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A model for risky cash flows and tax shields

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Abstract

We extend the WACC and APV frameworks by incorporating risky cash flows and the potential loss of tax shields. A closed-form solution is derived for the expected effective tax shields. Our model explains the under-leverage puzzle, and provides better estimates for the required equity return through the improved WACC and APV formulae. It offers four empirically testable predictions.

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Notes

  1. Such as perpetual debt with fixed value, and personal and corporate borrowing and lending rates are the same, and no bankruptcy.

  2. For example, when calculating the value of debt, Sarkar (2008) overlooks two important factors, 1) the coupons are taxed (40%, the rate he uses); and 2) capital gains or loss tax liabilities when default happens. Both factors are essential to how bond prices are affected by personal taxes and very substantial in value. For a correct and more complete treatment of personal tax effect on risky debt, see for example Liu et al. (2006).

  3. This type of models has to assume bankruptcy happens when asset value falls below a threshold of V D . This is why similar works such as Leland (1994) and Leland and Toft (1996) assume tax shields are (completely) lost when asset value V falls below a threshold V T rather than profits falling below a threshold because the profit-based bankruptcy technically cannot be handled unless further assumptions are made.

  4. It is the common wisdom that the APV method applies to constant debt level while the WACC method applies to constant leverage ratio situations (see e.g., Brealey et al. 2005; Ehrhardt and Daves 1999). Ruback (2002) proposed the capital cash flow (CCF) valuation method which is an extension of the APV to consider the case where debt varies in proportion with the total firm value (i.e., constant leverage), thus the discount rate for the tax shields is the asset rate of return (r A ). He proved that valuation by the WACC and CCF yields the same results. Thus, in this paper, we focus on the WACC and APV approaches.

  5. See, for example, Leland (1998) and Subramanian (2003) for discussions on partial loss of tax shields and (François 2006) for a discussion on the valuation of tax shields.

  6. This is equivalent to assuming away the loss carry-forwards and loss carry-backs. This oversimplifies the limitations on them.

  7. To avoid confusion, we note that the potential loss of tax shields does have an impact on the firm value V L regardless of the debt policy. The effect simply happens to cancel out only for the levered equity return r E under the constant leverage ratio debt policy.

  8. Of course, there are other factors that may reduce the optimal leverage, such as lease financing, asymmetric information, corporate governance concerns, etc.

  9. However, some studies question this crowding-out effect of the non debt tax shields. For example, Downs (1993) claims that the crowding-out does not actually occur. Alderson and Betker (1995) do not find evidence supporting the crowding-out effect.

  10. We note that this risk-seeking-like effect is only one factor. Higher earnings volatility is also associated with higher discount premium and higher default probability. Thus, the overall effect depends on the relative strength of these two opposite effects. We leave this investigation to a separate research project.

  11. The firm can lose part of or the entire tax shield if it does not have enough income to deduct tax interest payments.

  12. See e.g., Cooper and Nyborg (2006).

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Correspondence to Dean Johnson.

Appendices

Appendix A. WACC, APV and capital structure policy

1.1 Constant leverage

Assume the firm follows a debt policy that it maintains a constant leverage ratio forever. The tax shield each year r D shall fluctuate with firm value V L , given the debt policy.Footnote 11 As argued in Ruback (2002) in the spirit of the APV approach, the proper discount rate is the opportunity cost of capital r A (also referred to as the asset return) for both FCF and the tax shields. Therefore, the firm value is given by

$$ {V_L} = \frac{{FCF}}{{{r_A}}} + \frac{{{r_D}D\tau }}{{{r_A}}}. $$
(A1)

Since the cash flows are not constant, it is important to view them as the expected amounts. According to the WACC methodology, we have

$$ {V_L} = \frac{{FCF}}{{{r_A}}} + \frac{{{r_D}D\tau }}{{{r_A}}} \equiv \frac{{FCF}}{{WACC}}. $$
(A2)

Equation A2 says that instead of each cash flow stream separately, one can gross up the tax shields in an overall discount factor called WACC such that we will arrive at the same firm value V L . Therefore, to satisfy Eq. A2, WACC must take the following form,

$$ WACC = {r_A} - {r_D}\tau \times \frac{D}{{D + S}} = {r_A} - {r_D}\tau \times l. $$
(A3)

where \( l = \frac{D}{{{V_L}}} \) is the leverage ratio. According to the definition of WACC,

$$ WACC = {r_D}\left( {1 - \tau } \right)\frac{D}{{S + D}} + {r_E}\frac{S}{{S + D}}, $$
(A4)

where S is equity value and r E is the required rate of return on equity. Equations A3 and A4 imply that the equity holders would require a rate of return given below,

$$ {r_E} = {r_A} + \left( {{r_A} - {r_D}} \right)\frac{D}{S}. $$
(A5)

Equations A4 and A5 are (1) and (2) in the paper, respectively. As long as the firm does not deviate from its capital structure policy, the debt-equity ratio will remain constant, and thereby r E will remain the same as time goes.

