SESSIONS 3 & 4

8-2

Capital Budgeting

• Analysis of potential projects• Long-term decisions• Large expenditures• Difficult/impossible to reverse• Determines firm’s strategic direction

Keep in mind: VALUE CREATION

8-3

Required

Replacement

Expansion

Diversification

Capital Investment Process

Identification Evaluation Selection Implementation and follow-up

Riskier &

D

ifficult

Type of Investment

8-4

Net Present Value

How much value is created from undertaking an investment?

Step 1: Estimate the expected future cash flows.

Step 2: Estimate the required return for projects of this risk level. (Also called opportunity cost).

Step 3: Find the present value of the cash flows and subtract the initial investment to arrive at the

Net Present Value.

8-5

Net Present Value Sum of the PVs of all cash flows

Initial cost often is CF0 and is an outflow.

NPV = ∑n

t = 0

CFt

(1 + R)t

NPV = ∑n

t = 1

CFt

(1 + R)t- CF0

NOTE: t=0

8-6

NPV – Decision Rule

• If NPV is positive, accept the project• NPV > 0 means:

– Project is expected to add value to the firm– Will increase the wealth of the owners

• NPV is a direct measure of how well this project will meet the goal of increasing shareholder wealth.

8-7

Sample Project Data• You are looking at a new project and have

estimated the following cash flows, net income and book value data:– Year 0: CF = -165,000– Year 1: CF = 63,120 NI = 13,620– Year 2: CF = 70,800 NI = 3,300– Year 3: CF = 91,080 NI = 29,100– Average book value = $72,000

• Your required return for assets of this risk is 12%.• This project will be the example for all problem

exhibits in this chapter.

8-8

Computing NPV for the Project

• Using the formula:

NPV = -165,000/(1.12)0 + 63,120/(1.12)1 + 70,800/(1.12)2 + 91,080/(1.12)3 = 12,627.41

Capital Budgeting Project NPVRequired Return = 12%

Year CF Formula Disc CFs0 (165,000.00) =(-165000)/(1.12)^0 = (165,000.00)1 63,120.00 =(63120)/(1.12)^1 = 56,357.142 70,800.00 =(70800)/(1.12)^2 = 56,441.333 91,080.00 =(91080)/(1.12)^3 = 64,828.94

12,627.41

n

0tt

t

)R1(

CFNPV

8-9

Display You Enter C00 165000 C01 63120 F01 1 C02 70800 F02 1 C03 91080 F03 1 I 12 NPV 12,627.41

Cash Flows:

CF0 = -165000

CF1 = 63120

CF2 = 70800

CF3 = 91080

Computing NPV for the ProjectUsing the TI BAII+ CF Worksheet

8-10

Calculating NPVs with Excel• NPV function: =NPV(rate,CF01:CFnn)

– First parameter = required return entered as a decimal (5% = .05)– Second parameter = range of cash flows beginning with year 1

• After computing NPV, subtract the initial investment (CF0)

2

3

4

5

6

7

8

9

10

11

A B C D

Required Return = 12%Year CF Formula Disc CFs

0 (165,000.00) =(-165000)/(1.12)^0 = (165,000.00)1 63,120.00 =(63120)/(1.12)^1 = 56,357.142 70,800.00 =(70800)/(1.12)^2 = 56,441.333 91,080.00 =(91080)/(1.12)^3 = 64,828.94

12,627.41

EXCEL =NPV(D2,B5:B7) 177,627.41NPV + CF0 12,627.41

8-11

Net Present Value Sum of the PVs of all cash flows.

n

0tt

t

)R1(

CFNPV << CALCULATOR

<< EXCEL0

n

1tt

t CF)R1(

CFNPV

8-12

Rationale for the NPV Method

• NPV = PV inflows – Cost NPV=0 → Project’s inflows are “exactly

sufficient to repay the invested capital and provide the required rate of return”

• NPV = net gain in shareholder wealth

• Rule: Accept project if NPV > 0

8-13

NPV Method

–Meets all desirable criteria• Considers all CFs• Considers TVM• Adjusts for risk• Can rank mutually exclusive projects

–Directly related to increase in VF

–Dominant method; always prevails

8-14

Payback Period• How long does it take to recover the initial

cost of a project?• Computation

– Estimate the cash flows– Subtract the future cash flows from the initial

cost until initial investment is recovered– A “break-even” type measure

• Decision Rule – Accept if the payback period is less than some preset limit

8-15

Computing Payback for the Project

• Do we accept or reject the project?

