Profit shifting and the attractiveness of Advance Pricing Agreements

Abstract

This study investigates tax-induced profit shifting and the effect of the availability of Advance Pricing Agreement (APA) rules on the shifting behaviour of firms. We provide a theoretical model which demonstrates that an APA eliminates the transfer pricing uncertainty related to the calculation of the arm’s length parameter, and it eliminates the incidence of a tax penalization rate. In this case, APA may be attractive for firms focusing on a profit shifting strategy, as it reduces the implicit costs related to the expected tax amendment to be imposed by tax authorities. On an empirical application of our model, we find that firms have an incremental volume of intrafirm transactions with related parties located in low-tax countries with available APA, which is consistent with our theoretical prediction.

Appendix

1.1 Definitions and equivalent relations

For this Appendix Section only, we assume the general real-valued functions f,g: R $\to$ R with respect to any real variables x,y,z $\in$ R

General product rule is ∂(fg)/∂x = f’g + fg’, and the general chain rule is ∂f[g(x)]/∂x = f’(g) · g’

The absolute value function | · |: R $\to$ R₊ maps any non-zero real variable to its absolute value, with the definition |0|= 0. The sign function sgn ( ·): R $\to$ {− 1, 1} maps any non-zero real variable to its sign, with the definition sgn(0) = 0. The sign function has a multiplicative identity property such that sgn(x) = 1/sgn(x). Any real number x is a combination of an absolute value and a sign, for we have

$$x=sgn\left(x\right)\cdot \left|x\right|=\frac{|x|}{sgn(x)}.$$

(A1)

The derivative of the sign function is defined for any non-zero real number and it is equal to zero, ∂sgn(x)/∂x = 0, since ∂1/∂x = 0, ∂(− 1)/∂x = 0. Using the product rule, the derivative of the absolute value function |x| is equal to the sign function sgn(x), for we have

$$\frac{\partial |x|}{\partial x}=\frac{\partial [sgn\left(x\right)\cdot x]}{\partial x}=\frac{\partial sgn(x)}{\partial x}\cdot x+sgn\left(x\right)\cdot \frac{\partial x}{\partial x}=0\cdot x+sgn\left(x\right)\cdot 1=sgn\left(x\right).$$

(A2)

For any difference x – y, we clearly have ∂(x – y)/∂x = ∂x/∂x = 1, ∂(x – y)/∂y = ∂(− y)/∂y =  − 1, therefore the derivative of the difference x – y with respect to either x or y is equal to the sign of the argument $\in$within the difference, i.e. for any value z $\in$ {x,y}, x ≠ y, we have

$$\frac{\partial (x-y)}{\partial z}=sgn\left( \frac{\partial x}{\partial z} - \frac{\partial y}{\partial z} \right), z\in \left\{x,y\right\},x\ne y.$$

(A3)

The derivative of the absolute value of the difference |x – y| with respect to the max value between  x > 0, y > 0 is intuitively equal to 1. From Eqs. (A1)−(A3) and using the chain rule, we can show that this is the case, for we have

$$\frac{\partial |x-y|}{\partial z}=\frac{\partial |x-y|}{\partial \left(x-y\right)}\cdot \frac{\partial \left(x-y\right)}{\partial z}=sgn\left(x-y\right) \cdot sgn\left( \frac{\partial x}{\partial z} - \frac{\partial y}{\partial z} \right)=1, z = max(x>0,y>0).$$

(A4)

Results in Eqs. (A3) and (A4) have equivalent notations using the indicator function defined for real variables as I: R $\to$ {0, 1}, where the indicator is equal to one I_A(x) = 1 if the real variable x satisfies some general property A assigned to the function I_A, and zero otherwise.²⁸ Eq. (A3) is equivalent to

$$\frac{\partial (x-y)}{\partial z}=I\left(z=x\right)\cdot \frac{\partial \left(x-y\right)}{\partial x}+I\left(z=y\right)\cdot \frac{\partial \left(x-y\right)}{\partial y}=sgn\left( \frac{\partial x}{\partial z} - \frac{\partial y}{\partial z} \right).$$

