Calculus Logic Function in Tax Risk Avoidance in Different Stages of Enterprises

Tax risk belongs to the category of risk. Before understanding tax risk, it is necessary to clarify what is risk. In 1895, the American scholar Hayes gave a definition of risk from the perspective of economics. He believed that risk is “the possibility of loss”; uncertainty of probability random events is risk. Risks will only occur when there are two or more possibilities for an activity, and the occurrence of risks usually brings some form of loss. Therefore, tax risk is an uncertainty faced by enterprises in all tax-related activities, such as enterprises being inspected by tax authorities, bearing excessive tax liabilities, or incurring liabilities for tax reimbursement and fines. For the production and operation of an enterprise, the pursuit of profit is an eternal theme, but the profit of the enterprise must be controlled within an acceptable risk range. If the profit has a high risk, the enterprise will have the hidden danger of failure [4].

Tax risk has tax uncertainty. Uncertainty is the basis for tax risks. Without uncertainty, risks will not arise, and tax risks will be even more difficult to talk about. In the process of production and operation, enterprises are faced with a variety of decisions. Each decision is the result of a cost-benefit balance. The tax liability is the embodiment of each decision in tax law. Different decisions cause different tax burdens, and even the same decision will have different tax burdens due to different subjective reasons of the taxpayer and objective reasons of the external environment [5]. It should be noted that less tax and delayed taxation have tax risks, and more taxation and early taxation also have tax risks. Less tax payment and delayed tax payment means that failure to pay taxes on time and amount will be punished by tax authorities for late fees and fines, resulting in outflow of economic benefits and reduction of corporate value. By the same token, multiple taxation and early taxation will directly lead to the outflow of economic benefits and time value, reduce corporate value, and cause tax risks.

The generation of corporate tax risks is not caused overnight, but is a process of long-term accumulation and gradual development. In the entire development process of corporate tax risks, although these risks will show different tax characteristics, this performance may not be very significant and clear, and some may even be potential, but once it develops rapidly, it will It may enter a complete tax risk state immediately, leading to serious consequences, so the tax risk early warning requires research on the tax-related characteristics exhibited by the enterprise at different stages [6].

The formation of corporate tax risks should be divided into the following four main stages, and exhibit different characteristics. That is: tax risk incubation stage, tax risk development stage, tax risk deterioration stage and tax risk explosion stage. The different characteristics of these four stages can be described with the following figure.

In college physical education, martial arts have attracted more and more college students because of its unique charm and exercise value. In martial arts, it is easy to cause ligament damage and meniscus injury and aging. After severe meniscus injury or aging degeneration, a total meniscectomy is required. In order to prevent the degeneration of articular cartilage and subchondral bone in meniscectomy patients, try their best to maintain the normal biomechanical function of the knee joint. Plate transplantation has gradually become a new treatment approach. In this meniscus transplantation, in addition to the type matching of the meniscus transplantation, the reasonable fixation of the meniscus transplantation is another key issue in the transplantation operation. Related studies have confirmed that insufficient fixation of the anterior and posterior angles of the meniscus transplantation will cause Knee function degeneration after meniscus transplantation [7].

Tax risk belongs to the category of risk. Before understanding tax risk, it is necessary to clarify what is risk. In 1895, the American scholar Hayes gave a definition of risk from the perspective of economics. He believed that risk is “the possibility of loss”; uncertainty of probability random events is risk. Risks will only occur when there are two or more possibilities for an activity, and the occurrence of risks usually brings some form of loss. Therefore, tax risk is an uncertainty faced by enterprises in all tax-related activities, such as enterprises being inspected by tax authorities, bearing excessive tax liabilities, or incurring liabilities for tax reimbursement and fines. For the production and operation of an enterprise, the pursuit of profit is an eternal theme, but the profit of the enterprise must be controlled within an acceptable risk range. If the profit has a high risk, the enterprise will have the hidden danger of failure [8].

Tax risk has tax uncertainty. Uncertainty is the basis for tax risks. Without uncertainty, risks will not arise, and tax risks will be even more difficult to talk about. In the process of production and operation, enterprises are faced with a variety of decisions. Each decision is the result of a cost-benefit balance. The tax liability is the embodiment of each decision in tax law. Different decisions cause different tax burdens, and even the same decision will have different tax burdens due to different subjective reasons of the taxpayer and objective reasons of the external environment. It should be noted that less tax and delayed taxation have tax risks, and more taxation and early taxation also have tax risks. Less tax payment and delayed tax payment means that failure to pay taxes on time and amount will be punished by tax authorities for late fees and fines, resulting in outflow of economic benefits and reduction of corporate value. By the same token, multiple taxation and early taxation will directly lead to the outflow of economic benefits and time value, reduce corporate value, and cause tax risks.

