Mô hình thẩm định giá doanh nghiệp: Định giá nhất quán, Biến dạng thuế do lạm phát, và Tốc độ tăng trưởng lợi nhuận.Mô hình thẩm định giá doanh nghiệp: Định giá nhất quán, Biến dạng thuế do lạm phát, và Tốc độ tăng trưởng lợi nhuận.Mô hình thẩm định giá doanh nghiệp: Định giá nhất quán, Biến dạng thuế do lạm phát, và Tốc độ tăng trưởng lợi nhuận.Mô hình thẩm định giá doanh nghiệp: Định giá nhất quán, Biến dạng thuế do lạm phát, và Tốc độ tăng trưởng lợi nhuận.Mô hình thẩm định giá doanh nghiệp: Định giá nhất quán, Biến dạng thuế do lạm phát, và Tốc độ tăng trưởng lợi nhuận.Mô hình thẩm định giá doanh nghiệp: Định giá nhất quán, Biến dạng thuế do lạm phát, và Tốc độ tăng trưởng lợi nhuận.Mô hình thẩm định giá doanh nghiệp: Định giá nhất quán, Biến dạng thuế do lạm phát, và Tốc độ tăng trưởng lợi nhuận.Mô hình thẩm định giá doanh nghiệp: Định giá nhất quán, Biến dạng thuế do lạm phát, và Tốc độ tăng trưởng lợi nhuận.Mô hình thẩm định giá doanh nghiệp: Định giá nhất quán, Biến dạng thuế do lạm phát, và Tốc độ tăng trưởng lợi nhuận.Mô hình thẩm định giá doanh nghiệp: Định giá nhất quán, Biến dạng thuế do lạm phát, Tốc độ tăng trưởng lợi nhuận.Ministry of Education and Training UNIVERSITY OF ECONOMICS HO CHI MINH CITY Nguyen Kim Duc Business valuation models Consistent valuation, Inflation related tax distortions, and Earnings growth rates.
Ministry of Education and Training UNIVERSITY OF ECONOMICS HO CHI MINH CITY Nguyen Kim Duc Business valuation models: Consistent valuation, Inflation-related tax distortions, and Earnings growth rates Major: Development Economics Code: 62.31.05.01 SUMMARY OF Ph.D THESIS Ho Chi Minh City, Tuesday 11th April, 2023 Ministry of Education and Training UNIVERSITY OF ECONOMICS HO CHI MINH CITY Nguyen Kim Duc Business valuation models: Consistent valuation, Inflation-related tax distortions, and Earnings growth rates Major: Development Economics Code: 62.31.05.01 Supervisor: Assoc Prof Pham Khanh Nam SUMMARY OF Ph.D THESIS Ho Chi Minh City, Tuesday 11th April, 2023 The thesis is carried out at: University of Economics Ho Chi Minh City Academic supervisors: Assoc Prof Pham Khanh Nam Reviewer : Reviewer : Reviewer : The thesis will be defended at the Doctorate Thesis Committee of Examiners of the University of Economics Ho Chi Minh City at on The thesis can be referred at the library: ii Publications Journal Articles • Kim-Duc, N and Nam, P K (2023) Inflation-related tax distortions in business valuation models: A clarification North American Journal of Economics and Finance, 66, 101907 Conference Papers • Kim-Duc, N (2022) Consistent valuation: Extensions from bankruptcy costs and tax integration with time-varying debt Poland: EBES (Eurasia Business and Economics Society) ISBN: 978-605-80042-8-3 38th EBES Conference - Program and Abstract Book, 52 • Kim-Duc, N (2022) Inflation-related tax distortions in business valuation models Poland: EBES (Eurasia Business and Economics Society) ISBN: 978-605-80042-8-3 38th EBES Conference - Program and Abstract Book, 59 • Kim-Duc, N (2022) Earnings Growth Rates in Business Valuation Models: Impossible Quaternity Turkey: EBES (Eurasia Business and Economics Society) ISBN: 978-605-71739-0-4 40th EBES Conference - Program and Abstract Book, 124 • Kim-Duc, N (2019) Do firms face a higher real tax burden in the presence of inflation? An analysis based on business valuation theory Thailand: 22nd ASEAN Valuers Association Congress, 20-22 October 2019 iii Chapter General Introduction 1.1 A need for analyzing business valuation models For business valuation, the income-based approach, in general, and the discounted cash flow (DCF) technique, in particular, are the most popular tools for most valuation purposes, for example, M&A and IPO (see Damodaran, 2012; Koziol, 2014) DCF valuation provides the intrinsic value of a firm or equity as the present value of its expected future cash flows (Pinto et al., 2015) Although there is a large body of literature in academia and a vast experience in practice, the DCF technique is still an intensive discussion Many authors analyze, modify, and develop components in the DCF framework to make firm value or equity value more appropriate (see, e.g., Taggart, 1991; Buchenroth and Pilotte, 2004; Dempsey, 2013; Koziol, 2014; Krause and Lahmann, 2016; Kolari, 2018; Cooper and Nyborg, 2018; Dempsey, 2019) All of them imply that business valuation is the complex field; and analyzing, modifying, or developing current valuation models are necessary to provide an appropriate result The thesis extends existing valuation models under the DCF technique by considering inflation-related tax distortions under macroeconomic contexts 1.1.1 A need for study Consistent valuation between the CoC and APV methods Under the DCF framework, the CoC and APV methods has long been viewed as the most effective tools to valuing a business (Ross, 2006; PwC, 2017) Theoretically, these two methods achieve consistent valuation, although each incorporates the levels of risk and the effects of debt financing in