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Does tax convexity matter for risk? A dynamic study of tax asymmetry and equity beta

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Abstract

The purpose of this study is to explore the effect of tax convexity on firms’ market risk, where tax convexity measures the progressivity of firms’ tax function. We examine the relation between equity beta and tax convexity based on a standard contingent-claims model, in which firms face nonlinear tax schedules. We verify that in the presence of default and growth options, the effect of tax convexity on beta is significant and depends on several countervailing forces. Tax convexity has a direct, positive effect on beta, as well as two indirect countereffects through default and growth options. The overall effect is ambiguous and quantitatively significant. As asymmetric tax schedules are used in most countries, assuming a linear tax schedule in the valuation framework may misestimate beta and thus fail to assess risk accurately. Our theoretical model shows that tax convexity should be taken into consideration when estimating equity beta.

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Fig. 1

Notes

  1. Continuity means the equity value is a continuous function of the profit flow. There is no jump in this function. Real option literature states that continuity is a valid and appropriate setting on the valuation of real options.

  2. Smooth pasting is a kind of boundary condition in real option theory. In our paper, this condition is to ascertain that at the point of bankruptcy, X = XBG, the function of equity value is differentiable, and the slope should be zero so that the whole equity function can be derived.

  3. Based on the base parameter values and setting δ = 0, the leverage ratios are 56.89, 28.07, and 19.29 % for X = 2, 4, and 6, respectively. The high-leverage firm has a lower change compared with the low-leverage firm. When tax asymmetry is present, all leverage ratios increase. To obtain the leverage ratios, pre-expansion debt value is also derived using the same method as in Sects. 3 and 4.

  4. We use the base case parameter values suggested by Hong and Sarkar (2007) to achieve the new result. We also verify that the other results of this model are consistent with the literature.

  5. Supported by empirical evidence, the two papers state that growth opportunity is a key determinant for beta. Specifically, Hong and Sarkar (2007) provide a detailed study of the effect of the growth option on equity beta. Empirically, they find that a positive relation holds. At the same time, they offer a simulation case to illustrate that the growth option might negatively affect equity beta once the firm’s leverage ratio is very high.

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Acknowledgments

We would like to thank an anonymous referee and the editor (Lee, Cheng-Few) for their valuable suggestions, which greatly improved the paper. We thank Deng Jie for her excellent research assistance. We gratefully acknowledge financial support from the University of Macau.

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Correspondence to Adrian C. H. Lei.

Appendix

Appendix

1.1 Part 1. Deriving M1, M2, and M3 and Proof of Proposition 1

M1, M2, and M3 are the constants of the default option when the firm is in a normal condition and in distress. It is necessary to derive mathematical expressions of the three constants in order to obtain the function of equity value and continue the study of the default trigger. Equations (29) to (32) provide mathematical equations of the three constants.

Multiplying β by Eq. (10) and XBG by Eq. (11) and subtracting the resulting equations yields

$$ M_{1} = (1 - \tau + \delta )(1 + g)\left( {\frac{\beta - 1}{\alpha + \beta }} \right)\left[ {\left( {\frac{\beta }{\beta - 1}} \right)\frac{CF}{r} - \frac{{X_{BG} }}{r + \lambda \rho \sigma - \mu }} \right]X_{BG}^{\alpha } $$
(29)

Multiplying α by Eq. (10) and XBG by Eq. (11) and adding the resulting equation yields

$$ M_{2} = (1 - \tau + \delta )(1 + g)\left( {\frac{1 + \alpha }{\alpha + \beta }} \right)\left[ {\left( {\frac{\alpha }{1 + \alpha }} \right)\frac{CF}{r} - \frac{{X_{BG} }}{r + \lambda \rho \sigma - \mu }} \right]X_{BG}^{ - \beta } $$
(30)

M1 plus M2 are the constant terms of the value of the default option for a given set of parameter values. Multiplying α by Eq. (10) and CF by Eq. (11) and adding the resulting equations yields

$$ \delta (1 + g)\left( {\frac{r + \alpha \mu - \alpha \lambda \rho \sigma }{r + \lambda \rho \sigma - \mu }} \right)\frac{CF}{r} + (\alpha + \beta )M_{2} (CF)^{\beta } = 0 $$
(31)

Equation (31) is the last step of the proof of Proposition 1. By putting (30) into (31) we yield Proposition 1.

Substituting Eq. (30) into Eq. (31) yields Eq. (15). Finally, we solve M3 by either Eq. (10) or (11) to yield

$$ M_{3} = (1 - \tau + \delta )(1 + g)\left( {\frac{\beta - 1}{\alpha + \beta }} \right)X_{BG}^{\alpha } \quad \times \left\{ {\left( {\frac{\beta }{\beta - 1}} \right)\frac{CF}{r} - \frac{{X_{BG} }}{r + \lambda \rho \sigma - \mu } + \left( {\frac{\alpha + 1}{\beta - 1}} \right)\left( {\frac{{\beta - {r / {(\mu - \lambda \rho \sigma )}}}}{{\alpha + {r / {(\mu - \lambda \rho \sigma )}}}}} \right)\left[ {\frac{{X_{BG} }}{r + \lambda \rho \sigma - \mu } - \left( {\frac{\alpha }{1 + \alpha }} \right)\frac{CF}{r}} \right]\left[ {\frac{CF}{{X_{BG} }}} \right]^{\alpha + \beta } } \right\} $$
(32)

M3 is the constant for the default option when X ∈ [CF, ∞), i.e., when the firm is healthy.