The generation of corporate tax risks is not caused overnight, but is a process of long-term accumulation and gradual development. In the entire development process of corporate tax risks, although these risks will show different tax characteristics, this performance may not be very significant and clear, and some may even be potential, but once it develops rapidly, it will It may enter a complete tax risk state immediately, leading to serious consequences, so the tax risk early warning requires research on the tax-related characteristics exhibited by the enterprise at different stages [9].

The formation of corporate tax risks should be divided into the following four main stages, and exhibit different characteristics. That is: tax risk incubation stage, tax risk development stage, tax risk deterioration stage and tax risk explosion stage. The different characteristics of these four stages can be described with the following figure.

Figure 1

Through the description of the above four stages of tax risk, although not all the companies used have behaved this way most of the companies in financial crisis are similar, and they are widely universal. It's just that different enterprises have different reasons for generating tax risk crises, so the performance of these crisis characteristics are different, or they focus more on the performance of one or more crisis characteristics.

The stochastic model originated from the study of optimal cash holdings in financial management. In tax risk early warning, this model is also instructive. In the random mode shown in the figure below, the horizontal axis represents time, and the vertical axis represents the size of tax risk. As time changes, tax risk is high and low. For enterprises, risk and return are positively related. The higher the risk, the better, and not the lower the better. Theoretically, there should be an optimal level of tax risk. The constant degree of tax risks can only control tax risks within a feasible range. Assume that the straight-line R is the optimal amount of tax risk for the enterprise, the straight-line H is the upper limit of the tax risk allowed by the enterprise, and the straight-line L is the lower limit of the tax risk that the enterprise is willing to bear. When the tax risk reaches H, the early warning system issues a warning, and the enterprise adjusts the tax risk to the optimal position R. When the tax risk reaches L, the early warning system also issues an alarm, and the enterprise adjusts the tax risk to the state R.

Figure 2

As a kind of management behaviour of an enterprise, the tax risk early warning mechanism should serve the overall objective of the enterprise-maximizing the value of the enterprise. As mentioned earlier, because the situation of each enterprise is different, it is impossible to force each enterprise to establish a sound tax risk early warning mechanism. When considering whether and how to establish a tax risk early warning mechanism, we should pay attention to the actual situation of the enterprise and Consider the cost-benefit principle. In order to understand the realization of the objectives of the tax risk early warning mechanism, the following three questions need to be answered: First, should the enterprise establish a tax risk early warning mechanism? Second, how complete should the tax risk warning mechanism itself be? Third, what level should the tax risk early-warning mechanism control the tax risk of the enterprise? Figure 3 shows the scheme of tax risk early warning mechanism [10].

Figure 3

Convergence and Stability of Ordinary Differential Equation Algorithm (1) dxdy=f(x,y), x∈[a,b] {{dx} \over {dy}} = f\left({x,y} \right),\,x \in \left[{a,b} \right] (2) y(x0)=y0 y\left({{x_0}} \right) = {y_0}

The general form of the explicit one-step method of the above formula is (3) yn+1=yn+hφ(xn,yn,h)n=0,1,⋯, N−1 \matrix{{{y_{n + 1}} = {y_n} + h\varphi \left({{x_n},{y_n},h} \right)} \hfill & {n = 0,1, \cdots,\,N - 1} \hfill \cr}

Where h=b−aN h = {{b - a} \over N} , x_n = a + nh,. In this way, we use the solution y_n of the initial value problem (3) of the difference equation as the approximate value of the solution y(x_n) at x = x_n, that is, y(x_n) ≈ y_n. Therefore, the solution of (4) y(x+h)−y(x)h−φ(x,y(x),h) {{y\left({x + h} \right) - y\left(x \right)} \over h} - \varphi \left({x,y\left(x \right),h} \right)