(A5)

The indicator function I is a jump function, for the derivative of I(x) with respect to x is not defined in the continuous case. The usual convention is to define ∂I/∂x = 0, x $\in$ R. Nonetheless, the indicator function I is still useful for the specification of the derivative of a general real-valued function f with respect to some real variable x, regarding whether f is a function of x. For a general f(z), the derivative with respect to x is equal to

$$\frac{\partial f(z)}{\partial x}=\frac{\partial [I\left(z,x\right)\cdot f\left(z\right)]}{\partial x}=0\cdot f+I\left(z,x\right)\cdot {f}^{\mathrm{^{\prime}}}=I\left(z,x\right)\cdot {f}^{\mathrm{^{\prime}}},$$

(A6)

where we have I(z = x) = 1 and zero otherwise, i.e. Eq. (A6) is clearly equal to zero if f is independent of x, z ≠ x

On the other hand, it makes sense to refer to the difference operation of some function f(I) with respect to the indicator I, which is also called discrete differentiation. From the definition of the derivative in the usual sense, the discrete differentiation of f(I) with respect to I is equal to

$$\frac{\Delta f(I)}{\Delta I}=\frac{f\left(I=1\right)-f(I=0)}{1-0}=f\left(I=1\right)-f\left(I=0\right).$$

(A7)

The derivatives of the absolute value function and the sign function as in Eqs. (A2)−(A4) are applied directly for the solution in Sect. 2, Eqs. (5) and (6). The use of indicator functions I as in Eqs. (A5) and (A6) is applied directly for the solution in Sect. 2, Eqs. (9) and (10). The discrete differentiation defined in Eq. (A7) is applied directly for the solution in Sect. 2, Eqs. (18), (26) and (27)

1.2 Derivation of the key equations for the research hypotheses H1 and H2

We present the full derivation of the key equations in Sect. 2, which are the basis of our empirical hypotheses H1 and H2. Variables and equations used in this Appendix Section are the same as in Sect. 2, and all definitions are presented throughout that same Sect. 2

Let x* and m* be the maximum transfer price and the maximum internal output respectively, so they define the implicit functions equal to x* = x*(T) and m* = m*(T). The maxima levels in Eqs. (7) and (8) are equal to

$$T = sgn(T)\tau_{i} (1 + z_{i} ) \cdot [p_{i}^{*} + p_{i}^{{*\prime}} \cdot |x^{*} - \overline{x}|m^{*} ];$$

(A8)

$$T\overline{x }=\left(1-{\tau }_{1}\right){C}_{1}^{*\mathrm{^{\prime}}}-\left(1-{\tau }_{2}\right){C}_{2}^{*\mathrm{^{\prime}}},$$

(A9)

where the maxima imply the sign equality sgn(T) = sgn(x* – $\overline{x }$) for any LTP or HTP cases. Differentiating Eq. (A9) with respect to either one of the tax rates τ_i, τ_j and solving for ∂m*/∂T, we have

$$\left( {I_{2} - I_{1} } \right)\overline{x} = sgn \left( T \right)\overline{x} = - I_{1} \cdot C_{1}^{*{\prime}} + \left( {1 - \tau_{1} } \right)C_{1}^{*{\prime\prime}} \cdot \frac{{\partial m^{*} }}{\partial T} \cdot sgn \left( T \right) + I_{2} \cdot C_{2}^{*{\prime}} + \left( {1 - \tau_{2} } \right)C_{2}^{*{\prime\prime}} \cdot \frac{{\partial m^{*} }}{\partial T} \cdot sgn \left( T \right);$$

(A10)

$$sgn\left(T\right)\overline{x }=\frac{\partial {m}^{*}}{\partial T}\cdot sgn\left(T\right)\cdot {C}^{*{{\prime}}{{\prime}}}-{I}_{1}\cdot {C}_{1}^{*{{\prime}}}+{I}_{2}\cdot {C}_{2}^{*{{\prime}}};$$

$$\frac{\partial {m}^{*}}{\partial T}=\frac{\overline{x }+sgn\left(T\right)[{I}_{1}\cdot {C}_{1}^{*{{\prime}}}-{I}_{2}\cdot {C}_{2}^{*{{\prime}}}]}{{C}^{*{{\prime}}{{\prime}}}},$$

where we assume that both I₁ = 1 and I₂ = 1 do not occur simultaneously, therefore τ_i = τ₁ implies ∂T/∂τ_i = -1, τ_i = τ₂ implies ∂T/∂τ_i = 1, ∂T/∂τ_i = I₂ – I₁ = sgn(T). Eq. (A10) refers to Eq. (9) in Sect. 2.

