1. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 7-1 Developed By: Dr. Don Smith, P.E. Department of Industrial Engineering Texas A&M University College Station, Texas Executive Summary Version Chapter 7 Rate of Return Analysis: Single Alternative

2. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 7-2 LEARNING OBJECTIVES 1. Definition of ROR 2. ROR using PW and AW 3. Calculations about ROR 4. Multiple RORs 5. Composite ROR 6. ROR of bonds ROR = Rate of Return

3. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 7-3 Sct 7. 1 Rate of Return - Introduction  Referred to as ROR or IRR (Internal Rate of Return) method  It is one of the popular measures of investment worth  DEFINITION -- ROR is either the interest rate paid on the unpaid balance of a loan, or the interest rate earned on the unrecovered investment balance of an investment such that the final payment or receipt brings the terminal value to exactly equal “0”  The ROR of found using a PW or AW relation. The rate determined is called i*

4. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 7-4 Unrecovered Investment Balance  ROR is the interest rate earned/charged on the unrecovered balance of a loan or investment project  ROR is not the interest rate earned on the original loan amount or investment amount (P)  The i* value is compared to the MARR --  If i* > MARR, investment is justified  If i* = MARR, investment is justified (indifferent decision)  If i* < MARR, investment is not justified  See example 7.1

5. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 7-5 Valid Ranges for usable i* rates Mathematically, i* rates must be: 1.An i* = -100% signals total and complete loss of capital 2.One can have a negative i* value (feasible) but not less than –100% 3.All values above i* = 0 indicate a positive return on the investment

6. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 7-6 Sct 7.2 Calculation of i* using PW or AW Relations Set up an ROR equation using either PW or AW relations and equate to zero  0 = - PW of disbursements + PW receipts = - PW D + PW R  0 = - AW of disbursements + AW receipts = - AW D + AW R

7. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 7-7 i* by Trial and Error by Hand Using a PW Relation i* by Trial and Error by Hand Using a PW Relation 1.Draw a cash flow diagram 2.Set up the appropriate PW equivalence equation and set equal to 0 3.Select values of i and solve the PW equation 4.Repeat for values of i until “0” is bracketed, i.e., the equation is balanced 5.May have to interpolate to find the approximate i* value

8. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 7-8 ROR using Present Worth 0 1 2 3 4 5 -$1,000 +$500 +$1,500 Consider (Figure 7.2): 1000 = 500(P/F, i*,3) +1500(P/F, i*,5) Assume you invest $1,000 at t = 0; receive $500 @ t = 3 and $1500 at t = 5. What is the ROR of this project? The above PW expression must be solved by trial and error Guess at a rate and try it Adjust accordingly Bracket Interpolate i* approximately 16.9% per year on the unrecovered investment balances

9. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 7-9 Spreadsheet Methods Excel supports ROR analysis with 2 functions: =RATE(n,A,P,F) when A series is present =IRR(first_cell:last_cell, guess) when cash flows vary RATE is used when an investment (P) is made followed by “n” equal, end of period cash flows (A) This is a special case for annuities

10. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 7-10 The IRR Excel Function  When cash flows vary from period to period  Entries: Enter the cash flow values into contiguous cells (including any $0 amounts)  Enter the IRR function  =IRR(first_cell:last_cell,guess)  “guess” is an optional starting value the user feels is in the “vicinity” of the true i* value  If omitted, Excel assumes a starting value of 10%  Refer to Example 7.2

11. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 7-11 Sct 7.3 Cautions When Using ROR  When applied correctly, ROR method will always result in a good decision and should be consistent with PW, AW, or FW methods.  However, for some types of cash flows the ROR method can be computationally difficult and/or lead to erroneous decisions  Reinvestment assumption is at i* for ROR method; not the MARR. If MARR is far from i*, must use composite rate (Sct 7.5)  Some cash flows will result in multiple i* values. Raises questions as to which, if any, i* value is proper value

12. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 7-12 Special ROR Procedure for Multiple Alternatives  For analysis of two or more alternatives using ROR, resort to a different analysis approach as opposed to regular PW or AW method  Must apply an incremental analysis approach to guarantee a correct decision, i.e., same as PW or AW

13. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 7-13 Sct 7.4 Multiple Rates of Return  A class of ROR problems exist that will possess multiple i* values  Capability to predict the potential for multiple i* values  Two tests can be applied prior to the analysis

14. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 7-14 Tests for Multiple i* values 1. Cash Flow Rule of Signs  The total number of real value i*’s is always less than or equal to the number of sign changes in the original cash flow series 2. Cumulative Cash Flow Rule of Signs  Form the cumulative cash flow of the investment and count the number of sign changes in the cumulative cash flow series  Must perform both tests to be sure of one i* > 0 Predicting the likelihood of multiple i* values

15. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 7-15 Test 1 -- Cash Flow Rule of Signs  Examples of sign test for maximum i* values  Signs on cash flows by year 123456 Max i* values - +++ -- 2 + - + - ++4 - ++++ + 1

16. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 7-16 Test 2 -- Cumulative Cash Flow (CCF) Signs  A sufficient, but not necessary, condition for a single positive i* value is:  Initial cash flow has negative sign  The CCF value at year n is > 0  and there is exactly one sign change in the CCF series

17. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 7-17 Sct 7.5 Composite Rate of Return: Removing Multiple i* Values  This section introduces the concept of the composite rate of return or the external rate of return  Assume that any over-recovered funds from an investment can be invested at some interest rate termed “ c ”

18. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 7-18 Composite ROR Approach Consider the following investment -$10,000 0 1 2 3 4 5 +$8,000 +$9,000 Determine the ROR

19. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 7-19 Composite ROR Approach - Example The ROR analysis reveals: i* = 16.82%/year on the unrecovered investment balances over 5 years

20. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 7-20 Composite ROR Approach: IB’s The Investment Balances at i* are: All IB’s are < 0 for years 0 to 4 and it is 0 after 5 years Conventional (pure) investment

21. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 7-21 Composite ROR Approach i* = 16.82% -$10,000 0 1 2 3 4 5 +$8,000 +$9,000 Question: Is it reasonable to assume that the +8,000 can be invested forward at 16.82%? Reinvest forward at some interest rate

22. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 7-22 Composite ROR Approach  Remember … ROR assumes reinvestment at the calculated i* rate  What if it is not practical for the +$8,000 to be reinvested forward one year at 16.82%? Answer: Apply a reinvestment rate more in line with a 1-year investment return (apply this rate only to the positive investment balances)

23. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 7-23 Reinvestment Rates  Most firms can reinvest surplus funds at some conservative market rate of interest in effect at the time the surplus funds become available.  Often, the current market rate is less than a calculated ROR value  What then is the firm to do with the +$8,000 when it comes into the firm from the investment?

24. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 7-24 Composite ROR Approach  Consider a reinvestment rate that is closer to the current market rate for reinvestment of the $8,000 for the next time period(s)  Assume a reasonable market rate is 8% per year  Call this rate an external rate -- c

25. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 7-25 The external rate -- c  The external interest rate, c, is a rate that the firm can reinvest surplus funds for at least one time period at a time.  c is often set to equal current MARR rate

26. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 7-26 Composite ROR Approach  Procedure does the following:  Find i* given c when multiple ROR’s exist  If multiple ROR values present, the analysis determines a single, unique i* for a stated c  Technique is called net-investment procedure  Result is called the composite rate of return or i’

27. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 7-27 Composite ROR Approach  Finding i’ is a much more involved process than finding i*  Example 7.6 illustrates a manual approach  Prior to computer solution, only very small (n <= 4-5 time periods) could be manually evaluated

28. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 7-28 Sct 7.6 Rate of Return of a Bond Investment  Review Chapter 5, Sct 5.8 on bonds and their PW computation  Bond problems represent a classical ROR-type problem where a unique ROR exists  Invest P now, receive dividend payments for n periods and full investment P in the last period. What is the expected rate of return?

29. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 7-29 Typical Bond Cash Flow From the issuing company’s perspective P 0 is invested Net proceeds to company from sale of a bond A = the periodic bond interest payments from the firm to bond holders n periods F n is payment to bondholder to redeem the bond P 0 = A(P/A,i%,n) + F n (P/F i%,n)

30. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 7-30 ROR for Bond Investment: Ex 7.8  Purchase Price: P = $800/bond  Bond interest at 4% paid semiannually for $1,000 face value  Life = 20 years  Question: If you pay the $800 per bond, what is the ROR (yield) on this investment?

31. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 7-31 Ex. 7.8 -- Cash Flow Diagram …. …. …. 0 1 2 3 4 39 40 $800 F 40 = $1000 A= $1000(0.04/2) = $20.00 every 6 months for 20 years A = +$20/6 months From the bond purchaser’s perspective Pay $800 per bond to receive the $20each 6-months in interest cash flow plus $1,000 at the end of 40 time periods. What is the ROR of this cash flow?

32. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 7-32 Ex 7.8 -- Closed Form Setup Setup is:  0 = -$800 +20(P/A,i*,40 + $1000(P/F,i*,40)  Solve for i*  Manual or computer solution yields: i*=2.87%/6 months (intermediate answer)  Nominal ROR/year = (2.87%)(2) = 5.74%/yr  Effective ROR/year: (1.0287) 2 – 1 = 5.82%/yr

33. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 7-33 Summary  ROR analysis is often used but not always well understood by practitioners  ROR can be computationally difficult manually; a spreadsheet model helps reduce solution time  ROR problems may involve multiple i* rates  Requires the determination of the composite rate to solve  If an exact ROR is not necessary use the PW, AW, or FW methods

34. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 7-34 CHAPTER 7 End of Slide Set