This document discusses internal rate of return (IRR) and how it is used to evaluate investment projects. It defines IRR as the interest rate that sets the net present value of an investment to zero. IRR is calculated through trial and error or graphically. The document provides an example calculation of IRR for two projects, finding the IRR of Project A to be 10.8% and Project B to be 12.6%, showing Project B is preferable since its IRR is higher than the investor's required rate of return of 10%. It also discusses how investments can be classified as simple or non-simple based on the number of sign changes in their cash flows and presents rules for evaluating projects under IRR criteria.

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Dr. Hassan Ashraf
Engineering Economics _ CU Islamabad _ Wah Campus _ Civil
Engineering Department
Sequence 6_ Rate of Return
Method_ Engineering Economics
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2. Internal Rate of Return
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Internal Rate of Return or Discounted Cash Flow Rate of
Return (DCROR).
The IRR is the rate of return earned by a particular
individual’s or company’s investment.

3. Internal Rate of Return
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The IRR is defined as the interest rate which discounts a
series of cash flows to an NPV value of zero:
It should be noted that one cannot normally solve explicitly
for the IRR from the above equation. Therefore, a trial and
error solution is usually required. Graphically, the
relationship between NPV, interest rate, and IRR is
demonstrated in the figure presented in next slide.

4. Relationship between i, NPV, and IRR
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Consider the two investment opportunities examined in 3.3.
The investor’s MARR is 10% and investor only has enough
funds to invest in one of the projects. What are the IRRs for
each project?
Project A:
Project B:
0 1 2 3 4 5
-800 215 215 215 215 215
0 1 2 3 4 5
-800 100 215 100 100 900

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Calculations of IRR usually involves a trial and error
approach. While the NPV versus interest rate curves is not
a straight line, it is generally accurate enough to bracket
the IRR solution within 5% and then linearly interpolate for
the answer.
Project A: NPV for Project A = -800 + 215 (P/A)
Interest Rate NPV
0.0 275.0
10.0 15.0
15.0 -79.3

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Interpolating for IRR:
IRR=10.0 + (( 15-0)/15-(-79.3)) (15-10) = 10.8%
Project B:
NPV for Project B= -800 + 100 (P/A)+800(P/F)
Interest Rate, % NPV
0.0 100
10 75.8
15 -67.0

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Interpolating for IRR:
IRR=10.0 + (( 75.8-0)/75.8-(-67.0)) (15-10) = 12.6%
Both projects are acceptable as IRR>MARR. However, as
IRR of Project B> Project A, project B is more preferable.

9. Simple versus non-simple investments
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We can classify 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). We can then establish the
following categories.

10. Simple versus non-simple investments
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A simple ( or conventional) investment is simply when one
sign change occurs in the net cash flow series.
A non-simple (or non conventional) investment is an
investment in which more than one sign change occurs in
the cash flow series.
Multiple i*s occur only in non-simple investments. If there
is no sign change in the entire cash flow series, no rate of
return exists. The different types of investment possibilities
may be illustrated as follows:

11. Simple versus non-simple investments
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Investm
ent type
Cash Flow sign at period The
number
of sign
changes
0 1 2 3 4 5
Simple - + + + + + 1
Simple - - + + 0 + 1
Non-
Simple
- + - + + - 4
Non-
Simple
- + + - 0 + 3

12. Project Selection Rules under the IRR Criterion
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13. Project Selection Rules under the IRR Criterion
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14. Project Selection Rules under the IRR Criterion
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Evaluating a single project
If IRR> MARR, accept the project
If IRR = MARR , remain indifferent
If IRR< MARR, reject the project.

15. Project Selection Rules under the IRR Criterion
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Evaluating a mutually exclusive project
When we have to compare mutually exclusive investment
projects, we need to apply the incremental analysis
approach.

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Thank You
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