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Depreciation
By:
John Powell and Alan Sun
Mathematics of Finance
Dr.
Janusz Kawczak
Abstract
Depreciation is the decline in value of
an object over time and can be calculated in many
different methods. However, these
methods are all estimates
of an objects
value or “Book Value”,
and can differ
widely. Due to the importance of Depreciation in
today’s financial world, we
decided to
utilize each method of Depreciation and find the
most effective method. To achieve
this,
we used a test case and used its variable to find
the annual depreciations that would occur for
each method. Once the results were
found, the comparison showed that the Straight
Line Method
of Depreciation, where
there is an uniform annual decline of value over a
set period of time, is the
most
efficient and beneficial method. This conclusion
reinforces the current beliefs in the
financial world.
Background
Depreciation is the term used to
describe the decline in value of a physical asset
over a
certain span of time. Most
objects lose value as they are used due to damage
from use, aging,
obsolescence, and
impairment. However the exact amount of value that
is lost and the current
“book value” of
an asset cannot be easily known. Depreciation is
thus used as an estimate of this
decline a
nd is used for
several important purposes in today’s business
world.
Depreciation has been
a factor in human economics ever since the advent
of trade. At its
most basic,
depreciation states that an older object is
usually not as valuable as a newer one.
However, with the formation of large
corporations and the creation of our modern
economic
structure, depreciation has
taken on a greater importance with a genuine need
for accurate
calculations of
depreciation.
Today most companies
need a precise estimate o
f their
physical assets’ depreciation for
two
main reasons. First, companies must compare the
expenses in using and buying equipment
and materials to the revenue generated
in that year. Without an accurate estimate for
depreciation,
a company’s profit and
current value could not be known.
Secondly, companies must ascertain
the
value of their assets and determine that objects
are not appraised as being more costly than
their actual worth. In fact, the value
of such assets directly affects the taxes that
must be paid each
year and depreciation
is used to find the exact amount that is
deductible.
The exact amount
depreciated annually is based on several factors.
The main criteria is
the method used in
assessing an object's depreciation. The main
methods range from deducting a
set
amount for a certain period of time to evaluating
the exact hours that and object is used to
generate revenue. Also, different
physical assets can be evaluated in drastically
dissimilar
manners. In the United
States, objects can have a depreciable life span
ranging from 5 years to 50
years or
more depending on several factors. For example,
computers and electronics that become
obsolete quickly will undergo faster
depreciation than trucks or factories that have
much longer
life-spans. Most countries
today have set laws and regulations that determine
the calculation of
depreciation. In
America, the law uses Straight Line along with
Declining Balanace as an
orthodox
measure to make sure that all the companies and
corporations in the United States
conform with each other when reporting
their company’s data and filing their tax
reports.
Research Problem
Depreciation has become a complex
operation in today’s business world. With our
current economical situation, the
precise estimate of capital and assets becomes
crucial. Still,
there are many
differing ways of calculating depreciation,
ranging from the simplistic straight line,
to amortization that includes monetary
inflation. Using a specific method can result in
drastically
different results, and
there can be methods that are more beneficial to a
company. In fact, the
proper method of
depreciation can lower taxable income and portray
larger assets. We thus
decided to
research the methods of depreciation and determine
which one creates the best results.
Through setting up a test case and
calculating depreciation for each different type
we can
discover the method that
ameliorates a company’s financial
situation
Methods for
Finding Depreciation
1.1- Straight-Line
Method
The Straight-Line Method is the
most popular and simplistic way to find
depreciation. In this
method, it is
assumed that depreciation occurs at a constant
rate per unit of time. The constant
rate yields a linear equation appearing
as straight line when graphed. In
Eq.
1
R
is the
depreciation allowance,
W
is the wearing value,
which is original value minus scrap value,
and
n
is the
number of useful years that the item can be used.
Below is a
Basic Notation for
Depreciation
table.
(Guthrie, 2004)
R
?
W
n
1
Basic Notation for Depreciation
C
Original cost of the
capital asset
S
Scrap
Value
W
Wearing Value,
W
?
C
?
S
R
t
Depreciation
allowance for year t
B
t
Book Value
at the end of year t
r
Depreciation rate per year, per service hour, or
production unit.
n
Useful
life of capital asset in years, service hours, or
production units
Example
1.1:
Your family buys a car for $$20,000
and it has an estimated life of 8 years with a
trade in value of $$3,500. Find the
straight-line depreciation allowances and make a
schedule.
R
?
End
of Year
0
1
2
3
4
5
6
7
8
W
20,000
?
3,500
16500
?