Capital Budgeting Project

Year CF Cum. CFs0 (165,000)$ (165,000)$ 1 63,120$ (101,880)$ 2 70,800$ (31,080)$ 3 91,080$ 60,000$

Payback = year 2 ++ (31080/91080)

Payback = 2.34 years

8-16

Advantages and Disadvantages of Payback

• Advantages– Easy to understand– Adjusts for

uncertainty of later cash flows

– Biased towards liquidity

• Disadvantages– Ignores the time value of money– Requires an arbitrary cutoff point– Ignores cash flows beyond the cutoff date– Biased against long-term projects, such as

research and development, and new projects

ASKS THE WRONG QUESTION!

8-17

Internal Rate of Return

• Most important alternative to NPV• Widely used in practice • Intuitively appealing• Based entirely on the estimated cash

flows • Independent of interest rates

8-18

IRR Definition and Decision Rule

• Definition: – IRR = discount rate that makes the

NPV = 0

• Decision Rule: – Accept the project if the IRR is greater

than the required return

8-19

NPV vs. IRR

NPV)R1(

CFn

0tt

t

IRR: Enter NPV = 0, solve for IRR.

NPV: Enter r, solve for NPV

0)IRR1(

CFn

0tt

t

8-20

Computing IRR For The Project

• Without a financial calculator or Excel, this becomes a trial-and-error process.

• Calculator– Enter the cash flows as for NPV– Press IRR and then CPT– IRR = 16.13% > 12% required return

• Do we accept or reject the project?

8-21

Display You Enter C00 165000 C01 63120 F01 1 C02 70800 F02 1 C03 91080 F03 1 IRR 16.1322

Cash Flows:

CF0 = -165000

CF1 = 63120

CF2 = 70800

CF3 = 91080

Computing IRR for the ProjectUsing the TI BAII+ CF Worksheet

8-22

Calculating IRR with Excel

• Start with the cash flows as you did to solve for NPV

• Use the IRR function– Enter the range of cash flows, beginning with

the initial cash flow (Cash flow 0)– You can enter a guess, but it is not necessary– The default format is a whole percent

8-23

Calculating IRR with Excel

1

2

3

4

5

6

7

8

A B C

Year CF0 (165,000.00)1 63,120.002 70,800.003 91,080.00

EXCEL =IRR(B3:B6) 16.13%

IRR

8-24

NPV Profile For The Project

-20,000

-10,000

0

10,000

20,000

30,000

40,000

50,000

60,000

70,000

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22

Discount Rate

NP

V IRR = 16.13%

8-25

IRR - Advantages • Preferred by executives

– Intuitively appealing– Easy to communicate the value of a project

• If the IRR is high enough, may not need to estimate a required return

• Considers all cash flows• Considers time value of money

8-26

IRR - Disadvantages

• Can produce multiple answers• Cannot rank mutually exclusive projects• Reinvestment assumption flawed

8-27

Summary of Decisions for the Project

Net Present Value Accept*

Payback Period ???

Internal Rate of Return Accept*

*For a normal, single project these two criteria always agree

8-28

NPV vs. IRR• NPV and IRR will generally give the same

decision• Exceptions

– Non-conventional cash flows • Cash flow sign changes more than once

– Mutually exclusive projects• Initial investments are substantially different• Timing of cash flows is substantially different• Will not reliably rank projects

8-29

IRR & Non-Conventional Cash Flows

• “Non-conventional” – Cash flows change sign more than once– Most common:

• Initial cost (negative CF)• A stream of positive CFs• Negative cash flow to close project.• For example, nuclear power plant or strip mine.

– More than one IRR …. – Which one do you use to make your decision?

8-30

Multiple IRRs• Descartes Rule of Signs

• Polynomial of degree n→n roots– When you solve for IRR you are solving for the

root of an equation– 1 real root per sign change– Rest = imaginary (i2 = -1)

0)IRR1(

CFn

0tt

t

8-31

Non-Conventional Cash Flows

• Suppose an investment will cost $90,000 initially and will generate the following cash flows:– Year 1: 132,000– Year 2: 100,000– Year 3: -150,000

• The required return is 15%.• Should we accept or reject the project?

