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##### Page 10
10
Deﬁni8on for IRR
“IRR is the interest rate that makes both the PW and
the AW equal to 0.”
1. The IRR is based only on a project’s cash ﬂows, not
other things, thus the
internal
2. For the IRR to exist,
both
the beneﬁts and costs
must be deﬁned
3. If only the beneﬁts are deﬁned, both PW and AW
will be posi8ve
4. If only costs are known, they both are nega8ve (AC)
5. Only if both costs and beneﬁts are deﬁned can both
PW and AW be equal to 0 for some interest rate

##### Page 11
11
Issue with IRR
One serious issue is that unique solu8ons may
not exist when mul8ple sign changes take
place.
E.g.:
0
1
2
3
4
5
6
7
8
9
40
125
125
125
300
300
300
200
200
200

12

##### Page 13
13
Assump8ons
Our ﬁrst assump8ons match those of the PW and AW
measures:
1. Cash ﬂows occur at the end of periods 1 through N
2. Cash ﬂows are known, certain, values. The study
period length
N
is given and there is no uncertainty
3. The interest rate (discount rate or MARR)
i
is given
The ﬁrst of the above two is necessary, the last is a
suﬃcient condi8on
4. Both posi8ve and nega8ve cash ﬂows must be part
of the equa8on
5. The sequence of cash ﬂows should have only one
sign change between posi8ve & nega8ve cash ﬂows

##### Page 14
14
More on Assump8ons
Now let us look at the eﬀects of these assump8ons.
All three examples on the following slide sa8sfy the
ﬁrst 4 assump8ons
The ﬁrst 2 cases illustrate the common cash ﬂow
pa\erns of investments and loans and also sa8sfy
assump8on 5
But the third one changes signs more than once.
The examples show:
a) a
loan
b) an
investment
c) a
project
with a
nega0ve
salvage value

##### Page 15
15
Finding IRR
All three examples
satisfy the first 4
assumptions except the
last one violates the 5-th
assumption it has a
double sign change
\$50K
\$5,000
\$1,318.99
1
2
4
3
5
Loan
\$90K
1
2
4
3
5
\$180K
\$120K
\$80K
\$90K
\$150K
6
7
Investment
1
2
4
3
5
\$180K
6
7
\$60K
Project with a negative salvage value

##### Page 16
16
Finding the IRR
As interest rate increases, later payments
become less important
We borrow a \$5,000 mortgage at 10%. What is it
worth if MARR is 5% or 20%?
(Note: 5000*(A/P,
10%,5)=\$1319)
Cash ﬂows
PW@5%
PW@10%
PW@20%
0
5,000
5,000
5,000
5,000
1
1,319
1,256
1,199
1,047
2
1,319
1,196
1,090
831
3
1,319
1,139
991
659
4
1,319
1,085
901
523
5
1,319
1,033
819
415
PW =
 709
0
1,525

##### Page 17
17
Calcula8ng IRR
Find the interest rate that equates the equivalent
worth of an alterna8ve’s cash inﬂows (receipts or
savings) to its cash ouqlows (expenditures)
PW (inﬂows) = PW (ouqlows)
AW (inﬂows) = AW (ouqlows)
FW (inﬂows) = FW (ouqlows)
Mostly, you start oﬀ with an "educated" guess and
then by trial and error ﬁnd the closest two interest
rates
Then interpolate either mathema8cally, or graphically
to get the accurate interest rate
The most convenient method is using a spreadsheet
but with the caveat that double sign change can give
you more than one solu8on

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