1. Chapter 7 Rate of Return Analysis

2. 2  Rate of Return  Methods for Finding ROR  Internal Rate of Return (IRR) Criterion  Incremental Analysis  Mutually Exclusive Alternatives

3. 3 Rate of Return Definition: A relative percentage method which measures the yield as a percentage of investment over the life of a project Other terms may be used including yield, internal rate of return and marginal efficiency of capital.

4. 4 In 1970, when Wal-Mart Stores, Inc. went public, an investment of 100 shares cost $1,650. That investment would have been worth $12,283,904 on January 31, 2002. What is the rate of return on that investment? Example: Meaning of Rate of Return

5. 5 Solution: 0 32 $12,283,904 $1,650 Given: P = $1,650 F = $12,283,904 N = 32 Find i: $12,283,904 = $1,650 (1 + i ) 32 i = 32.13% Rate of Return

6. 6 Suppose that you invested that amount ($1,650) in a savings account at 6% per year. Then, you could have only $10,648 on January, 2002. What is the meaning of this 6% interest here? This is your opportunity cost if putting money in savings account was the best you can do at that time!

7. 7 So, in 1970, as long as you earn more than 6% interest in another investment, you will take that investment. Therefore, that 6% is viewed as a minimum attractive rate of return (or required rate of return). So, you can apply the following decision rule, to see if the proposed investment is a good one. ROR (32.13%) > MARR(6%)

8. 8 Why ROR measure is so popular? This project will bring in a 15% rate of return on investment. This project will result in a net surplus of $10,000 in NPW. Which statement is easier to understand?

9. 9 Return on Investment Definition 1: Rate of return (ROR) is defined as the interest rate earned on the unpaid balance of an installment loan. Example: A bank lends $10,000 and receives annual payment of $4,021 over 3 years. $10,000 = $4,021 (P/A,i,3)  i=10% The bank is said to earn a return of 10% on its loan of $10,000.

10. 10 Loan Balance Calculation: A = $10,000 (A/P, 10%, 3) = $4,021 Unpaid Return onUnpaid balance unpaidbalance at beg. balancePaymentat the end Yearof year(10%)receivedof year 01230123 -$10,000 -$6,979 -$3,656 -$1,000 -$698 -$366 +$4,021 -$10,000 -$6,979 -$3,656 0 A return of 10% on the amount still outstanding at the beginning of each year

11. 11 Rate of Return: Definition 2: Rate of return (ROR) is the break-even interest rate, i *, which equates the present worth of a project’s cash outflows to the present worth of its cash inflows. Mathematical Relation: PW(i*) (1+i*) N = FW(i*) = 0 AE (i*) = 0 i* of a project is the rate of interest that equates the present worth, future worth and annual equivalent worth of the entire series of cash flows to zero.

12. 12 Reconsider the loan transaction 0 1 23 $10,000 $4,021 Rate of Return = 10%

13. 13 Return on Invested Capital Definition 3: Return on invested capital is defined as the interest rate earned on the unrecovered project balance of an investment project. It is commonly known as internal rate of return (IRR). Example: A company invests $10,000 in a computer and results in equivalent annual labor savings of $4,021 over 3 years. The company is said to earn a return of 10% on its investment of $10,000.

14. 14 Project Balance We see that 10% is earned (or charged) on $10,000 during year one, 10% is earned on $6,979 during year two, and 10% is earned on $3,656 during year three. Or, the firm earns a 10% rate of return on funds that remain internally invested in the project, hence the IRR.

15. 15 Methods for Finding Rate of Return  Direct Solution Method  Trial-and-Error Method  Using Excel’s Financial Command  Cash Flow Analyzer – Online Financial Calculator from the book website at http://www.prenhall.com/park

16. 16 Example 7.2 Finding i* by Direct Solution: Two Flows and Two Periods nProject 1Project 2 0-$3,000-$2,000 10$1,300 20$1,500 30 4$4,500

17. 17  Project 1 Solution:  Project 2

18. 18 Finding i* by Trial and Error nCash Flow 0-$75,000 124,400 227,340 355,760 0 1 23 $55,760 $27,340 $24,400 $75,000

19. 19 Solution:  Step 1: Guess an interest rate, say, i = 15%  Step 2: Compute PW(i) at the guessed i value. (We are aiming zero) PW (15%) = $3,553  Step 3: If PW(i) > 0, then increase i. If PW(i) < 0, then decrease i. PW(18%) = -$749  Step 4: If you bracket the solution, you use a linear interpolation to approximate the solution. 3,553 0 -749 15% i 18%