8-32

Non-Conventional Cash FlowsSummary of Decision Rules

• NPV > 0 at 15% required return, so you should Accept

• IRR =10.11% (using a financial calculator), which would tell you to Reject

• Recognize the non-conventional cash flows and look at the NPV profile

I = 15%YR CF0 -$90,0001 $132,0002 $100,0003 -$150,000

NPV $1,769.54 > 0IRR-1 10.11% < 15%IRR-2 42.66% > 15%

8-33

NPV Profile

($10,000.00)

($8,000.00)

($6,000.00)

($4,000.00)

($2,000.00)

$0.00

$2,000.00

$4,000.00

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55

Discount Rate

NP

V

IRR = 10.11% and 42.66%

When you cross the x-axis more than once, there will be more than one return that solves the equation

8-34

Independent versus Mutually Exclusive Projects

• Independent–The cash flows of one project are

unaffected by the acceptance of the other.

• Mutually Exclusive –The acceptance of one project precludes

accepting the other.

8-35

Reinvestment Rate Assumption

• IRR assumes reinvestment at IRR• NPV assumes reinvestment at the firm’s

weighted average cost of capital(opportunity cost of capital)– More realistic – NPV method is best

• NPV should be used to choose between mutually exclusive projects

8-36

Example of Mutually Exclusive Projects

Period Project A Project B

0 -500 -400

1 325 325

2 325 200

IRR 19.43% 22.17%

NPV 64.05 60.74

The required return for both projects is 10%.

Which project should you accept and why?

8-37

NPV Profiles

($40.00)

($20.00)

$0.00

$20.00

$40.00

$60.00

$80.00

$100.00

$120.00

$140.00

$160.00

0 0.05 0.1 0.15 0.2 0.25 0.3

Discount Rate

NP

V AB

IRR for A = 19.43%

IRR for B = 22.17%

Crossover Point = 11.8%

8-38

Two Reasons NPV Profiles Cross

• Size (scale) differences. – Smaller project frees up funds sooner for

investment. – The higher the opportunity cost, the more

valuable these funds, so high discount rate favors small projects.

• Timing differences. – Project with faster payback provides more

CF in early years for reinvestment. – If discount rate is high, early CF especially

good

8-39

Conflicts Between NPV and IRR

• NPV directly measures the increase in value to the firm

• Whenever there is a conflict between NPV and another decision rule, always use NPV

• IRR is unreliable in the following situations:– Non-conventional cash flows– Mutually exclusive projects

8-40

Profitability Index

• Measures the benefit per unit cost, based on the time value of money– A profitability index of 1.1 implies that for

every $1 of investment, we create an additional $0.10 in value

• Can be very useful in situations of capital rationing

• Decision Rule: If PI > 1.0 Accept

8-41

Profitability Index

• For conventional CF Projects:

PV(Cash Inflows)

Absolute Value of Initial Investment0

n

1tt

t

CF)r1(

CF

PI

8-42

Advantages and Disadvantages of Profitability Index

• Advantages– Closely related to

NPV, generally leading to identical decisions• Considers all CFs• Considers TVM

– Easy to understand and communicate

– Useful in capital rationing

• Disadvantages– May lead to

incorrect decisions in comparisons of mutually exclusive investments (can conflict with NPV)

8-43

Profitability IndexExample of Conflict with NPV

A BCF(0) (10,000.00) (100,000.00)PV(CIF) 15,000.00 125,000.00PI 1.50 1.25NPV 5,000.00 25,000.00

8-44

Capital Budgeting In Practice

• Consider all investment criteria when making decisions

• NPV and IRR are the most commonly used primary investment criteria

• Payback is a commonly used secondary investment criteria

• All provide valuable information

8-45

Summary

Calculate ALL -- each has value

Method What it measures Metric

NPV $ increase in VF $$

Payback Liquidity Years

IRR E(R), risk %

PI If rationed Ratio

9-46

Evaluating Projects

• Steps in Evaluation:–Determine the relevant cash flows

for a proposed investment–Analyze the project’s projected

cash flows–Calculate and interpret estimated

NPV

9-47

Relevant Cash Flows

• Include only cash flows that will only occur if the project is accepted

• Incremental cash flows• The stand-alone principle allows us

to analyze each project in isolation from the firm simply by focusing on incremental cash flows

