Download Research on Mathematical Models of Tax Planning for

Research on Mathematical Models of Tax Planning for
Depreciation Methods of Fixed Assets
YANG Qi
Accounting Department, Xiamen University, P.R.China, 361005
[email protected]
Abstract: Choosing an appropriate depreciation method is of great importance in modern enterprises’
tax planning which aims at minimizing the income tax burden. Firstly, this paper describes the taxable
profits before depreciation during the future years as continuous random variables characterized by
normal distribution. Secondly, based on probability statistics, higher mathematics and higher algebra, it
sets up the mathematical models of the expected value for present value of income tax burden during the
future years. Finally, it defines a criterion parameter and the related judgment rules which can be utilized
as an accurate and reliable judgment tool for tax planning for depreciation methods of fixed assets.
Keywords: Tax planning, Depreciation methods, Mathematical model, Criterion parameter
1 Introduction
The tax legislation that allows the depreciation expenses of fixed assets to be deducted in calculating the
taxable profits provides opportunities for tax planning. Due to the different effects of the different
depreciation methods and the influence of the factor of time value of money, the present value of income
tax burden of enterprises may differ under the different depreciation methods. Obviously, choosing an
appropriate depreciation method is very important in modern enterprises’ tax planning.
Many scholars at home and abroad [1-7]conduct research on tax planning for depreciation methods based
on the hypothesis assuming future taxable profits determinate and positive, aim at minimizing the
present value of future income tax burden and show that the accelerated depreciation method is always
superior to the straight-line depreciation method. However, these literatures fail to analyze the impact of
the uncertainty risk of future taxable profits before depreciation. On the contrary, some literatures[8-9]
focusing on the choice of accounting policies regard the risk of future events as an important decision
factor, and YANG Qi [10-11] deals with the general principle of how to incorporate the risk element into
tax planning. Especially for tax planning of the depreciation methods, future taxable profits before
depreciation are certainly not determinate, so the risk element should be considered undoubtedly.
Berg & Moore[12] consider the uncertainty of future cash flows from operation in comparing the
accelerated depreciation method and the straight-line depreciation method, derive for the depreciation
method that should be adopted under different circumstances, and show that when the company is not
allowed to carry forward losses for tax purposes, the straight-line depreciation method is likely to be
preferable, for instance, the discount factor for future tax payments and /or the mean of future cash
flows are high. However, the results of Berg & Moore[12 ] are not complete. Firstly, the equation that cash
flows from operation minus depreciation equal to taxable profits is not always accurate, because there
are other non-cash expenses which can be deducted in calculating taxable profits. Secondly, its results
mostly derive from simplified two-period models, while in reality, tax planning of the depreciation
methods usually covers many years. Thirdly, compared to continuous random variables, describing
future cash flows as discrete random variables may result in some information loss. Finally, the derived
condition that the straight-line depreciation method may be preferable is not a definite quantitative
interval so that it can not be easily applied in tax planning practice.
This paper proceeds as follows. The next section sets up the mathematical models of the expected value
for present value of income tax burden during the future years, defines a criterion parameter and raises
the related judgment rules for tax planning of the depreciation methods, which overcome the limitation
of Berg & Moore[12] mentioned above. Section 3 analyzes the actual application and verification of the
models brought forward in section 2. The final section provides the conclusion of this paper.
461
2 The Set up of the Mathematical Models of Tax Planning for Depreciation
Methods of Fixed Assets
2.1 The mathematical expressions of the depreciation expenses
2.1.1 Accelerated depreciation method
Under the accelerated depreciation method, the depreciation expenses decline during the depreciated
periods. In practice, the accelerated depreciation methods mostly used include the double-declining
balance method and the sum-of-the-year’s-digits method.
When the double-declining balance method is applied, the depreciation expenses in the year j and the
last two years are as follows:
j −1

