5-1McGraw-Hill/Irwin Copyright © 2013 by The McGraw-Hill Companies, Inc. All rights reserved.

Net Present Value and Other Investment Rules

Chapter 5

5-2

Key Concepts and Skills

• Focus on capital budgeting, i.e., the decision making process for accepting or rejecting long-term investment projects.

• Come up with investment choice criteria.• Be able to compute net present value and understand why

it is the best decision criterion• Be able to compute payback and discounted payback and

understand their shortcomings• Be able to compute the internal rate of return and

profitability index, understanding the strengths and weaknesses of both approaches

5-3

Chapter Outline

5.1 Why Use Net Present Value?5.2 The Payback Period Method5.3 The Discounted Payback Period Method5.4 The Internal Rate of Return5.5 Problems with the IRR Approach5.6 The Profitability Index5.7 The Practice of Capital Budgeting

5-4

5.1 The Net Present Value (NPV) Rule

• Net Present Value (NPV) = Total PV of future CF’s - Initial Investment

• Estimating NPV (i.e., value created from undertaking investment):1. Estimate future cash flows: how much? and when?2. Estimate discount rate: the return that one can expect to earn on a

financial asset of comparable risk (i.e., the discount rate is an opportunity cost); difficult to determine appropriate discount rate.

3. Estimate initial costs

• Minimum Acceptance Criteria: Accept if NPV > 0• e.g., invest $100 to get $105 in a year; discount rate=10%. NPV=-

100+105/1.1=-$4.55<0 → do not invest. Alternative pays 10% while this project pays (105-100)/100=5%.

• Ranking Criteria: Choose the highest NPV project

5-5

Why Use Net Present Value?

• Accepting positive NPV projects benefits shareholders.

• The value of the firm rises by the NPV of the project, because the value of the firm is the sum of the values of different projects within the firm.

• NPV uses all the cash flows of the project• NPV discounts the cash flows properly

5-6

Calculating NPV with Financial Calculator

• What is the NPV of the following investment?• Cost=$165,000,

Year 1 CF=$63,120,Year 2 CF=$70,800,Year 3 CF=$91,080,Discount rate r=12%.

• CF, CF0=−165,000, C01=63,120, F01=1, C02=70,800, F02=1, C03=91,080, F03=1, NPV, I=12, ↓NPV, CPT, NPV=$12,627.41

5-7

Calculating NPV with Spreadsheets

• Spreadsheets are an excellent way to compute NPVs, especially when you have to compute the cash flows as well.

• Using the NPV function correctly:– The first component is the required return entered as a

decimal.– The second component is the range of cash flows

beginning with year 1. (If you entered the range of values beginning with year 0, instead, NPV would calculate the PV!!!)

– Add the initial investment after computing the NPV (because NPV only calculates PV!!!)

5-8

5.2 The Payback Period Method

• How long does it take the project to “pay back” its initial investment?

• Payback Period = number of years to recover initial costs

• Minimum Acceptance Criteria: – Set by management

• Ranking Criteria: – Set by management

5-9

Payback Period

• Computation– Estimate the cash flows– Subtract the future cash flows from the initial cost

until the initial investment has been recovered• Decision Rule: accept if the payback period is less

than some preset limit

5-10

Payback Period Example

• Assume we will accept the project if it pays back within 2 years. Cost of the project=$165,000, with cash flows $63,120 in year 1, $70,800 in year 2 and $91,080 in year 3 (same example as before for NPV).– Year 1: 165,000 – 63,120 = 101,880 still to recover– Year 2: 101,880 – 70,800 = 31,080 still to recover– Year 3: 31,080 – 91,080 = -60,000 project pays back during year

3– Payback = 2 years + 31,080/91,080 = 2.34 years

• Do we accept or reject the project?• Reject the project.

5-11

The Payback Period Method• Disadvantages:

– Ignores the time value of money (it does not discount the cash flows depending on timing)

– Ignores cash flows after the payback period (all that matters is how soon you get your money back)

– Hence, biased against long-term projects– Requires an arbitrary acceptance criterion– A project rejected based on the payback method may have a

positive NPV• Advantages:

– Getting their money back quickly is important for corporations– Easy to understand– Biased toward liquidity

• Conclusion: Good measure for small businesses that are cash constrained, or for large business when they are making small decisions.

5-12

5.3 The Discounted Payback Period

• How long does it take the project to “pay back” its initial investment, taking the time value of money into account?