various ways The reason is that different valuation methods apply varying discount rates, and these discount rates can offset that Factors such as conditions related to the level of debt, tax integration, or costs of financial distress are considered for both methods However, standard versions of textbooks only consider simplified scenarios Recent studies make constant efforts to remove the current simplifying assumptions gradually, to provide a more general equivalence However, the current CoC-APV equivalence versions have not yet fully captured the change of the CoC and APV methods in recent years They only remove one or some assumptions and continuously keep the rest Practically, valuers expect all valuation methods to provide identical values for the target firm being assessed However, as mentioned before, current versions of consistent valuation only deal with the different assumptions in the overall picture This makes it difficult for valuers to understand the existing equivalence versions to apply current valuation models As a result, valuers often face challenges in explaining the disparity in firm or equity value, when using valuation methods with different assumptions We consider all significant conditions to provide the more generalized CoC-APV equivalence, unlike several previous articles This equivalence allows the stochastic debt and considers the trade-off between corporate income taxes (CIT) and personal income taxes (PIT), and the trade-off between tax benefits and costs of financial distress 1.1.2 A need for study Inflation-related tax distortions in business valuation models In recent years, business valuation models have begun to reveal many limitations leading to inconsistent results and these drawbacks are explained by the lack of macro variable measurements (i.e., inflation and tax) (Zurita et al., 2019) Theoretically, macro-environment analysis is a required step of the whole procedure, but, in fact, it is only at qualitative level through professional judgment of appraisers instead of specific quantity (see Damodaran, 2012; Pinto et al., 2015; Pratt and Niculita, 2008; Ross et al., 2013; Trugman, 2017) This limitation not only directly impacts the price management policy and financial budget of governments at macro level but also causes difficulty for valuers in explaining the valuation results to clients at micro level Hence, extending the current business valuation models, which consider the distortions of macro variables, is highly beneficial for practitioners in the new context There is still one drawback in analyzing tax distortions in the DCF framework when countries experience inflation despite ongoing discussions This shortcoming comes from the lack of distinguishing between tax and accounting perspectives The current CIT expense occupies just one line item in the income report, but it is a unique line item following tax rules without financial reporting rules (see Hanlon and Heitzman, 2010; Scholes et al., 2015) Thus, tax distortions (if available) affect cash flows through only CIT expenses or the effective rates of CIT, rather than other components of cash flows, for example, depreciation Inflation-related tax distortions of depreciation only exist in taxable income for tax purposes, leading to a change in cash flows In contrast, there is no role for tax distortions, due to inflation for depreciation in cash flow formulas Besides, under the environment with CIT, companies can gain benefits, so-called the value of expected tax shields (V ET S), from interest expense deductions However, the more firms reap the benefits of tax deductions of interest payment firms, the more challenging it is for financing firms to handle the costs of raising debt financing, so-called the value of bankruptcy costs (V EBC) Thus, besides CIT distortions, the presence of inflation also make these costs and benefits distorted (Dhaliwal et al., 2015; Zurita et al., 2019) More importantly, empirical evidence related to the impacts of inflation on tax costs and tax benefits has been largely mixed and inconclusive (e.g., Anderson et al., 1983; Matolcsy, 1984; Bernard, 1986; Kaul, 1987; Ferris and Makhija, 1988; Lee, 2010; Dhaliwal et al., 2015; Finocchiaro et al., 2018) In addition, International Accounting Standards system (IAS) and International Financial Reporting Standards system (IFRS) have been changed a lot over the past decade IFRS 13—Fair Value Measurement—has been effective since 2013 but some countries (e.g., ASEAN) have just adopted IFRS in recent years Adopting IFRS 13 plays the vital role in mitigating inflation-related tax distortions in countries following the real tax basis By contrast, inflation makes more distortions if nations follow the nominal tax base The current business valuation models not reflect this leading to the necessity of the extended valuation models to consider the tax-inflation relationship in the new context 1.1.3 A need for study Earnings growth rates in business valuation models Under the CoC and APV methods, the