Differentiating Eq. (A8) with respect to either one of the tax rates τ_i, τ_j, we find an intricate expression equal to

$$sgn \left( T \right) = I_{i} \cdot sgn \left( T \right)\left( {1 + z_{i} } \right) \cdot \left[ {p_{i}^{*} + p_{i}^{*{\prime}} \cdot \left| {x^{*} - \overline{x}} \right|m^{*} } \right] + sgn \left( T \right)\tau_{i} \left( {1 + z_{i} } \right) \cdot \left[ {2p_{i}^{*{\prime}} \cdot sgn \left( T \right) \cdot \frac{{\partial x^{*} m^{*} }}{\partial T} \cdot sgn \left( T \right) - \overline{x} \cdot \frac{{\partial m^{*} }}{\partial T} \cdot sgn \left( T \right)} \right.\left. { + p_{i}^{*{\prime\prime}} \cdot sgn \left( T \right) \cdot \left[ {\frac{{\partial x^{*} m^{*} }}{\partial T} \cdot sgn \left( T \right) - \overline{x} \cdot \frac{{\partial m^{*} }}{\partial T} \cdot sgn \left( T \right)} \right] \cdot \left| {x^{*} - \overline{x}} \right|m^{*} } \right].$$

(A11)

Notice that for any non-negative value m* ≥ 0, the absolute value function implies |x* – $\overline{x }$|m* =|x*m* – $\overline{x }$ m*|. Simplifying the impact of sgn(T)²⁹ and substituting Eq. (A8) into Eq. (A11), we obtain

$$1 = I_{i} \cdot \frac{\left| T \right|}{{\tau_{i} }} + \tau_{i} \left( {1 + z_{i} } \right) \cdot \left[ {2p_{i}^{*{\prime}} \cdot \frac{{\partial x^{*} m^{*} }}{\partial T} - \overline{x} \cdot \frac{{\partial m^{*} }}{\partial T}} \right.\left. { + p_{i}^{*{\prime\prime}} \cdot \left[ {\frac{{\partial x^{*} m^{*} }}{\partial T} - \overline{x} \cdot \frac{{\partial m^{*} }}{\partial T}} \right] \cdot \left| {x^{*} - \overline{x}} \right|m^{*} } \right].$$

(A12)

Solving Eq. (A12) for ∂x*m*/∂T, we finally have

$$\frac{\partial {{x}^{*}m}^{*}}{\partial T}=\frac{1-{I}_{i}\cdot \frac{|T|}{{\tau }_{i}}}{{H}^{*}}+\overline{x}\cdot \frac{\partial {m }^{*}}{\partial T},$$

(A13)

which refers to Eq. (10) in Sect. 2, with H* = τ_i(1 + z_i)[2p_i*’ + p_i*’’|x* – $\overline{x }$|m*].

Now, let x* and m* be maxima, so they define the implicit functions equal to x* = x*(I_a) and m* = m*(I_a) respectively. The maxima levels in Eqs. (16) and (17) are equal to

$$T=sgn\left(T\right){\tau }_{i}\left[1+{z}_{i}\left(1-{I}_{y}\right)\right]\cdot \left[{p}_{i}^{*}+{p}_{i}^{*{{\prime}}}\cdot \left|{x}^{*}-X\right|{m}^{*}\right]+{I}_{y}\cdot b;$$

(A14)

$$X\left(T-{I}_{y}\cdot b\right)=\left(1-{\tau }_{1}\right){C}_{1}^{*{{\prime}}}-\left(1-{\tau }_{2}\right){C}_{2}^{*{{\prime}}}.$$