• Discounted payback period=number of years to recover initial costs, taking account of PVs (i.e., use present values of cash flows)

• Decision rule: Accept the project if it pays back on a discounted basis within the specified time.

• Drawbacks: arbitrary cutoff period, ignoring all cash flows after that date

• By the time you have discounted the cash flows, you might as well calculate the NPV.

5-13

5.4 The Internal Rate of Return• This is the most important alternative to NPV.• It is used in practice more often than NPV by CFOs and is

intuitively appealing.• IRR provides a single number summarizing the merits of

a project. The number is internal or intrinsic to the project and does not depend on the interest rate (discount rate) prevailing in the market.

• IRR=the discount rate that sets NPV to zero:0 = −C0+CF1/(1+IRR)+CF2/(1+IRR)2+…+CFT/(1+IRR)T • Minimum Acceptance Criterium:

– Accept if the IRR exceeds the required return (say, r, the discount rate prevailing in the capital market)

– Reject if the IRR is less than the required return• Ranking Criterium:

– Select alternative with the highest IRR

5-14

IRR: Example

Consider the following project:

0 1 2 3

$50 $100 $150

-$200

The internal rate of return for this project is 19.44%, in other words, IRR=19.44% solves the equation:

NPV=0= −200+50/(1+IRR)+100/(1+IRR)2+150/(1+IRR)3

5-15

IRR: Example (continued)

0% $100.004% $73.888% $51.11

12% $31.1316% $13.5220% ($2.08)24% ($15.97)28% ($28.38)32% ($39.51)36% ($49.54)40% ($58.60)44% ($66.82)

If we graph NPV versus the discount rate, we can see the IRR as the x-axis intercept.

IRR = 19.44%

($100.00)

($50.00)

$0.00

$50.00

$100.00

$150.00

-1% 9% 19% 29% 39%

Discount rate

NPV

5-16

IRR: Example with Financial Calculator

• Without a financial calculator this becomes a trial-and-error process

• With a financial calculator:1) Press CF and enter the cash flows as you did with

NPV2) Press IRR and then CPT

5-17

IRR: Second Example

• What is the IRR of the following investment?• Cost=$165,000,

Year 1 CF=$63,120,Year 2 CF=$70,800,Year 3 CF=$91,080, (i.e., same numbers as e.g. used before for NPV and Payback Period). The required return is 12%.

• Calculator yields IRR=16.13%>12%• Hence, accept the project.

5-18

IRR: Second Example (continued)

-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

5-19

Calculating IRR with Spreadsheets

• You start with the same cash flows as you did for the NPV.

• You use the IRR function:– You first enter your range of cash flows, beginning

with the initial cash flow.– You can enter a guess, but it is not necessary.– The default format is a whole percent – you will

normally want to increase the decimal places to at least two.

5-20

Reprise on Hypothetical Project: Summary of Decisions

• Cost=$165,000,Year 1 CF=$63,120,Year 2 CF=$70,800,Year 3 CF=$91,080

• Net Present Value (r=12%), Accept• Payback Period (2 year benchmark), Reject• Internal Rate of Return (12% benchmark),

Accept

5-21

Internal Rate of Return (IRR)• Advantages:

– Easy to understand and communicate– Knowing the intrinsic return is intuitively appealing– It is a simple way to communicate the value of a project

to someone who doesn’t know all the estimation details– If the IRR is high enough, you may not need to estimate a

required return, which is often a difficult task

• Disadvantages:– Does not distinguish between investing and financing projects– Problems with mutually exclusive investments– The scale problem– The timing problem– IRR may not exist, or there may be multiple IRRs

5-22

Investing vs Financing Projects• In an investing project, which is the typical type of project, the firm

initially pays out money and receives positive cash flows later on• In a financing project, the firm receives funds first and then pays

out funds later• Examples of the latter type are rare but do exist, for instance, a

corporation conducting a seminar where attendees pay in advance and the corporation pays out funds (incurs expenses) later

• In this case, the IRR is really a borrowing rate and lower is better• Hence, the Minimum Acceptance Criterion is reversed: accept if the

IRR is smaller than the required return and reject otherwise

5-23

Mutually Exclusive vs. Independent• Mutually Exclusive Projects: only one of several potential projects can

be chosen, e.g., acquiring an accounting system or building different types of buildings on the same lot. – Rank all alternatives, and select the best one. How?– Choose the project with the highest IRR provided that the project’s IRR is larger

than the required return– Reject any project whose IRR does not exceed the firm’s required return– NPV and IRR can produce conflicting conclusions when choosing between

mutually exclusive projects but, when conflict occurs, the NPV criterion is generally superior

• Independent Projects: accepting or rejecting one project does not affect the decision of the other projects. Then IRR and NPV line up (provide the same result).– Accept all projects that meet the Minimum Acceptance Criterion (i.e., IRR>r)

5-24

Example of Mutually Exclusive Projects

• A firm is planning on buying a new machine. It only needs one such machine. The cost of capital is 12%. Which machine should be purchased?

