1. Principles of Engineering Economic Analysis, 5th edition Chapter 15 Capital Budgeting

2. Principles of Engineering Economic Analysis, 5th edition The Classical Capital Budgeting Problem Independent and Indivisible Investments

3. Principles of Engineering Economic Analysis, 5th edition Systematic Economic Analysis Technique 1. Identify the investment alternatives 2. Define the planning horizon 3. Specify the discount rate 4. Estimate the cash flows 5. Compare the alternatives 6. Perform supplementary analyses 7. Select the preferred investment

4. Principles of Engineering Economic Analysis, 5th edition When deciding which investment opportunities to fund wholly (versus not at all), the optimum portfolio can be obtained by solving a binary linear programming problem with an objective of maximizing the present worth of the portfolio

5. Principles of Engineering Economic Analysis, 5th edition Mathematical Programming Formulation of the Capital Budgeting Problem Maximize PW 1 x 1 + PW 2 x 2 +... + PW n-1 x n-1 + PW n x n (15.1) subject to c 1 x 1 + c 2 x 2 +... + c n-1 x n- 1 + c n x n < C(15.2) x j = (0,1)j = 1,..., n(15.3) Establish an investment portfolio that maximizes the present worth of the portfolio without exceeding a constraint on the amount of investment capital available. The investment opportunities are independent and non- divisible, i.e., either the investment is pursued in total or not at all – no partial investments.

6. Principles of Engineering Economic Analysis, 5th edition Example 15.1 Recall the IRR example from Chapter 8 which includes 5 mutually exclusive investment alternatives, each of which returns the initial investment at any time the investor desires. Suppose each investment lasts for exactly 10 years and the investor can choose as many of the investment options as she or he wants, so long as no more than the total invested does not exceed $100,000. Which ones should be chosen? (Cannot choose multiples of the same investment.)

7. Principles of Engineering Economic Analysis, 5th edition Data for Example 15.1 Capital available: $100,000 MARR: 18%

8. Principles of Engineering Economic Analysis, 5th edition Mathematical Programming Formulation for Example 15.1 Maximize $4,718.79x 1 + $2,247.00x 2 + $9,212.88x 3 + $10,111.69x 4 + $7,415.24x 5 subject to $15,000x 1 + $25,000x 2 + $40,000x 3 + $50,000x 4 + $70,000x 5 < $100,000 x j = (0,1)j = 1,..., 5

9. Principles of Engineering Economic Analysis, 5th edition

12. Solving a BLP Using Enumeration Recall, in Chapter 1 (Example 1.5), we enumerated all possible investment alternatives when there were 3 investments available. Specifically, with m investment proposals there are 2 m possible mutually exclusive investment alternatives, including the “Do Nothing” alternative. In Example 1.5, m = 3; therefore, there were 8 possible alternatives.

13. Principles of Engineering Economic Analysis, 5th edition Solving Example 15.1 with Enumeration With m = 5, there are 2 5 = 32 possible investment alternatives. Shown below is a binary table, similar to Table 1.1, giving all possible investment alternatives. Investment alternatives that violate the capital constraint of $100,000 are eliminated, as shown. (Half of the possible investment alternatives are eliminated.)

14. Principles of Engineering Economic Analysis, 5th edition Solving Example 15.1 with Enumeration

15. Principles of Engineering Economic Analysis, 5th edition Solving Example 15.1 with Enumeration

16. Principles of Engineering Economic Analysis, 5th edition Solving Example 15.1 with Enumeration Of the 16 feasible investment alternatives, combination 7 has the greatest present worth ($19,324.57). Investments 3 and 4 are to be made. The same solution was obtained using Excel® SOLVER tool to solve the BLP problem.

17. Principles of Engineering Economic Analysis, 5th edition Adding Constraints  Mutually Exclusive  Contingent

18. Principles of Engineering Economic Analysis, 5th edition Example 15.2 Suppose the investment portfolio to be optimized consists of a mixture of independent and dependent investments. In particular, in the previous example, suppose investment 3 is contingent on investment 2 being selected (in other words, you cannot choose 3 without choosing 2). To solve the linear programming problem, a further constraint is required, x 3 < x 2, or D7 < C7.

