Chambers and Lacey – Modern Corporate Finance Advanced Topics in Capital Budgeting – Model Answers 4th Edition Chapter 7 and 5th Edition Chapter10 Problem 1. 1.
Project Jethro has an NPV of $100,000 and is expected to last 4 years. Using a discount rate of
10%, convert this NPV into an equivalent annuity.
Excel: “=PMT(0.10,4,100000,,0)”
Problem 2. 2.
Using a discount rate of 12%, convert the following NPVs to equivalent annuities based upon the
project lives.
a. Project Carter with an NPV of $10,000 lasting 1 year. Excel: “=PMT(0.12,1,10000,,0)”
b. Project Reagan with an NPV of $25,000 lasting 2 years. Excel: “=PMT(0.12,2,25000,,0)”
c. Project Roosevelt with an NPV of $30,000 lasting 4 years. Excel: “=PMT(0.12,2,25000,,0)”
d. If the White House Corporation can select only one of the projects and if the corporation believes that there will be opportunities to begin new projects with similar NPVs once a project terminates, which of the projects should be accepted? Why? Select Project Reagan because the projects are replicable and Project Reagan has the highest
equivalent annuity of benefits.
Problem 3. 3.
Project Enquirer has an NPV of $875,000 and is expected to last 5 years. Project Star has an
NPV of $1.225 million and is expected to last 7 years. Both projects have a discount rate of 15%.
Use the equivalent annuity approach to determine which project should be accepted.
Enquirer
Star
Excel: “=PMT(0.15,5,875000,,0)”
Excel: “=PMT(0.15,7,1225000,,0)”
Project Enquirer has the highest Equivalent Annual Annuity and should be the project selected,
on this basis.
Problem 4. 4.
Bundy Corporation has two projects under consideration that would use the same facilities and
management team: Project Al and Project Peg. The cash flows of the projects are given as
follows:
Project
Initial
Cash Inflows
Name
Cost
Year 1
Year 2
Al
10,000
7,000
6,000
Peg
20,000
8,000
9,000
Year 3
Year 4
10,000
12,000
Bundy Corporation realizes that the shorter life of Project Al would be an advantage because it
would enable the firm to begin another profitable project quicker than if Project Peg were
accepted because Project Peg takes longer to finish. Both projects have a discount rate of 10%.
Use the equivalent annuity approach to determine which project should be accepted.
Project Al
Excel: “=-10000+NPV(0.10,7000,6000)”
Excel: “=PMT(0.10,2,1322.31,,0)”
Project Peg
Excel: “=-20000+NPV(0.10,8000,9000,10000,12000)”
Excel: “=PMT(0.10,4,10420.05,,0)”
Project Al has an equivalent annual annuity of $ 761.90 and Project Peg has an equivalent annual
annuity of $ 3,287.22. Project Peg is therefore a better project as its equivalent annual annuity is
the higher.
Problem 5. 5.
Island Corporation has two projects under consideration, both of which fill the same niche in its
product line: Project Thomas and Project Croix. The cash flows of the projects are:
Project
Initial
Cash Inflows
Name
Cost
Year 1
Year 2
Thomas
200,000
90,000
160,000
Croix
200,000
80,000
90,000
Year 3
190,000
Island Corporation asks you to select the better project using the equivalent annuity approach.
Both projects have a discount rate of 8%. Determine which project should be accepted.
Project Thomas
Excel: “=-200000+NPV(0.08,90000,160000)”
Excel: “=PMT(0.10,2,-20507.54,,0)”
Project Croix
Excel: “=-200000+NPV(0.08,80000,90000,190000)”
Excel: “=PMT(0.10,3,-102062.69,,0)”
Project Thomas has an equivalent annual annuity of $ 11,500.00 and Project Croix has an
equivalent annual annuity of costs of $39,603.74. Project Croix is therefore a better project as its
equivalent annual annuity is the higher.
Problem 6. 6.
Paterno Corporation specializes in building production facilities in cold and remote locations.
The production facilities manufacture equipment used to reel in fishing lines and nets (known as
"linebackers" in the industry). The firm is con-sidering two projects for which it has been asked
to work. The first project, labeled "Ground," would involve shipping materials by ground and
would take 4 years to complete. The other project, labeled "Air” would be much quicker, but
more expensive. The cash flows are given as:
Cash Inflows
Project Name
Initial Cost
Year 1
Year 2
Year 3
Year 4
Ground
800,000
300,000
300.000
400,000
500,000
Air
2,000,000
2,000,000
900,000
Select the better project using the equivalent annuity approach. Both projects have a discount
rate of 12%.