20. 20 Period (n) Cash Flow 0-$1,000 1-500 2800 31,500 42,000 Finding Rate of Return on Excel =IRR(B3:B7,10%) Cell range Initial guess

21. 21 Using Cash Flow Analyzer Period (n) Cash Flow 0-$1,000 1-500 2800 31,500 42,000 Output

22. 22 Relationship to the PW Analysis  PW Analysis: If PW(i) > 0, accept, where i is MARR.  ROR Analysis: If IRR > MARR, accept IRR (i*) PW(i) 0 MARR(i) Accept Reject

23. 23 Basic Decision Rule: If ROR > MARR, Accept This rule does not work for a situation where an investment has multiple rates of return

24. 24 Simple versus Nonsimple Investments Simple investment  An investment in which the initial cash flows are negative and only one sign change occurs in the net cash flow series – a unique rate of return Nonsimple investment  An investment in which the initial cash flows are negative and more than one sign change occurs in the net cash flow series – a possibility of having more than one rate of return Note: We ignore a zero cash flow in determining the change.

25. 25 Investment Classification Simple (conventional) Investment Def: Initial cash flows are negative, and only one sign change occurs in the net cash flow series. Example: -$100, $250, $300 (-, +, +) ROR: A unique ROR Nonsimple (nonconventional) Investment Def: Initial cash flows are negative, but more than one sign changes in the remaining cash flow series. Example: -$100, $300, -$120 (-, +, -) ROR: A possibility of multiple RORs

26. 26 Example 7.1 Investment Classification

27. 27 Project Selection Rules under the IRR Criterion

28. 28 Decision Rule for Simple Investment  A unique ROR  IRR = i*  Decision Rule:  If IRR > MARR, Accept  If IRR = MARR, Indifferent  If IRR < MARR, Reject 0 PW(i) i*i* i* = IRR IRR > MARR IRR < MARR

29. 29 Decision Rules for Nonsimple Investment  A possibility of multiple RORs.  If the PW (i) plot looks like this, then, IRR = i*. If IRR > MARR, Accept  If the PW(i) plot looks like this, then, IRR  i*.  Find the true IRR by using the procedures in Appendix 7A or,  Abandon the IRR method and use the PW method. PW (i) i*i* i i i*i* i*i*

30. 30 Example 7.6 Turbo Image Processing’s Investment Decision Problem Find the rate(s) of return: $1,000 $2,300 $1,320 MARR = 15%

31. 31 Calculating i*s

32. 32 NPW Plot for a Nonsimple Investment with Multiple Rates of Return

33. 33 n = 0n = 1n = 2 Beg. Balance Interest Payment-$1,000 -$200 +$2,300 +$1,100 +$220 -$1,320 Ending Balance-$1,000+$1,100$0 i* =20% Cash borrowed (released) from the project is assumed to earn the same interest rate through external investment as money that remains internally invested. Project Balance Calculation

34. 34  If the firm’s MARR is exactly 20%, the answer is “yes”, because it represents the rate at which the firm can always invest the money in its investment pool. Then, the 20% is also true IRR for the project.  Suppose the firm’s MARR is 15% instead of 20%. The assumption used in calculating i* is no longer valid. In order to calculate i*, we assumed that all cash released from the project can be invested at the i* instead of MARR.  Conclusion: Neither 10% nor 20% is a true IRR, since the firm’s MARR is 15%. Conceptual Issue: Can the firm be able to invest the money released from the project at 20% externally in Period 1?

35. 35 At MARR = 15% PW(15%) = -$1,000 + $2,300 (P/F, 15%, 1) - $1,320 (P/F, 15%, 2 ) = $1.89 > 0 Accept the investment  Alternative 1: If you encounter multiple rates of return, abandon the IRR analysis and use the PW criterion.  Alternative 2: If you want to find the true rate of return (or return on invested capital) to the project, follow the procedure outlined in Chapter 7A. How to Proceed?

36. 36 Example Investing your funds in the purchase and operation of a parking lot is under consideration. It requires $50,000 for land and $5,000 for building. Operating and other expenses are estimated as $12,500/year and income as $20,000/year. What is the prospective rate of return on this investment, if you plan to operate it indefinitely?

