Matakuliah
Tahun
: D0762 – Ekonomi Teknik
: 2009
RATE OF RETURN ANALYSIS
Course Outline 7
Outline
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•
•
•
•
Next
Definition
ROR Facts Next
ROR for Single project Next
ROR for Multiple project Next
Spreadsheet Next
Refererences
- Engineering Economy – Leland T. Blank, Anthoy J.
Tarquin p.200-246
- Engineering Economic Analysis, Donald G. Newman, p.
163-196
2
- Engineering Economy, William G. Sulivan, p.137-194, p.
Definition
• Synonym: IRR (Internal Rate of Return)
• Popular measurement on investment worth
• Which one represent the correct interpretation of ROR?
• Rate of Return on the un-recovered balance
• Rate of Return on the initial balance
• ROR (i*) is the interest rate earned on the un-recovered
balance or the interest rate paid on the unpaid balance
of a loan in which the final payment or receipt brings the
terminal value to exactly equal “0”
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ROR Facts
• If i…
• > MARR, investment is justified
• = MARR, investment is justified (indifferent decision)
• < MARR, investment is not justified
• i is ranges …. –100% < i ≤ +
• –100%: means total lost of capital
• >0%: means positive return on investment
• Some CF might have multiple ROR
• If there is a reinvestment option, use the composite rate
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Single Project
• Equation for computing ROR
– Present Worth of benefits – PW of costs = 0
– PW of benefits/PW of costs = 1
– Net Present Worth = 0
– EUAB – EUAC = 0
– PW of cost = PW of benefits
Note :
EUAB : equivalent uniform annual benefit
EUAC : equivalent uniform annual cost
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Example. 1
An investment $8200 investment returned $2000 per year over a five – year
useful life.
What was the rate of return on the investment ?
Solution
PW of benefits
1
PW of cost
2000( P / A, i,5)
1
8200
8200
( P / A, i,5) 
2000
 4 .1
From interest table
(P/A,i,5):
i
(P/A,i,5)
6%
4.212
7%
4.100
8%
3.993
The rate of return for the investment is 7%:
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ROR Calculation
• Trial & Error…
• Draw the CF Diagram
• Set up the equivalence equation and set equal to 0
• Select values of i and solve the equation
• Repeat until you find the i which give a balanced
equation
• Sometimes might need to interpolate to find the
approximate value of i*
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Example 7.2
The table shows an investment cash flow
Year
Cashflow
0
-$7000
1
+100
2
+175
3
+250
4
+325
Find rate of return for the investment above
There are two different interest
Solution :
factor. Solve the equation by
trial and error
• EUAB –EUAC = 0
• 100 + 75(A/G,i, 4) -7000(A/P,i,4) = 0
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Solution for Example 7.2.
Try i = 5%
EUAB –EUAC = 0
100 + 75(G/A,5%, 4) -7000(A/P,5%,4) = 0
100 + 75(1.439) -7000(0.282) = 0
208 – 197 = +6
The EUAB –EUAC > 0, too low. If interest rate is increased, EUAC will increase.
Try i = 8%
EUAB –EUAC = 0
100 + 75(A/G,8%, 4) -7000(A/P,8%,4) = 0
100 + 75(1.404) -7000(0.3019) = 0
205 – 211 = -6
The EUAB –EUAC < 0, too high
Try i = 7%
EUAB –EUAC = 0
100 + 75(A/G,7%, 4) -7000(A/P,7%,4) = 0
100 + 75(1.416) -7000(0.2952) = 0
206 – 206 = 0
The Rate Of Return = 7%
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Multiple Values of ROR
• So far we have learnt IRR not ERR (External Rate of Return)
• The difference between IRR and ERR is…
Un-recovered balance versus positive CF generated becomes
released/external funds
• Solve it by…
Basic guesses (must performed both!):
• Descartes’ rule: sign change in the series of net CF
• Norstrom’s rule: sign change in the series of cumulative
CF
Graphically
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Better way: Composite Rate
of Return (CRR)
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Example 101
• For the CF below, how many ROR at most we
could have? Use Descartes’ Rule!
1
2
3
4
5
6
Max i* values
-
+
+
+
-
-
2
+
-
+
-
+
+
4
-
+
+
+
+
+
1
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Composite Rate of Return (CRR or ERR)
• CRR/ERR/RIC: the unique ROR for a project which
assumes that net positive CF, which represent money
not immediately needed by the project are reinvested at
the reinvestment rate “c”
• To summarize… any positive CF available in year X
• Let’s consider the funds released from a project in
calculating the overall ROR of a project
• Reinvestment rate, “c”
• Composite rate of return =i’
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Equation for CRR
• Ft = Ft-1 (1+i) + Ct
Where
t
= 1, 2, …, n
n
= total years in project
Ct
= net CF in year t
i
= c,
if Ft-1 > 0
i’,
if Ft-1 < 0
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Example 1
+$9,000
• Find the ROR!
0
+$8,000
1
2
3
4
5
-$10,000
ROR = 16.82% on the un-recovered investment
balances over 5 years
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Example 2
• Reinvestment rate, c = 15%. What is the CRR?
