SME
Engineering Economics
Replacement and break-even analysis
Use this for “A pump worth $12,000 now will sell for $2,000 in 4 years and costs $9,000 a year to run. At 10%, keep it or buy one with an EUAC of $13,500?”
Chart
This is an example. Type your own values to start a new problem.
$
EUAC_D
=
$
MVD
(A/P,
i
%,
nD
)
−
$
SD
(A/F,
i
%,
nD
)
+
$
AOCD
The defender is the pump you have: SD is what it will sell for after its remaining nD years, and AOCD what it costs to run each year.
(A/P)
Factor (A/P, i%, nD)
calculated
(A/F)
Factor (A/F, i%, nD)
calculated
EUAC_C
Challenger EUAC
example · $
Formulas
(A/P)
=
(
i
100
)
÷
(1
−
(1 +
i
100
)
−nD
)
0.31547
=
(
10
100
)
÷
(1
−
(1 +
10
100
)
-4
)
(A/F)
=
(
i
100
)
÷
(
(1 +
i
100
)
n
_D
−
1)
0.21547
=
(
10
100
)
÷
(
(1 +
10
100
)
4
−
1)
EUAC_D = MVD × ((A/P)) − SD × ((A/F)) + AOCD
12,355 = 12,000 × 0.31547 − 2,000 × 0.21547 + 9,000
Assumptions
•
The defender’s original cost is sunk: its first cost now is what it would sell for, MVD.
•
Costs are positive and lower is better (FE’s signed convention writes them negative).
•
The challenger’s EUAC is typed at its own economic service life.
Step-by-step
WE KNOW
MVD = $12,000, SD = $2,000, AOCD = $9,000, nD = 4 yr, i = 10%, EUAC_C = $13,500
FIND
factor (A/P, i%, nD) ((A/P)), factor (A/F, i%, nD) ((A/F)), defender EUAC (EUAC_D)
Step 1 · Find factor (A/P, i%, nD): (A/P)
Use (A/P) = (i ÷ 100) ÷ (1 − (1 + i ÷ 100)−nD)
The capital recovery factor spreads $1 now over nD equal year-end payments.
(A/P)
=
(
10
100
)
÷
(1
−
(1 +
10
100
)
-4
)
(A/P)
=
10
100
÷
(1
−
(1 + 0.1)
-4
)
(A/P)
=
10
100
÷
(1
−
1.1
-4
)
(A/P)
=
10
100
÷
(1
−
0.683)
(A/P)
=
10
100
÷
0.317
(A/P)
=
0.1
0.317
(A/P) = 0.31547
Step 2 · Find factor (A/F, i%, nD): (A/F)
Use (A/F) = (i ÷ 100) ÷ ((1 + i ÷ 100)ⁿD − 1)
The sinking fund factor: the equal year-end deposit that grows to $1 in nD years.
(A/F)
=
(
10
100
)
÷
(
(1 +
10
100
)
4
−
1)
(A/F)
=
10
100
÷
(
(1 + 0.1)
4
−
1)
(A/F)
=
10
100
÷
(
1.1
4
−
1)
(A/F)
=
10
100
÷
(1.4641
−
1)
(A/F)
=
10
100
÷
0.4641
(A/F)
=
0.1
0.4641
(A/F) = 0.21547
Step 3 · Find defender EUAC (EUAC_D)
Use EUAC_D = MVD × ((A/P)) − SD × ((A/F)) + AOCD
Recover the market value over the remaining life, take back the salvage, and add the running cost: not above EUAC_C, so keep the defender.
EUAC_D = 12,000 × 0.31547 − 2,000 × 0.21547 + 9,000
EUAC_D = 3,785.64 − 430.94 + 9,000
EUAC_D = about $12,355
Check
Put the numbers back in. Both sides should match.
0.31547
=
(
10
100
)
÷
(1
−
(1 +
10
100
)
-4
)
✓
0.21547
=
(
10
100
)
÷
(
(1 +
10
100
)
4
−
1)
✓
12,355 = 12,000 × 0.31547 − 2,000 × 0.21547 + 9,000 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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