growth rate of earnings is the key component to forecast future earnings However, it is difficult to identify errors because valuers cannot cross-check growth at the valuation date Thus, forecasting future earnings or cash flows is a rich and demanding task with more challenges (Pinto et al., 2015) We were motivated to conduct this research upon take part in business valuation in practice We face another situation in which clients provide both financial statements in the past and earnings projected in the future In this case, we need to apply a growth formula to re-estimate future reinvestment conversely The difference with the first situation is that we can compare the available data and information to ensure consistency among the components in business valuation models Biases in the estimation of earnings growth persist when calculating the reinvestment rate, RIR, and return on invested capital, ROIC in growth-rate formulas are cross-referenced across different ways as when they are calculated independently This leads to reinvestment that contributes to earnings growth, which is inconsistent between RIR and ROIC Specifically, both RIR and ROIC are financial ratios with numerators and denominators Practitioners must ensure that both the numerator and the denominator are numbers from the entire year or at a snapshot in time when calculating ratios independently To sum up, all problems mentioned above lead to the necessity of advanced research to develop the current business valuation models that mentions the impact of inflation on tax distortions This is highly significant not only in academic terms but also in practical aspects As a result, we select "Business valuation models: Consistent valuation, Inflation-related tax distortions, and Earnings growth rates" as the title of the thesis submitted for the degree of Doctor of Philosophy in the UEH School of Economics, UEH College of Economics, Law and Government 1.2 Objectives and research questions In general, this thesis aims to contribute to the valuation literature by developing an extended business valuation model Specifically, each study in the thesis provides the newest formulas for consistent valuation, tax distortions in firm value, and earnings growth rates, compared to the latest business valuation models All studies in this thesis apply math methodology to develop valuation models The specific aims in each study are as follows: 1.2.1 Consistent valuation: Extensions from bankruptcy costs and tax integration with timevarying debt The research question answered in this study is as follows: • What is the consistent valuation between the CoC and APV methods at the highest level? 1.2.2 Macroeconomic contexts and business valuation models: Inflation-related tax distortions This study goes to answer these research questions: • What is the appropriate formula of ET R in the case of tax distortions due to inflation? • What is the difference in firm value under the DCF framework between with and without considering these tax distortions? • What is the general equation to independently estimate the value of inflation-related tax distortions? 1.2.3 Earnings growth rates in business valuation models: Impossible quaternity This study discusses two following research questions: • What should valuers pay attention to when estimating growth rate if they cross-reference across different ways in calculating RIR and ROIC? • What is the role of the timing of reinvestment in growth estimation formulas? 1.3 Contributions of the research 1.3.1 Theoretical contributions The dissertation is a theoretical study that follows and develops business valuation models based on the arguments of previous articles but with some adjustments Hence, the thesis makes the following theoretical contribution From a general perspective of the dissertation: • As far as we know, the thesis is the first study to provide the business valuation models with the quantification of macro variables—the impact of inflation on tax distortions—rather than the qualitative analysis, which has been commonly based on valuers’ professional judgments as in previous versions of valuation methods • Unlike previous studies, throughout this thesis, we put ourselves in the role of the registered valuers and the valuation profession, rather than finance professionals (e.g., investors, financial managers, financial analysts) Theoretically, valuation models are used similarly for both of them However, this may not be true in practice because of the requirement of compliance with country or state legal regulations governing the right to practice valuation (RICS, 2019) Licensed valuers must adapt and apply international and/or local valuation standards to assure credible and consistent valuation opinions In most cases, aside from breach of contract, registered valuers may be sued for ‘tort’—an umbrella term for all civil wrongs recognized by the law (see RICS, 2018) However, in some countries, registered valuers may also be convicted by criminal law (e.g., Vietnam, Thailand) All