 9 10 P 2 = $100( P/A,10%,8 ) = $100( 5.3349 ) = $533.49 n annuity = 8 time periods P 0 = $533.49( P/F,10%,2 ) = $533.49( 0.8264 ) = $440.88 P 2 = $100(P/A,10%,8)

18. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University18 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: P 2 = $100( P/G,10%,8 ) = $100( 16.0287 ) = $1,602.87 P 0 = $1,602.87( P/F,10%,2 ) = $1,602.87( 0.8264 ) = $1,324.61 P 2 = $100(P/G,10%,8)

19. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University19 3.3 Example: Final Solution For the Base Annuity P 0 = $440.88 For the Linear Gradient P 0 = $1,324.61 Total Present Worth: $440.88 + $1,324.61 = $1,765.49

20. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University20 3.3 Shifted Geometric Gradient Conventional Geometric Gradient 0 1 2 3 … … … n A1A1 Present worth point is at t = 0 for a conventional geometric gradient! P of G. Grad. Is at t = “0”

21. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University21 3.3 Shifted Geometric Gradient Shifted Geometric Gradient 0 1 2 3 … … … n A1A1 Present worth point is at t = 2 for this example

22. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University22 3.3 Geometric Gradient Example 0 1 2 3 4 5 6 7 8 A = $700/yr 12% Increase/yr i = 10%/year A 1 = $400 @ t = 5

23. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University23 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

24. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University24 3.3 The Gradient Amounts 56785678 Present Worth of the Gradient at t = 4 P 4 = $400{ P/A 1,12%,10%,4 } = P 0 = $1,494.70( P/F,10%,4) = $1,494.70( 0. 6830 ) P 0 = $1,020,88

25. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University25 3.3 The Annuity Present Worth PW of the Annuity 0 1 2 3 4 5 6 7 8 i = 10%/year P 0 = $700(P/A,10%,4) = $700( 3.1699 ) = $2,218.94 A = $700/yr

26. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University26 3.3 Total Present Worth Geometric Gradient @ t = P 0 = $1,020,88 Annuity P 0 = $2,218.94 Total Present Worth ” $1,020.88 + $2,218.94 = $3,239.82

27. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University27 3.4 Shifted Decreasing Linear Gradients Given the following shifted, decreasing gradient: 0 1 2 3 4 5 6 7 8 F 3 = $1,000; G=-$100 i = 10%/year Find the Present Worth @ t = 0

28. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University28 3.4 Shifted Decreasing Linear Gradients Given the following shifted, decreasing gradient: 0 1 2 3 4 5 6 7 8 F 3 = $1,000; G=-$100 i = 10%/year PW point @ t = 2

29. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University29 3.4 Shifted Decreasing Linear Gradients 0 1 2 3 4 5 6 7 8 F 3 = $1,000; G=-$100 i = 10%/year P 2 or, F 2 : Then take back to t = 0 P 0 here Use (P/F,10%,2)

30. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University30 3.4 Shifted Decreasing Linear Gradients 0 1 2 3 4 5 6 7 8 F 3 = $1,000; G=-$100 P 2 or, F 2 : Then take back to t = 0 P 0 here i = 10%/year Base Annuity = $1,000 (P/F,10%,2)

31. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University31 3.4 Time Periods Involved 0 1 2 3 4 5 6 7 8 F 3 = $1,000; G=-$100 i = 10%/year P 2 or, F 2 : Take back to t = 0 P 0 here 12345 Dealing with n = 5 periods.

32. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University32 3.4 Time Periods Involved 0 1 2 3 4 5 6 7 8 F 3 = $1,000; G=-$100 i = 10%/year 12345 P 2 = $1,000( P/A,10%,5 ) – 100( P/G,10%.5 ) $1,000 G = -$100/yr P 2 = $1,000( 3.7908 ) - $100( 6.8618 ) = $3,104.62 P 0 = $3,104.62( P/F,10%,2 ) = $3104.62( 0.8264 ) = $2,565.65

33. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University33 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

34. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University34 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.

35. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th Edition. Ch.3 Authored by Dr. Don Smith, Texas A&M University35 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