9-48

Relevant Cash Flows:Incremental Cash Flow for a Project

Corporate cash flow with the project

Minus

Corporate cash flow without the project

9-49

Relevant Cash Flows

• “Sunk” Costs ………………………… N• Opportunity Costs …………………... Y• Side Effects/Erosion……..…………… Y• Net Working Capital………………….. Y• Financing Costs….………..…………. N• Tax Effects ………………………..….. Y

9-50

Pro Forma Statements and Cash Flow

• Pro Forma Financial Statements – Projects future operations

• Operating Cash Flow:OCF = EBIT + Depr – Taxes

OCF = NI + Depr if no interest expense

• Cash Flow From Assets:CFFA = OCF – NCS –ΔNWC

NCS = Net capital spending

9-51

Shark Attractant Project• Estimated sales 50,000 cans• Sales Price per can $4.00• Cost per can $2.50• Estimated life 3 years• Fixed costs $12,000/year• Initial equipment cost $90,000

– 100% depreciated over 3 year life• Investment in NWC $20,000• Tax rate 34%• Cost of capital 20%

9-52

Pro Forma Income StatementTable 9.1

Sales (50,000 units at $4.00/unit) $200,000

Variable Costs ($2.50/unit) 125,000

Gross profit $ 75,000

Fixed costs 12,000

Depreciation ($90,000 / 3) 30,000

EBIT $ 33,000

Taxes (34%) 11,220

Net Income $ 21,780

9-53

Projected Capital RequirementsTable 9.2

Year

0 1 2 3

NWC $20,000 $20,000 $20,000 $20,000

Net Fixed Assets

90,000 60,000 30,000 0

Total Investment

$110,000 $80,000 $50,000 $20,000

NFA declines by the amount of depreciation each year

Investment = book or accounting value, not market value

9-54

Projected Total Cash FlowsTable 9.5

Year

0 1 2 3

OCF $51,780 $51,780 $51,780

NWC -$20,000 20,000

Capital Spending

-$90,000

CFFA -$110,00 $51,780 $51,780 $71,780

Note: Investment in NWC is recovered in final year

Equipment cost is a cash outflow in year 0

9-55

Shark Attractant Project

OCF = EBIT + Depreciation – Taxes

OCF = Net Income + Depreciation (if no interest)

Year 0 1 2 3Sales 200,000 200,000 200,000Variable Costs 125,000 125,000 125,000Gross Profit 75,000 75,000 75,000Fixed Costs 12,000 12,000 12,000Depreciation 30,000 30,000 30,000EBIT 33,000 33,000 33,000Taxes 11,220 11,220 11,220Net Income 21,780 21,780 21,780

Operating Cash Flow 51,780 51,780 51,780Changes in NWC -20,000 20,000Net Capital Spending -90,000Cash Flow From Assets -110,000 51,780 51,780 71,780

Net Present Value $10,647.69IRR 25.76%

Pro Forma Income Statement

Cash Flows

9-56

Display You Enter ‘' C00 110000 S!#C01 51780 !# F01 2 !# C02 71780 !# F02 1 !#( I 20 !#NPV %10647.69 ) %25.76

Cash Flows:

CF0 = -110000

CF1 = 51780

CF2 = 51780

CF3 = 71780

Computing NPV for the ProjectUsing the TI BAII+ CF Worksheet

9-57

Making The Decision

• Should we accept or reject the project?

Operating Cash Flow 51,780 51,780 51,780Changes in NWC -20,000 20,000Net Capital Spending -90,000Cash Flow From Assets -110,000 51,780 51,780 71,780

Net Present Value $10,647.69IRR 25.76%

Cash Flows

9-58

The Tax Shield Approach to OCF

• OCF = (Sales – costs)(1 – T) + Deprec*T OCF=(200,000-137,000) x 66% + (30,000 x .34)

OCF = 51,780

• Particularly useful when the major incremental cash flows are the purchase of equipment and the associated depreciation tax shield – i.e., choosing between two different machines

9-59

Changes in NWC• GAAP requirements:

– Sales recorded when made, not when cash is received• Cash in = Sales - ΔAR

– Cost of goods sold recorded when the corresponding sales are made, whether suppliers paid yet or not• Cash out = COGS - ΔAP

