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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

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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

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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*

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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 ”

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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

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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

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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

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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

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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)

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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?

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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

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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

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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’

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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

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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?

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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)

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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?

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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?

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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

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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

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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

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