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4.CapitalbudgetingII.pptx4. Capital budgeting II: Investment decision rules
Chapter 10
Previously,
Where cash flows come from for an investment opportunity
Some accounting basics
Financial reports
Important accounting concepts and items: depreciation, tax, etc.
NPV of each project given cash flows and discount rate
Now
Investment decision rules: rules we use to decide whether to take a project
More importantly, which one to take facing constraints?
Outline
For one project, how to decide whether to accept it or not?
NPV (Net Present Value)
IRR (Internal Rate of Return), MIRR (Modified Internal Rate of Return)
Payback period
For mutually exclusive projects, how to decide which one to take?
Highest NPV
Highest IRR? – not necessarily
Diff in scale
Diff in cash flow patterns
What if resources are limited?
PI (Profitability Index)
1. When deciding whether to take a single project
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Take it or leave it? – Rule #1: The Net Present Value (NPV) Rule
Two ways to think about the NPV:
1. Think of the project as an asset that generates a stream of cash flows (either + or -)
NPV = sum of PV from each piece
2. Think of the project in a costs-and-benefits way
NPV = PV(Benefits) – PV(Costs)
The difference between the PV of benefits and PV of costs from a project
Decision?
Invest when NPV is positive! (NPV > 0)
Don’t invest otherwise
If can invest in multiple projects at the same time assuming unlimited resources
Choose all NPV>0 ones!
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The Net Present Value (NPV) Rule
Example
Using a 10% discount rate,
Therefore, invest!
t = 0
1
2
3
28M
4
-$81.6M
28M
28M
28M
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The Net Present Value (NPV) Rule
Example 3
You own a piece of land near UCR. You are considering to open a bar that costs $200,000 and will generate $40,000 for 10 years. At the end of the 10th year, you expect to sell it to a large coffee franchise for $330,000. Similar coffee shops have a cost of capital of 10%.
In excel, use =PV(0.1,10,-40000,-330000)-200000
Should you invest? Yes
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NPV, discount rates, & IRR
NPV is basically a present value calculation
Hence it changes when the discount rate changes
The internal rate of return (IRR) is the discount rates that makes NPV=0 IRR is the solution for r when we set NPV=0
t = 0
1
2
3
28M
4
-$81.6M
28M
28M
28M
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NPV, discount rates, & IRR
Measuring sensitivity with IRR
In order to compute NPV, you need an estimate for the discount rate, or cost of capital
But it will only be an estimate
Important to know how sensitive your analysis and decision will be to errors in this estimate
IRR can help
Look at how far your estimate of discount rate is from the IRR
If it’s too close, then your decision to invest could be flawed!!
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Rule #2: The Internal Rate of Return (IRR) Rule
If you can choose multiple projects
Invest in project whenever IRR exceeds the cost of capital
What does this mean?
The return on the investment is greater than that of other alternatives with equivalent risk and maturity (textbook)
Another way to think about it: only choose projects with return higher than what is demanded by investors (my understanding)
In a lot of cases (but not all), it also means NPV > 0
But doesn’t always work!
For it to work, NPV has to be downward sloping with respect to the discount rate (Safely so when first cash flow is an outlay, and all future cash flows are positive)
But there are some weird cases
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The Internal Rate of Return (IRR) Rule
Delayed investments (outflows)
When all the benefits (inflows) from an investment occur at time 0 and before the costs (outflows), NPV is upward sloping in the discount rate
t = 0
1
2
3
$1,000,000
500,000
500,000
500,000
Invest
Not Invest
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The Internal Rate of Return (IRR) Rule
Multiple IRRs
When the signs of cash flows change more than once, multiple IRRs can occur (multiple roots)
t = 0
1
2
3
$1,000,000
500,000
500,000
500,000
10
600,000
…
Invest
Not Invest
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How do we decide the number of roots (IRRs) from cash flow pattern?
Single or multiple?
Hard to tell multiple roots but easy to identify single root
2 cases for sure: NPV is either strictly upward sloping or downward sloping
Why multiple IRRs?