2
 2 
d
j = 1, 2, K , m − 2
=
1
−
 j  
 V
m
m




(1)

m− 2


1
2


d = d =
 1 −  V − S 
m
 m−1
2
 m 


where V denotes the original cost of fixed assets S denotes the estimated net salvage value, and
m denotes the estimated useful life.
When the sum-of-the-year’s-digits method is applied, the depreciation expense in the year j satisfies
2 ( m − j + 1)
dj =
j = 1, 2,K , m
(2)
(V − S )
m ( m + 1)
，
2.1.2 Straight-line depreciation method
Under the straight-line depreciation method, the depreciation expenses every year are the same, i.e.
dj =
V −S
≡b
m
(3)
2.2 The set up of the mathematical models of the expected value for present value of income tax
burden during the future years
Undoubtedly, risk exists due to the uncertainty of the future taxable profits before depreciation. In
practice, the management could not decide the accurate value but estimate the value range and related
probability for the future taxable profits before depreciation. Thus, describing the future taxable profits
before depreciation as continuous random variables X is reasonable. Accordingly, tax planning
should not aim at the minimization of present value of future income tax burden but the minimization of
the expected value for present value of future income tax burden. Here the probability density of the
random variable X is assumed to satisfy the most common normal probability density distribution, i.e.
−
1
f ( x) =
e
2πσ
( x − µ )2
2σ 2
−∞ < x < +∞
(4)
When losses remedy is not allowed for tax purposes, annual income tax is only paid on annual taxable
profits separately. In other words, if the current taxable profits are negative, no income tax is paid;
otherwise, the income tax is determined by the current taxable profits multiplied by the income tax rate t .
Then the expected value for present value of income tax burden during the future m years is given by
{
}
E max ( X − d j ) , 0 
j
j =1
(1 + r )
m
E (T ) = t ∑
Where r denotes the discount rate.
462
(5)
If the depreciation expenses d j in (5) is calculated according to (1) or (2), then E ( T ) is the expected
value for present value of income tax burden under the accelerated depreciation method, denoted
by E (Ta ) . If d j is calculated according to (3), then E ( T ) is the expected value for present value
of income tax burden under the straight-line depreciation method, denoted by E ( Ts ) .
For
the
convenience
{
of
deducing
the
expected
value
(
，
}
)
of max  X − d j , 0 ,

i.e.
E max ( X − d j ) , 0  , the subscript j is omitted which wouldn’t influence the results. The
definition of the expected value for random variables in probability statistics theory implies that
{
}
∞
E max ( X − d ) , 0  = ∫ ( x − d ) f ( x )dx
d
(6)
Put (4) into (6), then
{
( x − µ )2
}
−
∞
1
2σ 2
−
E max ( X − d ) , 0  =
x
d
e
dx
(
)
∫
d
2πσ
Transform the integrand factor ( x − d ) in (7) into ( x − µ + µ − d ) , then
{
}
1
E max ( X − d ) , 0  =
2πσ
Let
(x − µ)
w=
2σ 2
∞
∫ ( x − µ )e
−
( x − µ )2
2σ
d
2
µ −d
dx +
2πσ
(7)
∫
∞
d
−
e
( x − µ )2
2σ 2
dx
(8)
2
, from the first integral term in (8), we obtain
1
2πσ
∞
∫ ( x − µ )e
−
( x − µ )2
d
The same goes in the second integral term in (8), let
µ −d
2πσ
∫
∞
d
−
e
( x − µ )2
2σ 2
( d − µ )2
σ − 2σ
dx =
e
2π
x−µ
u=
，then
σ
2σ 2
2
(9)
 µ −d 
dx = ( µ − d ) Φ 

 σ 
(10)
 µ −d 
 is the distribution function value of standard normal distribution.
 σ 
Put (9) and (10) into (8), the expected value of max ( X − d ) , 0  is thus
where Φ 
σ −
E {max ( X − d ) , 0 } =
e
2π
( µ − d )2
2σ 2
µ −d 
+ (µ − d )Φ 

 σ 
(11)
According to (5) and (11), the mathematical models of the expected value for present value of income
tax burden during the future years under the accelerated depreciation method and the straight-line
depreciation method are shown as follows respectively
( µ j −d j )

−
 µj −dj
2σ j 2
 σj
+(µj − d j )Φ
 2π e
 σ
j


2
m
E (Ta ) = ∑
j =1
t
(1 + r )
j
463


 