37. 37 Solution Initial Investment: $50,000 + $5,000 = $55,000 Net Income: $20,000 - $12,500 = $7,500 You expect the business to go on forever: P=A/i  rate of return = $7,500 / $55,000 = 13.6% per year

38. 38 Example Assume that parking lot was operated for 4 years with the following incomes and expenditures and the land was sold for $66,000. YearIncomeExpensesNet income Salvage 119,000 12,000 7,000 220,000 12,500 7,500 321,000 11,500 9,500 420,000 12,000 8,000 66,000

39. 39 Solution

40. 40 Predicting Multiple i*s Net Cash Flow Rule of Signs The number of real i*s for a project is never greater than the number of sign changes in the sequence of the cash flows. A zero cash flow is ignored. Accumulated Cash Flow Sign Test If the sequence of accumulated cash flow series starts negatively and changes sign only once, then a unique positive i* exists

41. 41 Predicting Multiple RORs - 100% < i * < infinity Net Cash Flow Rule of Signs No. of real RORs (i*s) < No. of sign changes in the project cash flows

42. 42 Example n Net Cash flowSign Change 01234560123456 -$100 -$20 $50 0 $60 -$30 $100 111111 No. of real i*s ≤ 3 This implies that the project could have (0, 1, 2, or 3) i*s but NOT more than 3.

43. 43 Accumulated Cash Flow Sign Test Find the accounting sum of net cash flows at the end of each period over the life of the project Period Cash Flow Sum n A n S n If the series S starts negatively and changes sign ONLY ONCE, there exists a unique positive i*.

44. 44 Example nA n S n Sign change 01234560123456 -$100 -$20 $50 0 $60 -$30 $100 -$100 -$120 -$70 -$10 -$40 $60 1 No of sign change = 3 No of sign change in S= 1, indicating a unique i*. i* = 10.46%

45. 45 Example A.2 $2,145 $3,900 $5,030 $1,000 0 1 2 3 Is this a simple investment? How many RORs (i*s) can you expect from examining the cash flows? Can you tell whether or not this investment has a unique rate of return?

46. 46 Example nA n S n Sign change 01230123 -$1,000 $3,900 -$5,030 $2,145 -$1,000 $2,900 -$2,130 $15 111111 No of sign change in A = 3  Not a simple investment No of sign change in S = 3  Not a unique i*

47. 47 Resolution of Multiple RORs (Net Investment Test) (Chapter 7A) Pure vs Mixed Investment Pure investment: Project balances computed at the project’s i*s are all less than or equal to zero. Mixed investment: Any investment that is not pure. All simple investments are pure. Nonsimple investments can be pure or mixed.

48. 48 Example 7A.1 nABCD 0-$1,000 1 $1,600$500$3,900 2 $2,000-$300-$500-$5,030 3 $1,500-$200-$2,000$2,145 i*i*33.64%21.95%29.95%(10%,30%,50%)

49. 49 A: Simple investment  pure investment nAnAn PB(i*) n 0-$1,000 1-$1,000 (1+0.3364)-$1,000-$2,336.40 2 -$2,336.40 (1+0.3364) $2,000-$1,122.36 3-$1,122.36 (1+0.3364) $1,5000

50. 50 B: mixed investment nAnAn PB(i*) n 0-$1,000 1-$1,000 (1+0.2195) $1,600$380.50 2 $380.50 (1+0.2195)-$300$164.02 3$164.02 (1+0.2195)-$2000 Even for a nonsimple investment with one positive rate of return, the project may be a mixed investment. The unique i* may not be the true rate of return.

51. 51 C: pure investment nAnAn PB(i*) n 0-$1,000 1-$1,000 (1+0.2995)$500-$799.50 2 -$799.50 (1+0.2995)-$500-$1,538.95 3-$1,538.95 (1+0.2995)-$2,0000 Nonsimple but pure investment, apply simple investment procedure and accept if 29.95% > MARR

52. 52 D: mixed investment nAnAn PB(i*) n 0-$1,000 1-$1,000 (1+0.5)$3,900$2400 2 $2400 (1+0.5)-$5,030-$1,430 3-$1,430 (1+0.5)$2,1450 We can use any of the three interest rates of return for the net- investment test. If one fails, all of them will fail, if one passes, all of them will pass.

53. 53 Net investment test (Example 7.6) A positive project balance indicates that at some time during the project life, the firm acts as a borrower. Then, this is a mixed investment.

54. 54 Decision Rule for Pure Investment Decision Criterion for a Single Project:  If IRR > MARR, accept the project.  If IRR = MARR, remain indifferent.  If IRR < MARR, reject the project.

55. 55 Example : Investment Decision for a Pure Investment (MARR=18%)

56. 56 Solution:

57. 57 External interest rate When PB(i*)>0, firm acts as a borrower rather than investor in the project. The borrowed money is assumed to be invested at an external interest rate (MARR). If the investment is mixed,  apply the following procedure to find the true interest rate (return on invested capital-RIC),  select the investment if RIC > MARR.

