Rate of Return
Definition
The Rate of Return (ROR) is:
• A percentage (or interest rate) that describes the
merit of an investment.
•
(Return on investment during a year)/(Amount Invested)
•
The interest rate than makes the cash flows of
income equivalent to the cash flows of cost
Usage
•
We use the ROR to evaluate investments
because
– percentage rates are familiar
– percentage rates are dimensionless
– they are commonly used as business measures
•
Synonyms
– ROR: Rate of Return
– ROI: Return on Investment
– IRR: Internal Rate of Return
Single Project
•
The ROR is the interest rate that makes
– NPW = PW Benefits - PW Costs = 0, or
– NAW = AW Benefits - AW Costs = 0
Example 1:
•
Find the ROR of an investment of 100 at time 0
and a return of 250 at time 10.
250
0
0
-100
1
2
3
4
5
6
7
8
9
10
NPW = -100 + 250 (P/F, i, 10) = 0
Example 1: Exact Computation
•
•
•
•
•
•
•
•
Set NPW = 0
NPW
= -100 + 250 (P/F, i, 10) = 0
=> (P/F, i, 10) = 100/250 = 0.4
=> 1/(1+i)10 = 0.4
=> (1+i)10 = 1/0.4 = 2.5
=> (1+i)10(0.1) = (2.5)0.1
=> i = (2.5)0.1 - 1 = 0.09595
Therefore, the ROR = 9.595 %
Example 1: Trial and Error
•
•
•
•
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Set NPW = 0
NPW
= -100 + 250 (P/F, i, 10) = 0
Try 9% : -100 + 250 (P/F, 0.09, 10) = 5.6063
Try 10%: -100 + 250 (P/F, 0.10, 10) = -3.614
Linearly Interpolating:
ROR
= 0.09 + [(5.603)/(5.603- (-3.614))](0.1-0.09)
= 0.09608 or 9.608%
Linear Interpolation
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•
•
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•
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Shape ratio of pale rectangle: (A-B) / (y-x)
Shape ratio of smaller rectangle: (A-0) / (i-x)
Since shapes are the same:
(A-B) / (y-x)) = (A) / (i-x)
=> i-x = [ A / (A-B) ] (y-x)
=> i = x + [ A / (A-B) ] (y-x)
A
0
B
x
i
y
NPW (or NAW) as a function of i
For an investment
NPW
$150.00
$100.00
$50.00
($100.00)
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
($50.00)
0.02
$0.00
0
•
Example 2
•
Find the ROR of an investment of $200 at time 0 and
returns of $150 at time 1 and $175 at time 2.
15 0
17 5
0
0
-20 0
1
2
Example 2: Exact Computation
•
•
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Set NPW = 0
NPW = -200 + 150/(1+i) + 175/(1+i)2 = 0
Let x = 1/(1+i) and the expression becomes
175x2 + 150x -200
=0
150  1502  4(175)(200) 150  403.113


2(175)
350
•
So x = 1/(1+i) = 0.72318 => i = 0.3828 or 38.28%
Example 3
•
Find the ROR of an investment of $100 with a
revenue of $16 a year for 10 years.
16
0
0
-100
1
2
3
4
5
6
7
8
9
10
Example 3:
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•
•
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NAW = - 100(A/P, i, 10) + 16 = 0
(A/P, i, 10) = 0.16
or [ i (1 + i)10]/[(1 + i)10 - 1)] = 0.16
Difficult to solve for i using because of the
nonlinear factor
Example 3: Trial and Error
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•
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•
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Use trial and error
NAW = - 100(A/P, i, 10) + 16
Try 9%:
NAW = - 100(A/P, 0.09, 10) + 16 = 0.418
Try 10%:
NAW = - 100(A/P, 0.10, 10) + 16 = -0.275
Linear Interpolating: ROR  9.604%
Example 4
•
Find the ROR an investment of $16 a year for 10
years with a return of $250 at year 10
250
0
0
-16
1
2
3
4
5
6
7
8
9
10
Example 4: Trial and Error
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•
•
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Set FW = 0
FW
= -16 (F/A, i, 10) + 250 = 0
Try 8% :
-16 (14.4866) + 250 = 18.2144
Try 10%:
-16 (15.9374) + 250 = -4.9984
Interpolating:
ROR = 0.08 + [18.2144 /(18.2144+4.9984)](0.1-0.08)
= 0.09569 or 9.569%
ROR is approximately 9.569%
Example 5
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•
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Find the Rate of Borrowing associated with borrowing 100
and paying back 250 after 10 years.
ROR here is approximately 9.6%
ROR of return is actually the cost borrowing.
100
0
1
•NPW
0
2
3
4
5
6
7
8
9
10
= 100 - 250 (P/F, i, 10) =
-250
Example 6: Complex Example



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A machine costs 2000.
We expect a return of $600 per year for ten years.
The machine is then sold with a salvage of $400.
Operating cost is 100 in the first year and increases
by $50 per year thereafter.
600
400
0
0
1
2
3
4
5
6
7
8
9
10
-100 -150 -200
-550
-2000
Example 6: Trial and Error
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•
•
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NPW = -2000 + 500(P/A, i, 10) + 400(P/F, i, 10)50(P/G, i, 10)
Try i = 0.05, NPW = 523.83
Try i = 0.1, NPW = 81.93
Try i = 0.12, NPW = -58.8
Use linear interpolation to compute a value
between 10% and 12%
Example 7: Non-simple Investment
•
•
A0 = -100, A1 = 405, A2 = -500, A3 = 200,
A4 = -100, A5 = 100
40 5
20 0
10 0
0
0
1
-10 0
2
3
-10 0
-50 0
4
5
Example 7:
•
This is an example of a non-simple investment since
– the initial cash flow is negative, but
– more than one sign change occurs in the net cash flow
series.
•
•
•
•
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NPW = -100 + 405(P/F,i,1)
- 500(P/F,i,2)
+ 200(P/F,i,3)
- 100(P/F,i,4)
+ 100 (P/F,i,5)
Example 7: Graphically
$6 .00
$5 .00
$4 .00
$3 .00
NPW
$2 .00
$1 .00
$0 .00
($1 .00)
($2 .00)
($3 .00)
0.00
0.20
0.40
0.60
0.80
1.00
1.20
Simple Case 1
100
-100
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Total revenue = total cost
ROR = 0
Simple Case 2
50
50
50
150
-100
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Uniform inflow with Capital entirely recovered
ROR = Inflow/Investment = A/P
Simple Case 3
50
50
50
50
-100
•
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Uniform inflow lasting forever
ROR = inflow/Investment = A/P
50
Simple Case 4
50
50
50
50
-100
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One factor involved
Solve for factor value and use the tables
50
Making Decisions with ROR
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When Investing
– Accept the project if ROR ≥ MARR
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When Borrowing
– Accept the project if Rate of Borrowing ≤ MARB