19. Principles of Engineering Economic Analysis, 5th edition

22. Solving Example 15.2 with Enumeration Given the reduced binary table from Example 5.1, we now eliminate investment alternatives that violate the contingency constraint. Blue lines are used to show the alternatives that are eliminated.

23. Principles of Engineering Economic Analysis, 5th edition Solving Example 15.2 with Enumeration

24. Principles of Engineering Economic Analysis, 5th edition Solving Example 15.2 with Enumeration

25. Principles of Engineering Economic Analysis, 5th edition Solving Example 15.2 with Enumeration Of the 14 feasible investment alternatives, combination 27 has the greatest present worth ($17,077.53). Investments 1, 2, and 4 are to be made. The same solution was obtained using Excel® SOLVER tool to solve the BLP problem.

26. Principles of Engineering Economic Analysis, 5th edition Example 15.3 Extending the previous example, suppose investments 2 and 4 are mutually exclusive. To add a mutually exclusive constraint, it is necessary to ensure that either the product of x 2 and x 4 equals zero or their sum is less than or equal to 1.0 As shown on the following slide, the sum of x 2 and x 4 is entered in cell E11 and a constraint is added that E11 < 1.

27. Principles of Engineering Economic Analysis, 5th edition

30. Solving Example 15.3 with Enumeration Given the reduced binary table from Example 5.2 we now eliminate investment alternatives that violate the mutually exclusive constraint. Black lines are used to show the alternatives that are eliminated.

31. Principles of Engineering Economic Analysis, 5th edition Solving Example 15.3 with Enumeration

32. Principles of Engineering Economic Analysis, 5th edition Solving Example 15.3 with Enumeration

33. Principles of Engineering Economic Analysis, 5th edition Solving Example 15.3 with Enumeration Of the 12 feasible investment alternatives, combination 29 has the greatest present worth ($16,178.71). Investments 1, 2, and 3 are to be made. The same solution was obtained using Excel® SOLVER tool to solve the BLP problem.

34. Principles of Engineering Economic Analysis, 5th edition Example 15.4 Now, consider 6 investment opportunities, with MARR = 10%, C = $100,000, and the data shown below. Investments 1 and 2 are mutually exclusive and investment 6 is contingent on either or both of investments 3 and 4 being funded. To add an “either/or” contingent constraint, we set D14 equal to the sum of D9 and E9 and add the constraint: G9 <= D14, which is the same as x 6 < x 3 + x 4.

35. Principles of Engineering Economic Analysis, 5th edition

38. Solving Example 15.4 with Enumeration With m = 6, there are 2 6 = 64 possible investment alternatives. Shown below is a binary table listing all possible investment alternatives. Investment alternatives that violate the capital constraint of $100,000 are eliminated using red lines. Of the remaining investments, those that violate the mutually exclusive constraint are eliminated using blue lines. Of those that remain, those that violate the either/or constraint are eliminated using black lines.

39. Principles of Engineering Economic Analysis, 5th edition Solving Example 15.4 with Enumeration

40. Principles of Engineering Economic Analysis, 5th edition Solving Example 15.4 with Enumeration

41. Principles of Engineering Economic Analysis, 5th edition Solving Example 15.4 with Enumeration

42. Principles of Engineering Economic Analysis, 5th edition Solving Example 15.4 with Enumeration

43. Principles of Engineering Economic Analysis, 5th edition Solving Example 15.4 with Enumeration Of the 34 feasible investment alternatives, combination 46 has the greatest present worth ($12,265.06). Investments 1, 3, 4, and 6 are to be made. The same solution was obtained using Excel® SOLVER tool to solve the BLP problem.

