principles of engineering economic analysis, 5th edition chapter 15 capital budgeting

Principles of Engineering Economic Analysis, 5th edition

Chapter 15Chapter 15

Capital Budgeting


The Classical Capital Budgeting Problem

Independent and Indivisible Investments


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


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


Mathematical Programming Formulation of the Capital Budgeting Problem

Maximize PW1x1 + PW2x2 + ... + PWn-1 xn-1 + PWn xn (15.1)

subject to c1x1 + c2x2 + ... + cn-1 xn-1 + cn xn < C (15.2)

xj = (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.


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.)


Data for Example 15.1

Investment Opportunity 1 2 3 4 5

Initial Investment $15,000.00 $25,000.00 $40,000.00 $50,000.00 $70,000.00Annual Return $3,750.00 $5,000.00 $9,250.00 $11,250.00 $14,250.00Salvage Value $15,000.00 $25,000.00 $40,000.00 $50,000.00 $70,000.00Present Worth $4,718.79 $2,247.04 $9,212.88 $10,111.69 $7,415.24Internal Rate of Return 25.00% 20.00% 23.13% 22.50% 20.36%

Capital available: $100,000

MARR: 18%


Mathematical Programming Formulation for Example 15.1

Maximize $4,718.79x1 + $2,247.00x2 + $9,212.88x3 + $10,111.69x4 + $7,415.24x5

subject to $15,000x1 + $25,000x2 + $40,000x3 + $50,000x4 + $70,000x5 < $100,000

xj = (0,1) j = 1, ..., 5


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 2m possible mutually exclusive investment alternatives, including the “Do Nothing” alternative. In Example 1.5, m = 3; therefore, there were 8 possible alternatives.


Solving Example 15.1 with Enumeration

With m = 5, there are 25 = 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.)


Combination x1 x2 x3 x4 x5 Cum Inv Cum Ret PW1 0 0 0 0 0 $0 $0 $0.002 0 0 0 0 1 $70,000 $14,250 $7,415.243 0 0 0 1 0 $50,000 $11,250 $10,111.694 0 0 0 1 1 $120,000 $25,500 $17,526.945 0 0 1 0 0 $40,000 $9,250 $9,212.886 0 0 1 0 1 $110,000 $23,500 $16,628.127 0 0 1 1 0 $90,000 $20,500 $19,324.578 0 0 1 1 1 $160,000 $34,750 $26,739.819 0 1 0 0 0 $25,000 $5,000 $2,247.04

10 0 1 0 0 1 $95,000 $19,250 $9,662.2911 0 1 0 1 0 $75,000 $16,250 $12,358.7412 0 1 0 1 1 $145,000 $30,500 $19,773.9813 0 1 1 0 0 $65,000 $14,250 $11,459.9214 0 1 1 0 1 $135,000 $28,500 $18,875.1615 0 1 1 1 0 $115,000 $25,500 $21,571.6116 0 1 1 1 1 $185,000 $39,750 $28,986.86



Combination x1 x2 x3 x4 x5 Cum Inv Cum Ret PW17 1 0 0 0 0 $15,000 $3,750 $4,718.7918 1 0 0 0 1 $85,000 $18,000 $12,134.0319 1 0 0 1 0 $65,000 $15,000 $14,830.4820 1 0 0 1 1 $135,000 $29,250 $22,245.7321 1 0 1 0 0 $55,000 $13,000 $13,931.6722 1 0 1 0 1 $125,000 $27,250 $21,346.9123 1 0 1 1 0 $105,000 $24,250 $24,043.3624 1 0 1 1 1 $175,000 $38,500 $31,458.6025 1 1 0 0 0 $40,000 $8,750 $6,965.8326 1 1 0 0 1 $110,000 $23,000 $14,381.0827 1 1 0 1 0 $90,000 $20,000 $17,077.5328 1 1 0 1 1 $160,000 $34,250 $24,492.7729 1 1 1 0 0 $80,000 $18,000 $16,178.7130 1 1 1 0 1 $150,000 $32,250 $23,593.9531 1 1 1 1 0 $130,000 $29,250 $26,290.4032 1 1 1 1 1 $200,000 $43,500 $33,705.65




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.


Adding Constraints

Mutually ExclusiveContingent


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, x3 < x2, or D7 < C7.



