SME
Engineering Economics · Problem type
Benefit–cost ratio
Use this for “A highway saves users $900,000 a year, with $150,000 a year of disbenefits. It costs $5,000,000 plus $120,000 a year O&M. At 6% over 20 years, find B/C.”
Chart
This is an example. Type your own values to start a new problem.
$
PWB
=
($
B
−
$
D
)
×
(P/A,
i
%,
n
)
$
PWC
=
$
C
+
$
M
×
(P/A,
i
%,
n
)
B/C
=
$
$
=
D is the yearly disbenefit, the harm the project does to the public; M is its yearly operation and maintenance (O&M) cost.
(P/A)
Factor (P/A, i%, n)
calculated
Formulas
(P/A)
=
(1
−
(1 +
i
100
)
−
n
)
÷
(
i
100
)
11.4699
=
(1
−
(1 +
6
100
)
-20
)
÷
(
6
100
)
PWB
=
(
B
−
D
)
×
((P/A))
8,602,441 = (900,000 − 150,000) × 11.4699
PWC
=
C
+
M
×
((P/A))
6,376,391 = 5,000,000 + 120,000 × 11.4699
B
C
=
PWB
PWC
1.349
=
8,602,441
6,376,391
Assumptions
•
Conventional B/C: disbenefits are taken from the benefits, and O&M is counted as a cost.
•
Benefits and costs to the public are discounted at one rate i; amounts come at each year’s end.
•
B/C ≥ 1 means the project pays: its benefits are worth at least its costs.
Step-by-step
WE KNOW
B = $900,000, D = $150,000, C = $5,000,000, M = $120,000, i = 6%, n = 20 yr
FIND
factor (P/A, i%, n) ((P/A)), PW of net benefits (PWB), PW of costs (PWC), benefit–cost ratio (B/C)
Step 1 · Find factor (P/A, i%, n): (P/A)
Use (P/A) = (1 − (1 + i ÷ 100)⁻ⁿ) ÷ (i ÷ 100)
Add the present worth of $1 at the end of each year: the factor sums them.
(P/A)
=
(1
−
(1 +
6
100
)
-20
)
÷
(
6
100
)
(P/A)
=
(1
−
(1 + 0.06)
-20
)
÷
(
6
100
)
(P/A)
=
1 −
1.06
-20
6 ÷ 100
(P/A)
=
1 − 0.3118
0.06
(P/A)
=
0.6882
0.06
(P/A) = 11.469921
Step 2 · Find PW of net benefits (PWB)
Use PWB = (B − D) × ((P/A))
Take the disbenefits from the benefits, then bring the net amount of every year to today.
PWB = (900,000 − 150,000) × 11.469921
PWB = 750,000 × 11.4699
PWB = about $8,602,441
Step 3 · Find PW of costs (PWC)
Use PWC = C + M × ((P/A))
The
first
cost
is
already
at
time
0;
bring
each
year’s
O&
M
to
today
and
add
it.
PWC = 5,000,000 + 120,000 × 11.469921
PWC = 5,000,000 + 1,376,390.52
PWC = about $6,376,391
Step 4 · Find benefit–cost ratio (B/C)
Use B/C = PWB ÷ PWC
Divide what the public gains by what the project costs, both today: 1 or more, it pays.
B
C
=
8,602,441
6,376,391
B
C
=
1.349
Check
Put the numbers back in. Both sides should match.
11.469921
=
(1
−
(1 +
6
100
)
-20
)
÷
(
6
100
)
✓
8,602,441 = (900,000 − 150,000) × 11.469921 ✓
6,376,391 = 5,000,000 + 120,000 × 11.469921 ✓
1.349
=
8,602,441
6,376,391
✓
Topic check
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