Drop
A, and lose $9 in NPV
Drop
B, and lose $8
Drop
C and D, and lose 11
Drop
D and E, and lose 7.5
Solutions
to Practice Problems – Chapter 5
1.
First compute IRR for each project.
Any project with IRR < required return can be dropped right away. Only projects have IRR > required return need to be
compared by looking at the IRR of incremental cash flows.
Project A : IRR is given
by: -100 +
65/(1+IRR) + 52/(1+IRR)2
= 0
Using
a financial calculator, IRR = 11.6% < 12%
=> A
has a negative NPV and need not be considered further
Project B : IRR is given
by: -99.2 +
67.4/(1+IRR) + 50.6/(1+IRR)2
= 0
Using
a financial calculator, IRR = 13.1% > 12%
=> B
has a positive NPV; it stays in the running.
Project C : IRR is given
by: -50 +
32/(1+IRR) + 31/(1+IRR)2
= 0
Using
a financial calculator, IRR = 17% > 12%
=> C
has a positive NPV; it stays in the running.
So now we need to compare B and C.
The incremental cash flows for B - C are conventional cashflows:
-49,200, 35,400 and 19,600
IRR is given by:
- 49.2 + 35.4/(1+IRR) + 19.6/(1+IRR)2 = 0
Using a financial calculator, IRR = 8.6% < 12%
Since the IRR for B – C is less than the required return, the NPV of B
– C is negative. B isn’t better
than C, and so we should accept C.
2
a) The IRR for project A is the discount rate that solves the equation:
-250 + 149/(1+IRRA) + 136/(1+ IRRA)2 = 0
Using
a financial calculator (or the formula for a quadratic equation), IRRA
= 9.349%
Similarly,
for B: -200 +
120/(1+IRRB) + 125.6/(1+ IRRB)2 = 0
=> IRRB = 14.735%
For
C: -300 +
150/(1+IRRC) + 250/(1+ IRRC)2 = 0
=> IRRC = 19.649%
For
D: -480 +
370/(1+IRRD) + 220/(1+ IRRD)2 = 0
=> IRRD = 16.444%
b) At a required return of 8%, both projects have a positive
NPV. (Since they have conventional
cashflows, and IRR > required return). Check
if the NPV curves intersect:
Sum of A's cashflows = -250
+ 149 + 136 = 35
Sum of B's cashflows = -200
+ 120 + 125.6 = 45.6.
Since the project with the
higher IRR has the higher y-intercept, the NPV curves do not intersect.
So we can choose project B.
c) Both projects have conventional cashflows.
Project D has an IRR less than the required return of 17% => D has a
negative NPV. Project C has an IRR
greater than the required return of 17% => C has a positive NPV. Therefore, accept project C.
d) At a required return of 10%, both projects have a positive NPV.
Consider the incremental cashflows (D - C):
C0
C1
C2
D - C
-180,000 220,000
-30,400
These
are non-conventional cashflows. IRR
is given by
-180 + 220 - 30 =
0
=>
-18x2 + 22x - 3 = 0
x
x2
=>
x = -22 ±
[222 - 4*18*3]0.5
=>
x = 1.0659 or 0.1564
-36
Thus,
IRR = 6.59% and -84.36% (which is ignored, since it is negative)
If
we had conventional cashflows, we could say this is a lending situation, IRR
< required return, so NPV of D – C is negative.
But since the cashflows are non-conventional, all bets are off.
We do not know whether it is good or bad for IRR to be less
than required return. Have to
consider what the NPV curve may look like. It
intersects the x-axis at only one positive discount rate; the question is does
it intersect from above or from below?
The
sum of the cashflows = 10,000 => the y-axis intercept of the NPV curve is
positive => it intersects the x-axis from above.
Since it intersects the x-axis at 6.59%, the NPV at 10% will be negative.
So the NPV of D-C is negative => NPV of C > NPV of D. Hence, we should choose C.