Project Ground
Excel: “=-800000+NPV(0.12,300000,300000,400000,500000)”
Excel: “=PMT(0.12,4,309486.44,,0)”
Project Air
Excel: “=-2000000+NPV(0.12,2000000,900000)”
Excel: “=PMT(0.12,2,503188.77,,0)”
Project Ground has an equivalent annual annuity of $ 101,893.59 and Project Air has an
equivalent annual annuity of costs of $ 297,735.84. Project Air is therefore a better project as its
equivalent annual annuity is the higher.
Problem 7. 7.
Three alternatives produce identical benefits to the firm but have the following costs:
Project
Initial
Annual Cost
Name
Cost
Year 1
Year 2
Year 3
Year 4
A
100,000
10,000
10,000
10,000
10,000
B
120,000
8,000
6,000
4,000
2,000
C
40,000
40,000
40,000
40,000
40,000
Using a discount rate of 14%, compute the total present value of each alternative. Which is
cheapest?
A
Excel: “=-100000+NPV(0.14, -10000, -10000, -10000, - 10000)”
B
Excel: “=-120000+NPV(0.14, -8000, -6000, -4000, -2000)”
C
Excel: “=-40000+NPV(0.14, -40000, -40000, -40000, - 40000)”
Project A is the cheapest project.
Problem 8. 8.
Jordan Airlines is considering two alternatives for maintaining its fleet of small private jets. The
first alternative is to contract with a major airline to perform the necessary maintenance for an
annual cost of $1.2 million. The second alternative is for Jordan to construct its own facilities
and establish its own maintenance pro-gram. The second, "do-it-yourself" alternative will cost
less each year (only $500,000), but has large costs to get started-management estimates that it
will involve an initial investment of $9 million. The time horizon of the problem is 20 years
because a new fleet of jets will be purchased and the facilities and program will have to be
completely revised by that time. Which alternative has the lowest present value of costs, given a
13% discount rate?
We know that the annual maintenance cost is $1.2 million. We need to calculate the Equivalent
Annual Annuity of the “do it yourself” program in order to be able to compare them.
Project Do-It-Yourself
Excel: “=-9000000+NPV(0.13,-500000, -500000, -500000, -500000, -500000, -500000, 500000, -500000, -500000, -500000, -500000, -500000, -500000, -500000, -500000, -500000, 500000, -500000, -500000, -500000)”
Excel: “=PMT(0.13,20,12512375.79,,0)”
The Equivalent Annual Annuity of the “Do-It-Yourself” program is $1,781,184.00 which is
$581,184 more per year than purchasing the necessary maintenance.
Problem 9. 9.
Rose Resorts is debating which type of slot machine to buy for its new casino in Atlantic City.
There are two types: a high-quality model that costs $12,000 and is expected to require $1,500
per year of maintenance, and a low-quality model that costs $8,000 and is expected to require
$2,000 per year in maintenance. Both machines are expected to last 10 years. Using a discount
rate of 15%, compute the total present value of the costs of the two alternatives.
Project High Quality
Excel: “=-12000+NPV(0.15,-1500,-1500,-1500,-1500,-1500,-1500,-1500,-1500,-1500,-1500)”
Excel: “=PMT(0.15,10,19524.15,,0)”
Project Low Quality
Excel: “=-8000+NPV(0.15,-2000,-2000,-2000,-2000,-2000,-2000,-2000,-2000,-2000,-2000)”
Excel: “=PMT(0.15,10,18037.54,,0)”
Project High Quality has an equivalent annual annuity of the costs of $ 3,891.02 and Project Low
Quality has an equivalent annual annuity of costs of $ 3,594.02. Project Low quality is
marginally better based on its equivalent annual annuity of costs.
Problem 10. 10.
Three alternatives produce identical benefits to the firm but have unequal lives and different
costs:
Project
Initial
Annual Cost
Name
Cost
Year 1
Year 2
Short
100,000
10,000
10,000
Medium
160,000
10,000
8,000
6,000
Long
200,000
40,000
40,000
40,000
Year 3
Year 4
40,000
Using a discount rate of 10%, compute the total present value of each alternative and then
convert them into equivalent annuity costs. Which is cheapest?
