Here are the solutions to the homework problems: 1. To find the net present value at 18% MARR, discount each cash flow back to period 0: A0 = -1,500 A1 = 4,200 / (1.18) = 3,566.10 A2 = -3,500 / (1.18)^2 = -2,937.29 A3 = 1,890 / (1.18)^3 = 1,460.17 NPV = -1,500 + 3,566.10 - 2,937.29 + 1,460.17 = 589.98 Since NPV is positive, the project is acceptable at the 18%

1. Rate-of-Return
Analysis

2. Learning Objectives
● Evaluate project cash flows with the internal rate of return
(IRR) measure.
● Plot a project’s present worth (PW) against the interest
rate.
● Use an incremental rate of return analysis to evaluate
competing alternatives.

3. Rate of Return
There are several ways of defining the
concept of rate of return on investment. We
will show two of them: the first is based on a
typical loan transaction, and the second is
based on the mathematical expression of the
present-worth function.
Definition 1: Rate of Return is the interest
earned on the unpaid balance of amortized
loan.
Definition 2: Rate of return is the break even
interest rate i* at which the net present worth
of a project is zero
PW(i*) = PWcash inflow - PWcash outflow = 0

4. Return on Invested Capital
Investment projects can be viewed as
analogous to bank loans.
We will now introduce the concept of rate of
return based on the return on invested capital
in terms of a project investment.
A project's return is referred to as the internal
rate of return (IRR), or the yield promised by
an im·cstment project over its useful life.
Definition 3. The internal rate of return is the
interest rate charged to the unrecovered
project balance of the investment such that,
when the project terminates, the unrecovered
project balance is zero.

5. Methods for Finding Rate of
Return
Simple versus Nonsimple
Investments
We can classify an investment
project by counting the number of
sign changes in its net cash flow
sequence. A change from either "-
"to"+" or"+" to"-" is counted as
one sign change. We ignore a
zero cash flow.)

6. Simple versus Nonsimple
Investments
• A simple (or conventional) investment is
simply when one sign change occurs in the
net cash flow series. If the initial cash flows are
negative, we call them simple-investment
cash flows. If the initial flows are positive, we
call them simple-borrowing cash flows.
• A nonsimple (or nonconventional)
investment is an investment in which more
than one sign change occurs in the cash flow
series.

7. Computational Method
Once we identify the type of an
investment cash flow, there are several
ways to determine its rate of return. We
will discuss some of the most practical
methods here. They are as follows:
• Direct-solution method
• Trial-and-error method
• Excel method

8. Sample Problem
A cash flow profile with two
interest periods consists of
A0 = +1,200, A1 = —2,700,
and A2 = +1,500. Find the
internal rate of return per
interest period

9. Sample Problem
The Maxwell Manufacturing Company
plans to invest $77,000 in an energy
saving device with the expectation of
receiving benefits for the next 5 years
in the amounts of $38,000, $32,000,
$26,000, $20,000, and $14,000 at the
end of first, second, third, fourth, and
fifth years, respectively. What is the
internal rate of return for investing in
this device?

10. Internal Rate of Return Criterion
Relationship to the PW Analysis
PW analysis is dependent on the
rate of interest used for the PW
computation.
A different rate may change a
project from being considered
acceptable to being considered
unacceptable, or it may change the
priority of several projects.

12. Evaluating a Single Project
For a simple investment, i* is indeed the IRR of
the investment. Because firms typically wish to
do better than break even recall that at PW = 0,
we were indifferent to the project), a minimum
acceptable rate of return (MARR) is indicated by
company policy, management, or the project
decision maker. the decision rule for a simple
project is as follows:
If IRR > MARR, accept the project.
If IRR = MARR, remain indifferent.
If IRR < MARR, reject the project.

13. Sample Problem
An energy firm is betting on wind power's long-term
viability in Texas and plans to erect what would be one of
the biggest wind farms in the world with 200 wind turbines
costing some $1.69 million each. Energy companies
investing in wind power are also expecting governments
to toughen rules relating to traditional energy sources, as
part of long-term efforts to reduce global-warming
emissions. But generating power from wind is not
profitable for companies without government tax breaks.
The following financial and technical data have been
compiled for further consideration:
Number of wind turbines to be built: 200 units: Power
capacity: 310,000 kW ; Capital investment required:
$338,000,000 ; Project life: 20 years; Salvage value of the
wind turbines after 20 years: $0 ; Annual net cash flows
{after all deductions): $41,391,160
According to the data provided, answer the following
questions: (a) What is the projected IRR on this investment?
(b) If the company's MARR is known to be 10%, is the
investment justified?

14. Evaluating Mutually Exclusive Projects
When we have to compare
mutually exclusive investment
projects, we need to apply the
incremental analysis approach

15. Incremental Analysis
When there are two alternatives, rate of return
analysis is performed by computing the
incremental rate of return—IRR—on the difference
between the alternatives.
Since we want to look at increments of investment,
the cash flow for the difference between the
alternatives is computed by taking the higher
initial-cost alternative minus the lower initial-cost
alternative. If IRR is the same or greater than the
MARR, choose the higher-cost alternative. If IRR is
less than the MARR, choose the lower-cost
alternative.

16. Sample Problem
You must select one of two mutually exclusive
alternatives. (Note: Engineering economists often use the
term “mutually exclusive alternatives” to emphasize that
selecting one alternative precludes selecting any other.)
The alternatives are as follows:
Year Alt. 1 Alt. 2
0 −$1000 −$2000
1 +1500 +2800
Any money not invested here may be invested elsewhere
at the MARR of 6%. If you can choose only one alternative
one time, using the internal rate of return (IRR) analysis
method, which one would you select?

18. Homework # 3.
1. Find the net present value of the following unconventional investment
cash flow profile: A0 = -1,500, A1 = +4,200, A2 = -3,500, and A3 = +1,890.
The specified MARR is 18% per period.
2. Suppose you deposited $1,500 five years ago and another $500 three
years ago in your savings account. If your current account balance shows
$3,200, what is the rate of return on your deposit?
3. You spend $1000 and in return receive two payments of $1100—one at
the end of 3 years and the other at the end of 6 years. Calculate the
resulting rate of return.