3) IRR comes from solving the following equation:
-50 + 29/(1+IRR) + 10.4/(1+ IRR)2 =
0
=>
50 x2 - 29 x - 10.4 = 0
(where x = 1 + IRR)
x = 29 ±
[292 - 4 * 50 * (- 10.4)]0.5
100
=>
x = -0.2505 and 0.8305
=>
IRR = -125.05%, or –16.95%
Both IRRS are negative, which means IRR does not exist.
The NPV curve for A-B does not intersect the x-axis at any positive
discount rate. Summing up the
cashflows shows that at a discount rate of zero, the NPV is –10,600.
Since the NPV is negative at a zero discount rate it is negative at all
positive discount rates.
So NPVA-B < 0,
which implies NPVA < NPVB , hence we should pick
project B
4)
IRR comes from solving the following equation, where x = 1+IRR: 4x2
- 8.5x + 4 = 0
=>
x = -8.5 ±
[8.52
- 4 * 4 * 4]0.5
=> x =
1.4215 or 0.7035
8
Thus,
IRR = 42.15% and -29.65% (which is ignored, since it is negative)
The
NPV curve intersects the x-axis at 42.15%; once again, we have to consider
whether it intersects the x-axis from above or below.
This time the sum of the cashflows is negative => intersects from
below => at 13% NPV is still negative => NPV of A – B is negative =>
NPV of B is greater than that of A => choose B.
5
a) All 3 projects have conventional cashflows.
C has an IRR less than the required return.
Hence, its NPV is negative, so we drop project C.
Both A and B have positive NPVs. To
choose between them, we compute the IRR of the incremental cashflows A-B
Project
C0 C1 C2
A - B
-5,000
11,850
-7,000
IRRA-B
is given by: 5x2
- 11.85x + 7 = 0
=> x = 11.85 ±
(11.852 - 4 * 5 * 7)0.5
= 1.12, 1.25 =>
IRR = 12% and 25%
2*5
Thus,
the NPV curve for A - B intersects the x-axis at two points: 12% and 25%.
The y-axis intercept is -150. NPV
is negative below 12%, positive between 12 and 25% and again negative above 25%.
At a discount rate of 11%, the NPV of A-B is negative => B has a
higher NPV than A => choose B
b)
At a discount rate of 13%, the NPV of A-B is positive => A has a higher NPV
than B => choose A
6
a)
Computing the IRR of each project, we get:
For
A: -240x2
+ 218x + 62.704 = 0 =>
IRRA = 13.79%
For
B: -180x2 +
80x + 142 = 0 =>
IRRB = 13.78%
For
C: -250x2 +
145x + 145 = 0 =>
IRRC = 10.49%
All three projects have positive NPV at the required return of 10%.
Consider the sum of the cashflows:
A's sum = 40.704
B's sum = 42
C's sum = 40
A and B have intersecting NPV curves, but B and C do not. So we can rule out C; since B has a higher IRR and the curves do not intersect, B has a higher NPV than C.
To
choose between A and B, consider the incremental cashflows for B -A:
Project
C0
C1
C2
B - A
60,000
-138,000
79,296
IRR
is given by: 60x2 –
138x + 79.296 = 0
=>
x = 138 ±
[1382 – (4 * 60 * 79.296)]0.5
= 1.12, 1.18 =>
IRR = 12% and 18%
2*60
The
sum of the cashflows is positive. Thus,
NPV starts out positive, turns negative at 12% and becomes positive again at
18%. Since NPV of B-A is positive at
10%, we choose B.
b) Resist the impulse to say that since the NPV of B-A turns positive again at 18%, we still choose B. Looking at the original IRRs for A and B, both projects have a negative NPV at 20%, so we wouldn't take either one. B's NPV may be less negative than A's but we're still not going to take it.
