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 tradeoff 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.
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Notes
Therefore, the CoC method is also known as the weighted average cost of capital (WACC) method, the adjusted discount rate (ADR) method, or the free cash flow to the firm (FCFF) method (see, e.g., Taggart 1991).
The term “costs of financial distress” may be more appropriate than “bankruptcy costs” (see Ross et al. 2013). In this paper, we use these two terms interchangeably, assuming they convey the same meaning.
Under international valuation standards (IVSs), there are three approaches in the application for valuing a business, including marketbased, costbased, and incomebased approaches (IVSC 2021). Each of these approaches includes different valuation methods of application. The incomebased approach is the most popular approach for most valuation purposes, for example, M&A and IPO (see Damodaran 2012; Koziol 2014). A survey by Ross (2006) reveals that 29.1% of respondents typically prefer a DCF valuation as their first choice. Similarly, a valuation methodology survey in Africa reported by PwC (2017), indicates that 64% of respondents primarily use the incomebased approach for their valuations. The dividend discount (DDM), flowstoequity (FTE), capital cash flow (CCF), cost of capital (CoC), and adjusted present value (APV) methods are five wellknown models within the incomebased approach to value a business. The first two methods directly estimate equity value, whereas the others calculate the firm value directly.
From a valuer perspective, the CoC and APV methods are more frequently used in practice than other methods to ensure crosschecking and to minimize valuation errors. Ross (2006) also shows that 63% of respondents typically or almost always use the CoC method, while the APV method proves most useful in complex valuation situations.
Altman of NYU’s Stern School of Business estimates the risk probability of default as part of an annual series (see Damodaran 2012).
RICS (2021) is effective from January, 31 2022. RICS (Royal Institution of Chartered Surveyors) is an internationally acknowledged professional organization that promotes and enforces the highest professional standards in developing, managing, and valuing land, real estate, construction, infrastructure, projects, businesses, and intangible assets.
Effective from January 31, 2022.
These works have significantly influenced the new era of finance and valuation literature. Modigliani and Miller (1958) argue that optimal capital structures do not exist, and there is no difference in value between a levered and an unlevered company in complete and perfect capital markets. When CITs are introduced, but PITs are still assumed to be equal zero, the value of tax shields becomes positive. This increase leads to the value of levered firms being higher than that of unlevered firms, due to the contribution of tax savings (Modigliani and Miller 1963). However, there are certain implications when analyzing the 1963 MM theory: (i) the model only considers tax shields with a zero risk of financial distress, leading to maximum firm value when corporations finance with 100% debt, (ii) a linear relationship exists between capital structure and firm value because of the positive value of tax shields, and (iii) the model does not account for PITs, which may distort the present value of tax shields due to dividends paid from earnings after taxes.
Several factors identified in this literature have influenced contemporary capital structure theories. These include the PIT (Miller 1977), the tradeoff theory's costs of financial distress (Kraus and Litzenberger 1973; Scott 1976), agency costs with the agency theory (Jensen and Meckling 1976; Myers 1977), and the peckingorder theory's timing (Myers 1984).
For example, Sharpe (1964) and Lintner (1965) introduce the capital asset pricing model (CAPM) for calculating the discount rate (i.e., the cost of capital) based on the portfolio theory of Markowitz (1952). Myers (1974) develops the APV method based on the idea of tax shields of Modigliani and Miller (1963). Later studies have attempted to provide consistent valuation between the CoC and APV methods (e.g., Sick 1990; Taggart 1991; Schultze 2004; Dempsey 2013; Cooper and Nyborg 2018; Kolari 2018).