Project Short
Excel: “=-100000+NPV(0.10,-10000,-10000)”
Excel: “=PMT(0.10,2,117355.37,,0)”
Project Medium
Excel: “=-160000+NPV(0.10,-10000,-8000,-6000)”
Excel: “=PMT(0.10,3,180210.37,,0)”
Project Long
Excel: “=-200000+NPV(0.10,-40000,-40000,-40000,-40000)”
Excel: “=PMT(0.10,4,326794.62,,0)”
Project short is the cheapest with an Equivalent Annual Annuity of $67,619.04.
Problem 11. 11.
Returning to Problem 9, if Rose Resorts determines that the high-quality machine would last 15
years and that the low-quality machine would last only 10 years (with initial costs and annual
maintenance costs unchanged), which machine would have the lower equivalent annuity cost?
Project High Quality
Excel: “=-12000+NPV(0.15,-1500,-1500,-1500,-1500,-1500,-1500,-1500,-1500,-1500,-1500,1500,-1500,-1500,-1500,-1500)”
Excel: “=PMT(0.15,15,19524.15,,0)”
We calculated the equivalent annual annuity for Project Low Quality in our answer to question 9.
It was $ 3,594.02.
If the High Quality Machines will last 15 years then they will cost significantly less than the
Low Quality machines, only the equivalent of $1,842.03 each year.
Problem 12. 12.
Shark Entertainment Corporation has two alternatives under consideration that would provide
major entertainment and considerable benefit to the firm. The benefits of the alternatives are
equal, but their lives and costs are different. One alternative, code-named "Transfer," will last 2
years. The second alternative, code-named "Senior," will last 4 years. The costs of each of the
projects are:
Project
Initial
Name
Cost
Annual Cost
Year 1
Year 2
Year 3
Year 4
Transfer
100,000
90,000
160,000
Senior
50,000
80,000
120,000
180,000
240,000
Shark Entertainment Corporation asks you to select the better project using the equivalent
annuity approach. Both projects have a discount rate of 15%. Determine which should be
accepted.
Project Transfer
Excel: “=-100000+NPV(0.15,-90000,-160000)”
Excel: “=PMT(0.15,2,299243.86,,0)”
Project Senior
Excel: “=-50000+NPV(0.15,-80000,-120000,-180000,-240000)”
Excel: “=PMT(0.15,4,465876.16,,0)”
The equivalent annual annuity of the costs of Senior, $163,180.28, are less than the equivalent
annual annuity of the costs of Transfer, $184,069.77, so we should select project Senior.
Problem 13. 13.
Inflation rates can be computed for overall indexes or for individual items. Compute the inflation
rate implied by each of the following pairs of prices.
Item
Year
Price
Year
Price
Consumer price index
1940
42.0
1950
72.1
Consumer price index
1950
72.1
1960
88.7
Consumer price index
1960
88.7
1970
116.3
Consumer price index
1970
116.3
1980
246.8
Consumer price index
1980
246.8
1990
385.0
Consumer price index
1990
391.4
2000
515.8
Consumer price index
2000
515.8
2006
603.9
Milk
1965
$1.00
2000
$2.25
Gasoline
1965
$0.35
2000
$1.25
Beer
1975
$1.25
2000
$3.00
House
1950
$5,000
2000
$125,000
1985
$5,000
2000
$1,500
1975
$2,000
2000
$5,000
Computer
Tuition (public)
⎛ FV n ⎞
We can calculate the inflation rate using: r = ⎜
⎟
⎝ PV ⎠
Item
Consumer price index
1
n
−1
Year
Price
Year
Price
1940
42.0
1950
72.1
So for example:
1
r = ⎛⎜ FV n ⎞⎟ − 1 = ⎛⎜ 72.1 ⎞⎟
⎝ PV ⎠
⎝ 42 ⎠
n
1
10
−1 =
(1.716666667 )
Or 5.552517851% as in
Or using TVM_Solver
Or using Excel: “=RATE(10,0,-42,72.1,0,0.05)”
1
10
− 1 = 0.055525179
Year
1940
1950
1960
1970
1980
1990
2000
Price
42
72.1
88.7
116.3
246.8
391.4
515.8
Year
1950
1960
1970
1980
1990
2000
2006
Price
72.1
88.7
116.3
246.8
385
515.8
603.9
Number
of years
10
10
10
10
10
10
6
Milk
Gasoline
Beer
1965
1965
1975
1
0.35
1.25
2000
2000
2000
2.25
1.25
3
35
35
25
2.34399%
3.70399%
3.56391%
House
Computer
Tuition (public)
1950
1985
1975
5000
5000
2000
2000
2000
2000
125000
1500
5000
50
15
25
6.64949%
-7.71281%
3.73316%
Item
Consumer Price
Consumer Price
Consumer Price
Consumer Price
Consumer Price
Consumer Price
Consumer Price
Index
Index
Index
Index
Index
Index
Index
Inflation
5.55252%
2.09367%
2.74616%
7.81434%
4.54700%
2.79833%
2.66300%
Problem 14. 14.