(12)
( µ j −b )


−
 µ j − b 
2σ j 2
 σj
+ ( µ j − b) Φ 
 2π e
 σ  
j




2
m
E (Ts ) = ∑
j =1
t
(1 + r )
j
(13)
2.3 The definition of the criterion parameter rp and the set up of the equation of rp
Finding the equilibrium point is a basic technique in tax planning[4]. Consequently, in order to
conveniently apply the mathematical models (12) and (13) in tax planning, the criterion parameter rp is
defined as the discount rate with which the expected value for present value of income tax burden
during future years under the accelerated depreciation method equals to the one under the straight-line
depreciation method. With the criterion parameter, the management can easily decide the best
depreciation method by comparing the actually used discount rate r and the criterion parameter rp .
，
The following problem is to set up the mathematical equation for solving rp . Let (12) equals to (13) let
z = (1 + rp ) and put it into the equation composed of (12) and (13). After the proper equation
transformations, the mathematical equation for solving rp is given by
2
2

(µ j −d j ) 
 − ( µ j −b )
−
 µj −b 
 µj − dj
 σ  2σ j 2
2σ j 2 
F ( z ) = ∑  j e
b
d
µ
µ
−e
+
−
Φ
−
−
Φ



(
)
(
)
j
j
j

 σ 
 σ
j =1  2π
j
j








m
Because the actually used discount rate r is on the interval of
( 0,1) ， z

  m− j
 z = 0


(14)
in the equation (14)
generally is on the interval of (1, 2 ) . For a given problem in practice, put the known data
µ j 、σ j and
the calculated data d j and b from (1) or (2) and (3) into (14), then the number of the positive roots
can be easily obtained by applying algebra theory and Descartes’ rule of signs. With Newton’s method,
，
z on the interval of (1, 2 ) can be solved then rp can be obtained from z = (1 + rp ) .
Certainly, if the root of F ( z ) = 0 is not on interval of (1, 2 )
，i.e. the criterion parameter
rp is not
on the interval of ( 0,1) , then actual discount rate r would always be larger or smaller than rp so
that there must be one depreciation method that is always the best.
2.4 The judgment rules in the tax planning for depreciation methods
In the tax planning for depreciation methods, the criterion parameter can be used to decide the
depreciation method which results in the minimum expected value of present value of income tax
burden. The judgment rules are as follows:
① If r < rp , then E ( Ts ) < E ( Ta ) , i.e. the straight-line depreciation method is superior to the
accelerated depreciation method.
② If r > rp , then E ( Ts ) > E ( Ta ) , i.e. the accelerated depreciation method is superior to the
straight-line depreciation method.
③If rp ≤ 0 , then r is always larger than rp , so the accelerated depreciation method is always
superior to the straight-line depreciation method.
464
④If rp ≥ 1 , then
r is always smaller than rp , so the straight-line depreciation method is always
superior to the accelerated depreciation method.
⑤If r = rp , then E ( Ts ) = E ( Ta ) , i.e. the straight-line depreciation method can be equated to the
accelerated depreciation method and either will do.
3 The Application and Verification of the Mathematical Models of Tax Planning
for Depreciation Methods of Fixed Assets
3.1 Case 1 of tax planning: the tax planning between the accelerated depreciation method and the
straight-line depreciation method
A Ltd. owns a fixed assets and its management intends to conduct tax planning between the
double-declining balance method and the straight-line method. Suppose the original cost V is 21
million Yuan, the estimated net salvage value S is 3 million Yuan, the useful life m is 6 years and the
income tax rate t is 25%. It’s estimated that the taxable profits before depreciation during the future 6
，
years are normally distributed, i.e. X
～ N (µ ,σ
2
) . Annual µ j and σ j are shown in table 1.
Table 1 The Annual Parameters of Normal Distribution
2
3
4
j
µj
1
9
12
15
σj
9
9
9
5
6
15
15
15