Approach (5) y′−f(x,y(x))=0 {y^{'}} - f\left({x,y\left(x \right)} \right) = 0

It is possible to approximate the solution to the problem. Therefore, we expect that for any fixed x ∈ [a, b], (6) limh→0[y(x+h)−y(x)h−φ(x,y(x),h)]=0 \mathop {\lim}\limits_{h \to 0} \left[{{{y\left({x + h} \right) - y\left(x \right)} \over h} - \varphi \left({x,y\left(x \right),h} \right)} \right] = 0

If the increment function φ(x, y, h) is continuous with respect to h, then (7) φ(x,y,h)=f(x,y) \varphi \left({x,y,h} \right) = f\left({x,y} \right)

If relation (8) φ(x,y,h)≠f(x,y) \varphi \left({x,y,h} \right) \ne f\left({x,y} \right)

If it is true, it is said that the one-step method (3) is compatible with the initial value problem of differential equations, as shown in Figure 4 [11].

Figure 4

It is easy to verify that both the Euler method and the improved Euler method satisfy the compatibility conditions. In fact, for the Euler method, the incremental function is (9) φ(x,y,h)=f(x,y) \varphi \left({x,y,h} \right) = f\left({x,y} \right)

The compatibility condition is naturally satisfied, and the incremental function of the improved Euler method is (10) φ(x,y,h)=12[f(x,y)+f(x+h,y+hf(x,y))] \varphi \left({x,y,h} \right) = {1 \over 2}\left[{f\left({x,y} \right) + f\left({x + h,y + hf\left({x,y} \right)} \right)} \right]

Because f(x, y), Thus having (11) φ(x,y,h)=12[f(x,y)+f(x,y)]=f(x,y) \varphi \left({x,y,h} \right) = {1 \over 2}\left[{f\left({x,y} \right) + f\left({x,y} \right)} \right] = f\left({x,y} \right)

So, Euler's method and modified Euler's method are compatible with the initial value problem. Generally, if the single-step method is shown to have p-order accuracy (p > 0), its local error is [12] (12) y(x+h)−[y(x)+hφ(x,y(x),h)]=O(hp+1) y\left({x + h} \right) - \left[{y\left(x \right) + h\varphi \left({x,y\left(x \right),h} \right)} \right] = O\left({{h^{p + 1}}} \right)

Divide both ends of the above equation by h to get ɛ_{n + 1} → 0 and h → 0. If φ(x, y, h), then (13) y′(x)−φ(x,y,h)=0 {y^{'}}\left(x \right) - \varphi \left({x,y,h} \right) = 0 which is (14) φ(x,y,h)=f(x,y) \varphi \left({x,y,h} \right) = f\left({x,y} \right)

Therefore, the display single-step method of p > 0 is compatible with the initial value problem. Therefore, the RK method of each order is compatible with the initial value problem. A numerical method is called convergence. If there is any initial value y = 0 and any x ∈ (a, b], (15) limh→0yn=y(x)(x=a+nh) \matrix{{\mathop {\lim}\limits_{h \to 0} {y_n} = y\left(x \right)} \hfill & {\left({x = a + nh} \right)} \hfill \cr}

Where y(x) is the exact solution to the initial value problem?

The convergence of a numerical method needs to be determined based on the global truncation error ɛ_{n + 1} of the method. The global truncation error of the known Euler method has an estimated formula (16) |εn+1|≤hM2L|eL(b−a)−1|=O(h) \left| {{\varepsilon _{n + 1}}} \right| \le {{hM} \over {2L}}\left| {{e^{L\left({b - a} \right)}} - 1} \right| = O\left(h \right)

When h → 0, ɛ_{n + 1} → 0,, the Euler method converges.

It is assumed that the explicit single-step method has p-order accuracy, and its incremental function φ(x, y, h) satisfies Lipschitz's condition with respect to y. The problem is accurate. If y(x₀) = y₀, the overall truncation error of the explicit single-step method is (17) en+1=y(xn+1)−yn+1=O(hp) {e_{n + 1}} = y\left({{x_{n + 1}}} \right) - {y_{n + 1}} = O\left({{h^p}} \right)

Assuming that the increment function φx, y, h) is continuous over the region S, and that y satisfies the Lipschitz condition, then the sufficient and necessary condition for the explicit one-step convergence is that the compatibility condition holds, that (18) φ(x,y,h)=f(x,y) \varphi \left({x,y,h} \right) = f\left({x,y} \right)

If there are normal numbers h₀ and C, such that for any initial starting value y₀, y₀, the corresponding exact solution y_n, y_n of the one-step method (18), for all 0 < h ≤ h₀, there is always (19) |yn−y˜n|≤C|y0−y˜0|nh≤b−a \matrix{{\left| {{y_n} - {{\tilde y}_n}} \right| \le C\left| {{y_0} - {{\tilde y}_0}} \right|} \hfill & {nh \le b - a} \hfill \cr}

It is said that the single step method is stable.