58. 58 Return on Invested Capital (True IRR) Step 1: Identify MARR (or external interest rate) Step 2: Calculate project balance, PB(i,MARR) n PB(i, MARR) 0 =A 0 (So, if the net balance is positive, the firm is a borrower, and the interest rate is MARR. If it is negative, the firm is an investor, and the interest rate is i.) Step 3: Determine i by solving PB(i, MARR) N =0......

59. 59 Computational Logic for RIC

60. 60 Example 7A.2 Multiple Rates of Return Problem MARR=15% i*=10% and 20% Mixed investment $1,000 $2,300 $1,320

61. 61 Example 7A.2 PB (i,15%) 0 =  $1,000 < 0 PB (i,15%) 1 =  $1,000(1+i)+$2,300 = $1,300  $1,000i =$1,000(1.3-i) Case 1: i 0 PB (i,15%) 2 = $1,000(1.3-i)(1+0.15)  $1,320 = $175  $1,150i=0 IRR = 15.22% Case 2: i>1.3  PB (i,15%) 1 <0 PB (i,15%) 2 = $1,000(1.3-i)(1+i)  $1,320 = -$20+$300i  $1,000i 2 =0 IRR = 0.1 or 0.2<1.3, violates the initial assumption IRR = 15.22% > MARR = 15%  Accept the project.

62. 62 Example RIC for a Mixed Investment by Trial and Error nAnAn 01230123 -$1,000 3,900 -5,030 2,145 External interest rate = MARR = 6% Find the RIC for the project.

63. 63 Solution: Guess i = RIC at 8%: Guess i = RIC at 6.13%: The net investment is negative at the end of the project, indicating that our trial i =8% is in error. We lower the guess value and try again. RIC =6.13% > MARR, Select the investment

64. 64 Summary of IRR Criterion

65. Incremental Analysis

66. 66 Comparing Mutually Exclusive Alternatives Based on IRR Issue: Can we rank the mutually exclusive projects by the magnitude of its IRR? Suppose you have exactly 5000 to invest. nA1A2 0 1 IRR -$1,000-$5,000 $2,000$7,000 100% > 40% $818<$1,364 PW (10%)

67. 67 Comparing Mutually Exclusive Alternatives Based on IRR Issue: The IRR measure ignores the scale of the project. Prefer the second project (higher PW). nA1A2 0 1 IRR -$1,000-$5,000 $2,000$7,000 100% > 40% $818<$1,364 PW (10%)

68. 68 Who Got More Pay Raise? BillHillary 10%5%

69. 69 Can’t Compare without Knowing Their Base Salaries BillHillary Base Salary$50,000$200,000 Pay Raise (%)10%5% Pay Raise ($)$5,000$10,000 For the same reason, we can’t compare mutually exclusive projects based on the magnitude of its IRR. We need to know the size of investment and its timing of when to occur.

70. 70 Incremental Analysis PW, FW and AE are absolute measures of investment worth. IRR is a relative (percentage) measure and ignores the scale of the investment. We have to apply incremental analysis when comparing mutually exclusive alternatives.

71. 71 Incremental Investment Assuming a MARR of 10%, you can always earn that rate from other investment source, i.e., $4,400 at the end of one year for $4,000 investment. By investing the additional (incremental) $4,000 in A2, you would make additional $5,000, which is equivalent to earning at the rate of 25%. Therefore, the incremental investment in A2 is justified, as the incremental IRR > MARR. nProject A1Project A2 Incremental Investment (A2 – A1) 0101 -$1,000 $2,000 -$5,000 $7,000 -$4,000 $5,000 ROR PW(10%) 100% $818 40% $1,364 25% $546

72. 72 Incremental Analysis (Procedure) Step 1:Compute the cash flow for the difference between the projects (A,B) by subtracting the cash flow of the lower investment cost project (A) from that of the higher investment cost project (B). Step 2:Compute the IRR on this incremental investment (IRR B-A ). Step 3:Accept the investment B if and only if IRR B-A > MARR NOTE: Make sure that both IRR A and IRR B are greater than MARR. If not, eliminate that alternative from comparison.

