ECON308 Quantitative Financial Economics
Lecture Notes Chapter Two



by


Dr. Ruijun Bu



Semester One


2

Economic Review of Capital Budgeting

Objectives

To understand the implication of Fisher separation theorem on corporate policies

To understand the correct definition of shareholders’ wealth

To understand the essential properties of the rules of investment decisions

To be able to use the correct capital budgeting techniques to make investment
decisions for the certainty case

To understand the reinvestment assumption in capital budgeting

3
Key Contents

Fisher Separation and Investment Decisions

Shareholder Wealth Maximization

Essential Properties for the Best Capital Budgeting Techniques

Capital Budgeting Techniques

Comparison of NPV with IRR

Reinvestment assumption in IRR, ROI, and GAR
4
Fisher Separation of Utility Preferences from Investment Decisions

We have seen from the previous session that if capital markets are perfect Fisher
separation theorem obtains. It follows that the investment decisions can be
delegated to managers. This is known as the unanimity principle.








Individual 2
w0
D
B
w1
C0
C1
Separation of Shareholder Preferences
from the Production/Investment
Decision
Individual 1
A
Capital Market Line
Slope = - (1 + r)

5
Recall that the optimal investment decision requires the investors to invest in
such an amount that they can get onto the highest capital market line, because
on the highest capital market line, they can respectively maximize their
individual utility by getting onto their respective the highest indifference curve.

It is important to understand that getting onto the highest capital market line
is equivalent to maximizing the present value of the shareholders’ wealth.

At the investment step, this rule is the same for all investors which is
independent of shareholders’ time preferences for consumption.

Once the shareholders’ wealth is maximized, in the decision step investors can
retrieve their individual shares and decide individually how much to
borrow/lend in order to maximize their individual utility.
6
Shareholder Wealth Maximization

Shareholders’ wealth is the discounted value of (after-tax) cash flows paid out
by the firm. After-tax cash flows available for consumption can be shown to be
the same as the stream of dividends paid to shareholders. The discounted value
of the stream of dividends is

     T
s
TT
T
t
t
s
t
t
t
s
t
k
CGDiv
k
Div
k
Div
S






 



 111
1
11
0


0
S is the present value of shareholders’ wealth;
s
k is the market determined
required rate of return on equity capital; TCG is the capital gain at time T
Note that it is a multi-period formula which assumes that future cash flows are
known for certain and that the market determined discount rate is constant.
7
Capital Budgeting Techniques

Having established that maximizing shareholders’ wealth is equivalent to
maximizing the discounted cash flows provided by investment projects, we now
turn our attention to a discussion of investment decision rules.

We assume for the time being that the stream of cash flows provided by a project
can be estimated without error (no uncertainty) and that the opportunity cost of
funds provided to the firm (cost of capital) is also known.

8
Three Major Problems for Managers When Making Decisions

 Search out new opportunities in the marketplace and develop new
technologies. These are the basis of growth. Theory of finance cannot help
with the problem!

 Expected cash flows from projects have to be estimated. Various forecasting
techniques may be used.

 Projects have to be evaluated according to sound decision rules, usually
referred to as capital budgeting techniques. This session investigates the
best capital budgeting technique.
9
Essential Properties for the Best Capital Budgeting Techniques

The best technique will possess the following essential property: It will
maximize shareholders’ wealth. This essential property can be broken into
separate criteria:

 All cash flows should be considered.

 Cash flows should be discounted at the opportunity cost of funds.

 The technique should select, from a set of mutually exclusive projects, the
one that maximizes shareholders’ wealth.

 Managers should be able to consider one project independently from all
others. This is known as the value-additivity principle.
10
Value-additivity Principle: The value of the firm is the sum of the value of
separate projects accepted by management. It means that projects can be
considered on their own merits without the necessity of looking at them in a
large number of different combinations.

Four Widely Used Capital Budgeting Techniques

 the payback period method

 the accounting rate of return

 the net present value

 the internal rate of return

11
An Examination of the Four Methods

The following table lists the estimates of cash flows for four mutually exclusive
projects, each of which has a five-year life.

