Consistent valuation: extensions from bankruptcy costs and tax integration with time-varying debt

Abstract

This study introduces a new version of the adjusted present value (APV) method and ensures its consistent valuation with the cost of capital (CoC) method at the highest level of generalization. The newly developed APV version and equivalent formulae consider stochastic debt and the trade-off between corporate income taxes (CIT) and personal income taxes (PIT), as well as tax benefits and financial distress costs. The value of expected bankruptcy costs aligns with the valuation aspect, enabling practical application of the formulae by valuers. The equivalence also reflects the differing perspectives of tax shields between stockholders and debt holders when PITs are introduced. Ultimately, the results demonstrate that the equivalence in this study aligns with, and can reduce to, previous standard formulae, under their stringent assumptions.

Appendices

Appendix A: Summary of notation

Symbol	Description
$V^{A\ell }$	Intrinsic value of the levered firm
$V^{Au}$	Intrinsic value of the unlevered firm
CIT	Corporate income tax
PIT	Personal income tax
CoC	Cost of capital
APV	Adjusted present value
EBIT	Earnings before interest and taxes
FCFF	Free cash flow to the firm
TS	Tax savings from tax deductions of expected interest payments
BC	Bankruptcy costs
VETS	Present value of expected tax shields
VEBC	Present value of expected bankruptcy costs
E	Value of the firm’s (levered) equity
D	Value of the firm’s debt
$\triangle$	Change
$\tau _{cs}$	Statutory corporate income tax rate
$\tau _{pd}$	Personal tax rate on debt (interest) income
$\tau _{pe}$	Personal tax rate on equity income (i.e., a blended dividend and capital gains tax rate)
$\kappa ^A$	Expected rate (or cost) of capital (i.e., weighted average cost of capital, WACC)
$\kappa ^{E\ell }$	Expected rate (or cost) of levered equity (i.e., cost of equity of the levered firm)
$\kappa ^{Eu}$	Expected rate (or cost) of unlevered equity (i.e., cost of equity of the unlevered firm)
$\kappa ^{D}$	Expected rate (or cost) of debt
$\kappa ^{Dp}$	Promised yield on the debt
$\kappa ^{TS}$	Expected rate (or cost) of tax shield component
$\kappa ^{TS,e}$	Discount rate to discount tax shields from the point of view of equity holders
$\kappa ^{TS,d}$	Discount rate to discount tax shields from the point of view of debt holders
$\kappa ^{BC}$	Discount rate to discount bankruptcy costs
$\rho$	Risk probability of default
$\phi$	Indirect bankruptcy costs (the excess promised yield)
$\psi$	Ratio of direct bankruptcy costs in year t to debt value in year $t-1$
$\Upsilon ^{TS}$	Present value of $\triangle D$ in the future in a world without PIT, discounted at $\kappa ^{TS}$
$\Upsilon ^{BC}$	Present value of $\triangle D$ in the future in a world without PIT, discounted at $\kappa ^{BC}$
$\Phi ^{TS}$	Percentage of tax shields (at time t) to debt (at time $t-1$), i.e., $\Phi ^{TS}=\kappa ^{Dp}\tau _{cs}$
$\Phi ^{BC}$	Percentage of bankruptcy costs (at time t) to debt (at time $t-1$), i.e., $\Phi ^{BC}=\rho \phi +\rho \psi$
$\hbar ^{(.),D}$	Total discount factor for a company under a fixed level of debt in perpetuity [(.) includes TS and BC]
$\hbar ^{(.),\Upsilon }$	Total discount factor for the change in debt in the future [(.) includes TS and BC]
$\hbar ^{(.)}$	Total discount factor for (.) (i.e., TS and BC); $\hbar ^{(.)}=\hbar ^{(.),D}+\hbar ^{(.),\Upsilon }$
$\widetilde{(.)}$	Notations (.) in the presence of PIT
$\widetilde{\tau }$	Equivalent tax rate that properly accounts for integration between CIT and PIT
$\widetilde{TS}^e$	Gross tax shields available to levered shareholders
$\widetilde{TS}^d$	PIT paid by debt holders
$\widetilde{\kappa }^{TS,e}$	After-PIT $\kappa ^{TS}$ from the equity holders’ perspective [i.e., $\widetilde{\kappa }^{TS,e}=\kappa ^{TS}(1-\tau _{pe})$]
$\widetilde{\kappa }^{TS,d}$	After-PIT $\kappa ^{TS}$ from the debt holders’ perspective [i.e., $\widetilde{\kappa }^{TS,d}=\kappa ^{TS}(1-\tau _{pd})$]
$\widetilde{\kappa }^{BC,e}$	After-PIT $\kappa ^{BC}$ from the equity holders’ perspective [i.e., $\widetilde{\kappa }^{BC,e}=\kappa ^{BC}(1-\tau _{pe})$]
$\widetilde{\Upsilon }^{TS,e}$	Present value of $\triangle D$ in the future in a world with PIT, discounted at $\widetilde{\kappa }^{TS,e}$
$\widetilde{\Upsilon }^{TS,d}$	Present value of $\triangle D$ in the future in a world with PIT, discounted at $\widetilde{\kappa }^{TS,d}$
$\widetilde{\Upsilon }^{BC,e}$	Present value of $\triangle D$ in the future in a world with PIT, discounted at $\widetilde{\kappa }^{BC,e}$
$\widetilde{\Phi }^{TS,e}$	Percentage of tax shields after PIT (at time t) to debt (at time $t-1$) from the point of view of equity holders, i.e., $\widetilde{\Phi }^{TS,e}=\kappa ^{Dp}\left[ \tau _{cs}(1-\tau _{pe})+\tau _{pe}\right]$
$\widetilde{\Phi }^{TS,d}$	Percentage of tax shields after PIT (at time t) to debt (at time $t-1$) from the point of view of debt holders, i.e., $\widetilde{\Phi }^{TS,d}=\kappa ^{Dp}\tau _{pd}$
$\widetilde{\Phi }^{BC,e}$	Percentage of bankruptcy costs (at time t) to debt (at time $t-1$), i.e., $\widetilde{\Phi }^{BC,e}=(\rho \phi +\rho \psi )(1-\tau _{pe})$
$\widetilde{\hbar }^{(.),D}$	Total discount factor for a company under a fixed level of debt in perpetuity with PIT [(.) includes $TS^e$, $TS^d$, and BC]
$\widetilde{\hbar }^{(.),\Upsilon }$	The total discount factor for the change in debt in the future with PIT [(.) includes $TS^e$, $TS^d$, and BC]
$\widetilde{\hbar }^{(.)}$	Total discount factor for $\widetilde{(.)}$ for a company in which the level of debt may change over time according to a fixed schedule, but it considers the CIT-PIT integration; $\widetilde{\hbar }^{(.)}=\widetilde{\hbar }^{(.),D}+\widetilde{\hbar }^{(.),\Upsilon }$

