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##### Page 18
18
IRR of a Project  Example 1
What is the IRR to buy a bike which costs \$500
and would save \$160 per year in rental
charges over ﬁveyears?
PW=500 + 160(P/A,r,5)
Try r = 10% → PW = 106.52
Try r = 20% → PW = 21.50
Interpolate: r = 18.3%
Actual r = 18.03%

##### Page 19
19
Graphical Solu8on
Use the investment from before:
\$90K
1
2
4
3
5
\$180K
\$120K
\$80K
\$90K
\$150K
6
7
PW in Thousands
IRR
0%
5%
10%
15%
20%
Interest Rate
\$250K
\$200K
\$150K
\$100K
\$50K
0
-50K
-100K

##### Page 20
20
IRR of a Project  Example 1
Con8nuing with the investment example from before,
we calculate the IRR using EE factors:
PW = 90K 180K(P/F,i,1)90K(P/F,i,2)+80K(P/F,i,3)+
120K(P/A,i,3)(P/F,i,3) + 150K(P/F,i,7)
PW(12%) = \$7,479
PW(13%) = \$4202
Now interpolate:
IRR = 0.12 + 0.01*7479/(7479+4204) = 0.1264 = 12.6%
\$90K
1
2
4
3
5
\$180K
\$120K
\$80K
\$90K
\$150K
6
7

##### Page 21
21
Mul8ple IRR
One problem with using IRR is that there may
be more than one internal rate of return.
0=\$1,000\$5,000(P/F,i*,1) + \$6,000(P/F,i*,2)

##### Page 22
22
IRR  2 Sign Changes Example
One of the alterna8ves for improving an
opera8on is do nothing for 2 years and then
spend \$10,000 on improvements. The immediate
gain is \$3000, annual income should be \$2000
per year for 4 years. What rate of return can be
expected?
PW = 3000 10,000 (P/F,i,2) + 2000 (P/A,i,4) (P/
F,i,2)
PW = 0
i = 51%, i = 9.4%
2000
10,000
3000
0
6

##### Page 23
23
Tests for Mul8ple IRR
If the investment is not a
simple investment
The present worth of the project is zero for
several interest rates
-50.0
0.0
50.0
100.0
150.0
200.0
250.0
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
i
- interest rate
Present Worth

##### Page 24
24
External Rate of Return
We can test for mul8ple IRRs:
If the investment is not a
simple investment
The present worth of the project is zero for several
interest rates
But the best way is to calculate the External Rate of
Return (ERR)
ERR is the Rate of return of the money associated with
a project but is invested outside of the project at the
same interest MARR because we assume that money
not used now can be invested at the same rate.
For a given, explicit interest rate, a project can have
only one ERR
Must be used when there are mul8ple IRRs
A project is acceptable if ERR ≥ MARR

##### Page 25
25
Approximate ERR calcula8on
Since the exact ERR it is usually a very complex
calcula8on, we can use the "approximate"
ERR calc'n.
FW(inﬂows at MARR) = FW(ouqlows at ERR
approx
)
Note: the approximate ERR is always between
actual ERR and MARR
ERR Calcula8on.xls

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