The firm value in the valuation report is the sum of three components: (i) \(V^{A\ell }_t\); (ii) the value of operating assets that creates cash flows, but these cash flows have not yet been estimated in the FCFF (i.e., \(V^{A\ell }_t\)); and (iii) the value of nonoperating assets. Some researchers and practitioners in valuation regard the last two components as the value of nonoperating assets. Most valuation textbooks focus on \(V^{A\ell }_t\) for the incomebased approach. Indeed, when applying the DCF techniquebased valuation, valuers take more time to estimate the first component. Therefore, we concentrate solely on valuation models to estimate \(V^{A\ell }_t\). However, we note that these three components are independent, primarily because of the summation calculator. This implies that all items used to estimate \(V^{A\ell }_t\) (e.g., earnings, cash flows, assets, reinvestment) must be independent and not related to any items in the remaining components, such as the current financial investment, to prevent overlap in the estimation.
There are three main arguments related to the period between cash flows and discount rates. The first is the same period for both FCFF and \(\kappa\), that is, \(FCFF_i\) is discounted at \(\kappa _i\) (e.g., Miles and Ezzell 1980). The second is that \(\kappa\) lags one period behind cash flows, that is, \(\kappa _i\) is used to discount \(FCFF_{i+1}\) (e.g.,Kruschwitz and Löffler 2006; Husmann et al. 2006; Bade 2009; Dempsey 2019). The third is the constant cost of capital, i.e., \(\kappa _i=\kappa\) (e.g., Molnár and Nyborg 2013; Damodaran 2012; Pinto et al. 2015; Cooper and Nyborg 2018). Because there is no significant difference between these approaches, for simplicity, we apply the third option. The results are not affected by this assumption. We can replace \((1+\kappa )^{i}\) for the third idea with \(\prod \limits _{j=t+1}^{t+i}(1+\kappa _j)\) for the first idea or \(\prod \limits _{j=t+0}^{t+i1}(1+\kappa _j)\) for the second idea, whenever needed.
The debt concept in this study includes debt and debtlike liabilities.
Eq. (2) is the standard equation referred to as the textbook WACC under the assumption of riskfree debt (Dempsey 2013). Some studies provide discount rate formulae under the alternative condition of risky debt (e.g., Miles and Ezzell 1980; Taggart 1991; Cooper and Nyborg 2008). However, Cooper and Nyborg (2008) argue that the textbook WACC assumption leads only to minor errors, in general.
This method's results are also used to crossverify those from other methods in the incomebased approach.
Note that while bankruptcy costs decrease firm value, the firm's value may continue to decline even without VEBC, owing to factors such as lawyers' advisory fees. Consequently, financial distress costs might be a more accurate concept than bankruptcy costs (Ross et al. 2013). However, our study focuses on valuation and anticipates the application of valuation models in the real world. Therefore, we use the term “bankruptcy costs” because of the substantial empirical evidence related to this cost.
Numerous subsequent studies introduce various costs to balance in a costbenefit analysis related to debt, such as the agency cost presented in the agency theory developed by Jensen and Meckling (1976).
The notations above are similar to that of KimDuc and Nam (2023).
Note that other indirect costs may apply, such as customer loss or increased employee turnover.
Examples provided by Cooper and Davydenko (2007) show that the difference between \(\kappa ^{Dp}\) and the riskfree rate, \(\kappa ^F\), is 3%, whereas the risk premium (i.e., \(\kappa ^{D}\kappa ^F\)) in \(\kappa ^{D}\) is only 1%.
Because \(\kappa ^{E\ell }= \dfrac{E+DVETS+VEBC}{E}\kappa ^{Eu} +\dfrac{VETS}{E}\kappa ^{TS} \dfrac{VEBC}{E}\kappa ^{BC} \dfrac{D}{E}\kappa ^{Dp}\).
These notations were originally introduced in the manuscript of this study in 2022, which had not yet been published in a journal at that time. Later, KimDuc and Nam (2023) used these notations, crediting the source of these notations as this study, which was still in the form of a working paper.
See KimDuc and Nam (2023) for a discussion of the \(\Upsilon ^{TS}_t\) and \(\Upsilon ^{BC}_t\) formulae.