44. Principles of Engineering Economic Analysis, 5th edition Example 15.5 In the previous example, suppose at most 3 and at least 2 investments must be made. “at most” implies or H9 <=3 “at least” implies H9 >=2 As shown in Figure 15.6, the optimum investment portfolio is {2,4,6} with PW = $11,846.63 and IRR = 15.70%

45. Principles of Engineering Economic Analysis, 5th edition

48. Solving Example 15.5 with Enumeration Given the reduced binary table from Example 5.4 we now eliminate investment alternatives that violate the constraint that at most 3 and at least 2 investments must be made. Green lines are used to show the alternatives that are eliminated.

49. Principles of Engineering Economic Analysis, 5th edition Solving Example 15.4 with Enumeration

50. Principles of Engineering Economic Analysis, 5th edition Solving Example 15.4 with Enumeration

51. Principles of Engineering Economic Analysis, 5th edition Solving Example 15.4 with Enumeration

52. Principles of Engineering Economic Analysis, 5th edition Solving Example 15.4 with Enumeration

53. Principles of Engineering Economic Analysis, 5th edition Solving Example 15.4 with Enumeration Of the 25 feasible investment alternatives, combination 22 has the greatest present worth ($11,846.63). Investments 2, 4, and 6 are to be made. The same solution was obtained using Excel® SOLVER tool to solve the BLP problem. (For m > 6, enumeration is not reasonable. Use SOLVER.)

54. Principles of Engineering Economic Analysis, 5th edition Sensitivity Analysis  Capital Available  MARR

55. Principles of Engineering Economic Analysis, 5th edition Example 15.6 Recall Example 15.1. What effect does the capital limit have on the optimum investment portfolio? For example, what if the capital limit is raised to $105,000. What would be the impact on PW and IRR?

56. Principles of Engineering Economic Analysis, 5th edition

58. IRR = ($3,750 + $9,250 + $11,250)/$105,000 = 0.2310 or 23.10%

59. Principles of Engineering Economic Analysis, 5th edition Sensitivity Analysis of the Optimum Investment Portfolio

60. Principles of Engineering Economic Analysis, 5th edition Example 15.7 Consider the cash flow profiles given below, with a MARR of 10% and capital limit of $100,000. How sensitive is the optimum portfolio to MARR values in the interval [0%,26%]?

61. Principles of Engineering Economic Analysis, 5th edition

65. The Capital Budgeting Problem Independent and Divisible Investments

66. Principles of Engineering Economic Analysis, 5th edition Mathematical Programming Formulation of the Capital Budgeting Problem with Divisible Investments Maximize PW 1 p 1 + PW 2 p 2 +... + PW n-1 p n-1 + PW n p n (15.1) subject to c 1 p 1 + c 2 p 2 +... + c n-1 p n- 1 + c n p n < C(15.2) 0 < p j < 1j = 1,..., n(15.3) Establish an investment portfolio that maximizes the present worth of the portfolio without exceeding a constraint on the amount of investment capital available. The investment opportunities are independent and divisible, i.e., a percentage of an investment can be pursued.

67. Principles of Engineering Economic Analysis, 5th edition when partial funding of investments is allowed, to obtain the optimum investment portfolio, (a) rank the investment opportunities on their internal rates of return, and (b) form the portfolio by “filling the investment bucket,” starting with the opportunity having the greatest internal rate of return and proceeding sequentially until the “bucket” is full.

68. Principles of Engineering Economic Analysis, 5th edition Example 15.8 Recall Example 15.1. Now, suppose the investments are divisible, i.e., you can choose to make fractional investments. When investments are independent and divisible, the optimum investment portfolio is obtained by rank ordering the investments based on IRR and investing, beginning with the investment having the greatest IRR, until “your money runs out,” as shown on the following chart.

69. Principles of Engineering Economic Analysis, 5th edition

70. Optimum Divisible Portfolio

71. Principles of Engineering Economic Analysis, 5th edition

74. Two Observations With divisible investments, the investments are rank ordered on IRR, which is contrary to everything we learned regarding mutually exclusive investment alternatives and non- divisible, independent investments. Notice, rank ordering the investments on PW, which we do with mutually exclusive investment alternatives and non-divisible, independent investments, will not yield the optimum portfolio—as shown on the following charts.

