SME
Engineering Economics
Rate of return and benefit–cost analysis
Use this for “A $50,000 machine saves $12,000 a year for 6 years and sells for $5,000 at the end. What is its rate of return? Is it worth it at a 10% MARR?”
Chart
This is an example. Type your own values to start a new problem.
0
=
−$
C
+
$
A
(P/A,
i*
%,
n
)
+
$
S
(P/F,
i*
%,
n
)
Formulas
C
=
A
×
(1
−
(1 + i* ÷ 100)
−
n
)
÷
(i*
÷
100)
+
S
×
(1 + i* ÷ 100)
−
n
50,000
=
12,000
×
(1
−
(1 +
13.2558
100
)
-6
)
÷
(
13.2558
100
)
+
5,000
×
(1 +
13.2558
100
)
-6
Assumptions
•
One first cost, then equal net returns each year and a salvage at the end: one sign change, so one i*.
•
Accept the project when i* ≥ MARR (minimum attractive rate of return).
•
For two alternatives, type the differences ΔC, ΔA and ΔS: i* is then the incremental rate of return.
Step-by-step
WE KNOW
C = $50,000, A = $12,000, S = $5,000, n = 6 yr
FIND
rate of return (i*)
Step 1 · Find rate of return (i*)
Use C = A × (1 − (1 + i* ÷ 100)⁻ⁿ) ÷ (i* ÷ 100) + S × (1 + i* ÷ 100)⁻ⁿ
No rearrangement puts i* on its own, so it is found by trial: each next try is where a straight line through the last two meets the target.
Try
i*
=
10:
12,000
×
(1
−
(1 +
10
100
)
-6
)
÷
(
10
100
)
+
5,000
×
(1 +
10
100
)
-6
=
55,085.5
(want
50,000)
Try
i*
=
20:
12,000
×
(1
−
(1 +
20
100
)
-6
)
÷
(
20
100
)
+
5,000
×
(1 +
20
100
)
-6
=
41,580.6
(want
50,000)
Try
i*
=
13.7657:
12,000
×
(1
−
(1 +
13.7657
100
)
-6
)
÷
(
13.7657
100
)
+
5,000
×
(1 +
13.7657
100
)
-6
=
49,271.2
(want
50,000)
Try
i*
=
13.1749:
12,000
×
(1
−
(1 +
13.1749
100
)
-6
)
÷
(
13.1749
100
)
+
5,000
×
(1 +
13.1749
100
)
-6
=
50,117.3
(want
50,000)
Try
i*
=
13.2568:
12,000
×
(1
−
(1 +
13.2568
100
)
-6
)
÷
(
13.2568
100
)
+
5,000
×
(1 +
13.2568
100
)
-6
=
49,998.6
(want
50,000)
Try
i*
=
13.2558:
12,000
×
(1
−
(1 +
13.2558
100
)
-6
)
÷
(
13.2558
100
)
+
5,000
×
(1 +
13.2558
100
)
-6
=
50,000.1
(want
50,000)
Try
i*
=
13.2559:
12,000
×
(1
−
(1 +
13.2559
100
)
-6
)
÷
(
13.2559
100
)
+
5,000
×
(1 +
13.2559
100
)
-6
=
49,999.9
(want
50,000)
i* = 13.2558%
Check
Put the numbers back in. Both sides should match.
50,000
=
12,000
×
(1
−
(1 +
13.2558
100
)
-6
)
÷
(
13.2558
100
)
+
5,000
×
(1 +
13.2558
100
)
-6
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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