(A15)

where the maxima imply the sign equality sgn(T) = sgn(x* – X) for any LTP or HTP cases. Differentiating Eq. (A15) with respect to I_a and solving for ∆m*/∆I_a, we have

$$\begin{gathered}- I_{y|a} \cdot X_{y} b = \left( {1 - \tau_{1} } \right)C_{1}^{*{\prime\prime}} \cdot \frac{{\Delta m^{*} }}{{\Delta I_{a} }} + \left( {1 - \tau_{2} } \right)C_{2}^{*{\prime\prime}} \cdot \frac{{\Delta m^{*} }}{{\Delta I_{a} }} \le 0;\\ \frac{{\Delta m^{*} }}{{\Delta I_{a} }} = - I_{y|a} \cdot \frac{{X_{y} b}}{{C^{*{\prime\prime}} }} \le 0;\end{gathered}$$

(A16)

where the variable X_y states explicitly the position where the random arm’s length parameter X is evaluated under the APA rules y. Equation (A16) is always non-positive since we have X_y ≥ 0, b ≥ 0, C’’ ≥ 0. In Eqs. (A16), the random arm’s length price X_y is clearly evaluated under the APA rules y, since the complete equation is equal to zero if the MNE chooses for no APA, I_y|a = 0. Eqs. (A16) refers to Eqs. (26) in Sect. 2.

Now for the Eqs. (A14), the full expression of the maximum is equal to

$$T = sgn \left( T \right)\tau_{i} \left( {1 + z_{i} } \right) \cdot \left[ {p_{i}^{*} + p_{i}^{*{\prime}} \cdot \left| {x^{*} - X} \right|m^{*} } \right] - I_{y} \cdot \left[ {sgn \left( T \right)\tau_{i} \left( {1 + z_{i} } \right) \cdot \left[ {p_{i}^{*} + p_{i}^{*{\prime}} \cdot \left| {x^{*} - X} \right|m^{*} } \right]} \right.\left. { - sgn \left( T \right)\tau_{i} \left[ {p_{iy}^{*} + p_{iy}^{*{\prime}} \cdot \left| {x^{*} - X_{y} } \right|m^{*} } \right] - b} \right],$$

(A17)

where we have p_iy = p_i(|x – X_y|m), and we use the indicator function I_y for the complete extended notation. Differentiating Eqs. (A17) with respect to I_a, we obtain the cumbersome expression

