SME
Engineering Economics · Problem type
Declining balance and DDB
Use this for “An $84,000 machine is depreciated by double declining balance over 7 years. What are its depreciation and book value in year 3?”
Chart
BV(3) ≈ 30,610
This is an example. Type your own values to start a new problem.
d
%
=
k
n
$
BVⱼ
=
$
C
(1
−
%)
BV_(j−1)
Book value at the start of year j
calculated · $
Dⱼ
Depreciation in year j
calculated · $
Formulas
d
=
100
×
k
n
28.5714
=
100
×
2
7
BVⱼ
=
C
×
(1 −
d
100
)
j
30,612.24
=
84,000
×
(1 −
28.5714
100
)
3
BV_(
j
−1)
=
C
×
(1 −
d
100
)
j
− 1
42,857.14
=
84,000
×
(1 −
28.5714
100
)
3 − 1
Dⱼ
=
d
100
×
(BV_(
j
−1))
12,244.90
=
28.5714
100
×
42,857.14
BVⱼ
=
BV_(
j
−1)
−
Dⱼ
30,612.24 = 42,857.14 − 12,244.90
Assumptions
•
Each year takes the same fraction d of the book value left at the start of that year.
•
Salvage is not taken off first, and the book value never goes below salvage: stop, or switch to straight line, there.
•
The multiple k is 2 for double declining balance (DDB) and 1.5 for 150% declining balance.
Step-by-step
WE KNOW
C = $84,000, n = 7 yr, k = 2, j = 3 yr
FIND
depreciation rate (d), book value after year j (BVⱼ), book value at the start of year j (BV_(j−1)), depreciation in year j (Dⱼ)
Step 1 · Find depreciation rate (d)
Use d = 100 × k ÷ n
Straight
line
would
take
1
n
of
the
cost
each
year;
declining
balance
takes
k
times
that
rate.
d
=
100
×
2
7
d
=
200
7
d
=
28.57143%
Step 2 · Find book value after year j (BVⱼ)
Use BVⱼ = C × (1 − d ÷ 100)ʲ
Each
year
keeps
1
−
d
of
the
book
value,
so
j
years
multiply
the
cost
by
(1
−
d
)
j
times.
BVⱼ
=
84,000
×
(1 −
28.57143
100
)
3
BVⱼ
=
84,000
×
(1 − 0.2857)
3
BVⱼ
=
84,000
×
0.7143
3
BVⱼ = 84,000 × 0.3645
BVⱼ = about $30,612.24
Step 3 · Find book value at the start of year j (BV_(j−1))
Use BV_(j−1) = C × (1 − d ÷ 100)ʲ⁻¹
Year
j
starts
with
the
value
left
after
j
−
1
years
of
keeping
1
−
d
.
BV_(
j
−1)
=
84,000
×
(1 −
28.57143
100
)
3 − 1
BV_(
j
−1)
=
84,000
×
(1 − 0.2857)
2
BV_(
j
−1)
=
84,000
×
0.7143
2
BV_(
j
−1)
=
84,000
×
0.5102
BV_(
j
−1)
=
about
$42,857.14
Step 4 · Find depreciation in year j (Dⱼ)
Use Dⱼ = d ÷ 100 × (BV_(j−1))
The
year
takes
d
of
the
book
value
it
starts
with.
Dⱼ
=
28.57143
100
×
42,857.14
Dⱼ = 0.28571 × 42,857.14
Dⱼ = about $12,244.90
Check
Put the numbers back in. Both sides should match.
28.57143
=
100
×
2
7
✓
30,612.24
=
84,000
×
(1 −
28.57143
100
)
3
✓
42,857.14
=
84,000
×
(1 −
28.57143
100
)
3 − 1
✓
12,244.90
=
28.57143
100
×
42,857.14
✓
30,612.24 = 42,857.14 − 12,244.90 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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Use this for “A machine brings in $30,000 a year before tax and is depreciated $12,000 this year. At a 21% tax rate, what is the after-tax cash flow?”