Four Mutually Exclusive Projects
Cash Flows
Year A B C D PV Factor at 10%
0 -1000 -1000 -1000 -1000 1.000
1 100 0 100 200 0.909
2 900 0 200 300 0.826
3 100 300 300 500 0.751
4 -100 700 400 500 0.683
5 -400 1300 1250 600 0.621
12
The Payback Method

The payback period for a project is simply the number of years it takes to recover
the initial cash outlay on a project. The payback periods for the four projects are:
Project A, 2 years; Project B, 4 years; Project C, 4 years; Project D, 3 years.

If management were adhering strictly to the payback method, it would choose
project A, which has the shortest payback period. Casual inspection of the cash
flows shows that this is clearly wrong. The difficulty with the payback method
is that it does not consider all cash flows and it also ignores the time value of
money. We reject the payback method because it violates at least the first two of
the four properties.


13
The Accounting Rate of Return

The accounting rate of return (ARR) is the average after-tax profits divided by
the initial cash outlay. It is very similar to (and in some uses exactly the same
as) the return on assets (ROA) or the return on investment (ROI).

 The first deficiency of ARR is that it uses after-tax profits rather than cash
flows. The after-tax profit is an accounting definition of profit which
considers, for instance, depreciation which is not flow of funds in real sense.

 Secondly, it also ignores the time value of money, even if it uses the correct
streams of cash.

14
Assume for conveniences that the numbers in the table are after-tax profits
(which in general would be different from cash flows). Then the average after-
tax profits for A would be

80
5
4001001009001001000




and the ARR would be

%8
1000
80





outlayInitial
profittaxafterAverage
ARR


15
The ARRs for the four projects are

Project A, ARR = - 8 %;
Project B, ARR = 26 %;
Project C, ARR = 25 %;
Project D, ARR = 22 %;

If the managers were using the ARR, they would choose project B as the best.
The problem with the ARR is that it uses accounting profits instead of cash flows
and does not consider the time value of money. It at least violates criteria one
and two.


16
Net Present Value

The net present value (NPV) is computed by discounting the cash flows at the
firm’s opportunity cost of capital. Mathematically,

 
0
1 1
I
k
FCF
NPV
N
t
t
t 

 



where
t
FCF is the free cash flow in time period t;
0
I is the initial cash outlay;
k is the firm’s cost of capital and N is the number of years in the project. The
NPV criterion will accept projects that have an NPV greater than zero.


17
Assuming that the cost of capital is 10%, the NPVs for the four projects are:

Project A, NPV = -407.30;
Project B, NPV = 510.70;
Project C, NPV = 530.85;
Project D, NPV = 519.20;

If these projects are mutually independent, we would select projects B, C and D,
because positive NPV increases the shareholders’ wealth. Since they are
mutually exclusive, we select the project with the greatest NPV, project C. The
NPV of the project is exactly the same as the increase in shareholders’ wealth.
This fact makes it the correct decision rule for capital budgeting purposes.


18
Internal Rate of Return

The internal rate of return (IRR) on a project is defined as the rate which equates
the present value of the cash outflows and cash inflows. In other words, it is the
rate that makes the computed NPV exactly zero.

Mathematically, we solve for the rate of return where the NPV equals zero:

 
0
1
0
1


 

I
IRR
FCF
NPV
N
t
t
t


We can solve for the IRR on a project by either analytically or numerically.

19
The figure below gives a graphical example of how to solve for the IRR for
Project C, where it shows that when IRR = 22.8 %, the NPV is equal to zero.




















IRR = 22.8%
10% 20% 30%
500
1000
1500
NPV
Discount Rate
20
The IRRs for the four projects are

Project A, IRR = -200%;
Project B, IRR = 20.9%;
Project C, IRR = 22.8%;
Project D, IRR = 25.4%;

If we use the IRR criterion and the projects are independent, we accept any
project that has an IRR greater than the opportunity cost of capital, which is
assumed to be 10% in the example. Therefore, we would accept projects B, C,
and D. However, since these projects are mutually exclusive, the IRR rule leads
us to accept project D as the best.

21
The NPV and the IRR favor conflicting project choices. Both techniques
consider all cash flows and time value of money.

However, we must choose from among the four mutually exclusive projects the
one that maximizes shareholders’ wealth.