$$\begin{gathered} 0 = sgn \left( T \right)\tau _{i} \left( {1 + z_{i} } \right) \cdot \left[ {2p_{i}^{{*'}} \cdot sgn \left( T \right)\left\lceil {\frac{{\Delta x^{*} m^{*} }}{{\Delta I_{a} }} - X \cdot \frac{{\Delta m^{*} }}{{\Delta I_{a} }}} \right\rceil } \right. \hfill \\ \left. { + p_{i}^{{*''}} \cdot sgn \left( T \right)\left\lceil {\frac{{\Delta x^{*} m^{*} }}{{\Delta I_{a} }} - X \cdot \frac{{\Delta m^{*} }}{{\Delta I_{a} }}} \right\rceil \cdot \left| {x^{*} - X} \right|m^{*} } \right] \hfill \\ - ~I_{{y|a}} \cdot sgn \left( T \right)\tau _{i} \left[ {\left( {1 + z_{i} } \right) \cdot \left[ {p_{i}^{*} + p_{i}^{{*'}} \cdot \left| {x^{*} - X} \right|m^{*} } \right]} \right.\left. { - ~\left[ {p_{{iy}}^{*} + p_{{iy}}^{{*'}} \cdot \left| {x^{*} - X_{y} } \right|m^{*} } \right]} \right] \hfill \\ \cdot \left[ {\left( {1 + z_{i} } \right)\left[ {2p_{i}^{{*'}} \cdot sgn \left( T \right)\left\lceil {\frac{{\Delta x^{*} m^{*} }}{{\Delta I_{a} }} - X \cdot \frac{{\Delta m^{*} }}{{\Delta I_{a} }}} \right\rceil } \right.} \right.\left. { + p_{i}^{{*''}} \cdot \text{sgn} \left( T \right)\left\lceil {\frac{{\Delta x^{*} m^{*} }}{{\Delta I_{a} }} - X \cdot \frac{{\Delta m^{*} }}{{\Delta I_{a} }}} \right\rceil \cdot \left| {x^{*} - X} \right|m^{*} } \right] \hfill \\ - \left[ {2p_{{iy}}^{{*'}} \cdot \text{sgn} \left( T \right)\left[ {\frac{{\Delta x^{*} m^{*} }}{{\Delta I_{a} }}} \right.} \right.\left. {\left. { - X_{y} \cdot \frac{{\Delta m^{*} }}{{\Delta I_{a} }}} \right]\left. { + p_{{iy}}^{{*''}} \cdot sgn \left( T \right)\left\lceil {\frac{{\Delta x^{*} m^{*} }}{{\Delta I_{a} }} - X_{y} \cdot \frac{{\Delta m^{*} }}{{\Delta I_{a} }}} \right\rceil \cdot \left| {x^{*} - X_{y} } \right|m^{*} } \right]} \right] \hfill \\ + I_{{y|a}} \cdot b - sgn \left( T \right)\tau _{i} \left( {1 + z_{i} } \right) \cdot \left[ {2p_{i}^{{*'}} \cdot sgn \left( T \right)\left\lceil {\frac{{\Delta x^{*} m^{*} }}{{\Delta I_{a} }} - X \cdot \frac{{\Delta m^{*} }}{{\Delta I_{a} }}} \right\rceil } \right.\left. { + p_{i}^{{*''}} \cdot sgn \left( T \right)\left\lceil {\frac{{\Delta x^{*} m^{*} }}{{\Delta I_{a} }} - X \cdot \frac{{\Delta m^{*} }}{{\Delta I_{a} }}} \right\rceil \cdot \left| {x^{*} - X} \right|m^{*} } \right]. \hfill \\ \end{gathered}$$

(A18)

We go for the tedious solution of Eqs. (A18) with respect to ∆x*m*/∆I_a. First, we simplify the square of the sign sgn(T) · sgn(T) = 1, we offset the first and the last terms of the right-hand side of Eqs. (A18), and we isolate the terms sharing the discrete differential ∆I_a, so we obtain the equality

$$\begin{gathered} I_{{y|a}} \cdot b = I_{{y|a}} \cdot \tau _{i} \left[ {\left( {1 + z_{i} } \right) \cdot \left[ {p_{i}^{*} + p_{i}^{{*\prime }} \cdot \left| {x^{*} - X} \right|m^{*} } \right]} \right. \hfill \\ - \left. {\left[ {p_{{iy}}^{*} + p_{{iy}}^{{*\prime }} \cdot \left| {x^{*} - X_{y} } \right|m^{*} } \right]} \right] \cdot \left[ {\left( {1 + z_{i} } \right)\left[ {2p_{i}^{{*\prime }} } \right.} \right.\left. { + p_{i}^{{*\prime \prime }} \cdot \left| {x^{*} - X} \right|m^{*} } \right] \hfill \\ - \left. {\left[ {2p_{{iy}}^{{*\prime }} + p_{{iy}}^{{*\prime \prime }} \cdot \left| {x^{*} - X_{y} } \right|m^{*} } \right]} \right] \cdot \frac{{\Delta x^{*} m^{*} }}{{\Delta I_{a} }} - \left[ {X + I_{{y|a}} \cdot \left( {X_{y} - X} \right)} \right] \cdot \frac{{\Delta m^{*} }}{{\Delta I_{a} }}, \hfill \\ \end{gathered}$$