If φ(x, y, h) for x ∈ [a, b], h ∈ (0, h₀], and all real numbers y, with respect to lip satisfying the Lipschitz condition, then the one-step method (19) is stable.

For a given differential equation and a given step size h, if there is an error of size δ when calculating y_n by the single-step method, then y˜n=yn+δ {\tilde y_n} = {y_n} + \delta is calculated, and the subsequent change of the value y_m(m > n) is less than δ, then said that The single step method is absolutely stable [13].

Generally limited to typical differential equations (20) y′=μy {y^{'}} = \mu y

Consider the absolute stability of the numerical method, where μ is a complex constant (we are limited to the case where μ is a real number). If for all μh ∈ (α, β), the one-step method is absolutely stable, then (α, β) is called the absolute stability interval. According to the above definition we can The absolute stability interval of Euler's method is E (−2, 0), the absolute stability interval of trapezoidal formula is (−∞, 0), and the absolute stability interval of Runge-Kutta is (−2.78, 0).

The Euler method, improved Euler method, and Runge-Kutta were used to solve the initial value problem. (21) {y′=−2xy2(0≤x≤1.2)y(0)=1 \left\{{\matrix{{{y^{'}} = - 2x{y^2}\left({0 \le x \le 1.2} \right)} \cr {y\left(0 \right) = 1} \cr}} \right.

Figure 5

Take equal scores as N = 12, 24, 36, 120, respectively, and calculate the maximum error from the real solution. Euler's method algorithm: After calculation, we can get the maximum error between the numerical solution and the real solution when the initial value problem is not taken as shown in Table 1.

From the running results of the program recorded in Table 1, when the equal score N becomes larger, its error is decreasing. According to the definition we can prove that the method is convergent.

Improved Euler's method: After calculation, we can get the maximum error between the numerical solution and the real solution when the initial value problem is not taken as shown in Table 2.

From the program running results recorded in Table 2, when the equal score N becomes larger, its error is decreasing. According to the definition, we can prove that the method is convergent.

Runge-Kutta algorithm: After calculation, we can get the maximum error between the numerical solution and the real solution when the initial value problem is not taken as shown in Table 3.

From the program running results recorded in Table 3, when the equal score N becomes larger, its error is decreasing. According to the definition, we can prove that the method is convergent.

In order to compare the convergence speeds of the above three methods, we have calculated their minimum equal fraction N with an error accuracy of 10-5 as shown in Table 4 below.

From the running results of the programs recorded in Table 4, the Runge-Kutta method has the fastest convergence speed, followed by the improved Euler method, and the Euler method is the worst. From this point, the Runge-Kutta method is the most ideal numerical solution among them.

The author believes that the relationship between the tax risk attitude of corporate managers or shareholders and the early warning of tax risks is reflected in the following: Tax risk appetite is willing to accept higher tax risk levels and expect higher returns, and usually sets the threshold for tax risk early warning. Slightly higher; on the contrary, tax risk aversion is willing to bear a lower level of tax risk and can get lower expected returns, generally the tax risk early warning threshold is determined to be more conservative; while the tax risk neutral is between the two [14].

The requirements and changes of the macro-environment are the important reasons that cause enterprises to face tax risks. Therefore, companies should: 1)

Study the macro environment. Although the external macro-environment exists outside the enterprise and cannot be changed, the enterprise can adapt to changes in the environment by adjusting its own operating behaviour. For example, analysis and research on the changing external macro environment, especially analysis and research on national industrial policies, industry policies, relevant laws and regulations, market changes, new products, new technologies, and changes in the international situation, and keep abreast of changes. Understand the trends and laws of the environment, understand and analyse the impact of environmental changes on the production and operation of the enterprise, and fully consider the possibility, severity and impact of changes in the macro environment that make the enterprise face tax risks. If it affects the overall tax burden of the enterprise, how long will the existence of tax risks affect its performance How much influence the indicator has, and so on [15].