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.



10 0 1 0 0 1 $95,000 $19,250 $9,662.2911 0 1 0 1 0 $75,000 $16,250 $12,358.7412 0 1 0 1 1 $145,000 $30,500 $19,773.9813 0 1 1 0 0 $65,000 $14,250 $11,459.9214 0 1 1 0 1 $135,000 $28,500 $18,875.1615 0 1 1 1 0 $115,000 $25,500 $21,571.6116 0 1 1 1 1 $185,000 $39,750 $28,986.86




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.



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 x2 and x4 equals zero or their sum is less than or equal to 1.0

• As shown on the following slide, the sum of x2 and x4 is entered in cell E11 and a constraint is added that E11 < 1.



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.



10 0 1 0 0 1 $95,000 $19,250 $9,662.2911 0 1 0 1 0 $75,000 $16,250 $12,358.7412 0 1 0 1 1 $145,000 $30,500 $19,773.9813 0 1 1 0 0 $65,000 $14,250 $11,459.9214 0 1 1 0 1 $135,000 $28,500 $18,875.1615 0 1 1 1 0 $115,000 $25,500 $21,571.6116 0 1 1 1 1 $185,000 $39,750 $28,986.86



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

x6 < x3 + x4.

EOY CF(1) CF(2) CF(3) CF(4)] CF(5) CF(6)

0 -$15,000.00 -$18,000.00 -$20,000.00 -$25,000.00 -$30,000.00 -$40,000.001 $4,500.00 $3,000.00 $4,000.00 $4,500.00 $6,000.00 $15,000.002 $4,500.00 $4,500.00 $5,000.00 $4,500.00 $9,000.00 $15,000.003 $4,500.00 $6,000.00 $6,000.00 $4,500.00 $12,000.00 $25,000.004 $4,500.00 $7,500.00 $7,000.00 $4,500.00 $15,000.00 $0.005 $4,500.00 $9,000.00 $8,000.00 $4,500.00 $0.00 $0.00



With m = 6, there are 26 = 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.


Combination x1 x2 x3 x4 x5 x6 Cum Inv PW1 0 0 0 0 0 0 $0 $0.002 0 0 0 0 0 1 $40,000 $4,815.933 0 0 0 0 1 0 $30,000 $2,153.544 0 0 0 0 1 1 $70,000 $6,969.475 0 0 0 1 0 0 $25,000 $3,365.646 0 0 0 1 0 1 $65,000 $8,181.577 0 0 0 1 1 0 $55,000 $5,519.188 0 0 0 1 1 1 $95,000 $10,335.119 0 0 1 0 0 0 $20,000 $2,024.95

10 0 0 1 0 0 1 $60,000 $6,840.8811 0 0 1 0 1 0 $50,000 $4,178.4912 0 0 1 0 1 1 $90,000 $8,994.4213 0 0 1 1 0 0 $45,000 $5,390.5914 0 0 1 1 0 1 $85,000 $10,206.5215 0 0 1 1 1 0 $75,000 $7,544.1316 0 0 1 1 1 1 $115,000 $12,360.06



Combination x1 x2 x3 x4 x5 x6 Cum Inv PW17 0 1 0 0 0 0 $18,000 $3,665.0618 0 1 0 0 0 1 $58,000 $8,480.9919 0 1 0 0 1 0 $48,000 $5,818.6020 0 1 0 0 1 1 $88,000 $10,634.5321 0 1 0 1 0 0 $43,000 $7,030.7022 0 1 0 1 0 1 $83,000 $11,846.6323 0 1 0 1 1 0 $73,000 $9,184.2524 0 1 0 1 1 1 $113,000 $14,000.1725 0 1 1 0 0 0 $38,000 $5,690.0126 0 1 1 0 0 1 $78,000 $10,505.9427 0 1 1 0 1 0 $68,000 $7,843.5528 0 1 1 0 1 1 $108,000 $12,659.4829 0 1 1 1 0 0 $63,000 $9,055.6530 0 1 1 1 0 1 $103,000 $13,871.5831 0 1 1 1 1 0 $93,000 $11,209.1932 0 1 1 1 1 1 $133,000 $16,025.12




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.