Consequently, only one technique can be correct. We shall see that the NPV
criterion is the only one that is necessarily consistent with maximizing
shareholders’ wealth, given current set of assumptions.




22
Comparison of NPV with IRR

The IRR rule errs in several ways:

 Firstly, it compares alternative projects by looking at the rate of return only
but not the actual value they can bring to the shareholders.

 Secondly, the IRR rule assumes that funds invested in the projects have
opportunity costs equal to the IRR for the project.

 Thirdly, it does not obey the value-additivity principle. Consequently
managers using IRR cannot consider projects independently of each other.

 Finally, the IRR rule can lead to multiple rates of return if the sign of cash
flows change more than once.
23
Rate of Return versus Value of Project

The IRR compares alternative projects by considering only the rate of return
from projects. It ignores the fact that it is possible that a large project with
relatively low rate of return may offer higher total value than a high return
project which has small scale. Consider the following two projects.

Year
Project
A
Project
B
0 -10 -1000
1 100 10000


24
It can be easily verified that both projects would offer the same IRR = 900%. If
we were using the IRR rule Project A and B would be equally preferred.
Obviously, Project B would add much more value to the firm than Project A.
The NPV rule will choose Project B as long as the cost of capital is less than
900%. For instance, when the opportunity cost of capital is 10%, the NPVs for
project A and B are 80.91 and 8091.91, respectively.







25
The Reinvestment Rate Assumption

The following figure compares projects B, C and D under different discount
rates.

 For very low discount rates,
project B has the highest NPV;

 For intermediate discount rates,
project C has the highest NPV;

 For high discount rates,
project D has the highest NPV.

10% 20% 30%
500
1000
1500
NPV
Discount Rate
B
C
D
B
C
D 0
- 500
26

Important implication is that using different discount rates will result in different
choices even though the NPV criterion is used. Therefore, it is crucial when
evaluating a project the correct discount rate is used.

The NPV compares the projects at correct discount rate which is the market-
determined opportunity cost of capital. It assumes that time value of money is
the opportunity cost of capital determined by the slope of the capital market line.
This assumption has come to be called the reinvestment rate assumption.

The IRR rule does not discount cash flows at the opportunity cost of capital.
Instead, it implicitly assumes that the time value of money is the IRR of the
project itself, because all cash flows are discounted at this rate.

27
The correct interpretation for the reinvestment rate is that it is really the same
thing as the opportunity cost of capital. Both the NPV rule and the IRR rule
make implicit assumptions about the reinvestment rate. The NPV rule assumes
that shareholders can reinvest their money at the market-determined opportunity
cost of capital. The NPV is making the correct assumption. The projects in our
examples have the same risk (in fact no risk) therefore the cash flows of all
projects should be discounted at the same rate (10%).

The IRR rule assumes that investors can reinvest their money at the same rate
as the current project is offering!!! Therefore, in our example, it assumes that
shareholders can reinvest funds in project B at 20.9%, in project C at 22.8% and
in project D at 25.4%.

28
This assumption errs in the following two respects.

First of all, there are no such identical projects for the firm to ‘reinvest’ in. It
defies logic to assume that we can reinvest in an identical project whenever we
are evaluating one project.

Secondly, we have been told that all projects have the same risk (in fact no risk
in our example). Therefore their cash flows should be discounted at the same
rate, market determined opportunity cost. The IRR rule defies logic again by
assuming they should be discounted at different rates.

In summary, although the IRR does discount cash flow, it does not discount them
at the opportunity cost of capital. It violates the second of the four properties
and it also violates the Fisher separation theorem.
29
The Value-Additivity Principle

The value-additivity principle demands that managers be able to consider one
project independently of all others. It implies that the value of the firm is equal
to the sum of the values of each of its projects. In other words, same conclusion
should be made whether or not the management consider project independently
or collectively.