Let’s have some math fun
Think of NPV as a polynomial function of discount rate r with all cash flows as constants:
If we want to see how NPV changes with r, we take the first-order derivative of NPV by r:
Therefore, does not matter anymore in deciding the sign of the first-order derivative!
When to are all positive at the same time, ; NPV is strictly decreasing in r; single IRR!
When to are all negative at the same time, ; NPV is strictly increasing in r; single IRR!
When the sign of to are not consistent, NPV is not monotonic in r, meaning we may have multiple roots (IRRs).
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The Internal Rate of Return (IRR) Rule
Ex. Single or multiple?
When given a cash flow stream, you should have a sense if it has single or multiple IRRs, and whether we can directly use IRR (or MIRR in the next slide)
t = 0
1
2
3
$1 million
500
325
1.5
10
400,000
…
$1
$1
- $1
$1
- $1
$1
- $3
$0
$1
$1
Proj. A
Proj. B
Proj. C
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The Modified Internal Rate of Return (MIRR) Rule
Multiple IRRs
Modified IRR
Discount all negative cash flows to the present so that there is one negative cash flow at the beginning, and compound all positive cash flows afterwards to the future
t = 0
1
2
-$1,000
2,500
1,540
(1)
(2)
(1)
(2)
Not Invest
Invest
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Rule #3: The Payback Period Rule
A naïve rule
An opportunity that at least pays back its initial investment is good
How long does the investment take to pay back its initial investment?
Calculate payback period
Accept (reject) project if payback period is less (greater) than some pre-specified amount of time
Example 1
Payback period: 3 years
If you require a payback period of 2 years or less, you reject this project. Is this sound?
t = 0
1
2
3
M
4
-$10M
M
M
M
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The Payback Period Rule
Obvious flaws
Ignores time value of money
Discounted payback period rule: Compute payback period based on discounted cash flows
4 years in the previous example, using 10% as discount rate
Ignores cash flows after the payback period
Lacks decision criterion grounded in economics
Ex) What’s the right number of years one should require?
But it IS simple…
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2. When deciding mutually exclusive projects
When we choose one from many…
Mutually exclusive: meaning by picking one project to invest, we are rejecting all others
Example: the design for a company logo, the usage of a piece of land
NPV rule is simple: pick the biggest NPV
IRR rule is not the case – the highest IRR is NOT necessarily our best choice!
Effects of scale: IRR is a return while NPV is a dollar amount. When you compare two mutually exclusive projects, you want a bigger dollar impact on value
Effects of cash flow timing: A higher IRR over a short period of time could add less value than a lower IRR over a longer period of time
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Choosing from mutually exclusive projects: NPV vs. IRR
Differences in scale
t = 0
1
2
3
-$10,000
5,000
5,000
5,000
6,000
6,000
6,000
,
,
t = 0
1
2
3
-$10,000
5,000
5,000
5,000
25,000
25,000
25,000
,
,
Discount rate = 12%
Discount rate = 12%
-$10,000
-$50,000
(1)
(2)
(1)
(2)
(1)
(2)
(1)
(2)
(1)
(2)
(1)
(2)
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The Internal Rate of Return (IRR) Rule
Timing of cash flows
t = 0
1
2
…
25,000
25,000
25,000
-$50,000
…
65,000
-$50,000
(1)
(2)
(1)
(2)
Discount rate=12%
IRR=23.4%, NPV=$10,046
IRR=30%, NPV=$8,036
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Reality
Graham and Harvey (2001) CFO Survey
Surveys 400 CFOs
Around 75% use both NPV and IRR
Around 50% use the payback rule!!
Why??
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Reality
Most of time, NPV and IRR are substitutable.
But in reality, NPV requires a lot of assumptions
It requires we know the discount rates at each period
Simpler rules require less
IRR doesn’t require estimate for cost of capital to compute it
Payback rule doesn’t require estimate for cost of capital, nor cash flow estimates far into the future
Nonetheless, NPV has the distinct advantage of being much more informative and reliable
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IRR vs. NPV: Recap
So why do financial managers still use IRR so often?