1.2 Part 2. Deriving the derivative of the default trigger with respect to tax asymmetry, \( \frac{{dX_{BG} }}{d\delta } \)

Equations (33) to (37) give the derivation of the effect of tax asymmetry on the default trigger. Equation (33) differentiates each term in Eq. (14) with respect to delta, δ, in order to create the derivative, \( \frac{{dX_{BG} }}{d\delta } \). Equation (34) rearranges the terms of Eq. (33). Equations (35) and (36) further simplify (34) using the results in Eqs. (7) and (8) and yield Eq. (37), which gives the expression of the derivative of default trigger with respect to tax asymmetry, \( \frac{{dX_{BG} }}{d\delta }. \) Finally, Eq. (38) is used to show that Eq. (37) is always positive, i.e., an increase in tax asymmetry lifts up the default trigger.

Totally differentiating Eq. (14) with respect to δ yields

$$ \left( {\frac{r + \alpha \mu }{r - \mu }} \right)\frac{CF}{r} - (\alpha + 1)\left[ {\frac{{X_{BG} }}{r + \lambda \rho \sigma - \mu } - \left( {\frac{\alpha }{1 + \alpha }} \right)\frac{CF}{r}} \right]\left( {\frac{CF}{{X_{BG} }}} \right)^{\beta } - (1 - \tau + \delta )(\alpha + 1)\left[ {\left( {\frac{\alpha \beta }{\alpha + 1}} \right)\frac{CF}{{rX_{BG} }} - \frac{\beta - 1}{r + \lambda \rho \sigma - \mu }} \right]\left( {\frac{CF}{{X_{BG} }}} \right)^{\beta } \frac{{{\text{d}}X_{BG} }}{{{\text{d}}\delta }} = 0 $$
(33)

Using Eq. (14), we can write Eq. (33) as

$$ \frac{(1 - \tau + \delta )(\beta - 1)}{{X_{B} }}\left[ {\frac{\alpha \beta }{{\left( {\alpha + 1} \right)\left( {\beta - 1} \right)}}\frac{CF}{r} - \frac{{X_{B} }}{r + \lambda \rho \sigma - \mu }} \right]\frac{{{\text{d}}X_{B} }}{{{\text{d}}\delta }} = \left( {\frac{1 - \tau }{\delta }} \right)\left[ {\frac{{X_{B} }}{r + \lambda \rho \sigma - \mu } - \left( {\frac{\alpha }{1 + \alpha }} \right)\frac{CF}{r}} \right] $$
(34)

From Eqs. (7) and (8), we have

$$ \alpha \beta = \left( {\frac{1}{2} - \frac{(\mu - \lambda \rho \sigma )}{{\sigma^{2} }}} \right)^{2} + \frac{2r}{{\sigma^{2} }} - \left( {\frac{1}{2} - \frac{(\mu - \lambda \rho \sigma )}{{\sigma^{2} }}} \right)^{2} = \frac{2r}{{\sigma^{2} }} $$
(35)

and

$$ (1 + \alpha )(\beta - 1) = \alpha \beta + \beta - \alpha - 1 = \frac{2(r + \lambda \rho \sigma - \mu )}{{\sigma^{2} }} $$
(36)

Substituting Eqs. (35) and (36) into Eq. (34) and rearranging terms yields

$$ \frac{{{\text{d}}X_{B} }}{{{\text{d}}\delta }} = \frac{{(1 - \tau )(r + \lambda \rho \sigma - \mu )X_{B} }}{{\delta (1 - \tau + \delta )(\beta - 1)[CF - X_{B} ]}}\left[ {\frac{{X_{B} }}{r + \lambda \rho \sigma - \mu } - \left( {\frac{\alpha }{1 + \alpha }} \right)\frac{CF}{r}} \right] $$
(37)

Equation (37) is the expression of the derivative of default trigger with respect to tax asymmetry, \( \frac{{dX_{BG} }}{d\delta } \) in terms of the parameter values. So we can use this expression to estimate the effect of tax asymmetry on the default trigger for different sets of parameter values.

From Eq. (14), we have

$$ \frac{{X_{B} }}{r + \lambda \rho \sigma - \mu } \ge \left( {\frac{\alpha }{1 + \alpha }} \right)\frac{CF}{r} $$
(38)

where the equality holds only when δ = 0. It then follows from inequality Eq. (37) and Eq. (38) that dXB/dδ > 0 for all δ ∈ (0, τ). It also shows that the value M2 in Eq. (30) is negative.

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Lei, A.C.H., Yick, M.H.Y. & Lam, K.S.K. Does tax convexity matter for risk? A dynamic study of tax asymmetry and equity beta. Rev Quant Finan Acc 41, 131–147 (2013). https://doi.org/10.1007/s11156-012-0303-2

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