SME
Process Design · Problem type
Net present value and payback
Use this for “A $1,000,000 plant returns $200,000 a year for 10 years. At 10% interest, what is its NPV and its payback?”
Chart
NPV = PW − C₀
1,228,913 − 1,000,000 = 228,913
The flows in add and the flows out take away: NPV = 228,913.
This is an example. Type your own values to start a new problem.
(P/A)
=
(1
−
(1
+
%)
−
)
÷
i
%
$
NPV
=
$
A
×
(P/A)
−
$
C₀
tₚ (yr)
=
$
C₀
÷
$
A
PW
Present worth of cash
calculated · $
Formulas
(P/
A
)
=
(1
−
(1 +
i
100
)
−
n
)
÷
(
i
100
)
6.1446
=
(1
−
(1 +
10
100
)
-10
)
÷
(
10
100
)
PW
=
A
×
((P/
A
))
1,228,913 = 200,000 × 6.1446
NPV
=
PW
−
C₀
228,913 = 1,228,913 − 1,000,000
tₚ
=
C₀
A
5
=
1,000,000
200,000
Assumptions
•
The investment is spent at time 0; the same cash flow A comes at the end of each year.
•
One interest rate i (the minimum acceptable return) discounts every year; no taxes or salvage value.
•
Simple payback ignores interest: the years of cash that repay the investment.
Step-by-step
WE KNOW
C₀ = $1,000,000, A = $200,000, i = 10%, n = 10 yr
FIND
annuity factor ((P/A)), present worth of cash (PW), net present value (NPV), payback time (tₚ)
Step 1 · Find annuity factor ((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 annuity factor sums them.
(P/
A
)
=
(1
−
(1 +
10
100
)
-10
)
÷
(
10
100
)
(P/
A
)
=
(1
−
(1 + 0.1)
-10
)
÷
0.1
(P/
A
)
=
1 −
1.1
-10
0.1
(P/
A
)
=
1 − 0.38554
0.1
(P/
A
)
=
0.61446
0.1
(P/
A
)
=
6.144567
Step 2 · Find present worth of cash (PW)
Use PW = A × ((P/A))
Each year’s cash, times the factor.
PW = 200,000 × 6.144567
PW = about $1,228,913
Step 3 · Find net present value (NPV)
Use NPV = PW − C₀
Take
the
investment
from
the
present
worth:
above
0,
the
project
earns
more
than
i
.
NPV = 1,228,913 − 1,000,000
NPV = about $228,913
Step 4 · Find payback time (tₚ)
Use tₚ = C₀ ÷ A
Years of cash that pay back the investment.
tₚ
=
1,000,000
200,000
tₚ = 5 yr
Check
Put the numbers back in. Both sides should match.
6.144567
=
(1
−
(1 +
10
100
)
-10
)
÷
(
10
100
)
✓
1,228,913 = 200,000 × 6.144567 ✓
228,913 = 1,228,913 − 1,000,000 ✓
5
=
1,000,000
200,000
✓
Topic check
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