An Example of Value-Additivity

Consider the following example where project 1 and 2 are mutually exclusive
and project 3 is independent of both project 1 and 2. The cash flows as well as
the NPVs and IRRs for all possible combinations of investment projects are
given in the table below
30

Cash flows, NPVs, and IRRs for Five Possible Situations











Year 1 2 3 1+3 2+3
0 -100 -100 -100 -200 -200
1 0 225 450 450 675
2 550 0 0 550 0
NPV at
10%
354.55 104.55 309.09 663.64 413.64
IRR 134.52% 125.00% 350.00% 212.89% 237.50%
31
If we consider them independently, the NPV rule (assuming opportunity cost of
capital is 10%) will choose project 1 and 3 because NPV for project 3 is positive
and NPV for project 1 is greater than that of project 2. The IRR rule will also
choose project 1 and 3, because the IRR for project 3 is greater than the
opportunity cost of capital and the IRR for project 1 is greater than that of project
2.

If we alternatively consider possible combinations of projects, the NPV as
before will choose project 1+3 because it provides the highest NPV, however
the IRR will choose project 2+3 because when considered collectively project
2+3 provides higher IRR than project 1+3. Obviously, the NPV obeys the value-
additivity principle but the IRR does not.

32
Multiple Rate of Return

Another difficulty of the IRR rule is that it can result in multiple rates of return
if the stream of estimated cash flows changes signs more than once. Consider
the following example
Year Cash Flow
0 -1600
1 10000
2 -10000

     
0
1
10000
1
10000
1
1600
210









IRRIRRIRR
NPV

%400%25 orIRR 
33









The IRR rule would accept this project if the opportunity cost of capital is 10%,
but the NPV would reject it, since

     
55.773
%101
10000
%101
10000
%101
1600
210








NPV
Discount Rate
400%
500
1000
NPV
0
- 500
- 1000
- 1500
300% 200% 100%
34
Solution to Multiple Roots Problem

The investment process can be regarded as a process of fund exchanges between
investor and project in multiple periods. This is in the sense that investor puts
funds into project and receives funds from project at different times.

In such a process, investor lends funds to project expecting to earn a rate of
return on investment, R (to be determined), and the project lends funds to
investor at opportunity cost of capital, k (market-determined). The correct
value of R is such that at the end of the project no liability exists between
investor and the project.

Consider the above example once again in following steps. Pay attention to the
firm’s positions in each step.

35
Position One:
At the beginning, the firm invested (lent) 1600 to the project and expected to
earn a rate of R at the end of the first period. In this position, the firm lent to
project and mathematically it expected to earn  R11600 .

Position Two:
At the end of the first period, firm received 10000 from project. This 10000
covers: (1) what the firm has expected to earn from the initial 1600; (2) the
surplus funds  R 1160010000 is the amount that project lent to the firm at
the opportunity cost of capital assumed here to be k = 10%. Thus, the firm is
now in a position where it has received funds lent by project, and it is now
expected to pay back    10.011160010000  R at the end of the second
period to make present value of fund exchanges between two sides is zero.
36
Position Three
At the end of the second period, the firm actually put (paid) back 10000 to
project. Therefore, the correct the rate of return R should be such that
    1000010.011160010000  R . It is unique and equals -43.18%.

This way of looking at cash flows of project solves the multiple roots problem
and reveals the correct Rate of Return on Investment (ROI). Since the cash
flows lent to the firm provide a known rate of return (opportunity cost of capital),
it is possible to isolate the rate of return on funds invested in project. We should
reject the project since the return is less than the opportunity cost of capital. Rate
of return on investment calculated in this way gives the same conclusion as the
NPV rule does, but it still violates the value-additivity principle. Thus, the NPV
is still the only correct rule for investment decision making.
37
Summary

In frictionless capital markets, Fisher separation theorem implies that
investment decisions can be delegated to managers whose task is to maximize
shareholders’ wealth, which is equal to the discounted value of future cash flows
paid out by the firm.

The best capital budgeting techniques should consider all cash flows, use the
opportunity cost of capital to discount cash flows, be able to select from
mutually exclusive projects the one that maximizes shareholders’ wealth, and
allow the managers to be able to consider one project independently from all
others.

38
Among the four widely used capital budgeting techniques (payback method,
ARR, NPV and IRR), only the NPV is the correct decision rule which satisfy all
four conditions.

The IRR rule uses incorrect assumption of the reinvestment rate, violates the
value-additivity principle, and may have multiple solutions.


To be continued……