1.2 Constant debt

If the firm maintains the same debt level D in dollar amount, then the debt and the associated debt tax shields are certain cash flows. The proper discount rate for the tax shields becomes the cost of debt r D . Following the APV spirit, the firm value is given by

$$ {V_L} = \frac{{FCF}}{{{r_A}}} + \frac{{{r_D}D\tau }}{{{r_D}}} = \frac{{FCF}}{{{r_A}}} + D\tau . $$
(A6)

Rewrite Eq. A6,

$$ {V_L}\left( {1 - \frac{{D\tau }}{{{V_L}}}} \right) = {V_L}\left( {1 - l\tau } \right) = \frac{{FCF}}{{{r_A}}}. $$
(A7)

and by the definition of the WACC methodology,

$$ {V_L} = \frac{{FCF}}{{{r_A}\left( {1 - l\tau } \right)}} \equiv \frac{{FCF}}{{WACC}}, $$
(A8)

Thus the proper discount rate is

$$ r = WACC = {r_A}\left( {1 - l\tau } \right). $$
(A9)

Equation A9 was proposed by Modigliani and Miller and also referred to as the MM formula (e.g., see Brealey et al. (2005)). Combining Eqs. A4 and A8, we have

$$ {r_E} = {r_A} + \left( {{r_A} - {r_D}} \right)\left( {1 - \tau } \right)\frac{D}{S}. $$
(A10)

Equations A9 and A10 are (3) and (4) in the paper, respectively. Equation A10 is the MM Proposition II. Applying the CAPM to (A10), we can also get the levered equity beta formula which is normally referred to as the Hamada (1972) formula. Comparing Eqs. A5 and A10, one can see that r E is higher in the constant leverage ratio case given everything else being equal. This is because the present value of the tax shields is worth less when discounted by r A in Eq. A2 rather than a lower rate of r D in Eq. A6,Footnote 12 and equity holders bear this reduction in firm value, implying a higher expected equity required rate of return.

A major difference between Eqs. A5 and A10 is that r E in A10 changes over time due to a changing debt-equity ratio because the firm does not continuously rebalance its capital structure.

Appendix B. Value of effective tax shields—proof of Proposition I

In Appendix B, we derive the factor Q in Eq. 6 and prove Proposition I. Recognizing the possible loss of tax shields, the annual debt tax shield is given by

$$ \left\{ {\begin{array}{*{20}{c}} {D{r_D}\tau, \quad {\hbox{if}}\;E > K} \hfill \\{D{r_D}\tau \times \frac{E}{K},\quad {\hbox{if}}\;0 \leqslant E \leqslant K} \hfill \\{0,\quad {\hbox{if}}\;E < 0} \hfill \\\end{array} } \right., $$
(B.1)

(B.1) simplifies to

$$ {\hbox{Annual}}\,{\hbox{tax}}\,{\hbox{shield}} = D{r_D}\tau \times \max \left[ {\frac{{K - \max \left[ {K - E,0} \right]}}{K},0} \right]. $$
(B.2)

Assume the annual earnings before tax follows a normal distribution \( {\rm N}\left[ {\bar{E},\sigma_E^2} \right] \), then the expected effective annual tax shield is

$$ \begin{array}{*{20}{c}} {Q = \int\limits_{{ - \infty }}^{\infty } {\max \left[ {\frac{{K - \max \left[ {K - E,0} \right]}}{K},0} \right] \times \frac{1}{{\sqrt {{2\pi \sigma_E^2}} }}} {e^{{ - \frac{{{{\left( {E - \bar{E}} \right)}^2}}}{{2\sigma_E^2}}}}} \times {\hbox{d}}E} \\{ = \int\limits_0^{\infty } {\frac{{K - \max \left[ {K - E,0} \right]}}{K} \times \frac{1}{{\sqrt {{2\pi \sigma_E^2}} }}} {e^{{ - \frac{{{{\left( {E - \bar{E}} \right)}^2}}}{{2\sigma_E^2}}}}} \times {\hbox{d}}E} \\\end{array} . $$
(B.3)