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%


0 -$15,000.00 -$18,000.00 -$20,000.00 -$25,000.00 -$30,000.00 -$40,000.001 $4,500.00 $3,000.00 $4,000.00 $4,500.00 $6,000.00 $15,000.002 $4,500.00 $4,500.00 $5,000.00 $4,500.00 $9,000.00 $15,000.003 $4,500.00 $6,000.00 $6,000.00 $4,500.00 $12,000.00 $25,000.004 $4,500.00 $7,500.00 $7,000.00 $4,500.00 $15,000.00 $0.005 $4,500.00 $9,000.00 $8,000.00 $4,500.00 $0.00 $0.00

31

n

jjx



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.


Combination x1 x2 x3 x4 x5 x6 Cum Inv PW1 0 0 0 0 0 0 $0 $0.002 0 0 0 0 0 1 $40,000 $4,815.933 0 0 0 0 1 0 $30,000 $2,153.544 0 0 0 0 1 1 $70,000 $6,969.475 0 0 0 1 0 0 $25,000 $3,365.646 0 0 0 1 0 1 $65,000 $8,181.577 0 0 0 1 1 0 $55,000 $5,519.188 0 0 0 1 1 1 $95,000 $10,335.119 0 0 1 0 0 0 $20,000 $2,024.95

10 0 0 1 0 0 1 $60,000 $6,840.8811 0 0 1 0 1 0 $50,000 $4,178.4912 0 0 1 0 1 1 $90,000 $8,994.4213 0 0 1 1 0 0 $45,000 $5,390.5914 0 0 1 1 0 1 $85,000 $10,206.5215 0 0 1 1 1 0 $75,000 $7,544.1316 0 0 1 1 1 1 $115,000 $12,360.06





The same solution was obtained using Excel® SOLVER tool to solve the BLP problem. (For m > 6, enumeration is not reasonable. Use SOLVER.)


Sensitivity Analysis

Capital AvailableMARR


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?


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


Sensitivity Analysis of the Optimum Investment Portfolio

Capital Available

for Investment

Optimum Portfolio

Portfolio PW

Portfolio IRR

Capital Available

for Investment

Optimum Portfolio

Portfolio PW

Portfolio IRR

$5,000 {Ø} $0.00 0.00% $105,000 {1,3,4} $24,043.36 23.10%$10,000 {Ø} $0.00 0.00% $110,000 {1,3,4} $24,044.36 23.10%$15,000 {1} $4,718.79 25.00% $115,000 {1,3,4} $24,045.36 23.10%$20,000 {1} $4,718.79 25.00% $120,000 {1,3,4} $24,046.36 23.10%$25,000 {1} $4,718.79 25.00% $125,000 {1,3,4} $24,047.36 23.10%$30,000 {1} $4,718.79 25.00% $130,000 {1,2,3,4} $26,290.40 22.50%$35,000 {1} $4,718.79 25.00% $135,000 {1,2,3,4} $26,291.40 22.50%$40,000 {3} $9,212.88 23.13% $140,000 {1,2,3,4} $26,292.40 22.50%$45,000 {3} $9,212.88 23.13% $145,000 {1,2,3,4} $26,293.40 22.50%$50,000 {4} $10,111.69 22.50% $150,000 {1,2,3,4} $26,294.40 22.50%$55,000 {1,3} $13,931.67 23.64% $155,000 {1,2,3,4} $26,295.40 22.50%$60,000 {1,3} $13,932.67 23.64% $160,000 {3,4,5} $26,739.81 21.72%$65,000 {1,4} $14,830.48 23.08% $165,000 {3,4,5} $26,740.81 21.72%$70,000 {1,4} $14,831.48 23.08% $170,000 {3,4,5} $26,741.81 21.72%$75,000 {1,4} $14,832.48 23.08% $175,000 {1,3,4,5} $31,458.60 22.00%$80,000 {1,2,3} $16,178.71 22.50% $180,000 {1,3,4,5} $31,459.60 22.00%$85,000 {1,2,3} $16,179.71 22.50% $185,000 {1,3,4,5} $31,460.60 22.00%$90,000 {3,4} $19,324.57 22.78% $190,000 {1,3,4,5} $31,461.60 22.00%$95,000 {3,4} $19,325.57 22.78% $195,000 {1,3,4,5} $31,462.60 22.00%$100,000 {3,4} $19,326.57 22.78% $200,000 {1,2,3,4,5} $33,705.65 21.75%


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%]?