?
?
$$2062.50
n
8
8
Accumulated
Depreciation
…
$$2,062.50
$$4,125.00
$$6,187.50
$$8,250.00
$$10,312.50
$$12,375.00
$$14,437.00
$$16,500.00
Book Value of Car
$$20,000.00
$$17,937.50
$$15,875.00
$$13,812.50
$$11,750.00
$$9,687.50
$$7,625.00
$$5,562.50
$$3500.00
Annual Depreciation
Allowance
…
$$2,062.50
$$2,062.50
$$2,062.50
$$2,062.50
$$2,062.50
$$2,062.50
$$2,062.50
$$2,062.50
In the example above the car
depreciates at a rate of $$2062.50 a year for 8
years. Each year the
car depreciates
by another $$2,062.50 until it is traded in at the
end of the eight year for $$3,500.
The
car is not always traded in at the end of its
“estimated” life but can be used for longer
periods
.
However its “book
value” remains at $$3,500.
1.2- Service Hours Method
The Service Hours Method is similar to
the Straight-
Line Method except it
calculates “book
value” based on the
hours an object has been used instead of set
amount of depreciation for
each
year. When using this system of
depreciation a company has to keep up with all of
the hours of
usage so that it can
calculate the correct depreciation amount. This
makes it much more difficult
to obtain
than the straight-line method and makes it a lot
less practical in real life. (Guthrie 2004).
R
t
?
rH
t
2
In
Eq.
2
R
t
is the
depreciation allowance for year t,
r
is the rate in dollars per
service hour,
H
t
represents the
service hours expended in year t. (Guthrie 2004).
1.3-Production Units Method
The Production Units Method of
depreciation uses the number of actual units
produced to
determine the depreciation
allowance. To get the depreciation allowance the
depreciation rate is
multiplied by the
number of items produced. This method works well
when factory production is
up because
when a company produces more they can write off
larger amounts as depreciation.
This
method is also very similar to the straight-line
method except that
n
is the
estimated life in
units produced and
the rate of depreciation is expressed in dollars
per unit. (Book Depreciation
2004-2005)
R
t
?
rU
t
3
In
Eq. 3
r
is the rate in dollars per
unit,
U
t
are the
number of output units. (Guthrie 2004).
2.1-
Sum of the Year’s
Digits
The Sum of the Year’s
Digits method is a way to calculate depreciation
where the depreciation
rate diminishes
as it gets closer to the end of the estimated life
of the item. To calculate the rate
of
depreciation the number of years of the estimated
life of the item are added up and placed in
the denominator and the numerator is
the number of years that are left in the items
estimated life.
Then the rate of
depreciation is multiplied by the wearing value to
get the depreciation value.
Below is
Eq. 4
which shows the
depreciation formula. (Depreciation Schedule
2004-2005)
r
1
?
n
n
?
1
1
, … ,
r
n
?
,
r
2
?
1
?
2
?
...
?
n
1
?
2
?
...
?
n
1
?
2
?
...
?
n
R
t
< br>?
rW
t
4
In
Eq. 4
r
t
is the
depreciation rate per year and
W
is the wearing value.
Example 2.1:
A clothing
company buys several cash registers for $$5,000
each. Each cash
register has an
expected life of 6 years with a
trade-
in value of $$900. Find the sum of
the year’s
depreciation and make a
schedule.
Wearing Value:
W
?
C
?
S
?
$$5,000
?<
/p>
$$900
?
$$4,100
Denominator of the Depreciation Ra
te:
1
?
2
?
...
?
6
?
21
R
t
?
rW
?
R
1
?
t
End of Year
0
1
2
3
4
5
6
6
5
?
$$4,100
?
?
$$1171.43
R
2
?
?
$$4,100
?
?
$$976.19
< br>, and so on.
21
21
Annual
Depreciation
Allowance
…
$$1,171.43
$$976.19
$$780.95
$$585.71
$$390.48
$$195.24
Accumulated
Depreciation
…
$$1,171.43
$$2,147.62
$$2,928.57
$$3,514.28
$$3,904.76
$$4,100
Book Value of Car
$$5,000.00
$$3,828.57
$$2,852.38
$$2,071.43
$$1,485.72
$$1,095.24
$$900
2.2 Declining Balance Method
The Declining Balance Method of
depreciation uses a fixed rate of depreciation
that is multiplied
by the book value to
get the depreciation allowance. In this formula
the depreciation rate stays
constant
but since the depreciation allowance depends on
the book value as well, the amount of
money that is depreciated changes at
every step. The larger amounts of depreciation
will take
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