End of Chapter Exercises: Solutions
1. (a) What is meant by the term “mutually exclusive projects” ?
(b) Explain why the IRR decision rule could give the wrong result when
comparing mutually exclusive projects.
(a) Projects are said to be “mutually exclusive” when the selection of one
automatically eliminates the need for any of the alternatives; eg. choosing among a
number of alternative road designs between the same two points.
(b) It is possible that the ranking of projects using their NPVs changes when the
discount rate changes (“switching”). In this case one project will not be
unambiguously superior to another as a ranking by IRR would suggest.
2. The following net cash flows relate to two projects:
NET CASH FLOWS (IN $ 1,000)
YEAR 0 1 2 3 4 5 6
PROJECT A -60 20 20 20 20 20 20
PROJECT B -72 45 22 20 13 13 13
(i) Calculate the NPVs for each project, assuming 10% cost of capital.
(ii) Assuming that the two projects are independent, would you accept
them if the cost of capital is 15%?
(iii) What is the IRR of each project?
(iv) Which of the two projects would you prefer if they are mutually
exclusive, given a 15% discount rate?
(i) NPVA(10%) = 27.11; NPVB(10%) = 26.41
(ii) Yes. NPVA(15%)=15.69;NPVB(15%)=16.43
(iii) IRR(A)=24%; IRR(B)=26%
(iv) Prefer B – higher NPV(15%)
3. Using a spreadsheet generate your own set of Discount and Annuity Tables
for, say, all discount rates between 1% and 20% (at 1 percentage point
intervals) and for time periods 1 to 30 (at one time period intervals), as well as
time periods 50 and 100. You should generate these tables by inserting the
numbers for the time periods in the first column of each row and the discount
rates in the first row of each column, and then inserting the appropriate
formula into one cell of the table – year 1 at 1% - and then copying it to all
other cells in the matrix. (Hint: Do not forget to anchor the references to
periods and discount rates using the “$” symbol.)
Answer: Write the discount formula into spreadsheet as shown below and then copy
across 20 columns (headed 1% through 20%) and down 50 rows (headed 1 to
1% 2% 3%
1 =1/(1+B$2)^$A2 0.980 0.971
2 0.980 0.961 0.943
3 0.971 0.942 0.915
4. A firm has a capital budget of $100 which must be spent on one of two
projects, each requiring a present outlay of $100. Project A yields a return of
$120 after one year, whereas Project B yields $201.14 after 5 years.
(i) the NPV of each project using a discount rate of 10%;
(ii) the IRR of each project.
What are the project rankings on the basis of these two investment decision
rules? Suppose that you are told that the firm’s reinvestment rate is 12%,
which project should the firm choose?
(i)NPV(A) = 9.09; NPV(B) = 24.89, B>A
(ii) IRR(A) = 20%; IRR(B) = 15%, A>B
Using a reinvestment rate of 12% the terminal values are TV(A) = 188.82; TV(B) =
201.14, hence B>A. Alternatively calculate the IRR of (B-A): IRR(B-A) = 13.78% >
12%, hence undertake the “extra project” (B-A) ie. undertake B.
5. A firm has a capital budget of $100 which must be spent on one of two
projects, with any unspent balance being placed in a bank deposit earning
15%. Project A involves a present outlay of $100 and yields $321.76 after 5
years. Project B involves a present outlay of $40 and yields $92 after one
(i) the IRR of each project;
(ii) the B/C ratio of each project, using a 15% discount rate.
What are the project rankings on the basis of these investment decision
rules? Suppose that if Project B is undertaken its benefit can be reinvested at
17%; what project should the firm choose? Show your calculations
(spreadsheet printout is acceptable as long as entries are clearly labelled).
(i)IRR(A) = 26.3%; IRR(B) = 130%, B>A
(ii)BCR(A) = 1.6; BCR(B) = 2, B>A
Using a reinvestment rate of 17% the terminal values are: TV(A)=321.76;
TV(B)=293.08, hence A>B. Alternatively calculate the IRR of (A-B): IRR(A-
B)=18.51%>17%, hence undertake the “extra project” (A-B), ie. undertake A.
6. A firm has a capital budget of $30,000 and is considering three possible
independent projects. Project A has a present outlay of $12,000 and yields
$4, 281 per annum for 5 years. Project B has a present outlay of $10,000 and
yields $4,184 per annum for 5 years. Project C has a present outlay of
$17,000 and yields $5,802 per annum for 10 years. Funds which are not
allocated to one of the projects can be placed in a bank deposit where they
will earn 15%.
(a) Identify six combinations of project investments and a bank deposit which
exhaust the budget.
(b) Which of the above combinations should the firm choose:
(i) when the reinvestment rate is 15%?
(ii) when the reinvestment rate is 20%?
Explain your answer and show your calculations (spreadsheet printout is
acceptable as long as entries are clearly labelled).
To answer these questions it is probably best to compare future values after
10 years, bearing in mind that projects with a 5-year life will have terminal
value after 5 years which needs to compounded forward at the reinvestment
rate (15% and 20%) to year 10, and the residual deposited in the bank is
always compounded over 10 years at a 15% interest rate.
Future values are as follows:
A+$18,000 in bank $130,876 $152,092
B+ $20,000 in bank $137,652 $158,387
C+$13,000 in bank $170,394 $203,205
A+B+$8,000 in bank $147,161 $189,112
B+C+$3,000 in bank $186,679 $240,224
A+C+$1,000 in bank. $179,904 $233,929
At a reinvestment rate of 15% or 20% the combination of projects B+C would
7. A public decision-maker has a budget of $100 which must be spent in the
current year. Three projects are proposed, each of which is indivisible (it is not
possible to undertake less than the whole project) and non-reproducible (it is not
possible to construct two versions of the same project). The discount rate is
10% per annum. The project benefits and costs are summarized in the following
Project Cost ($) Benefits ($)
Year 0 Year 1 Year 2
A 30 40 0
B 30 0 50
C 70 0 100
(i) Work out the Net Present Value (NPV), Internal Rate of Return (IRR)
and Benefit/Cost Ratio (B/C) for each project;
(ii) Rank the projects according to the NPV, IRR and B/C investment
(iii) Which projects should be undertaken to spend the budget:
(a) if the reinvestment rate is 22% per annum;
(b) if the reinvestment rate is 28% per annum?
(In answering this question the student should assume that the full amount of
the budget must be exhausted on some combination of the three projects; ie.
it cannot be assumed that some part of the initial budget is invested at the
reinvestment rate. It should also be assumed that end of year 2 is the terminal
NPV IRR BCR
A $6.36 33% 1.21
B $11.32 29% 1.38
C $12.64 20% 1.18
(ii) rank by: NPV C, B, A
IRR A, B, C
(iii) Only project A is affected by the reinvestment rate.
(a) At 22% the future value of $40 at end of year 2 is $48.8.Therefore
select projects B and C to maximize return on $100 investment.
(b) At 28% the future value of $40 at end of year 2 is $51.2. Therefore
select projects A and C to maximize return on $100 investment.