Net Present Value
Profile
A graph showing the
relationship between a
project's
NPV
and
the
firm's cost
of
capital.

384
Part 4
Investing
in
Long-Term
Assets:
Capital Budgeting
Crossover Rate
The
cost
of
capital
at
which the
NPV
profiles
of
two
projects
cross
and,
thus,
at
which the proj-
ects
'
NPVs
are equal.
•
•
Project 8
NPV:
-$1,000
+
$1
00
/(
1.1
0)
1
+
$1
,300
/(
1.1
0)
2
=
$165
.
29
.
Alternatively, enter the
cash
flows into the financial calculator
as
follows:
CF
0
=
-1000;
CF
1
=
1 00;
CF
2
=
1300;
1/YR
=
1
0;
•
NPV
=
$165.29
.
IRR
: Enter the
cash
flows into the financial calculator
as
follows:
CF
0
=
-1000;
CF
1
=
100;
CF
2
=
1300;
.I
RR
=
19.13%
.
MIRR:
0
2
-$
1,000
$100
$1,300
I
X
1.10
•
$
110
- $1,000
TV= $1,410
Using a financial calculator, enter the following data: N
=
2;
PV
=
-1000;
PMT
=
0;
FV
=
141
0; and solve for
1/YR
=
MIRR
=
18.74%
.
b.
Here's a summary
of
the results. The project chosen under each method
is
highlighted.
Project A
Project B
NPV
$128.10
1
s
165
.2
91
IRR
123.12%
1
19.13%
MIRR
16.83%
1 18.74
%1
Using the
NPV
and
MIRR
criteria, you would select Project
B;
however,
if
you
use
the
IRR
criteria you would select Project
A.
Because
Project B adds the most value
to
the
firm, B should be chosen.
The IRRs are fixed,
and
S has the
higher
IRR regardless of the cost of capital.
However,
the NPVs
vary
depending
on
the actual cost of capital.
The two NPV profile lines cross
at
a cost of capital of 11.975%, which is
called the
crossover rate.
The crossover rate can be found
by
calculating the
IRR of the differences in the projects' cash flows, as
demonstrated
below:
0
2
3
4
ProjectS
-$1,000
$500
$400
$300
$1
00
-Project
L
-$1,000
$100
$300
$400
$675
6
=
CFs-
CFL
$
0
$400
$100
-$100
-$575
IRR
6
=
11.975%
=
Crossover rate
Project L
has
the
higher
NPV if the cost of capital is less
than
the crossover
rate,
butS
has the
higher
NPV if the cost of capital is greater
than
that rate.
Notice
that
Project L has the steeper slope, indicating that a given increase in
the cost of capital causes a larger decline in NPVL
than
in NPV
s.
To see
why
this is
so, recall that L's cash flows come in later
than
those of
S.
Therefore, L is a long-
term project
and
S is a short-term project. Next, recall the equation for the NPV:

Chapter 1 1
The
Basics
of
Capital Budgeting
385
FIGURE
11.5
NPV
Profile
for
Project
S
NPV
($)
5
At
r=
10
%,
NP
V > 0,
NPV=
0,
so
IRR
,=
14.489%
IRR
>r=
10
%,
so
accept
I
I
I
I
20
Cost
of
Capital
(%)
Cost
of
Capital
NPV
5
0%
$300.00
5
180.42
10
78.82
IRR
s = 14.489
0.00
15
-8.33
20
- 83.72
CF
1
CF
2
CFN
NPV =
CF
0
+---+---+
···
+---
(
1
+
r)
1
(
1
+
r)
2
(
1
+
r)N
Now
recognize that the impact of
an
increase
in
the cost of capital is
much
greater
on
distant
than
near-term cash flows, as
we
demonstrate here:
Effect
of doubling
ron
a
Year
1
cash
flow:
.
$100
PV
of
$100
due m
1
year
@ r
=
5%
:---
1
=
$95.24
(1.05)
.
$100
PV
of
$100
due m
1
year
@ r =
10% :
---
1
=
$90.
91
(1.10)
d
I
.
d
h' h
$95.24 -
$90.
91
Percentage
ec
me
ue to
19
er r
=
$
=
4.5%
95.24
Effect
of doubling
ron
a
Year
20
cash
flow:
.
$100
PV
of
$100
due m
20
years @ r =
5% :
20
=
$37.69
(1.05)
.
$100
PV
of
$100
due m
20
years @ r =
10
%:
20
=
$14.86
(1.10)
P
d
r
d
h' h
$37.69 -
$14.86
ercentage
ec
me
ue to
19
err
=
$
37
.
69
=
60.6%
Thus, a
doubling
of the discount rate results in only a 4.5% decline in the PV of
a Year 1 cash flow,
but
the same discount rate increase causes the PV of a Year
20
cash flow to fall
by
more
than
60%.