Machine A Machine BCF0 -500 -1200C01 200 500C02 200 600C03 300 500• IRR=17.50% IRR=15.86%• Hence, IRR picks Machine A• But Machine A has NPV=$51.54 and Machine B has NPV $80.64• Hence, pick Machine B

5-25

Example of Independent Projects

• A firm has the following independent projects under consideration. Which should be taken? r=12%

• A B C D• CF0 -500 -1200 -200 -300• C01 200 500 100 25• C02 200 600 60 25• C03 300 500 80 400• IRR 17.50% 15.86% 10.18% 15.50%• NPV $51.54 $80.64 $(5.94) $26.96• Hence, undertake projects A, B and D

5-26

Problems Specific to Mutually Exclusive Projects

• Some specific conceptual problems with IRR arise when there are mutually exclusive projects:

• The Scale Problem: IRR ignores issues of scale• The Timing Problem

5-27

The Scale Problem• e.g. Choose one of the two fictitious opportunities (with r=0%

because of the quick turnaround):

• You invest $1 at start of class, I pay you $1.50 at end of class. IRR =50%, NPV=$.50

• You invest $10 at start of class, I pay you $11 at end of class. IRR =10%, NPV=$1

• If you followed IRR, you would choose the first project even though the NPV of the second opportunity is higher (and we should choose the higher NPV opportunity with mutually exclusive projects). Put it simply, the higher return on opportunity one is offset by the fact that you can make more money with opportunity two on a bigger investment.

• Thus, IRR ignores the scale

5-28

The Timing Problem

0 1 2 3

$10,000 $1,000$1,000

-$10,000

Project A

0 1 2 3

$1,000 $1,000 $12,000

-$10,000

Project B

5-29

The Timing Problem (continued)• IRR: A: 16.04%, B: 12.94%• Pick A • NPV at 10%: A: $669, B: $751• Pick B • NPV at 15%: A: $109, B: -484• Pick A• The NPV of project B is higher with low discount rates, and the NPV of

project A is higher with high discount rates.• This is so because most of project A’s cash flows occur earlier than

project B’s, therefore, a low discount rate favors project B and a high discount rate favors project A (because we are implicitly assuming that the high early proceeds can be reinvested at that high rate)

• IRR DOES NOT capture the timing effects• The preferred project by NPV standards depends on the discount rate

5-30

The Timing Problem

($5,000.00)

($4,000.00)

($3,000.00)

($2,000.00)

($1,000.00)

$0.00

$1,000.00

$2,000.00

$3,000.00

$4,000.00

$5,000.00

0% 10% 20% 30% 40%

Discount rate

NP

V

Project A

Project B

10.55% = crossover rate

16.04% = IRRA12.94% = IRRB

The crossover rate is where two projects have the same IRR (and NPV). Crossover rate=IRR for project “A-B” or “B-A” (it does not matter)

5-31

Using IRR with Mutually Exclusive Projects

• One method that generally gets around these pitfalls of IRR (scale and timing problems) is:

• Obtain IRR on INCREMENTAL CASH FLOWS,• Called incremental IRR• Example: John Hollymovie is trying to choose

between big budget and low budget approaches to his next blockbuster, Revenge of the Cockroach. Discount rate = 20%. CFs in $ millions.

0 1 2 NPV IRR• Big Budget -25 35 20 18 83.6%• Low Budget -10 15 10 9.4 100

5-32

Using IRR with Mutually Exclusive Projects: Incremental IRR

View big budget as ADDITIONAL investment over low budget. So incremental cash flows are the CFs for the additional investment.

