SME
Engineering Economics · Problem type
Capitalized cost
Use this for “A $2,000,000 bridge costs $40,000 a year to keep up and $150,000 to resurface every 10 years. At 5%, what is its capitalized cost?”
Chart
CC = C + PA + PR
2,000,000 + 800,000 + 238,514 = 3,038,514
The flows in add and the flows out take away: CC = 3,038,514.
This is an example. Type your own values to start a new problem.
$
CC
=
$
C
+
$
%
+
$
(A/F,
%,
)
%
(A/F)
Factor (A/F, i%, k)
calculated
PA
Worth of annual cost
calculated · $
PR
Worth of periodic cost
calculated · $
Formulas
(
A
/F)
=
(
i
100
)
÷
(
(1 +
i
100
)
k
−
1)
0.079505
=
(
5
100
)
÷
(
(1 +
5
100
)
10
−
1)
PA
=
A
i
÷ 100
800,000
=
40,000
5 ÷ 100
PR
=
R
×
(
A
/F)
i
÷ 100
238,514
=
150,000
×
0.079505
5 ÷ 100
CC
=
C
+
PA
+
PR
3,038,514 = 2,000,000 + 800,000 + 238,514
Assumptions
•
The project lasts forever, so a yearly amount A is worth A ÷ i now.
•
A cost R every k years is first spread over its k years by (A/F, i%, k).
•
Costs count positive; C is paid now and the first R comes at the end of year k.
Step-by-step
WE KNOW
C = $2,000,000, A = $40,000, R = $150,000, k = 10 yr, i = 5%
FIND
factor (A/F, i%, k) ((A/F)), worth of annual cost (PA), worth of periodic cost (PR), capitalized cost (CC)
Step 1 · Find factor (A/F, i%, k): (A/F)
Use (A/F) = (i ÷ 100) ÷ ((1 + i ÷ 100)ᵏ − 1)
The
sinking-fund
factor
over
k
years:
the
yearly
deposit
that
grows
to
$1
by
the
time
R
is
due.
(
A
/F)
=
(
5
100
)
÷
(
(1 +
5
100
)
10
−
1)
(
A
/F)
=
5
100
÷
(
(1 + 0.05)
10
−
1)
(
A
/F)
=
5
100
÷
(
1.05
10
−
1)
(
A
/F)
=
5
100
÷
(1.629
−
1)
(
A
/F)
=
5
100
÷
0.629
(
A
/F)
=
0.05
0.629
(
A
/F)
=
0.0795046
Step 2 · Find worth of annual cost (PA)
Use PA = A ÷ (i ÷ 100)
Money
worth
A
i
now
pays
A
in
interest
every
year
without
end.
PA
=
40,000
5 ÷ 100
PA
=
40,000
0.05
PA = $800,000
Step 3 · Find worth of periodic cost (PR)
Use PR = R × ((A/F)) ÷ (i ÷ 100)
Spread
R
over
its
k
years
as
a
yearly
amount,
then
treat
that
amount
as
lasting
forever.
PR
=
150,000
×
0.0795046
5 ÷ 100
PR
=
150,000
×
0.0795
0.05
PR
=
11,925.69
0.05
PR = about $238,514
Step 4 · Find capitalized cost (CC)
Use CC = C + PA + PR
Every cost is now a present worth, so they add.
CC
=
2,000,000
+
800,000
+
238,514
CC
=
about
$3,038,514
Check
Put the numbers back in. Both sides should match.
0.0795046
=
(
5
100
)
÷
(
(1 +
5
100
)
10
−
1)
✓
800,000
=
40,000
5 ÷ 100
✓
238,514
=
150,000
×
0.0795046
5 ÷ 100
✓
3,038,514 = 2,000,000 + 800,000 + 238,514 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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