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Title of Presentation
FIN 614: Financial Management
Larry Schrenk, Instructor
1. What is the Internal Rate of Return ?
2. Calculating the Internal Rate of Return
3. Analysis of the Internal Rate of Return
4. Technical Issues
1. Comparing Projects
2. Multiple Sign Changes
IRR is the discount rate that makes
present value of all cash flows (including
any required investments) equal to zero.
IRR s.t. Investment = PV(Cash Flows)
IRR s.t. - Investment + PV(Cash Flows) = 0
T
IRR s.t. 
t=0
Ct
1+IRR 
t
=0
Rule: Do project if IRR > required rate of
return (r).
If the return on the project (IRR) is
greater than the return expected on
projects with this level of risk (r), then do
the project
-C0
C1
C2
C3
C4
-C0
PV(C1)
C1
C1/(1+IRR)
C2
C3
C4
-C0
PV(C1)
+
PV(C2)
C1
C1/(1+IRR)
C2/(1+IRR)2
C2
C3
C4
-C0
PV(C1)
+
PV(C2)
+
PV(C3)
C1
C2
C1/(1+IRR)
C2/(1+IRR)2
C3/(1+IRR)3
C3
C4
-C0
PV(C1)
+
PV(C2)
+
PV(C3)
+
PV(C4)
C1
C2
C3
C1/(1+IRR)
C2/(1+IRR)2
C3/(1+IRR)3
C4/(1+IRR)4
C4
-C0
PV(C1)
+
PV(C2)
+
PV(C3)
+
PV(C4)
=
Total PV
C1
C2
C3
C1/(1+IRR)
C2/(1+IRR)2
C3/(1+IRR)3
C4/(1+IRR)4
C4
-C0
PV(C1)
+
PV(C2)
+
PV(C3)
+
PV(C4)
=
Total PV
|-C0|
C1
C2
C3
C1/(1+IRR)
C2/(1+IRR)2
C3/(1+IRR)3
C4/(1+IRR)4
C4
-C0
PV(C1)
+
PV(C2)
+
PV(C3)
+
PV(C4)
C1
C2
C3
C1/(1+IRR)
C2/(1+IRR)2
C3/(1+IRR)3
C4/(1+IRR)4
=
Total PV
=
|-C0|
IRR is the discount rate
that makes
Total PV = |C0 |
C4
EXAMPLE (r = 10%):
0
-1,000
1
300
2
200
3
400
4
700
IRR Calculation:
-1,000 +
300.00
200.00
400.00
700.00
+
+
+
=0
2
3
4
1+IRR  1+IRR  1+IRR  1+IRR 
iff IRR = 18.10%
Result: 18.10% > 10% Good Project
Finance Apps #8
Screen will show
irr(
Function Syntax
irr(CF0, {CF1, CF2,…}, {Freq1, Freq2,…})
CFt = Cash Flow at Time t
Freqt = Frequency of Cash Flow at Time t
CF0 = -Initial Investment
When doing uneven cash flows, CF0 = 0
Consider again these cash flows:
irr(CF0, {CF1, CF2,…}, {Freq1, Freq2,…})
irr(-1000, {300, 200, 400, 700} and ENTER
Answer: 18.10%
Notes:
Cash Flows 1+ are not entered as negative (Unless they are
negative numbers).
Enter investment (CF0) as negative.
Assuming no technical problems occur,
NPV and IRR always give the same and
the correct answer about whether or
not to do one specific project.
NPV > 0 iff IRR > r
The IRR and MIRR rules cannot be
used to compare projects or select
among projects since they do not
meaningfully compare the absolute
advantage of one project over
another.
Instead, the NPV rule must be used to
compare or select among projects.
EXAMPLE (r = 10%):
Period
1
A -1,000
B -100
2
300
40
3
200
30
4
400
50
5
700
80
IRRA
300.00
200.00
400.00
700.00
-1,000 +
+
+
+
=0
2
3
4
1+IRR  1+IRR  1+IRR  1+IRR 
iff IRRA = 18.1%
IRRB
40.00
30.00
50.00
80.00
-100 +
+
+
+
=0
2
3
4
1+IRR  1+IRR  1+IRR  1+IRR 
iff IRRB = 29.6%
NPVA
300.00 200.00 400.00 700.00
-1,000 +
+
+
+
= $216.65
2
3
4
1.1
1.1
1.1
1.1
NPVB
40.00 30.00 50.00 80.00
-100 +
+
+
+
= $53.36
2
3
4
1.1 1.1 1.1 1.1
Results
IRRA = 18.1%
NPVA = $216.65
<
>
IRRB = 29.6%
NPVB = $53.36
Question: Would you rather have a higher
rate of return or a higher dollar return?
In the end it is the dollar return that counts
Project A increases firm value by $216.65.
Project B increases firm value by $53.36.
Project A is worth $163.29 more than B!
What is the IRR of the following cash flow?
0
-3
1
20
2
-16
3
-32
4
32
There are multiple correct answers!
20.00
-16.00
-32.00
32.00
-3 +
+
+
+
=0
2
3
4
1+IRR  1+IRR  1+IRR  1+IRR 
iff
IRR » 7%
OR IRR » 48%
OR IRR » 437%
This is possible whenever there is more
than one sign change in the cash flows!
The line crosses the x-axis at each IRR.
Multiple IRR's
$1.4
$1.2
$1.0
$0.8
$0.4
$0.2
-$0.6
-$0.8
Discount Rate
495%
469%
443%
417%
391%
365%
339%
313%
287%
261%
235%
209%
183%
157%
131%
79%
105%
-$0.4
53%
-$0.2
27%
$0.0
1%
NPV
$0.6
Advantages
Properly adjusts for time value of money
Uses cash flows rather than earnings
Accounts for all cash flows
Project IRR is a number with intuitive
appeal
Key Problems:
Cannot compare projects
Multiple IRR’s
FIN 614: Financial Management
Larry Schrenk, Instructor

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