of them require valuers to work at the highest responsibility within their profession This leads to different risk appetites in choosing the appropriate valuation approach(es) and method(s), as well as the different methods of calculating and adjusting components of valuation models compared to finance professionals For the first study: • The first is also the most important contribution, we generalize the CoC–APV equivalence that considers both PIT (i.e., the trade-off between CIT and PIT) and bankruptcy costs (i.e., the trade-off between tax benefits and costs of financial distress) Recent efforts have just focused on one of them, such as Schultze (2004); Kolari (2018); Dempsey (2019) for PIT, and Cooper and Nyborg (2018) for bankruptcy costs • Unlike Cooper and Nyborg (2018), we use the risk probability of default and the present value of bankruptcy costs to estimate the value of expected costs of bankruptcy (see Damodaran, 2012) The reasons are that there is a large body of literature in calculating the risk probability of default (e.g., Altman, 1968; Altman and Hotchkiss, 2006; Altman et al., 2017)1 and the present value of bankruptcy costs In addition, many studies provide empirical evidence about direct and indirect costs of bankruptcy (e.g., Warner, 1977; Weiss, 1990; Branch, 2002) This also ensures that valuers can apply our formulas practically • We generalize the CoC–APV equivalence, under the debt financing strategy that allows debt levels to change over time, according to a fixed schedule This means that it is the highest level of generalization for the first category, as discussed above • We employ the general discount rate named κT S , appropriate to tax savings, rather than specific discount rates (i.e., the cost of debt, the unleveraged cost of equity capital) for two scenarios of financing strategy, i.e., the model of Modigliani and Miller (1963) (hereafter MM), and the model of Miles and Ezzell (1980) (hereafter ME), respectively This is appropriate for the third contribution of this study—a scenario of time-varying debt—and makes our formulas more general and possible to collapse to standard versions of the MM and ME models • Kolari (2018) contributes to the literature by distinguishing between gross and net tax shields, applying this idea to the Miller (1977) and ME models, and providing a revised ME model Following Kolari (2018), our equivalence also captures the difference in the point of view of tax shields, between stockholders and debt holders However, we only adopt this idea for our general formulas instead of comparing the MM/Miller (1977) and ME models Moreover, we not apply the revised ME model of Kolari because we need further research to ensure that there are no confusions similar to those mentioned above For the second study: • We present inflation-related tax distortions by modified equations in valuation models Previous studies examine whether firms face a higher real CIT burden or a lower real tax benefit in the presence of inflation These papers, however, focus on providing empirical results rather than modifying or developing equations in valuation models For example, Dhaliwal et al (2015) find that the interaction between the U.S tax Altman of NYU’s Stern School of Business estimates the risk probability of default as part of an annual series (see Damodaran, 2012) base and inflation increases the real CIT tax burden That burden is more serious for capital- and inventory-intensive firms • We distinguish tax and accounting perspectives when analyzing tax distortions As mentioned above, tax distortions due to inflation (if available) only exist in CIT or CIT rates rather than other components of cash flows As a result, there is confusion if valuers analyze distortions for CIT and other components (see, e.g., Zurita et al., 2019) because the others follow accounting rules • We show these distortions in standard formulas of the DCF business valuation technique and provide a new equation to estimate the value of inflation-related tax distortions independently The value of inflation-related tax distortions is the difference between the true value estimated when valuers employ ET R and firm value in which the tax rate is ST R We have shown these distortions in F CF F and firm value through τ IT D and the k factor for two scenarios, with and without personal income tax, if enterprise value is correctly estimated, compared to valuers ignoring the ET R Thus, our analysis provides a general formula for estimating the value of inflation-related tax distortions independently For the third study: • The study shows that the current standard formula for growth rates is still correct, and the estimation of fundamental growth determinants is also valid if they are calculated independently However, this can lead to inconsistencies when all of them are used simultaneously in growth estimation formulas We illustrate that cross-reference in the calculation between RIR and ROIC leads to valuation errors The results show a negative linear