9
6
6
Firstly, d j and b can be calculated by put V , S and m into (1) and (3). Secondly, put
σ j in table 1 and d j , b
µj ,
into (14) , hence the equation for solving the criterion parameter rp can be
deduced as follows:
F ( z ) = 0.8948 z 5 + 0.4600 z 4 + 0.0339 z 3 − 0.2888 z 2 − 0.7990 z − 0.7990 = 0
The equation F ( z ) = 0 is a quintic equation of z . For one thing, following algebra theory, the order
of the equation is odd number, so there must be not less than one positive real root. For another,
following Descartes’ rule of signs, the number of sign differences between consecutive nonzero
coefficients is only one, so there must be not more than one positive root. As a result, the equation only
，the only
(1, 2 ) . Applying Newton’s method to solve the equation，
has one positive real root. Furthermore, due to F (1) = −0.4981 and F ( 2 ) = 32.7126
positive real root must be on the interval of
we can have z = 1.0822 and rp = z − 1 = 0.0822 .
According to the derived criterion parameter rp and the judgment rules
①, ② and ⑤，we can come to
the conclusion: For A Ltd, when actual discount rate r is smaller than 8.22%, the straight-line method
is preferable; when actual discount rate r is larger than 8.22%, the double-declining balance method is
preferable; when actual discount rate r equals to 8.22%, either will do.
The correctness of the above conclusion can be validated as follows. We put µ j , σ j , d j , b
respectively into (12) and (13), then we can have the changing expected value for present value of
income tax burden under the double-declining method and the straight-line method along with different
discount rates, as shown in table 2.
465
Table 2 The Relationship between E
(Ts ) , E (Ta )
r
and
r
E (Ta )
0.02
5.20
0.04
4.81
0.06
4.47
0.0822
4.13
0.10
3.88
0.12
3.63
0.14
3.39
0.16
3.19
E ( Ts )
5.11
4.76
4.44
4.13
3.90
3.66
3.45
3.25
In Table 2, it clearly shows: If r < rp ( = 0.0822 ) , then E ( Ts ) < E ( Ta ) , i.e. the straight-line
method is superior to the double-balance declining method; if r > rp , then E ( Ts ) > E ( Ta ) , i.e. the
double-balance declining method is superior to the straight-line method; if r = rp , then
E ( Ts ) = E ( Ta ) , i.e. either will do. Now the conclusion based on the models in part 2 is validated.
3.2 Case 2 of tax planning: the accelerated depreciation method is always superior to the
straight-line depreciation method
B Ltd. owns a fixed assets and its management intends to conduct tax planning between the
sum-of-the-year’s-digits method and the straight-line method. Suppose the original cost is 15.8 million
Yuan, the estimated net salvage value is 0.8 million Yuan, the useful life is 5 years and the income tax
rate is 25%. It’s estimated that the taxable profits before depreciation during the future 5 years are
，
normally distributed, i.e. X
j
µj
σj
～ N (µ ,σ
Table 3
1
2
) . Annual µ j and σ j are shown in table 3.
The Annual Parameters of Normal Distribution
2
3
4
5
10
8
6
4
2
3
3
2
2
1
Based on the above known conditions, we could have the equation for solving the criterion parameter as
follows according to (2), (3) and (14).
F ( z ) = 1.9485 z 4 + 0.9360 z 3 − 0.7709 z − 1 = 0
Following algebra theory and Descartes’ rule of signs, there is not more than one positive root.
Furthermore, F (1) > 0 and F ( 2 ) > 0 imply that there is no root on the interval of (1, 2 ) ,
while F ( 0 ) < 0 and F (1) > 0 imply that the root is on the interval of ( 0,1) , i.e. −1 < rp < 0 .
③，
Based on the judgment rule
we can draw the conclusion that no matter what discount rate B Ltd uses
in tax planning, the sum-of-the-year’s-digits method is always superior to the straight-line method.
3.3 Case 3 of tax planning: the straight-line depreciation method is always superior to the
Accelerated depreciation method
C Ltd. owns a fixed assets and its management intends to conduct tax planning between the
