Lecture slides to accompany Engineering Economy 8 th

Chapter 3 Combining Factors and Spreadsheet Functions ©Mc. Graw-Hill Education. All rights reserved. Authorized

LEARNING OUTCOMES 1. Shifted uniform series 2. Shifted series and single cash flows 3.

Shifted Uniform Series A shifted uniform series starts at a time other than period

Example Using P/A Factor: Shifted Uniform Series • 0 1 0 5 2 1

Example Using F/A Factor: Shifted Uniform Series How much money would be available in

Shifted Series and Random Single Amounts For cash flows that include uniform series and

Example: Series and Random Single Amounts (1) Find the present worth in year 0

Example: Series and Random Single Amounts (2) P A 0 9 1 10 2

Example Worked a Different Way (Using F/A instead of P/A for uniform series) The

Example: Series and Random Amounts • 0 7 1 2 8 3 0 5

Shifted Arithmetic Gradients Shifted gradient begins at a time other than between periods 1

Example: Shifted Arithmetic Gradient John Deere expects the cost of a tractor part to

Shifted Geometric Gradients Shifted gradient begins at a time other than between periods 1

Example: Shifted Geometric Gradient • ©Mc. Graw-Hill Education. (1)

Example: Shifted Geometric Gradient • ©Mc. Graw-Hill Education. (2)

Negative Shifted Gradients • • • For negative geometric gradients, change signs on both

Example: Negative Shifted Arithmetic Gradient • 0 1 2 0 6 3 1 700

Summary of Important Points P for shifted uniform series is one period ahead of

Slides: 19

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Lecture slides to accompany Engineering Economy, 8 th edition Leland Blank, Anthony Tarquin ©Mc. Graw-Hill Education. All rights reserved. Authorized only for instructor use in the classroom. No reproduction or further distribution permitted without the prior written consent of Mc. Graw-Hill Education.

Chapter 3 Combining Factors and Spreadsheet Functions ©Mc. Graw-Hill Education. All rights reserved. Authorized only for instructor use in the classroom. No reproduction or further distribution permitted without the prior written consent of Mc. Graw-Hill Education.

LEARNING OUTCOMES 1. Shifted uniform series 2. Shifted series and single cash flows 3. Shifted gradients ©Mc. Graw-Hill Education.

Shifted Uniform Series A shifted uniform series starts at a time other than period 1 The cash flow diagram below is an example of a shifted series Series starts in period 2, not period 1 Shifted series usually require the 4 2 3 0 1 5 use of multiple factors Remember: When using P/A or A/P factor, PA is always one year ahead of first A When using F/A or A/F factor, F A is in same year as last A ©Mc. Graw-Hill Education.

Example Using P/A Factor: Shifted Uniform Series • 0 1 0 5 2 1 3 4 5 2 3 4 Actual 6 year Series year A = $10, 000 • • ©Mc. Graw-Hill Education.

Example Using F/A Factor: Shifted Uniform Series How much money would be available in year 10 if $8000 is deposited each year in years 3 through 10 at an interest rate of 10% per year? Cash flow diagram is: Actual year 0 10 1 2 0 8 3 1 4 2 5 3 4 • ©Mc. Graw-Hill Education. 6 7 5 8 6 9 7 Series year

Shifted Series and Random Single Amounts For cash flows that include uniform series and randomly placed single amounts: Uniform series procedures are applied to the series amounts Single amount formulas are applied to the one-time cash flows The resulting values are then combined per the problem statement The following slides illustrate the procedure ©Mc. Graw-Hill Education.

Example: Series and Random Single Amounts (1) Find the present worth in year 0 for the cash flows shown using an interest rate of 10% per year. 0 9 1 10 2 3 4 5 6 ©Mc. Graw-Hill Education. 1 10 2 0 3 1 4 2 5 3 8 $2000 0 9 • 7 6 4 7 5 8 6 Actual 7 year 8 Series year

Example: Series and Random Single Amounts (2) P A 0 9 1 10 2 0 3 1 2 • ©Mc. Graw-Hill Education. 4 5 3 6 4 7 5 $2000 8 6 Actual year 7 8 Series year

Example Worked a Different Way (Using F/A instead of P/A for uniform series) The same re-numbered diagram from the previous slide is used 0 9 1 10 2 0 3 1 4 2 5 3 6 4 7 5 8 6 7 8 $2000 • As shown, there are usually multiple ways to work equivalency problems ©Mc. Graw-Hill Education.

Shifted Arithmetic Gradients Shifted gradient begins at a time other than between periods 1 and 2 Present worth PG is located 2 periods before gradient starts Must use multiple factors to find PT in actual year 0 To find equivalent A series, find PT at actual time 0 and apply (A/P, i, n) ©Mc. Graw-Hill Education.

Example: Shifted Arithmetic Gradient John Deere expects the cost of a tractor part to increase by $5 per year beginning 4 years from now. If the cost in years 1 -3 is $60, determine the present worth in year 0 of the cost through year 10 at an interest rate of 12% per year. 0 1 3 2 0 60 60 4 1 60 2 65 Actual years 10 5 3 8 Gradient years 70 95 • • Next, move P 2 back to • year 0 Next, find PA for the $60 amounts of years 1 and 2 Finally, add P 0 and PA to get PT in year 0 ©Mc. Graw-Hill Education. • •

Shifted Geometric Gradients Shifted gradient begins at a time other than between periods 1 and 2 Equation yields Pg for all cash flows (base amount A 1 is included) Equation (i ≠ g): • For negative gradient, change signs on both g values There are no tables for geometric gradient factors ©Mc. Graw-Hill Education.

Negative Shifted Gradients • • • For negative geometric gradients, change signs on both g values • • • All other procedures are the same as for positive gradients ©Mc. Graw-Hill Education.

Example: Negative Shifted Arithmetic Gradient • 0 1 2 0 6 3 1 700 4 2 650 3 600 5 4 550 500 6 5 Actual years 7 Gradient years 450 Solution: Gradient G first occurs between actual years 2 and 3; these are gradient years 1 and 2 PG is located in gradient year 0 (actual year 1); base amount of $700 is in gradient years 1 -6 • ©Mc. Graw-Hill Education.

Summary of Important Points P for shifted uniform series is one period ahead of first A; n is equal to number of A values F for shifted uniform series is in same period as last A; n is equal to number of A values For gradients, first change equal to G or g occurs between gradient years 1 and 2 • • ©Mc. Graw-Hill Education.