1. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 2-1 Developed By: Dr. Don Smith, P.E. Department of Industrial Engineering Texas A&M University College Station, Texas Executive Summary Version Chapter 2 Factors: How Time and Interest Affect Money

2. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 2-2 LEARNING OBJECTIVES 1.F/P and P/F factors 2.P/A and A/P factors 3.Interpolate for factor values 4.P/G and A/G factors 5. Geometric gradient 6. Calculate i 7. Calculate n 8. Spreadsheets

3. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 2-3 Sct 2.1 Single-Payment Factors (F/P and P/F)  Objective:  Derive factors to determine the present or future worth of a cash flow  Cash Flow Diagram – basic format 0 1 2 3 n-1 n P0P0 FnFn i% / period P 0 = F n 1/(1+i) n →(P/F,i%,n) factor: Excel: =PV(i%,n,,F) F n = P 0 (1+i) n →(F/P,i%,n) factor: Excel: =FV(i%,n,,P)

4. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 2-4 Sct 2.2 Uniform-Series: Present Worth Factor (P/A) and Capital Recovery Factor(A/P)  Cash flow profile for P/A factor.... 0 1 2 3 n-2 n-1 n $A per interest period i% per interest period Required: To find P given A Cash flows are equal, uninterrupted and flow at the end of each interest period Find P

5. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 2-5 (P/A) Factor Derivation  Setup the following:  Multiply by to obtain a second equation…  Subtract (1) from (2) to yield… (1) (2) (3)

6. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 2-6 (P/A) and (A/P) Factor Formulas  Simplify (3) to yield…  Solve (4) for A to get (A/P) factor (4) (P/A,i%,n) factor Excel: =PV(i%,n,A) (A/P,i%,n) factor Excel: =PMT(i%,n,P) (5)

7. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 2-7 ANSI Standard Notation for Interest Factors  Standard notation has been adopted to represent the various interest factors  Consists of two cash flow symbols, the interest rate, and the number of time periods  General form:(X/Y,i%,n)  X represents what is unknown  Y represents what is known  i and n represent input parameters; can be known or unknown depending upon the problem

8. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 2-8 Notation - continued  Example: (F/P,6%,20) is read as:  To find F, given P when the interest rate is 6% and the number of time periods equals 20.  In problem formulation, the standard notation is often used in place of the closed-form equivalent relations (factor)  Tables at the back of the text provide tabulations of common values for i% and n

9. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 2-9 Sct 2.3 Sinking Fund Factor and Uniform Series Compound Amount Factor (A/F and F/A)  Cash flow diagram for (A/F) factor  Start with what has already been developed.... 0 1 2 3 n-2 n-1 n A=? per interest period i% per interest period F = given Find A, given F

10. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 2-10 (F/A) factor from (A/F)  Given:  Solve for F in terms of A to yield (A/F,i%,n) factor Excel: =PMT(i%,n,,F) (F/A,i%,n) factor Excel: =FV(i%,n,A)

11. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 2-11 Sct 2.4 Interpolation in Interest Tables  When using tabulated interest tables one might be forced to approximate a factor that is not tabulated  Can apply linear interpolation to approximate  See Table 2-4  Factors are nonlinear functions, hence linear interpolation will yield errors in the 2-4% range  Use a spreadsheet model to calculate the factor precisely

12. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 2-12 Sct 2.5 Arithmetic Gradient Factors (P/G) and (A/G)  Cash flow profile 0 1 2 3 n-1 n A 1 +G A 1 +2G A 1 +( n-2)G A 1 +(n-1)G Find P, given gradient cash flow G CF n = A 1 ± (n-1)G Base amount = A 1

13. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 2-13 Gradient Example 0 1 2 3 4 5 6 7 $100 $200 $300 $400 $500 $600 $700 Gradients have two components: 1. The base amount and the gradient 2. The base amount (above) = $100/time period

14. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 2-14 Gradient Components …….. 0 1 2 3 n-2 n-1 n Base amount = A / period 0G 1G 2G (n-3)G (n-2)G (n-1)G  Present worth point is 1 period to the left of the 0G cash flow  For present worth of the base amount, use the P/A factor (already known)  For present worth of the gradient series, use the P/G factor (to be derived) Find P of gradient series

15. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 2-15 Gradient Decomposition  As we know, arithmetic gradients are comprised of two components 1. Gradient component 2. Base amount  When working with a cash flow containing a gradient, the (P/G) factor is only for the gradient component  Apply the (P/A) factor to work on the base amount component  P = PW(gradient) + PW(base amount)

16. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 2-16 Derivation Summary for (P/G)  Start with:  Multiply (1) by (1+i) 1 to create a second equation  Subtract (1) from the second equation and simplify  Yields… ( 1) (P/G,i,n) factor No Excel relation exists

17. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 2-17 Use of the (A/G) Factor 0 1 2 3 n-1 n G 2G ( n-2)G (n-1)G Find A, given gradient cash flow G CF n = (n-1)G Equivalent A of gradient series A A A... A A A = G(A/G,i,n)

18. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 2-18 Sct 2.6 Geometric Gradient Series Factor  Geometric Gradient  Cash flow series that starts with a base amount A 1  Increases or decreases from period to period by a constant percentage amount  This uniform rate of change defines…  A GEOMETRIC GRADIENT  Notation: g = the constant rate of change, in decimal form, by which future amounts increase or decrease from one time period to the next

19. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 2-19 Typical Geometric Gradient A1A1 A 1 (1+g) A 1 (1+g) 2.... 0 1 2 3 n-2 n-1 n A 1 (1+g) n-1 Required: Find a factor (P/A,g%,i%,n) that will convert future cash flows to a single present worth value at time t = 0 Given A 1, i%, and g%

20. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 2-20 Basic Derivation: Geometric Gradient ( 1) Subtract Eq. (2 ) from Eq. (3 ) to yield Solve for P g and simplify to yield…. Start with: Factor out A 1 out and re-write (2) (3) Multiply by (1+g)/(1+i) to obtain Eq. (3 )

21. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 2-21 Two Forms to Consider… Case: g = i  A 1 is the starting cash flow  There is NO base amount associated with a geometric gradient  The remaining cash flows are generated from the A 1 starting value  No tables available to tabulate this factor…too many combinations of i% and g% to support tables To use the (P/A,g%,i%,n) factor

22. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 2-22 Sct 2.7 Determination of Unknown Interest Rate  Class of problems where the interest rate, i%, is the unknown value  For simple, single payment problems (i.e., P and F only), solving for i% given the other parameters is not difficult  For annuity and gradient type problems, solving for i% can be tedious  Trial and error method  Apply spreadsheet models

23. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 2-23 The IRR Spreadsheet Function  Define the total cash flow as a column of values within Excel  Apply the IRR function:  = IRR(first_cell:last_cell, guess value)  If the cash flow series is an A value then apply the RATE function:  =RATE(number_years, A,P,F)  See examples 2.12 and 2.13

24. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 2-24 Sct 2.8 Determination of Unknown Number of Years  Class of problems where the number of time periods (years) is the unknown  In single payment type problems, solving for n is straight forward  In other types of cash flow profiles, solving for n requires trial and error or spreadsheet  In Excel, given A, P, and/or F, and i% values apply:  =NPER(i%,A,P,F) to return the value of n

25. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 2-25 Sct 2.9 Spreadsheet Application – Basic Sensitivity Analysis  Sensitivity Analysis is a process of determining what input variables really matter in a given problem formulation  Sensitivity analysis aids in evaluating certain what-If scenarios  Spreadsheet modeling is the best approach to formulate sensitivity analysis for a given problem

26. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 2-26 Sensitivity Analysis…  See Example 2.15  Illustratesa what-if situation for receiving money in three different time periods  Tabulates the associated rate of returns for the three situations  See Example 2.16  The evaluation of non-sequential cash flows

27. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 2-27 Summary  Interest factors exist to aid in determining economic equivalence of various cash flow patterns  Notation is introduced that is applied throughout the remainder of the text  Introduction of important Excel spreadsheet financial functions to aid in evaluation of engineering economy problems

28. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 2-28 End of Slide Set