73. 73 Example 7.7 IRR on Incremental Investment: Two Alternatives

74. 74 Example 7.8 IRR on Incremental Investment: Three Alternatives

75. 75 Example 7.10 Comparing Unequal- Service-Life Problems

76. 76 Solution:

77. 77 Practice Problem You are considering four types of engineering designs. The project lasts 10 years with the following estimated cash flows. The interest rate (MARR) is 10%. Which of the four is more attractive? (Internal rates of return for each alternative is greater than 10%.) ABCD Initial cost $150$220$300$340 Revenues/Year $115$125$160$185 Expenses/Year $70$65$60$80 Net income / year $45$60$100$105

78. 78 Solution YearB-AC-BD-C 0-$70-$80-$40 1-10$15$40$5 IRR16.95%49.08%4.28% Select BSelect C

79. 79 Example 7.9: Incremental Analysis for Cost-Only Projects (MARR = 15%, Lifetimes= 6 years) ItemsCMS OptionFMS Option Annual O&M costs: Annual labor cost$1,169,600$707,200 Annual material cost832,320598,400 Annual overhead cost3,150,0001,950,000 Annual tooling cost470,000300,000 Annual inventory cost141,00031,500 Annual income taxes1,650,0001,917,000 Total annual costs$7,412,920$5,504,100 Initial Investment$4,500,000$12,500,000 Net salvage value (end of 6th year) $500,000$1,000,000

80. 80 Example 7.9 Incremental Cash Flow (FMS – CMS) nCMS OptionFMS Option Incremental (FMS-CMS) 0-$4,500,000-$12,500,000-$8,000,000 1-7,412,920-5,504,1001,908,820 2-7,412,920-5,504,1001,908,820 3-7,412,920-5,504,1001,908,820 4-7,412,920-5,504,1001,908,820 5-7,412,920-5,504,1001,908,820 6-7,412,920-5,504,100 $2,408,820 Salvage+ $500,000+ $1,000,000

81. 81 Solution:

82. 82 Example: Unequal Service Lives AB First cost-$8,000-$13,000 Annual cost-$3,500-$1,600 Salvage value 0$2,000 Life105 MARR = 15%

83. 83 Solution PW(15%)=-5,000+1,900(P/A,15%,10)- 11,000(P/F,15%,5)+2000(P/F,10%,15)=-439.08 PW(10%)=-5,000+1,900(P/A,10%,10)- 11,000(P/F,10%,5)+2000(P/F,10%,10)=615.84 Interpolation: i*=12.65%  Is it unique? Apply net investment test YearB-A 0-5,000 1-101,900 52,000-13,000=-11,000 102,000 Decrease your guess

84. 84 Solution YearB-A 0-5,000 1-101,900 52,000-13,000=- 11,000 102,000 Since project balance at the end of the period 4 is positive, project is a mixed investment therefore perform RIC procedure. RIC =%12.72, Reject B Is there a simple way to decide ?

85. 85 Solution PW(15%)=-5,000+1,900(P/A,15%,10)- 11,000(P/F,15%,5)+2000(P/F,10%,15)=-439.08 YearB-A 0-5,000 1-101,900 52,000-13,000=-11,000 102,000 Is there a simple way to decide ? Whatever the RIC is, it must make PW of the incremental project equal to zero. Since PW(15%)<0, we exactly know that RIC is less than 15% -because the procedure requires to decrease the initial value for a negative PW-. Hence we reject to invest in the expensive project which requires more initial cost.

86. 86 Ultimate Decision Rule: If IRR > MARR, Accept This rule works for any investment situation In many situations, IRR = ROR but this relationship does not hold for an investment with multiple RORs.

87. 87 Summary Rate of return (ROR) is the interest rate earned on unrecovered project balances such that an investment’s cash receipts make the terminal project balance equal to zero. Rate of return is an intuitively familiar and understandable measure of project profitability that many managers prefer to NPW or other equivalence measures. Mathematically we can determine the rate of return for a given project cash flow series by locating an interest rate that equates the net present worth of its cash flows to zero. This break-even interest rate is denoted by the symbol i*.

88. 88 Internal rate of return (IRR) is another term for ROR that stresses the fact that we are concerned with the interest earned on the portion of the project that is internally invested, not those portions that are released by (borrowed from) the project. To apply rate of return analysis correctly, we need to classify an investment into either a simple or a nonsimple investment. A simple investment is defined as one in which the initial cash flows are negative and only one sign change occurs in the net cash flow, whereas a nonsimple investment is one for which more than one sign change occurs in the net cash flow series. Multiple i*s occur only in nonsimple investments. However, not all nonsimple investments will have multiple i*s either.

89. 89 For a simple investment, the solving rate of return (i*) is the rate of return internal to the project; so the decision rule is: If IRR > MARR, accept the project. If IRR = MARR, remain indifferent. If IRR < MARR, reject the project. IRR analysis yields results consistent with NPW and other equivalence methods. For a nonsimple investment, because of the possibility of having multiple rates of return, we need to calculate the true IRR, or known as “return on invested capital.” However, if your objective is to make an accept or reject decision, it is recommended the IRR analysis be abandoned and either the NPW or AE analysis be used to make an accept/reject decision. When properly selecting among alternative projects by IRR analysis, incremental investment must be used.