The discount rate at which to discount TS is still a controversial issue (see Graham 2003). For example, Miles and Ezzell (1985) argue that TSs have equity risk because debt is not fixed, and thus TS should be discounted at the return on assets, \(\kappa ^A\). If the D/E ratio is fixed in the first period, the discount rate is required to adjust, \(\kappa ^{TS}=\frac{1+\kappa ^A}{1+\kappa ^{D}}\) (Miles and Ezzell 1985).
Eq. (19) is identical to Eq. (18) when there is no PIT (i.e., \(\tau _{pd}=\tau _{pe}=0\)) or \(\tau _{pd}=\tau _{pe}\) (\(\ne 0\)). When \(\tau _{pd}>\tau _{pe}\), VETS under Miller (1977) is less than VETS under Modigliani and Miller (1963). This argument by Miller (1977) is strongly supported, because the facts typically show a lower \(\tau _{pe}\) relative to \(\tau _{pd}\). More importantly, in some situations, \(VETS \le 0\) if \(\tau _{pd}\) is large relative to \(\tau _{cs}\) and \(\tau _{pe}\) (Graham 2003).
\(\kappa ^{BC,e}\) in this section is equal to \(\kappa ^{BC}\) in previous sections. We add \(^e\) so that the notation of the discount rates in \(\widetilde{VEBC}\) are consistent with that in \(\widetilde{VETS}\).
Because \(\widetilde{\kappa }^{E\ell }= \dfrac{E+D\widetilde{VETS}^e+\widetilde{VETS}^d+\widetilde{VEBC}}{E}\widetilde{\kappa }^{Eu} +\dfrac{\widetilde{VETS}^e}{E}\widetilde{\kappa }^{TS,e} \dfrac{\widetilde{VETS}^d}{E}\widetilde{\kappa }^{TS,d} \dfrac{\widetilde{VEBC}}{E}\widetilde{\kappa }^{BC,e} \dfrac{D}{E}\widetilde{\kappa }^{Dp}\).
Similarly, these notations were originally introduced in the manuscript of this study in 2022, which had not yet been published in a journal at that time. Subsequently, KimDuc and Nam (2023) used these notations, crediting the source of these notations as this study, which was still in the form of a working paper.
Because of the same growth rate of earning in the first and stable period (i.e., \(g^E=4.5\%\)), Eq. (31) can be rewritten as \(V^{A\ell }_0=\underbrace{\frac{\widetilde{FCFF}_1}{\widetilde{\kappa }^{Eu}g}}_{\widetilde{V}^{Au}_0} +\underbrace{\left[ \frac{D_0}{\widetilde{\kappa }^{TS,e}}+\widetilde{\Upsilon }^{TS,e}_0\right] \widetilde{\Phi }^{TS,e} \left[ \frac{D_0}{\widetilde{\kappa }^{TS,d}}+\widetilde{\Upsilon }^{TS,d}_0\right] \widetilde{\Phi }^{TS,d}}_{\widetilde{VETS}_0} \underbrace{\left[ \frac{D_0}{\widetilde{\kappa }^{BC,e}}+\widetilde{\Upsilon }^{BC,e}_0\right] \widetilde{\Phi }^{BC,e}}_{\widetilde{VEBC}_0}\). Hence, Eq. (32) becomes \(V^{A\ell }_0=\frac{1,607}{11.7\%4.5\%} +\left[ \frac{4,000}{9\%}+117,384\right] 2.24\% \left[ \frac{4,000}{8.5\%}+135,886\right] 1.20\% \left[ \frac{4000}{8.1\%}+154,691\right] 0.03\% =\$23,686.\)
Because \(\widetilde{\kappa }^{E\ell }= \dfrac{E+D\widetilde{VETS}^e+\widetilde{VETS}^d+\widetilde{VEBC}^e\widetilde{VEBC}^d}{E}\widetilde{\kappa }^{Eu} +\dfrac{\widetilde{VETS}^e}{E}\widetilde{\kappa }^{TS,e} \dfrac{\widetilde{VETS}^d}{E}\widetilde{\kappa }^{TS,d} \dfrac{\widetilde{VEBC}^e}{E}\widetilde{\kappa }^{BC,e} +\dfrac{\widetilde{VEBC}^d}{E}\widetilde{\kappa }^{BC,d} \dfrac{D}{E}\widetilde{\kappa }^D\).
