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Published in: Review of Quantitative Finance and Accounting 2/2024

24-11-2023 | Original Research

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

Authors: Nguyen Kim-Duc, Pham Khanh Nam

Published in: Review of Quantitative Finance and Accounting | Issue 2/2024

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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.

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Appendix
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Footnotes
1
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).
 
2
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.
 
3
Under international valuation standards (IVSs), there are three approaches in the application for valuing a business, including market-based, cost-based, and income-based approaches (IVSC 2021). Each of these approaches includes different valuation methods of application. The income-based 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 income-based approach for their valuations. The dividend discount (DDM), flows-to-equity (FTE), capital cash flow (CCF), cost of capital (CoC), and adjusted present value (APV) methods are five well-known models within the income-based approach to value a business. The first two methods directly estimate equity value, whereas the others calculate the firm value directly.
 
4
From a valuer perspective, the CoC and APV methods are more frequently used in practice than other methods to ensure cross-checking 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.
 
5
Altman of NYU’s Stern School of Business estimates the risk probability of default as part of an annual series (see Damodaran 2012).
 
6
Most research on CoC-APV equivalence begins with these two well-known studies, Modigliani and Miller (1963)’s model and Miles and Ezzell (1980)’s model. The primary differences between them lie in their assumptions about financing strategy and the level of risk in tax savings.
 
7
Kolari (2018) contributes to the literature by differentiating between gross and net tax shields. This concept is applied to the Miller (1977) and ME models, yielding a revised ME model.
 
8
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.
 
9
Effective from January 31, 2022.
 
10
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.
 
11
Several factors identified in this literature have influenced contemporary capital structure theories. These include the PIT (Miller 1977), the trade-off 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 pecking-order theory's timing (Myers 1984).
 
12
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).
 
13
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 non-operating assets. Some researchers and practitioners in valuation regard the last two components as the value of non-operating assets. Most valuation textbooks focus on \(V^{A\ell }_t\) for the income-based approach. Indeed, when applying the DCF technique-based 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.
 
14
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 \(\prod \limits _{j=t+0}^{t+i-1}(1+\kappa _j)\) for the second idea, whenever needed.
 
15
The debt concept in this study includes debt and debt-like liabilities.
 
16
Eq. (2) is the standard equation referred to as the textbook WACC under the assumption of risk-free 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.
 
17
This method's results are also used to cross-verify those from other methods in the income-based approach.
 
18
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.
 
19
Numerous subsequent studies introduce various costs to balance in a cost-benefit analysis related to debt, such as the agency cost presented in the agency theory developed by Jensen and Meckling (1976).
 
20
The notations above are similar to that of Kim-Duc and Nam (2023).
 
21
See, for example, Damodaran (2012), Koller et al. (2015).
 
22
Eq. (3) in this study corresponds to Eq. (5) in the study by Kim-Duc and Nam (2023).
 
23
Note that other indirect costs may apply, such as customer loss or increased employee turnover.
 
24
See, for example, Dempsey (2013) or Cooper and Nyborg (2018).
 
25
See, for example, Cooper and Nyborg (2006), Dempsey (2013), or Cooper and Nyborg (2018).
 
26
In addition, recent studies provide general formulae for calculating VETS and establish the equivalence among costs of capital (see Cooper and Nyborg 2006, 2007, 2008, 2018; Dempsey 2013).
 
27
Examples provided by Cooper and Davydenko (2007) show that the difference between \(\kappa ^{Dp}\) and the risk-free rate, \(\kappa ^F\), is 3%, whereas the risk premium (i.e., \(\kappa ^{D}-\kappa ^F\)) in \(\kappa ^{D}\) is only 1%.
 
28
Because \(\kappa ^{E\ell }= \dfrac{E+D-VETS+VEBC}{E}\kappa ^{Eu} +\dfrac{VETS}{E}\kappa ^{TS} -\dfrac{VEBC}{E}\kappa ^{BC} -\dfrac{D}{E}\kappa ^{Dp}\).
 
29
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, Kim-Duc 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.
 
30
See Kim-Duc and Nam (2023) for a discussion of the \(\Upsilon ^{TS}_t\) and \(\Upsilon ^{BC}_t\) formulae.
 
31
See Kim-Duc and Nam (2023) for a detail description of the first and second terms on the right-hand side of Eqs. (9) and (10). Equations (9) and (10) in this study correspond to Eqs. (6) and (7) in Kim-Duc and Nam (2023).
 
32
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).
 
33
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).
 
34
\(\kappa ^{BC,e}\) 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 \(\widetilde{VETS}\).
 
35
Because \(\widetilde{\kappa }^{E\ell }= \dfrac{E+D-\widetilde{VETS}^e+\widetilde{VETS}^d+\widetilde{VEBC}}{E}\widetilde{\kappa }^{Eu} +\dfrac{\widetilde{VETS}^e}{E}\widetilde{\kappa }^{TS,e} -\dfrac{\widetilde{VETS}^d}{E}\widetilde{\kappa }^{TS,d} -\dfrac{\widetilde{VEBC}}{E}\widetilde{\kappa }^{BC,e} -\dfrac{D}{E}\widetilde{\kappa }^{Dp}\).
 
36
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, Kim-Duc 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.
 
37
Eqs. (24) and (25) in this study correspond to Eqs. (9) and (10) in Kim-Duc and Nam (2023).
 
38
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 \(V^{A\ell }_0=\frac{1,607}{11.7\%-4.5\%} +\left[ \frac{4,000}{9\%}+117,384\right] 2.24\% -\left[ \frac{4,000}{8.5\%}+135,886\right] 1.20\% -\left[ \frac{4000}{8.1\%}+154,691\right] 0.03\% =\$23,686.\)
 
39
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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Metadata
Title
Consistent valuation: extensions from bankruptcy costs and tax integration with time-varying debt
Authors
Nguyen Kim-Duc
Pham Khanh Nam
Publication date
24-11-2023
Publisher
Springer US
Published in
Review of Quantitative Finance and Accounting / Issue 2/2024
Print ISSN: 0924-865X
Electronic ISSN: 1573-7179
DOI
https://doi.org/10.1007/s11156-023-01217-5

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