Use the approximate formula for the Fisher effect to determine the missing values:
Nominal Rate
a. __________
b. __________
c. 10%
d. 8%
Real Rate
3%
2%
4%
___________
Expected Inflation Rate
5%
15%
_______________
6%
Nominal = Real + Inflation
a.
= 3% + 5% = 8%
b.
= 2% + 15% = 17%
Inflation = Nominal – Real
c.
= 10% - 4% = 6%
Real = Nominal – Inflation
d.
= 8% - 6% = 2% Problem 15. 15.
Use the precise formula for the Fisher effect in order to revise your answers:
(1 + Nominal) = (1 + Real) x (1 + Expected Inflation)
Nominal Rate
a. __________
b. __________
c. 10%
d. 8%
a.
b.
Expected Inflation Rate
5%
15%
_______________
6%
(1 + r ) = (1 + s ) * (1 + g ) = (1 + .03) *(1 + .05) = 1.0815
r = 0.0815 = 8.15%
(1 + r ) = (1 + s ) * (1 + g ) = (1 + .02) *(1 + .15) = 1.173
r = 0.173 = 17.3%
(1 + r ) = (1 + s ) * (1 + g ) ⇒
c.
Real Rate
3%
2%
4%
___________
(1 + g ) =
(1 + r ) = (1 + s ) * (1 + g ) ⇒
(1 + s )
(1 + s )
(1 + r ) = 1.10 = 1.057692308
(1 + s ) 1.04
g = 0.057692308 = 5.7692308%
(1 + r ) = (1 + s ) * (1 + g ) ⇒
d.
(1 + s ) =
(1 + r ) = (1 + s ) * (1 + g ) ⇒
(1 + g )
(1 + g )
(1 + r ) = 1.08 = 1.018867925
(1 + g ) 1.06
s = 0.018867925 = 1.8867925%
Problem 16. 16.
Keuka Corporation is considering a project that costs $1 million and will produce benefits for 5
years. The first year cash inflow can be rather accurately estimated at $240,000, measured in
today's dollars. If the actual first-through fifth-year cash flows are assumed to be the same size
($240,000), what would the NPV of the investment be if the appropriate discount rate is 15%?
Problem 17. 17.
Return to Keuka Corporation in Problem 16. Now assume that the cash inflow will rise each year
with inflation. Assume that the inflation rate is 10%. Thus the first-year cash inflow will be
found by multiplying the $240,000 by (1 + 10%) to obtain $264,000. After the first-year cash
inflow of $264,000, each cash inflow is expected to rise by another 10%. Compute:
a. The second‐year cash inflow = __ b. The third‐year cash inflow = __ c. The fourth‐year cash inflow = __ d. The fifth‐year cash inflow = __ (Hint: Each cash inflow must be determined by applying the inflation rate to the previous year's cash inflow.) e. Now compute the NPV using the cash inflows above, the original project cost of $1,000,000, and the discount rate of 15%. f. Why did the NPV change relative to problem 16? g. Which computation of NPV is more reliable? Problem 18. 18.