Appendix B: VEBC with PIT on interest income

Appendix B demonstrates that the indirect costs of bankruptcy should not differentiate between gross and net excess rates of the promised yield, and should not employ two types of discount rates like the idea of $\widetilde{VETS}$. In doing this, we begin by considering the CIT-PIT trade-off for the excess promised yield (i.e., $\phi$), similarly to the analysis of $\widetilde{VETS}$. In this situation, the indirect costs of bankruptcy at time t can be rewritten as

$$\begin{aligned} \widetilde{BC}^{[I]}_t =D_{t+i-1}\phi \left[ (1-\tau _{pd})-(1-\tau _{cs})(1-\tau _{pe})\right]. \end{aligned}$$

(B.1)

Following the idea that $\widetilde{TS}$ includes $\widetilde{TS}^e$ and $\widetilde{TS}^d$, we divide $\widetilde{BC}^{[I]}$ into two components, namely, the indirect costs of bankruptcy from the point of view of equity, $\widetilde{BC}^{[I],e}$, and that from the point of view of debt, $\widetilde{BC}^{[I],d}$. We define $\kappa ^{BC,e}$ and $\kappa ^{BC,d}$ as the discount rates to discount bankruptcy costs from these two viewpoints. Equation (22) in the main text becomes

$$\begin{aligned}&\widetilde{VEBC}_t =\underbrace{\rho _t\sum _{i=1}^\infty \dfrac{\overbrace{D_{t+i-1}\phi [\tau _{cs}(1-\tau _{pe})+\tau _{pe}]}^{\widetilde{BC}^{[I],e}_{t+i}}}{\left( 1+\kappa ^{BC,e}(1-\tau _{pe})\right) ^i}}_{\widetilde{VEBC}^{[I],e}_t}\nonumber \\&\quad +\underbrace{\rho _t\sum _{i=1}^\infty \frac{\overbrace{D_{t+i-1}\psi (1-\tau _{pe})}^{\widetilde{BC}^{[D]}_{t+i}}}{\left( 1+\kappa ^{BC,e}(1-\tau _{pe})\right) ^i}}_{\widetilde{VEBC}^{[D]}_t} -\underbrace{\rho _t\sum _{i=1}^\infty \dfrac{\overbrace{D_{t+i-1}\phi \tau _{pd}}^{\widetilde{BC}^{[I],d}_{t+i}}}{\left( 1+\kappa ^{BC,d}(1-\tau _{pd})\right) ^i}}_{\widetilde{VEBC}^{[I],d}_t}. \end{aligned}$$