When resources are limited, percentage returns can be more straightforward and intuitive to understand than NPV
50% return vs. NPV=$500,000
IRR provides a break-even cost of financing (the cross-over point in a NPV-discount rate graph)
The number itself is of interest to financial managers
“At what financing cost does the project break-even?”
It serves as a benchmark to evaluate how sensitive your estimate for the cost of capital is
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3. When deciding projects facing resource constraints
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Limited Resources and Investment Decisions
Even when resources are limited, we can use NPV to make investment decisions, much like IRR
Assume as a financial manager, you are working with a budget of $200M
Which project should you choose?
NPV rule: Project A
But this would use up all your resources
You can actually do better by choosing B and C
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Limited Resources and Investment Decisions
Profitability Index (PI) rule
Simply a rate of return on investment
Rank projects in order according to PI
Choose the highest ones until resources are depleted
You would first choose C, then B
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Limited Resources and Investment Decisions
Example 1
You have 7 projects with positive NPV to invest in. You have 190 employees to allocate in each project. Each employee can only work on one project, and you can hire no more of them. Given the projects’ NPVs and employee requirements, which projects should you choose to invest in?
| Project | NPV ($ millions) | Employee Headcount (EHC) |
| 1 | 17.7 | 50 |
| 2 | 22.7 | 47 |
| 3 | 8.1 | 44 |
| 4 | 14.0 | 40 |
| 5 | 11.5 | 61 |
| 6 | 20.6 | 58 |
| 7 | 12.9 | 32 |
| Total | 107.5 | 332 |
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Limited Resources and Investment Decisions
Example 1
You have 7 projects with positive NPV to invest in. You have 190 employees to allocate in each project. Each employee can only work on one project, and you can hire no more of them. Given the projects’ NPVs and employee requirements, which projects should you choose to invest in?
| Project | NPV ($ millions) | Employee Headcount (EHC) | Profitability Index (NPV per EHC) |
| 1 | 17.7 | 50 | 0.354 |
| 2 | 22.7 | 47 | 0.483 |
| 3 | 8.1 | 44 | 0.184 |
| 4 | 14.0 | 40 | 0.350 |
| 5 | 11.5 | 61 | 0.189 |
| 6 | 20.6 | 58 | 0.355 |
| 7 | 12.9 | 32 | 0.403 |
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Limited Resources and Investment Decisions
Problems with the Profitability Index (PI) rule
Sometimes it might make sense to use up what’s left after choosing high PI projects on a low PI one
Example
A firm faces three investment opportunities:
A: NPV = 3, INV = 1, PI = 3
B: NPV = 2, INV = 2, PI = 1
C: NPV = 2.5, INV = 3, PI = 0.803
Given a $4 investment resources, which one should the firm take?
A and C
Not so simple when there are multiple resource constraints
But that’s what computers are for
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Investment Decision Rules: Recap
| Criteria | Strength | Weakness |
| IRR | Gives same answers as the NPV rule whenever the NPV of a project is a smoothly declining function of the discount rate. | Doesn't hold when the NPV of a project is not a declining function of the discount rate. (When CF signs change.) Could have multiple or no IRR. Unreliable in ranking projects with different scales & different pattern of CFs. Hard to deal with the changes in the opportunity costs. (When the term structure of interest matters.) |
| Payback Period | Easy to calculate. Gives insights in terms of how fast the investment could be recovered. | Ignores CFs after the cutoff. Ignores time value of money. Tends to accept poor short-lived projects. |
| Discounted Payback | Strengths of pure payback period method. Take the time value of money into account. | Still Ignores CFs after the cutoff. |
| PI | Incorporates the benefits of NPV rule Also takes into account the rate of return when resources are limited | Occasionally fails to fully utilize resources More complicated under multiple resource constraints |
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