7. Both projects have a positive NPV (conventional cashflows, IRR > and required return). To choose between them, look at the incremental cashflows A – B:
C0
C1
C2
A - B
-5,000 12,250
-7,500
=>
50x2 – 122.5x + 75 = 0
x = 122.5 ±
[122.52 – (4*50*75)]0.5
100
=>
x = 1.20 and 1.25
=>
IRR = 20% and 25%
The
sum of the cashflows is negative. Thus,
NPV starts out negative, turns positive at 20% and becomes negative again at
25%. Since NPV of A – B is
negative at 15%, NPV of B is higher than NPV of A => choose B.
8. These are borrowing cashflows, so we will want to accept the project if IRR is less than the required return.
The equation for IRR is: 4,000 – 2,000/(1+IRR) –
2,310/(1+IRR)2 = 0
Solving by the quadratic formula, 4x2 – 2x – 2.31 = 0
=>
x = 2 +/- [22 + 4*100*(-2.31)]0.5
= 1.05, -0.55
8
=>
IRR = 5% (dropping the negative value of –155%)
Since the IRR
> required return, we reject this project.
9.
Picking in order of P.I. we would pick C first (using up 10 of our
budget), then E (we’ve now spent 15) followed by B (we’re up to 35) and D. At this stage we’ve spent $50, got total NPV of 8 + 6 + 5 +
2.5 = 21.5, and have $10 left going waste.
The
question is: is there a combination of projects where we use some of that wasted
$10 and get a total NPV of more than 21.5.
Ultimately, you just have to look at different combinations, and see what
works. Here, it will turn out that
A+B+C uses up exactly $60 and generates total NPV of $23, and that’s the best
possible choice.
There
are no simple rules to pick the best combination, but there are shortcuts that
cut down the number of combinations you need to look at.
Here, for example, the total investment for all 5 projects is $80.
Since you have $60 to spend, you can tell yourself that you have to drop
projects with combined investment of $20 or more (to stay within the budget),
and you’re looking to drop the combination with the lowest combined NPV.
Choices
would then be:
|
Drop
A, and lose $9 in NPV |
|
Drop
B, and lose $8 |
|
Drop
C and D, and lose 11 |
|
Drop
D and E, and lose 7.5 |
(anything
else will be worse)
So
we drop D and E => take A, B, C
10.
Investment PV
NPV
P.I.
A
55
85
28 0.51
B
35
60
25 0.71
C
45
65
20 0.44
D
20
35
15 0.75
E
20
30
10 0.50
a)
Picking in order of P.I., we take D first (spending 20), then B (which puts us
at 55) and then A, for a total investment of $110 and total NPV of $68.
Since picking by P.I. exactly used up the budget, this is guaranteed to
be the best combination.
b)
Eyeballing different combinations will show that you should pick B, C, D and E,
for a total investment of $120 and total NPV of $70. ($5 still goes waste, but
there's no better combination.)
Once
again, we say we need to drop projects with investment of $50 or more (since
total investment in all projects is $175 and the budget is $125).
Choices
would then be:
|
Drop
A, and lose $28 in NPV |
|
Drop
B + E, and lose $35 |
|
Drop
C + E, and lose $30 |
(anything
else will be worse)
So
we drop A => take B, C, D and E
11 a) Computing the Profitability Index for each project:
PROJECT
INVESTMENT
NPV
P.I.
A
250
45
.18
B
200
40
.20
C
180
31.5
.175
D
155
36
.23
E
300
45
.15
F
195
33
.17
Picking in order of P.I., we would first take D (spending 155), then
B (which puts us at 355), then A (now up to 605), followed by C (which exactly
uses up our budget of 785). Since
picking in order of P.I. fully uses up the budget this is guaranteed to be the
best combination. So we pick A, B, C, D
and the total NPV generated is 36+40+45+31.5 = 152.5.
b) Now that the budget is 800, we won’t fully use up the
budget picking in order of P.I. So
we need to eyeball different combinations and see if there is a better one.
One obvious choice is to drop C and pick up F instead. So we pick A, B, D, F. We would now be investing 800, and we would pick up a total NPV of 154. Checking other combinations, there is nothing better, so this is the best choice.