• Buy inventory/materials to support sales before any cash collected

9-60

Depreciation & Capital Budgeting

• Use the schedule required by the IRS for tax purposes

• Depreciation = non-cash expense–Only relevant due to tax affects

• Depreciation tax shield = DT–D = depreciation expense–T = marginal tax rate

9-61

Computing Depreciation• Straight-line depreciation

D = (Initial cost – salvage) / number of years

Straight Line Salvage Value

• MACRS Depreciate 0 Recovery Period = Class Life

1/2 Year Convention

Multiply percentage in table by the initial cost

9-62

After-Tax Salvage

• If the salvage value is different from the book value of the asset,

then there is a tax effect• Book value = initial cost –

accumulated depreciation• After-tax salvage = salvage –

T(salvage – book value)

9-63

Tax Effect on Salvage

Net Salvage Cash Flow = SP - (SP-BV)(T)

Where:SP = Selling PriceBV = Book ValueT = Corporate tax rate

9-64

Example: Depreciation and After-tax Salvage

• Car purchased for $12,000 • 5-year property• Marginal tax rate = 34%.

Depreciation 5-year Asset

Year Beg BV Depr % Deprec End BV1 12,000.00$ 20.00% 2,400.00$ 9,600.00$ 2 9,600.00$ 32.00% 3,840.00$ 5,760.00$ 3 5,760.00$ 19.20% 2,304.00$ 3,456.00$ 4 3,456.00$ 11.52% 1,382.40$ 2,073.60$ 5 2,073.60$ 11.52% 1,382.40$ 691.20$ 6 691.20$ 5.76% 691.20$ -$

100.00% 12,000.00$

9-65

Salvage Value & Tax EffectsDepreciation 5-year Asset

Year Beg BV Depr % Deprec End BV1 12,000.00$ 20.00% 2,400.00$ 9,600.00$ 2 9,600.00$ 32.00% 3,840.00$ 5,760.00$ 3 5,760.00$ 19.20% 2,304.00$ 3,456.00$ 4 3,456.00$ 11.52% 1,382.40$ 2,073.60$ 5 2,073.60$ 11.52% 1,382.40$ 691.20$ 6 691.20$ 5.76% 691.20$ -$

100.00% 12,000.00$

Net Salvage Cash Flow = SP - (SP-BV)(T)

If sold at EOY 5 for $3,000:NSCF = 3,000 - (3000 - 691.20)(.34) = $2,215.01

= $3,000 – 784.99 = $2,215.01If sold at EOY 2 for $4,000:

NSCF = 4,000 - (4000 - 5,760)(.34) = $4,598.40 = $4,000 – (-598.40) = $4,598.40

9-66

Evaluating NPV Estimates

• NPV estimates are only estimates• Forecasting risk:

–Sensitivity of NPV to changes in cash flow estimates • The more sensitive, the greater the

forecasting risk

• Sources of value • Be able to articulate why this project creates

value

9-67

Scenario Analysis

• Examines several possible situations:–Worst case

–Base case or most likely case

–Best case

• Provides a range of possible outcomes

9-68

Scenario Analysis Example

Note: “Lower” ≠ Worst

“Upper” ≠ Best

Base Lower UpperUnits 6,000 5,500 6,500 Price/unit 80.00$ 75.00$ 85.00$ Variable cost/unit 60.00$ 58.00$ 62.00$ Fixed cost/year 50,000$ 45,000$ 55,000$

BASE BEST WORST

Initial investment 200,000$ Depreciated to salvage value of 0 over 5 yearsDeprec/yr 40,000$

Project Life 5 yearsTax rate 34%Required return 12%

9-69

BASE WORST BESTUnits 6,000 5,500 6,500 Price/unit 80.00$ 75.00$ 85.00$ Variable cost/unit 60.00$ 62.00$ 58.00$ Fixed Cost 50,000$ 55,000$ 45,000$

Sales 480,000$ 412,500$ 552,500$ Variable Cost 360,000 341,000 377,000Fixed Cost 50,000 55,000 45,000Depreciation 40,000 40,000 40,000EBIT 30,000 (23,500) 90,500Taxes 10,200 (7,990) 30,770Net Income 19,800 (15,510) 59,730 + Deprec 40,000 40,000 40,000