• Example: John Hollymovie is trying to choose between big budget and low budget approaches to his next blockbuster, Revenge of the Cockroach. Discount rate = 20%. CFs in $ millions.0 1 2 NPV IRR

• Big Budget -25 35 20 18 83.6%• Low Budget -10 15 10 9.4 100• Incremental CF -15 20 10 8.6 72.1

• I.e., incremental CF from big budget has positive NPV and IRR greater than discount rate (72.1%>20%)

5-33

Using IRR with Mutually Exclusive Projects: Incremental IRR

• The incremental IRR and the crossover rate are always the same

• This is so because the incremental IRR is the rate that causes the incremental cash flows to have zero NPV. The incremental cash flows have zero NPV when the two projects have the same NPV.

5-34

Multiple IRRs

There are two IRRs for this project:

0 1 2 3

$200 $800

-$200

- $800

($100.00)

($50.00)

$0.00

$50.00

$100.00

-50% 0% 50% 100% 150% 200%Discount rate

NP

V

100% = IRR2

0% = IRR1

Which one should we use?

5-35

IRR and Nonconventional Cash Flows

• When the cash flows change signs more than once, there is more than one IRR

• The number of IRRs is equivalent to the number of sign changes in the cash flows

• When you solve for IRR, you are solving for the root of an equation and when you cross the x-axis more than once, there will be more than one returns that solve the equation

• If you have more than one IRR, which one do you use to make your decision?

• Ignore IRR in this case and use NPV

5-36

Nonconventional Cash Flows (cont.)

• 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?• NPV = 132,000 / 1.15 + 100,000 / (1.15)2 – 150,000 / (1.15)3 – 90,000 =

1,769.54• Calculator: CF0 = -90,000; C01 = 132,000; F01 = 1; C02 = 100,000; F02 =

1; C03 = -150,000; F03 = 1; NPV; I = 15; CPT NPV = 1,769.54• If you compute the IRR on the calculator, you get 10.11% because it is

the first one that you come to.• So, if you just blindly use the calculator without recognizing the uneven

cash flows, IRR would say to reject, but NPV would say to accept.

5-37

NPV at Different Discount Rates (NPV Profile)

5-38

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

i.e., positive NPV if required return is between 10.11% and 42.66%. You should accept the project if the required return is between 10.11% and 42.66%

IRR = 10.11% and 42.66%

5-39

Modified IRR (MIRR)• One proposal to deal with multiple IRRs is to

compute IRR of “modified cash flows”• Discounting Approach (Method 1) – Discount

future outflows to present and add to CF0• Reinvestment Approach (Method 2) - Compound

all CFs except the first one, forward to end• Combination Approach (Method 3) – Discount

outflows to present; compound inflows to end• MIRR will be a unique number for each method,

but is difficult to interpret; discount/compound rate is externally supplied

5-40

Example of MIRR• Project cash flows: (Required rate of return is 11%)• Time 0: -$500 today; Time 1: + $1,000; Time 2: -$100• Method 1: Discount future outflows to present and add to CF0

Time 0: -$500 + (-$100/(1.11)2 ) = -$581.16Time 1: +$1,000Time 2: +$0• Use this method and IRR = 72.06965%• Method 2: Compound all CFs except the first one to endTime 0: -$500 Time 1: +$0Time 2: -$100 + $(1000 X 1.11) = $1010.00• Use this method and IRR = 42.1267%

5-41

Example of MIRR (continued)• Project cash flows: (Required rate of return is 11%)• Time 0: -$500 today; Time 1: + $1,000; Time 2: -$100• Method 3: Discount outflows to present; compound inflows

to endTime 0: -$500 + (-$100/(1.11)2 ) = -$581.16Time 1: +$1,000Time 2: +$0• Time 0: PV (outflows) = -$500 + -$100/(1.11)2 = -$581.16• Time 1: $0• Time 2: FV (inflow) = $1,000 x 1.11 = $1,110• MIRR: N=2; PV=-581.16; FV=1,110; I/YR = MIRR = 38.2%

5-42

To sum up: NPV versus IRR

• NPV and IRR will generally give the same decision.

• Exceptions:– Non-conventional cash flows – cash flow signs

change more than once– Mutually exclusive projects

• Initial investments are substantially different• Timing of cash flows is substantially different

5-43

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, you should always use NPV

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

5-44

5.6 The Profitability Index (PI)

• Future Cash Flows=Cash Flows Subsequent to Initial Investment

• Minimum Acceptance Criteria: – Accept if PI > 1

• Ranking Criteria: – Select alternative with highest PI

Investent Initial

FlowsCash Futureof PVTotalPI

5-45

The Profitability Index

• Disadvantages:– Problems with mutually exclusive investments

• Advantages:– May be useful when available investment funds

are limited (rank projects by PI, take highest first, etc.)