0 -$15,000.00 -$18,000.00 -$20,000.00 -$22,000.00 -$35,000.00 -$40,000.001 $4,500.00 $10,000.00 $4,000.00 $6,500.00 $6,000.00 $10,000.002 $4,500.00 $7,500.00 $5,000.00 $6,000.00 $6,900.00 $10,000.003 $4,500.00 $5,000.00 $6,000.00 $5,500.00 $7,935.00 $10,000.004 $4,500.00 $2,500.00 $7,000.00 $5,000.00 $9,125.25 $10,000.005 $4,500.00 $5,000.00 $8,000.00 $8,500.00 $15,494.04 $15,000.00

MARR RangeOptimum Portfolio

IRR

[0.00%,8.10%] {2,3,4,6} 14.01%[8.11%,8.64%] {1,2,3,6} 14.48%

[8.65%,12.89%] {1,2,3,4} 15.98%[12.90%,13.45%] {1,2,3} 17.30%[13.46%,15.23%] {1,2} 20.17%[15.24%,25.07%] {2} 25.07%


The Capital Budgeting Problem

Independent and Divisible Investments


Mathematical Programming Formulation of the Capital Budgeting Problem with Divisible

Investments

Maximize PW1p1 + PW2p2 + ... + PWn-1pn-1 + PWnpn (15.1)

subject to c1p1 + c2p2 + ... + cn-1pn-1 + cnpn < C (15.2)

0 < pj < 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 divisible, i.e., a percentage of an investment can be pursued.


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.


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.


1 2 3 4 5IRR 25.00% 20.00% 23.13% 22.50% 20.36%Annual Return $3,750.00 $5,000.00 $9,250.00 $11,250.00 $14,250.00Investment $15,000.00 $25,000.00 $40,000.00 $50,000.00 $70,000.00

1 3 4 5 2IRR 25.00% 23.13% 22.50% 20.36% 20.00%Annual Return $3,750.00 $9,250.00 $11,250.00 $14,250.00 $5,000.00Investment $15,000.00 $40,000.00 $50,000.00 $70,000.00 $25,000.00

Economically Viable Investments

Economically Viable Investments Sorted by IRR

Investment Portfolio

Cumulative Funds Req'd

{B} $15,000.00{B,D} $55,000.00

{B,D,E} $105,000.00{B,D,E,F} $175,000.00

{F,D,E,F,C} $200,000.00


Optimum Divisible Portfolio


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.


B C D E FPresent Worth $4,718.79 $2,247.04 $9,212.88 $10,111.69 $7,415.24Annual Return $3,750.00 $5,000.00 $9,250.00 $11,250.00 $14,250.00Investment $15,000.00 $25,000.00 $40,000.00 $50,000.00 $70,000.00

E D F B CPresent Worth $10,111.69 $9,212.88 $7,415.24 $4,718.79 $2,247.04Annual Return $11,250.00 $9,250.00 $14,250.00 $3,750.00 $5,000.00Investment $50,000.00 $40,000.00 $70,000.00 $15,000.00 $25,000.00

Economically Viable Investments

Economically Viable Investments Sorted by Present Worth

Investment Portfolio

Cumulative Funds Req'd

{E} $50,000.00{E,D} $90,000.00

{E,D,F} $160,000.00{E,D,F,B} $175,000.00

{E,D,F,B,C} $200,000.00


Suboptimum Divisible Portfolio

$0.00

$50,000.00

$100,000.00

$150,000.00

$200,000.00

$250,000.00

{E} {E,D} {E,D,F} {E,D,F,B} {E,D,F,B,C}

investment capital available

Making 100% investments in E & D and investing $10,000 in F yields an

overall return of 22.54% and a present worth of $20,383.89


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.


Example 15.11In 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.


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 x3 to be less than or equal to the value of a cell in which you have entered x3 + x5

• For our example: E5 <= F8 when D5+F5 has been entered in cell F8




• x4 < x3 + x5




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?

• x1 + x2 + x3 > 2

• After entering B5+C5+D5 in F8, add a constraint to SOLVER: F8>=2


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 x3 + x4 < 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 x2 - x1 < 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

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