TOTAL CF 59,800 24,490 99,730

NPV 15,566 (111,719) 159,504

IRR 15.1% -14.4% 40.9%

Scenario Analysis Example

9-70

Problems with Scenario Analysis

• Considers only a few possible out-comes

• Assumes perfectly correlated inputs–All “bad” values occur together and all

“good” values occur together• Focuses on stand-alone risk,

although subjective adjustments can be made

9-71

Sensitivity Analysis

• Shows how changes in an input variable affect NPV or IRR

• Each variable is fixed except one– Change one variable to see the effect on

NPV or IRR• Answers “what if” questions

9-72

Sensitivity Analysis: Unit Sales

Base Units UnitsUnits 6,000 5,500 6,500 Price/unit 80$ 80 80Variable cost/unit 60$ 60 60Fixed cost/year 50,000$ 50,000 50,000

Initial investment 200,000$ Depreciated to salvage value of 0 over 5 yearsDeprec/yr 40,000$

Tax rate 34%Required Return 12%

BASE UNITS UNITSUnits 6,000 5,500 6,500 Price/unit 80$ 80$ 80$ Variable cost/unit 60$ 60$ 60$ Fixed cost 50,000$ 50,000$ 50,000$

Sales 480,000$ 440,000$ 520,000$ Variable Cost 360,000 330,000 390,000 Fixed Cost 50,000 50,000 50,000 Depreciation 40,000 40,000 40,000 EBIT 30,000 20,000 40,000 Taxes 10,200 6,800 13,600 Net Income 19,800 13,200 26,400 + Deprec 40,000 40,000 40,000

TOTAL CF 59,800 53,200 66,400

NPV 15,566$ (8,226)$ 39,357$

Unit Sales Sensitivity

$(8,226)

$15,566

$39,357

-20,000.00

-10,000.00

0.00

10,000.00

20,000.00

30,000.00

40,000.00

50,000.00

5,500 6,000 6,500

Unit Sales

NP

V

9-73

Sensitivity Analysis: Fixed Costs

Fixed Cost Sensitivity

$27,461

$15,566

$3,670

0.00

5,000.00

10,000.00

15,000.00

20,000.00

25,000.00

30,000.00

$45,000 $50,000 $55,000

Fixed Cost

NP

VBase Fixed Cost Fixed Cost

Units 6,000 6,000 6,000 Price/unit 80$ 80 80Variable cost/unit 60$ 60 60Fixed cost/year 50,000$ 55,000 45,000

Initial investment 200,000$ Depreciated to salvage value of 0 over 5 yearsDeprec/yr 40,000$

Tax rate 34%Required Return 12%

BASE FC FCUnits 6,000 6,000 6,000 Price/unit 80$ 80$ 80$ Variable cost/unit 60$ 60$ 60$ Fixed cost 50,000$ 55,000$ 45,000$

Sales 480,000$ 480,000$ 480,000$ Variable Cost 360,000 360,000 360,000 Fixed Cost 50,000 55,000 45,000 Depreciation 40,000 40,000 40,000 EBIT 30,000 25,000 35,000 Taxes 10,200 8,500 11,900 Net Income 19,800 16,500 23,100 + Deprec 40,000 40,000 40,000

TOTAL CF 59,800 56,500 63,100

NPV 15,566$ 3,670$ 27,461$

9-74

Sensitivity Analysis:• Strengths

– Provides indication of stand-alone risk.– Identifies dangerous variables.– Gives some breakeven information.

• Weaknesses– Does not reflect diversification.– Says nothing about the likelihood of change

in a variable, – Ignores relationships among variables.

9-75

Disadvantages of Sensitivity and Scenario Analysis

• Neither provides a decision rule. –No indication whether a project’s

expected return is sufficient to compensate for its risk.

• Ignores diversification. –Measures only stand-alone risk, which

may not be the most relevant risk in capital budgeting.

9-76

Managerial Options• Contingency planning• Option to expand

– Expansion of existing product line– New products– New geographic markets

• Option to abandon– Contraction– Temporary suspension

• Option to wait• Strategic options

9-77

Capital Rationing

• Capital rationing occurs when a firm or division has limited resources– Soft rationing – the limited resources are

temporary, often self-imposed– Hard rationing – capital will never be available

for this project• The profitability index is a useful tool when

faced with soft rationing