(A19)

where the term X + I_y|a · (X_y – X) indicates explicitly the impact of the APA choice I_y|a over the random arm’s length parameter X for the output variation ∆m*/∆I_a. From the solution of Eqs. (A16), we obtain the simplification

$$\left[ {X + I_{y|a} \cdot \left( {X_{y} - X} \right)} \right] \cdot \left( { - I_{y|a} } \right) \cdot \frac{{X_{y} b}}{{C^{*{\prime\prime}} }} = \left[ { - I_{y|a} \cdot X - I_{y|a} \cdot X_{y} + I_{y|a} \cdot X} \right] \cdot \frac{{X_{y} b}}{{C^{\prime\prime}}} = - I_{y|a} \cdot X_{y} \cdot \frac{{X_{y} b}}{{C^{*{\prime\prime}} }} = X_{y} \cdot \frac{{\Delta m^{*} }}{{\Delta I_{a} }},$$

(A20)

which means that the random arm’s length parameter X_y will always be evaluated under the APA rules y if the MNE chooses for the APA.

Substitute Eqs. (A20) into Eqs. (A19) and apply the notation

$$p_{\Delta }^{*} = \tau_{i} \left[ {\left( {1 + z_{i} } \right) \cdot \left[ {p_{i}^{*} + p_{i}^{*{\prime}} \cdot \left| {x^{*} - X} \right|m^{*} } \right]} \right.\left. { - \left[ {p_{iy}^{*} + p_{iy}^{*{\prime}} \cdot \left| {x^{*} - X_{y} } \right|m^{*} } \right]} \right] \cdot \left[ {\left( {1 + z_{i} } \right)\left[ {2p_{i}^{*{\prime}} } \right.} \right.\left. { + p_{i}^{*{\prime\prime}} \cdot \left| {x^{*} - X} \right|m^{*} } \right] - \left. {\left[ {2p_{iy}^{*{\prime}} + p_{iy}^{*{\prime\prime}} \cdot \left| {x^{*} - X_{y} } \right|m^{*} } \right]} \right].$$

(A21)

so we obtain

$${I}_{y|a}\cdot b= {I}_{y|a}\cdot {p}_{\Delta }^{*}\cdot \left[\frac{\Delta {x}^{*}{m}^{*}}{\Delta {I}_{a}}-{X}_{y}\cdot \frac{\Delta {m}^{*}}{\Delta {I}_{a}}\right].$$

(A22)

Solving Eqs. (A22) for ∆x*m*/∆I_a, with the rational property of the indicator such that I_y|a/I_y|a = 1, we finally have

$$\frac{\Delta {x}^{*}{m}^{*}}{\Delta {I}_{a}}=\frac{b}{{p}_{\Delta }^{*}}+{X}_{y}\cdot \frac{\Delta {m}^{*}}{\Delta {I}_{a}},$$

(A23)

which refers to the Eqs. (27) in Sect. 2.

1.3 Derivation of Eqs.  ( 31 )

Variables and equations used in this Appendix Section are the same as in Sect. 2, and all definitions are presented throughout that same Sect. 2.

The variation in probability p_∆* defined in Eqs. (28) is equal to

$$p_{\Delta }^{*} = \left[ {\left[ {p_{i}^{*} + p_{i}^{*{\prime}} \cdot \left| {x^{*} - X} \right|m^{*} } \right]\left( {1 + z_{i} } \right)} \right.\left. { - \left[ {p_{iy}^{*} + p_{iy}^{*{\prime}} \cdot \left| {x^{*} - X_{y} } \right|m^{*} } \right]} \right] \cdot \left[ {\left[ {2p_{i}^{*{\prime}} } \right.} \right.\left. { + p_{i}^{*{\prime\prime}} \cdot \left| {x^{*} - X} \right|m^{*} } \right]\left( {1 + z_{i} } \right) - \left. {\left[ {2p_{iy}^{*{\prime}} + p_{iy}^{*{\prime\prime}} \cdot \left| {x^{*} - X_{y} } \right|m^{*} } \right]} \right].$$

(A24)