(B.2)

We introduce some additional notations:

• ${\widetilde{\Upsilon }}^{BC,d}_t$ denotes the present value of the change of debt in the future period in a world with PIT, discounted at $\widetilde{\kappa }^{BC,d}$, that is, ${\widetilde{\Upsilon }}^{BC,d}_t =\sum \limits _{i=2}^\infty \frac{\sum \limits _{j=t+1}^{t+i-1}\triangle D_j}{\left( 1+\widetilde{\kappa }^{BC,d}\right) ^i}$;

• $\widetilde{\Phi }^{BC,e}$ denotes the percentage of bankruptcy costs (at time t) to debt (at time $t-1$) from the point of view of equity holders, that is, $\widetilde{\Phi }^{BC,e}=\rho \phi \left[ \tau _{cs}(1-\tau _{pe})+\tau _{pe}\right] +\rho \psi (1-\tau _{pe})$;

• $\widetilde{\Phi }^{BC,d}$ denotes the percentage of bankruptcy costs (at time t) to debt (at time $t-1$) from the point of view of debt holders, that is, $\widetilde{\Phi }^{BC,d}=\rho \phi \tau _{pd}$.

Equation (25) can be rewritten as

$$\begin{aligned} \begin{aligned} \widetilde{VEBC}_t =\underbrace{\dfrac{D_t\widetilde{\Phi }^{BC,e}}{\widetilde{\kappa }^{BC,e}}+{\widetilde{\Upsilon }}^{BC,e}_t\widetilde{\Phi }^{BC,e}}_{\widetilde{VEBC}^e_t} -\underbrace{\left[ \dfrac{D_t\widetilde{\Phi }^{BC,d}}{\widetilde{\kappa }^{BC,d}}+{\widetilde{\Upsilon }}^{BC,d}_t\widetilde{\Phi }^{BC,d}\right] }_{\widetilde{VEBC}^d_t}. \end{aligned} \end{aligned}$$

(B.3)

We have $E=[\widetilde{V}^{Au}+\widetilde{VETS}^e-\widetilde{VETS}^d-\widetilde{VEBC}^e+\widetilde{VEBC}^d-D]$ by combining the equation $V^{A\ell }=D+E$ of the CoC method and Eq. (3) of the APV method. Equation (23) in the main text becomes³⁹

$$\begin{aligned} \begin{aligned} \widetilde{\kappa }^{E\ell }&=\widetilde{\kappa }^{Eu} -\dfrac{\widetilde{VETS}^e}{E}\left( \widetilde{\kappa }^{Eu}-\widetilde{\kappa }^{TS,e}\right) +\dfrac{\widetilde{VETS}^d}{E}\left( \widetilde{\kappa }^{Eu}-\widetilde{\kappa }^{TS,d}\right) \\&+\dfrac{\widetilde{VEBC}^e}{E}\left( \widetilde{\kappa }^{Eu}-\widetilde{\kappa }^{BC,e}\right) -\dfrac{\widetilde{VEBC}^d}{E}\left( \widetilde{\kappa }^{Eu}-\widetilde{\kappa }^{BC,d}\right) +\dfrac{D}{E}\left( \widetilde{\kappa }^{Eu}-\widetilde{\kappa }^{Dp}\right). \end{aligned} \end{aligned}$$

(B.4)

To this end, we define

$$\begin{aligned} \widetilde{\hbar }^{BC,d}_t =\frac{\widetilde{VEBC}^d_t}{D_t\widetilde{\Phi }^{BC,d}} =\underbrace{\frac{1}{\widetilde{\kappa }^{BC,d}}}_{\widetilde{\hbar }^{BC,d,D}_t} +\underbrace{\frac{\widetilde{\Upsilon }^{BC,d}}{D_t}}_{\widetilde{\hbar }^{BC,d,\Upsilon }_t}. \end{aligned}$$