Substituting \( Z = \frac{{E - \bar{E}}}{{{\sigma_E}}} \) in (B.3), and recognizing that

$$ \max \left[ {K - \left( {{\sigma_E}Z + \bar{E}} \right),0} \right] \to Z \leqslant \frac{{K - \bar{E}}}{{{\sigma_E}}} $$

therefore, (B.3) becomes

$$ \begin{array}{*{20}{c}} {Q = {\rm N}\left[ {\frac{{\bar{E}}}{{{\sigma_E}}}} \right] - \int\limits_{{ - \frac{{\bar{E}}}{{{\sigma_E}}}}}^{{\frac{{K - \bar{E}}}{{{\sigma_E}}}}} {\left[ {1 - \frac{{{\sigma_E}}}{K}Z - \frac{{\bar{E}}}{K}} \right]} \frac{{{e^{{ - \frac{{{Z^2}}}{2}}}}}}{{\sqrt {{2\pi }} }} \times {\hbox{d}}Z} \\{ = {\rm N}\left[ {\frac{{\bar{E}}}{{{\sigma_E}}}} \right] + \left( {\frac{{\bar{E}}}{K} - 1} \right)\left\{ {{\rm N}\left[ {\frac{{K - \bar{E}}}{{{\sigma_E}}}} \right] - {\rm N}\left[ { - \frac{{\bar{E}}}{{{\sigma_E}}}} \right]} \right\} + \frac{{{\sigma_E}}}{K}\int\limits_{{ - \frac{{\bar{E}}}{{{\sigma_E}}}}}^{{\frac{{K - \bar{E}}}{{{\sigma_E}}}}} {Z \times {\hbox{d}}\left( {{\rm N}\left[ Z \right]} \right)} } \\\end{array} . $$

Integrating the last term by part, and simplify the notation with the substitutions \( a = \frac{{K - \bar{E}}}{{{\sigma_E}}};\quad b = \frac{{\bar{E}}}{{{\sigma_E}}};\quad d = \frac{{\bar{E}}}{K}, \) we have

$$ \begin{array}{*{20}{c}} {Q = {\rm N}\left[ b \right] + \frac{{ - da}}{b}\left( {{\rm N}\left[ a \right] - {\rm N}\left[ { - b} \right]} \right) + \frac{d}{b}\int\limits_{{ - b}}^a {z{\hbox{d}}{\rm N}\left[ z \right]} } \\{ = {\rm N}\left[ b \right] + \frac{{da}}{b}\left( {1 - {\rm N}\left[ b \right] - {\rm N}\left[ a \right]} \right) + \frac{d}{b}\left[ {z{\rm N}\left[ z \right]} \right]|_{{ - b}}^a - \frac{d}{b}\int\limits_{{ - b}}^a {{\rm N}\left[ z \right]{\hbox{d}}z} } \\{ = {\rm N}\left[ b \right] + \frac{{da}}{b}\left( {1 - {\rm N}\left[ b \right] - {\rm N}\left[ a \right]} \right) + \frac{d}{b}\left[ {a{\rm N}\left[ a \right] + b{\rm N}\left[ { - b} \right]} \right] - \frac{d}{b}\int\limits_{{ - b}}^a {{\rm N}\left[ z \right]{\hbox{d}}z} } \\{ = {\rm N}\left[ b \right] + \frac{{da}}{b}\left( {1 - {\rm N}\left[ b \right] - {\rm N}\left[ a \right]} \right) + \frac{d}{b}\left[ {a{\rm N}\left[ a \right] + b - b{\rm N}\left[ b \right]} \right] - \frac{d}{b}\int\limits_{{ - b}}^a {{\rm N}\left[ z \right]{\hbox{d}}z} } \\{ = {\rm N}\left[ b \right]\left( {1 - \frac{{da}}{b} - d} \right) + {\rm N}\left[ a \right]\left( {\frac{{da}}{b} - \frac{{da}}{b}} \right) + \left( {\frac{{da}}{b} + d} \right) - \frac{d}{b}\int\limits_{{ - b}}^a {{\rm N}\left[ z \right]{\hbox{d}}z} } \\{ = {\rm N}\left[ b \right] \times 0 + {\rm N}\left[ a \right] \times 0 + \left( {\frac{{da}}{b} + d} \right) - \frac{d}{b}\int\limits_{{ - b}}^a {{\rm N}\left[ z \right]{\hbox{d}}z} } \\{ = 1 - \frac{d}{b}\int\limits_{{ - b}}^a {{\rm N}\left[ z \right]{\hbox{d}}z} } \\\end{array} $$

Thus,

$$ Q = 1 - \frac{d}{b}\int\limits_{{ - b}}^a {{\rm N}\left[ z \right]{\hbox{d}}z} = 1 - \frac{{{\sigma_E}}}{K}\int\limits_{{ - \frac{{\bar{E}}}{{{\sigma_E}}}}}^{{\frac{{K - \bar{E}}}{{{\sigma_E}}}}} {{\rm N}\left[ z \right]{\hbox{d}}z}, $$
(B.4)

This proves Proposition I.

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Qi, H., Liu, S. & Johnson, D. A model for risky cash flows and tax shields. J Econ Finan 36, 868–881 (2012). https://doi.org/10.1007/s12197-010-9162-7

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