0. Net Present Value and Other Investment Rules
Chapter 5
Net Present Value and Other Investment Rules

1. 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
Now we are going to start talking about capital budgeting. While discussing key aspects of financial management, we talked about decisions regarding long term investments. Investment could be intangibles like human capital or tangible assets such as plant and equipment. This is called capital budgeting. One of the most important issues as they can determine a firm’s operations and products for years to come as these decisions could be hard to reverse. A firm can face a number of possible investments. Some are good for the firm and others are not. How should it go about deciding which one is a good investment? Now we learn how to decide which business ventures are worth taking?

2. Chapter Outline
5.1 Why Use Net Present Value? 5.2 The Payback Period Method 5.3 The Discounted Payback Period Method 5.4 The Internal Rate of Return 5.5 Problems with the IRR Approach 5.6 The Profitability Index 5.7 The Practice of Capital Budgeting

3. 5.1 The Net Present Value (NPV) Rule
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= /1.1=-$4.55<0 → do not invest. Alternative pays 10% while this project pays ( )/100=5%.
Ranking Criteria: Choose the highest NPV project
Note that although we add the initial investment, this value is a negative number.

4. 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
Note that the NPV recognizes the magnitude, risk, and timing of cash flows, which was an important description of why stock price maximization should be the primary corporate goal.

5. 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

6. 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!!!)
Click on the Excel icon to go to an embedded Excel worksheet that has example cash flows, along with the correct and incorrect ways to compute NPV. Click on the cell with the solution to show the students the difference in the formulas.
Again, note that when we add the initial investment, this is a negative number.

7. 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:

8. 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

9. 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.

10. 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.
Cash flows prior to the cutoff are implicitly discounted at a rate of zero, and cash flows after the cutoff are discounted using an infinite discount rate.

11. 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.
Similar advantages and disadvantages as standard payback method, with the exception that cash flows prior to the cutoff are not discounted at zero.

12. 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

13. IRR: Example Consider the following project: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

14. IRR: Example (continued)
If we graph NPV versus the discount rate, we can see the IRR as the x-axis intercept.
IRR = 19.44%

15. IRR: Example with Financial Calculator
Without a financial calculator this becomes a trial-and-error process
With a financial calculator:
Press CF and enter the cash flows as you did with NPV
Press IRR and then CPT

16. 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.

17. IRR: Second Example (continued)

18. 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.
Click on the Excel icon to go to an embedded spreadsheet so that you can illustrate how to compute IRR on a spreadsheet.

19. 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

20. 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

21. 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

22. 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)

23. 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 B CF C C C
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

24. Example of Independent Projects
A firm has the following independent projects under consideration. Which should be taken? r=12%
A B C D CF C C C
IRR % 15.86% 10.18% 15.50%
NPV $51.54 $80.64 $(5.94) $26.96
Hence, undertake projects A, B and D

25. 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

26. 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

27. The Timing Problem $10,000 $1,000 $1,000 Project A 0 1 2 3 -$10,000
$10, $1,000 $1,000
-$10,000
Project A
$1, $1, $12,000
-$10,000
Project B
The preferred project in this case depends on the discount rate, not the IRR.

28. 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
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

29. The Timing Problem 10.55% = crossover rate 12.94% = IRRB 16.04% = IRRA
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)
10.55% = crossover rate
16.04% = IRRA
12.94% = IRRB
The cross-over rate is the IRR of project A-B

30. 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 %
Low Budget

31. 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 %
Low Budget
Incremental CF
I.e., incremental CF from big budget has positive NPV and IRR greater than discount rate (72.1%>20%)

32. 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.

33. Multiple IRRs There are two IRRs for this project:
$ $800
-$200
- $800
Which one should we use?
100% = IRR2
It is good to mention that the number of IRRs is equivalent to the number of sign changes in the cash flows.
0% = IRR1

34. 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

35. 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 / ,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.

36. NPV at Different Discount Rates (NPV Profile)

37. NPV Profile
IRR = 10.11% and 42.66%
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%

38. 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
The alternative approach discussed in the text is to discount cash flows over a single period (or subset of time), then combine them as necessary to eliminate any sign differences. The approach presented in the slide is more generalized.

39. Example of MIRR Project cash flows: (Required rate of return is 11%)
Time 0: -$500 today; Time 1: + $1,000; Time 2: -$100
Method 1: Discount future outflows to present and add to CF0
Time 0: -$ (-$100/(1.11)2 ) = -$581.16
Time 1: +$1,000
Time 2: +$0
Use this method and IRR = %
Method 2: Compound all CFs except the first one to end
Time 0: -$500
Time 1: +$0
Time 2: -$100 + $(1000 X 1.11) = $
Use this method and IRR = %
We could have different rate for re-investment or the same rate.

40. Example of MIRR (continued)
Project cash flows: (Required rate of return is 11%)
Time 0: -$500 today; Time 1: + $1,000; Time 2: -$100
Method 3: Discount outflows to present; compound inflows to end
Time 0: -$ (-$100/(1.11)2 ) = -$581.16
Time 1: +$1,000
Time 2: +$0
Time 0: PV (outflows) = -$ $100/(1.11)2 = -$581.16
Time 1: $0
Time 2: FV (inflow) = $1,000 x 1.11 = $1,110
MIRR: N=2; PV= ; FV=1,110; I/YR = MIRR = 38.2%
Could have different rates for discounting and compounding. You get totally different MIRR and that is one shortcoming of this method.
Since we have required rate of return why not just calculate NPV. Also, this is not really internal rate of return as the answer is dependent on required rate of return.

41. 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

42. 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

43. 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

44. 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

45. Example of PI Project 1: C0=-$20, CF1=$70, CF2=$10
Project 1: PV of future cash flows= =70/ /(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)

46. 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.

47. 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 -$150 1 $200 $50 2 $800 $ $800 $150

48. Example of Investment Rules
Project A Project B CF0 -$ $ PV0 of CF1-3 $ $ NPV = $41.92 $90.80 IRR = 0%, 100% 36.19% PI =

49. Example of Investment Rules
Payback Period: Project A Project B Time CF Cum. CF CF Cum. CF Payback period for project A = 1 year. Payback period for project B = 2 years.

50. NPV and IRR Relationship
Discount rate NPV for A NPV for B -10% % % % % % % %

51. NPV Profiles NPV IRR 1(A) IRR (B) IRR 2(A) Project A Discount rates
$400
NPV
IRR 1(A)
$300
IRR (B)
IRR 2(A)
$200
$100 $0
-15% 0%
15%
30%
45%
70%
100%
130%
160%
190%
($100)
($200)
Project A
Discount rates
Cross-over Rate
Project B

52. 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

53. 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
There is an arbitrary cutoff period

54. Quick Quiz
Consider an investment that costs $100,000 and has a cash inflow of $25,000 every year for 5 years. The required return is 9%, and payback cutoff is 4 years.
What is the payback period?
4 years
What is the NPV? -$
What is the IRR?
7.93%
Should we accept the project?
NO!
What method should be the primary decision rule?
NPV
Payback period = 4 years
The project does not pay back on a discounted basis.
NPV =
IRR = 7.93%

55. Quick Quiz
1. Which one of the following indicates a project has a rate of return that exceeds its required return? a. a positive NPV b. a payback period that exceeds the required period c. a PI less than 1.0 d. a positive accounting rate of return 2. Which one of the following ignores the time value of money? a. net present value b. internal rate of return c. payback

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 NPV b. a payback period that exceeds the required period c. a PI less than 1.0 d. a positive accounting rate of return
2. Which one of the following ignores the time value of money?  a. net present value b. internal rate of return c. payback