(B.5)

Equation (29) in the main text becomes

$$\begin{aligned} \begin{aligned} \widetilde{\kappa }^{E\ell }_t&=\widetilde{\kappa }^{Eu} -\dfrac{D_t}{E_t}\widetilde{\hbar }^{TS,e}_t\widetilde{\Phi }^{TS,e}\left( \widetilde{\kappa }^{Eu}-\widetilde{\kappa }^{TS,e}\right) \\&+\dfrac{D_t}{E_t}\widetilde{\hbar }^{TS,d}_t\widetilde{\Phi }^{TS,d}\left( \widetilde{\kappa }^{Eu}-\widetilde{\kappa }^{TS,d}\right) \\&+\dfrac{D_t}{E_t}\widetilde{\hbar }^{BC,e}_t\widetilde{\Phi }^{BC,e}\left( \widetilde{\kappa }^{Eu}-\widetilde{\kappa }^{BC,e}\right) -\dfrac{D_t}{E_t}\widetilde{\hbar }^{BC,d}_t\widetilde{\Phi }^{BC,d}\left( \widetilde{\kappa }^{Eu}-\widetilde{\kappa }^{BC,d}\right) \\&+\dfrac{D_t}{E_t}\left( \widetilde{\kappa }^{Eu}-\widetilde{\kappa }^{Dp}\right). \end{aligned} \end{aligned}$$

(B.6)

In terms of the CIT-PIT trade-off, the difference between $\widetilde{\Phi }^{TS,e}$ and $\widetilde{\Phi }^{TS,d}$ reflects the trade-off for tax benefits, whereas the difference between $\widetilde{\Phi }^{BC,e}$ and $\widetilde{\Phi }^{BC,d}$ shows that for bankruptcy costs. The CoC-APV equivalence is still appropriate, because $\widetilde{\Phi }^{TS,e}$ (or $\widetilde{\Phi }^{BC,e}$) can be larger than or smaller than or equal to $\widetilde{\Phi }^{TS,d}$ (or $\widetilde{\Phi }^{BC,d}$).

For the TS-BC trade-off, the difference between $\widetilde{\Phi }^{TS,e}$ and $\widetilde{\Phi }^{BC,e}$ captures the trade-off from the point of view of equity in a world with PIT, whereas the difference between $\widetilde{\Phi }^{TS,d}$ and $\widetilde{\Phi }^{BC,d}$ implies that from the viewpoint of debt holders. When distinguishing between gross and net $\widetilde{VEBC}^{[I]}$ (i.e., the excess rates of the promised yield), the CoC-APV equivalence differs from that in sect. 5 in two ways. First, there is a new component in the equivalent formulae, $\widetilde{\Phi }^{BC,d}$, reflecting the excess rates of the promised yield from the debt holders’ viewpoint. The second is the difference in the formula of $\widetilde{\Phi }^{BC,e}$, that is, $\rho \phi \left[ \tau _{cs}(1-\tau _{pe})+\tau _{pe}\right] +\rho \psi (1-\tau _{pe})$, in this situation, rather than $(\rho \phi +\rho \psi )(1-\tau _{pe})$ in sect. 5.

Most importantly, two issues cause the mismatch in perspective from the TS-BC trade-off. First, $\widetilde{\Phi }^{TS,d}$ (i.e., $\kappa ^{Dp}\tau _{pd}$) is always larger than $\widetilde{\Phi }^{BC,d}$ (i.e., $\rho \phi \tau _{pd}$), because $\kappa ^{Dp}$ is often larger than $\phi$, and $\rho$ is always less than one. Hence, $\widetilde{\Phi }^{TS,d}-\widetilde{\Phi }^{BC,d}$ is unable to reflect the trade-off between tax benefits and indirect bankruptcy costs from the viewpoint of debt holders. This leads to the second issue that $\widetilde{\Phi }^{TS,e}$ is always larger than $\widetilde{\Phi }^{BC,e}$ if $\widetilde{\Phi }^{BC,e}$ considers only indirect bankruptcy costs. In other words, in this situation, $\widetilde{\Phi }^{TS,e}-\widetilde{\Phi }^{BC,e}$ captures only the TS-BC trade-off if and only if direct costs of financial distress are present. These confusions imply that $\widetilde{VEBC}$, in general, and $\widetilde{VEBC}^{[I]}$, in particular, should only follow the viewpoint of equity in the presence of PIT.