• Answer…
•
•
•
•
•
F0 = 50,
F1 = -142.50
F2 = -142.50 (1+i’) + 50
F3 = F2 (1+i’) + 100
Set F3 = 0 to find the i’
Year
Cash Flow, $
0
50
1
-200
2
50
3
100
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Example 3 - CF Diagram
• Purchase price: P - $800/bond. Bond interest at 4% paid semiannually
for $1000 face value. Life = 20 years. If you pay the $800 per bond,
what is the ROR (yield) on this investment?
F40 = $1000
From the bond purchaser’s perspective
A = +$20/6 months
0
$800
1
2
3
4
…. …. ….
39
40
Pay $800 per bond to receive the $20each 6-months in interest cash
flow plus $1,000 at the end of 40 time periods. What is the ROR of
this cash flow?
A= $1000(0.04/2) = $20.00 every 6 months for 20 years
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Example 3 (Cont’d)
• Equation:
0 = -$800 + $20(P/A, i*, 40) + $1000 (P/F, i*, 40)
• Solve for i*, we get 2.87% per semiannual
• Not done yet, thus find the …
Nominal ROR/year = (2.87%)(2) = 5.74%/year
Effective ROR/year = 5.82%/year
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Multiple Projects
•
•
•
•
Incremental Analysis, Introduction
ROR on Extra Investment
ROR Analysis
Multiple Alternatives
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Incremental Analysis
• MARR Definition
• Example:
A company uses a MARR of 16% per year. The company has
$90,000 available for investment and that two alternatives (A and
B) are being evaluated. Alternative A requires an investment of
$50,000 and will yield an IRR of 35% per year.
Alternative B requires $85,000 and will yield an IRR of 29% per
year. Which alternative will be the best?
• Overall ROR(A) = [50k (.35) + 40k (.16)]/90k = 26.6%
• Overall ROR(B) = [85k (.29) + 5k (.16)]/90k = 28.3%
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Tabulation for Incremental CF For 2 Alternatives
• Equal lives versus Unequal lives
• ROR Analysis on incremental CF:
• Need to use LCM (no matter what!)
• Larger initial investment  alternative B!
• Incremental CF = CFB – CFA
• Check the sign changes like in Descartes’ and
Norstorm’s rules
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ROR on Extra Investment
• Decision:
• Do-Nothing alternative
• Equivalent worth of the savings > equivalent worth of
the extra investment using company’s MARR
DO the extra investment
• If the extra investment is not justified by the savings
Choose LOWER first-cost proposal
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ROR Analysis Procedure
• Sort the alternative by initial investment in an ascending
order
• Develop CF and incremental CF series
• Draw if necessary
• Count the # of sign changes
• Set up PW equation for the incremental CF & find i*B-A
• If i*B-A < MARR:
choose A,
• Otherwise:
choose B
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Example 8.3
• A leather clothes manufacturer is considering the purchase of one new
industrial sewing machine, which is either semiautomatic or fully
automatic. Which machine should be selected if the MARR is 15% per
year? The estimates are listed in the table below.
Fully Automatic
Semiautomatic
First cost, $
13,000
8,000
Annual disbursement, $
1,600
3,500
Salvage value, $
2,000
0
Life, years
5
10
• SORT!  Incremental CF  # sign Δ (max #ROR)  PW Incremental CF
 trial & error
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• 12.65% < MARR  choose the lower-cost
(Semiautomatic)
Example 8.3
• What if…
• MARR is 12.65%, which alternative is better?
• MARR = 10%, which one will you choose?
• MARR = 20%, semiautomatic or full automatic?
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Multiple Alternatives
• Criteria:
Select one alternative that requires the largest
investment AND indicated that the extra
investment over another acceptable alternative is
justified
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Multiple Alternatives Procedure
1. Sort them in an ascending order
2. Determine the nature of the CF series
•
•
Some positive CF: do nothing (defender) vs. lowest-initial investment
alternative. Go to step 3
All negative CF: lowest-initial investment (defender) vs. next-higher investment
3. Find the i* of the defender
•
•
4.
5.
6.
7.
If i* < MARR, remove the lowest-investment alternative
Compute the next one. Repeat until i* ≥ MARR, this alternative  defender and
compares it with the next one
Find the annual incremental CF between the challenger and defender
Find i* using PW-based or AW
If i* ≥ MARR, challenger  new defender, o/w next challenger vs. defender
Repeat until only 1 alternative remains, OPTIMAL
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Example 8.7 (T. Blank, p. 247)
Four different prefab-building locations have been suggested, of which
only one will be selected. Cost and annual net cash-flow information
are detailed in table below. The annual net cash-flow series vary due
to differences in maintenance, labor costs, transportation charges, etc.
If the MARR is 10%, use ROR analysis to select the one
economically-best location
Location
Building cost, $
Annual Cash flow, $
Life, years
A
-200,000
+22,000
30
B
-275,000
+35,000
30
C
-190,000
+19,500
30
D
-350,000
+42,000
30
• Answer:
• Sort C, A, B, D
• Start comparing one by one, i = 9.63%; 10.49%, 17.28%, 8.55%
• Choose B
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Spreadsheet Example
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