sum-of-the-year’s-digits method and the straight-line method. Suppose the original cost is 57.9 million
Yuan, the estimated net salvage value is 2.9 million Yuan, the useful life is 10 years and the income tax
rate is 25%. It’s estimated that the taxable profits before depreciation during the future 10 years are
，
normally distributed, i.e. X
～ N (µ ,σ
Table 4
2
) . Annual µ j and σ j are shown in table 4.
The Annual Parameters of Normal Distribution
466
j
µj
1
2
3
4
5
6
7
8
9
10
2
2
2
4
5
6
7
8
9
10
σj
1
1
1
1
1
1
1
1
1
1
Based on the above known conditions, we could have the equation for solving the criterion parameter as
follows according to (2), (3) and (14).
F ( z ) = 0.0009 z 9 + 0.0009 z8 + 0.0009 z 7 + 0.0288 z 6 + 0.1146 z 5
−0.3855 z 4 − 1.4712 z 3 − 2.4980 z 2 − 3.4991z − 4.5 = 0
Following algebra theory and Descartes’ rule of signs, there is only one positive real root. In addition,
F (1) < 0 and F ( 2 ) < 0 imply that there is no root on the interval of (1, 2 ) , while F ( 4 ) > 0
and F ( 2 ) < 0 imply that the positive real root of the equation is on the interval of
1 < rp < 3 . Based on the judgment rule
( 2, 4 ) , i.e.
④，we can conclude that no matter what discount rate C Ltd
uses in tax planning, the straight-line method is always superior to the sum-of-the-year’s-digits method.
4 Conclusion
A summary of the main results of this paper is as follows:
Firstly, applying probability statistics, higher mathematics and higher algebra, this paper sets up the
mathematical models of expected value for present value of income tax burden during future years
without any approximate treatment so that the mathematical models are rigorous in theory.
Secondly, based on the mathematical models, this paper defines a criterion parameter, deduces the
equation for solving it, and raises the judgment rules, which can be utilized in deciding the best
depreciation method accurately and reliably in the tax planning practice.
Finally, the mathematical model can also be put into practice in the tax planning of transfer pricing or
other issues. These are left for future research.
References
[1]. Shao ruiqing. Research on the Measurement Model for Impact of Depreciation Methods on
Enterprises’ Income Tax. Shanghai Accounting, 2001, (1): 39 41 (in Chinese)
[2]. Yang feifei. Research on Income Tax Planning in the Operation Process of Enterprises. Finance
and Trade Research, 2004, (6): 109 110 (in Chinese)
[3]. Xiao xiaofei. On the Impact of the Choice of Accounting Methods on Enterprises’ Tax Planning.
The Theory and Practice of Finance and Economics, 2004, (25): 81 83 (in Chinese)
[4]. Gai Di Corporate Tax Planning Theory and Practice Dongbei University of Finance & Economics
Press 2005: 123 127 , 83 110(in Chinese)
[5]. Linhart, P. Some Analytical Results on Tax Depreciation. Bell Journal of Economics and
Management Science, 1970, (1): 82 112
[6]. Roemmich, R., G. L. Duke, and W. H. Gates. Maximizing the Present Value of Tax Savings from
Depreciation. Management Accounting, 1978, 56: 55 57
[7]. Wakeman, L. M., Optimal Tax Depreciation. Journal of Accounting and Economics, 1980, (1):
213 237
[8]. Baber, W. Budget-based Compensation and Discretionary Spending. The Accounting Review, 1985,
60: 1 9
[9]. Healy, P. The Effects of Bonus Schemes on Accounting Decisions. Journal of Accounting and
～
～
．
，
～
～
．
～
～
～
～
467
～
～
Economics, 1985, (7): 85 107
[10]. Yang Qi. Research on the risk entropy assessment of tax planning. Communication of Finance and
Accounting (Academic Edition), 2005,(2): 60 62 (in Chinese)
[11]. Yang Qi. Research on Applying Game Theory to the Decisions Making of Risky Tax Planning.
Statistics and Decision, 2007, (21): 36 38 (in Chinese)
[12]. Berg, M., and G. Moore. The Choice of Depreciation Method under Uncertainty, Decision Sciences,
1989, 20(4): 643 653
～
～
～
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