– Easy to understand and communicate– Correct decision when evaluating independent

projects

5-46

Example of PI

• Project 1: C0=-$20, CF1=$70, CF2=$10• Project 2: C0=-$10, CF1=$15, CF2=$40• r=12%• Project 1: PV of future cash flows= =70/1.12+10/(1.12)^2=70.5;

PI=70.5/20=3.53; NPV=50.5• Project 2: PV of future cash flows=45.3; PI=45.3/10=4.53;

NPV=35.3• If projects are independent: PI>1 iff NPV>0. Hence, accept an

independent project if PI>1 (i.e., accept both in this case)• If projects are mutually exclusive, NPV dictates that project 1 is

accepted because it has the higher NPV, but PI would lead to the wrong decision (i.e., would pick project 2)

5-47

5.7 The Practice of Capital Budgeting• The capital budgeting method companies are

using varies by industry.• The most frequently used technique for large

corporations is either IRR or NPV.• Over half of companies also use payback

method perhaps because of its simplicity.

5-48

Example of Investment Rules

Compute the IRR, NPV, PI, and payback period for the following two projects. Assume the required return is 10%. Year Project A Project B

0 -$200 -$1501 $200 $502 $800 $1003 -$800 $150

5-49

Example of Investment Rules

Project A Project BCF0 -$200.00 -$150.00

PV0 of CF1-3$241.92 $240.80

NPV = $41.92 $90.80IRR = 0%, 100% 36.19%PI = 1.2096 1.6053

5-50

Example of Investment Rules

Payback Period:Project A Project B

Time CF Cum. CF CF Cum. CF0 -200 -200 -150 -1501 200 0 50 -1002 800 800 100 03 -800 0 150 150

Payback period for project A = 1 year.Payback period for project B = 2 years.

5-51

NPV and IRR RelationshipDiscount rate NPV for A NPV

for B-10% -87.52 234.770% 0.00 150.0020% 59.26 47.9240% 59.48 -8.6060% 42.19 -43.0780% 20.85 -65.64100%0.00 -81.25120%-18.93 -92.52

5-52

Project AProject B

($200)

($100)

$0

$100

$200

$300

$400

-15% 0% 15% 30% 45% 70% 100% 130% 160% 190%

Discount rates

NP

V

IRR 1(A) IRR (B)

NPV Profiles

IRR 2(A)

Cross-over Rate

5-53

Summary – Discounted Cash Flow

• Net present value– Difference between market value and cost– Accept the project if the NPV is positive– Has no serious problems– Preferred decision criterion

• Internal rate of return– Discount rate that makes NPV = 0– Take the project if the IRR is greater than the required return– Same decision as NPV with conventional cash flows– IRR is unreliable with non-conventional cash flows or mutually exclusive

projects• Profitability Index

– Benefit-cost ratio– Take investment if PI > 1– Cannot be used to rank mutually exclusive projects– May be used to rank projects in the presence of capital rationing

5-54

Summary – Payback Criteria

• Payback period– Length of time until initial investment is recovered– Take the project if it pays back in some specified period– Does not account for time value of money, and there is an

arbitrary cutoff period

• Discounted payback period– Length of time until initial investment is recovered on a

discounted basis– Take the project if it pays back in some specified period– There is an arbitrary cutoff period

5-55

Quick Quiz• Consider an investment that costs $100,000 and has a cash

inflow of $25,000 every year for 5 years. The required return is 9%, and payback cutoff is 4 years.– What is the payback period?– 4 years– What is the NPV?– -$2758.72– What is the IRR?– 7.93%– Should we accept the project?– NO!

• What method should be the primary decision rule?• NPV

5-56

Quick Quiz

1. Which one of the following indicates a project has a rate of return that exceeds its required return?

a. a positive NPVb. a payback period that exceeds the required periodc. a PI less than 1.0d. a positive accounting rate of return

2. Which one of the following ignores the time value of money? a. net present valueb. internal rate of returnc. payback

5-57

Quick Quiz

1. Which one of the following indicates a project has a rate of return that exceeds its required return?

a. a positive NPVb. a payback period that exceeds the required periodc. a PI less than 1.0d. a positive accounting rate of return

2. Which one of the following ignores the time value of money? a. net present valueb. internal rate of returnc. payback