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Acknowledgements
We sincerely thank anonymous reviewers for many constructive comments and suggestions, which significantly enhance our paper’s exposition. We also thank the conference participants at the \(38^{th}\) EBES Conference  Warsaw hosted in Poland in 2022 for the helpful comments provided. This research is funded by the University of Economics Ho Chi Minh City (UEH), Vietnam.
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This research is funded by the University of Economics Ho Chi Minh City (UEH), Vietnam.
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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 \(t1\) 
\(\Upsilon ^{TS}\)  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 \(t1\)), i.e., \(\Phi ^{TS}=\kappa ^{Dp}\tau _{cs}\) 
\(\Phi ^{BC}\)  Percentage of bankruptcy costs (at time t) to debt (at time \(t1\)), 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 
\(\widetilde{\kappa }^{TS,e}\)  AfterPIT \(\kappa ^{TS}\) from the equity holders’ perspective [i.e., \(\widetilde{\kappa }^{TS,e}=\kappa ^{TS}(1\tau _{pe})\)] 
\(\widetilde{\kappa }^{TS,d}\)  AfterPIT \(\kappa ^{TS}\) from the debt holders’ perspective [i.e., \(\widetilde{\kappa }^{TS,d}=\kappa ^{TS}(1\tau _{pd})\)] 
\(\widetilde{\kappa }^{BC,e}\)  AfterPIT \(\kappa ^{BC}\) 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 \(t1\)) 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 \(t1\)) 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 \(t1\)), 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 CITPIT 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 CITPIT tradeoff 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
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
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, \({\widetilde{\Upsilon }}^{BC,d}_t =\sum \limits _{i=2}^\infty \frac{\sum \limits _{j=t+1}^{t+i1}\triangle D_j}{\left( 1+\widetilde{\kappa }^{BC,d}\right) ^i}\);

\(\widetilde{\Phi }^{BC,e}\) denotes the percentage of bankruptcy costs (at time t) to debt (at time \(t1\)) 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 \(t1\)) from the point of view of debt holders, that is, \(\widetilde{\Phi }^{BC,d}=\rho \phi \tau _{pd}\).
Equation (25) can be rewritten as
We have \(E=[\widetilde{V}^{Au}+\widetilde{VETS}^e\widetilde{VETS}^d\widetilde{VEBC}^e+\widetilde{VEBC}^dD]\) 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 becomes^{Footnote 39}
To this end, we define
Equation (29) in the main text becomes
In terms of the CITPIT tradeoff, the difference between \(\widetilde{\Phi }^{TS,e}\) and \(\widetilde{\Phi }^{TS,d}\) reflects the tradeoff for tax benefits, whereas the difference between \(\widetilde{\Phi }^{BC,e}\) and \(\widetilde{\Phi }^{BC,d}\) shows that for bankruptcy costs. The CoCAPV 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 TSBC tradeoff, the difference between \(\widetilde{\Phi }^{TS,e}\) and \(\widetilde{\Phi }^{BC,e}\) captures the tradeoff 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 CoCAPV 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 TSBC tradeoff. 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 tradeoff 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 TSBC tradeoff 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.
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KimDuc, N., Nam, P.K. Consistent valuation: extensions from bankruptcy costs and tax integration with timevarying debt. Rev Quant Finan Acc 62, 719–754 (2024). https://doi.org/10.1007/s11156023012175
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DOI: https://doi.org/10.1007/s11156023012175