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11ComparisonMethods2PDF.pdf
11ComparisonMethods2PDF.pdf
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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
Page 12
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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