Returning once again to Keuka Corporation (problem 16), let us compute the NPV using real
cash flows and real interest rates. We thus use the original cash flow estimate of $240,000 for
each of the 5 years. To start, we approximate the real rate of interest by subtracting the inflation
rate (10%) from the nominal in-terest rate (15%) and obtaining 5%.
a. Compute the NPV using these real cash flows and the real discount rate (5%). b. Compute the NPV using the real cash flows and a discount rate of 4.55%. c. Can you verify that the discount rate used in Problem 18(b) is found by com‐puting the real rate of interest using the precise form of the Fisher effect? d. Ignoring rounding errors, compare your answers to Problems 16, 17(e), 18(a), and 18(b). Which is (are) correct? Problem 19. 19.
Brooksmel Corporation is considering offering a pension alternative to its em-ployees that is
indexed to inflation. In other words, each employee would receive a pension that is increased
each year in order to keep up with the inflation rate as measured by the consumer price index. Of
course, employees who opt for this plan would have to be willing to accept a lower starting
pension. Brooksmel's CFO is attempting to compute the cost of this pension to the firm. Consider
Gene, who has worked for Brooksmel for years and is considering retirement soon. For math
simplification, consider that Gene's life expectancy is only 8 years.
a. Compute the present value of Gene's pension if he opts for a fixed pension of $8,000 per year and the interest rate is 10%. b. If Gene opts for a pension indexed to inflation, his "base pension" would be $5,000. All of his actual receipts would be higher than this figure and would depend upon the inflation rate. For example, if the inflation rate is 5%, then his first‐year pension would be $5,250, which is found by multiplying $5,000 by (1 + 5%). If the inflation rate continues at 5%, his second‐year pension would be $5,512.50, and so forth. Project his eight annual pension receipts, as‐suming that the inflation rate starts and stays at 6%. c. Let us now solve the problem in real terms. Compute the present value of Gene's pension using a fixed annual cash flow of $5,000 and a discount rate of 4%. This approximate real rate of interest was found by subtracting the 6% inflation rate from the nominal interest rate of 10%. d. In order to obtain an exact solution, find the real rate of interest using the pre‐cise form of the Fisher effect and solve Problem 19(c) again with the new rate. Problem 20. 20.
Plastex Corporation is considering expanding but wishes to control growth by imposing a limit
on capital expenditures of $500,000. The capital investment op-portunities are:
Investment Opportunities of Plastex Corporation (all dollar values are in thousands)
Profitability
First-Year
Project
Cost
NPV
Index
PI Rank
Cash Flow
1
$200
$50
1.25
1
$0
2
$200
$40
1.20
5
$5
3
$400
$99
1.2475
2
$10
4
$100
$17
1.17
7
$2
5
$100
$19
1.19
6
$3
6
$ 50
$12
1.24
3
$20
7
$ 50
$11
1.22
4
$15
a. Use the profitability index to select projects subject to the constraint that the firm can only invest $500,000. In other words, use the PI ranking to accept the best project, the next‐best project, and so forth until the money is completely used. If a project's cost brings the total cost over the $500,000 limit while money remains, then skip the project and attempt to add a cheaper one. b. Compute the total of the NPVs of the projects selected earlier. c. Use your common sense to figure out a better solution‐one that produces a higher total NPV but still uses only $500,000. Compute the sum of the NPVs of this better solution. d. Why do the answers to Problem 20 (a) and (c) differ and why is the profitabil‐ity index method flawed? Problem 21. 21.
[Based on the chapter Appendix] Using the following notation:
NPVi = the net present value of Project i
Xi = the decision variable produced by the software, which has a value of 1 if the project should
be accepted and 0 if it should be rejected
COi = initial cost of Project i
Cti = cash inflow (+) or outflow (-) in period T for Project i
a.
Express the objective function of Problem 20.
b.
Express the constraint that all the initial costs of the accepted projects must be less than
or equal to $500,000.
c.
Express the constraint that all the first-year cash flows of the accepted pro-jects must add
together to be at least $25,000.
d.
Express the constraint that Projects 3 and 4 use the same facilities, so only one of them
can be accepted.
e.
Express the constraint that the company is required by previous contracts to produce a
certain good, which means that the company must accept either Project 4 or 5 (or both).
f.
Express the constraint that no more than three projects can be accepted from the group of
projects which require supervisors (Projects 3-7).
Problem 22. 22.
Based on the chapter Appendix, use integer linear programming software to solve for the
optimal solution to Problem 20 without incorporating any of the additional constraints in
problem 21.