Expand Eqs. (A24) in a second-order Taylor series with respect to X around the mean E(X), for the full expansion is equal to

$$p_{\Delta }^{*} \approx \left[ {\left[ {p_{i}^{*} + p_{i}^{*{\prime}} \left| {x^{*} - E\left( X \right)} \right|m^{*} + \left( {\frac{{3p_{i}^{*{\prime\prime}} + p_{i}^{*{\prime\prime\prime}} \left| {x^{*} - E\left( X \right)} \right|m^{*} }}{2}} \right)Var\left( X \right)\left( {m^{*} } \right)^{2} } \right.} \right]\left( {1 + z_{i} } \right) - \left[ {\left. {p_{iy}^{*} + p_{iy}^{*{\prime}} \left| {x^{*} - E\left( {X|y} \right)} \right|m^{*} + \left( {\frac{{3p_{iy}^{*{\prime\prime}} + p_{iy}^{*{\prime\prime\prime}} \left| {x^{*} - E\left( {X|y} \right)} \right|m^{*} }}{2}} \right)Var(X|y)\left( {m^{*} } \right)^{2} } \right]} \right] \cdot \left[ {\left[ {2p_{i}^{*{\prime}} + p_{i}^{*{\prime\prime}} \left| {x^{*} - E\left( X \right)} \right|m^{*} + \left( {\frac{{4p_{i}^{*{\prime\prime\prime}} + p_{i}^{*{\prime\prime\prime\prime}} \left| {x^{*} - E\left( X \right)} \right|m^{*} }}{2}} \right)Var\left( X \right)\left( {m^{*} } \right)^{2} } \right.} \right]\left( {1 + z_{i} } \right) - \left[ {\left. {2p_{iy}^{*{\prime}} + p_{iy}^{*{\prime\prime}} \left| {x^{*} - E\left( {X|y} \right)} \right|m^{*} + \left( {\frac{{4p_{iy}^{*{\prime\prime\prime}} + p_{iy}^{*{\prime\prime\prime\prime}} \left| {x^{*} - E\left( {X|y} \right)} \right|m^{*} }}{2}} \right)Var\left( {X|y} \right)\left( {m^{*} } \right)^{2} } \right]} \right]$$

(A25)

Let the penalty rate be zero z_i = 0, and assume the conditional mean E(X|y) is such that it produces the approximation p_i ≈ p_iy within Eqs. (A25), while it preserves the sign equality sgn[x* – E(X)] = sgn[x* – E(X|y)]. We obtain

$$p_{\Delta }^{*} \left( {z_{i} = 0} \right) \approx [p_{i}^{*{\prime}} \left| {E\left( X \right) - E\left( {X|y} \right)} \right|m^{*} + D \cdot \left( {m^{*} } \right)^{2} \left. { \cdot [Var\left( X \right) - Var(X|y)]} \right] \cdot [p_{i}^{*{\prime\prime}} \left| {E\left( X \right) - E\left( {X|y} \right)} \right|m^{*} + D^{\prime}\left. { \cdot \left( {m^{*} } \right)^{2} \cdot \left[ {Var\left( X \right) - Var\left( {X|y} \right)} \right]} \right],$$

(A26)

with the temporary notation for the second-order Taylor coefficients equal to

$$D = \frac{{3p_{i}^{*{\prime\prime}} + p_{i}^{*{\prime\prime\prime}} \left| {E\left( X \right) - E\left( {X|y} \right)} \right|m^{*} }}{2},\,D^{\prime} = \frac{{4p_{i}^{*{\prime\prime\prime}} + p_{i}^{*{\prime\prime\prime\prime}} \left| {E\left( X \right) - E\left( {X|y} \right)} \right|m^{*} }}{2}$$

(A27)

Finally, assume the approximation E(X) ≈ E(X|y) within Eqs. (A26), so the absolute difference |E(X) – E(X|y)| ≈ 0 is approximate to zero. We obtain

$${p}_{\Delta }^{*}\left({z}_{i}=0\right)\approx 3{p}_{i}^{*{{\prime}}{{\prime}}}\cdot {p}_{i}^{*{{\prime}}{{\prime}}{{\prime}}}\cdot {\left({m}^{*}\right)}^{2}\cdot \left[Var\left(X\right)-Var\left(X|y\right)\right].$$

(A28)

which refers to Eqs. (31) in Sect. 2.