75. Principles of Engineering Economic Analysis, 5th edition

76. Suboptimum Divisible Portfolio

77. Principles of Engineering Economic Analysis, 5th edition A Third Observation The IRR for each investment alternative is independent of the MARR. Hence, the only effect a change in the MARR has on the optimum investment portfolio is the elimination of alternatives with an IRR less than the MARR. If all investment alternatives have IRR values greater than the MARR, then the optimum investment portfolio will be unchanged. However, the PW of the optimum investment portfolio will change with changes in the MARR.

78. Principles of Engineering Economic Analysis, 5th edition Example 15.11 In Example 15.10, suppose investments 1 and 2 are mutually exclusive and investments 3 and 4 are mutually exclusive. Further, suppose investment 5 is contingent on investment 3 being funded. Show how SOLVER can be used to determine the optimum investment portfolio.

79. Principles of Engineering Economic Analysis, 5th edition

82. Question With indivisible investments, how would you incorporate into SOLVER the following constraint: investment 4 is contingent on either investment 3 or investment 5? Add a constraint to SOLVER for x 3 to be less than or equal to the value of a cell in which you have entered x 3 + x 5 For our example: E5 <= F8 when D5+F5 has been entered in cell F8

83. Principles of Engineering Economic Analysis, 5th edition Question With indivisible investments, how would you incorporate into SOLVER the following constraint: investment 4 is contingent on either investment 3 or investment 5? x 4 < x 3 + x 5 Add a constraint to SOLVER for x 3 to be less than or equal to the value of a cell in which you have entered x 3 + x 5 For our example: E5 <= F8 when D5+F5 has been entered in cell F8

84. Principles of Engineering Economic Analysis, 5th edition Question With indivisible investments, how would you incorporate into SOLVER the following constraint: investment 4 is contingent on either investment 3 or investment 5? x 4 < x 3 + x 5 Add a constraint to SOLVER for x 4 to be less than or equal to the value of a cell in which you have entered x 3 + x 5 For our example: E5 <= F8 when D5+F5 has been entered in cell F8

85. Principles of Engineering Economic Analysis, 5th edition Another Question How would you incorporate into SOLVER the following constraint: at least two of the indivisible investments 1, 2, or 3 must be included in the investment portfolio? x 1 + x 2 + x 3 > 2 After entering B5+C5+D5 in F8, add a constraint to SOLVER: F8>=2

86. Principles of Engineering Economic Analysis, 5th edition Another Question How would you incorporate into SOLVER the following constraint: at least two of the indivisible investments 1, 2, or 3 must be included in the investment portfolio? x 1 + x 2 + x 3 > 2 After entering B5+C5+D5 in F8, add a constraint to SOLVER: F8>=2

87. Principles of Engineering Economic Analysis, 5th edition Another Question How would you incorporate into SOLVER the following constraint: at least two of the indivisible investments 1, 2, or 3 must be included in the investment portfolio? x 1 + x 2 + x 3 > 2 After entering B5+C5+D5 in F8, add a constraint to SOLVER: F8>=2

88. Principles of Engineering Economic Analysis, 5th edition Pit Stop #15—The Finish Line Is In Sight! 1.True or False: When several independent, indivisible investments are available, form the investment portfolio so that the present worth of the portfolio is maximized. True 2.True or False: If independent, indivisible investments 3 and 4 are mutually exclusive, then x 3 + x 4 < 1 is added as a constraint to the BLP formulation. True 3.True or False: If indivisible investment 2 is contingent on indivisible investment 1 being funded, then x 2 - x 1 < 0 is added as a constraint to the BLP formulation. True 4.True or False: When multiple independent, divisible investments are available, form the investment portfolio so that the internal rate of return is maximized. False 5.True or False: When multiple independent divisible investments are available, choose investments to add to the portfolio on the basis of their IRR and move from the largest to the smallest IRR until the capacity limit is reached. True