relationship between the timing of reinvestment and the incorrect growth rate Specifically, when capital this year is reinvested closer to the end of the year, the annual growth rate due to miscalculations will decrease • We clarify the formulas of earnings growth rates to avoid mistakes, as mentioned above, under two situations: reinvestment is at the end of each year, and reinvestment is at any time of each year The study also shows the time of capital reinvestment, where the false growth rate due to cross-referencing will be equal to the actual growth rate In addition, we also highlight the case where the incorrect growth rate is always higher than or lower than the real growth rate Numerical examples are used to illustrate that our models are correct and provide errors due to cross-referencing • We provide a way to apply our model in practice for two scenarios: valuers directly estimate earnings growth rates, and clients provide future earnings and related information • The research also shows the indirect role of the timing of reinvestment in fundamental growth Our results imply the principle of “impossible quaternity” in earnings growth rates of the DCF framework More precisely, it is impossible for a business valuation to have an available-expected growth rate, a fixed change in ROIC, independence in the timing of reinvestment, and a fixed level of actual reinvestment 1.3.2 Practical contributions Tax distortions have always been ignored because of the lack of the appropriate valuation models, and thus, the asset value has not been accepted by clients sometimes Therefore, the thesis also makes some contributions to valuation activity in practice From a general perspective of the dissertation: • For the development of valuation in the future, "Valuation is a science and an art " is the famous concept of Prof Lim Lan Yuan—Chairman of World Association of Valuation Organizations, WAVO In which, art is valuers’ abilities to give professional judgments Art exists due to not only the characteristics of valuation but also the lack of theories, models, or empirical evidence because the reality is usually diverse Therefore, the highest practical significance of the thesis is the contribution to a partial shift from art to science, and thus, over time, art will be due to only the characteristics of the valuation industry • For certified public valuers, the thesis also contributes to the current understanding of valuers to analyze and select appropriate viewpoints in the whole valuation process The study develops an extended model of business valuation by quantifying the role of macro variables including inflation and tax Thus, practitioners are able to apply them when valuing enterprises in the inflationary environment • For international valuation associations, the results of this study are also the background so that they (i.e., AVA, WAVO, IVS, RICS) and policy makers can consider to adjust current valuation models in the near future, especially AVA that has a project to provide a new version of the valuation standards system for ASEAN countries • For Vietnam Ministry of Finance, business valuation is the most difficult field because it requires valuers to have advanced knowledge related to macroeconomics, valuation, auditing, finance, and statistics (Damodaran, 2012) Hence, although valuation was established in Vietnam from 1997, the business valuation standard no 12 (Circular No 122/2017/TT-BTC) was only introduced in 2017 and became effective from 01 January 2018 This standard, however, shows many disadvantages and this leads to the valuation results to be inappropriate Most importantly, on 14 February 2020, Vietnam Ministry of Finance issued the official dispatch No 29/QLG-QLTDG to receive the comments related to the business valuation standard no 12 to adjust this standard, after only two years of validity The new version of the business valuation standard No 12 (Circular No 28/2021/TT-BTC) has just introduced in 2021 and became effective from 01 July 2021 This argues that business valuation is the complex field and the thesis is also the new document to support policy makers to revise the new standard after the year 2024 (after the law of price is to be revised) By reviewing literature and international valuation standards (IVSs), as well as discussing with certified public valuers working at Big-5 valuation firms (i.e., CBRE, Savills, Colliers, JLL, and Knight Frank), the research is the first document adjusting business valuation models professionally Hence, each study of the thesis makes the following practical contribution For the first study: Registered valuers can understand a general picture about the consistent valuation when they apply the CoC and APV methods in business valuation Besides, the assumptions in the valuation report will be also fully identified and presented to meet the valuation standards when one or more of the items in the consistent equation can be impossible in practice For the second study: Through the results of this paper, valuers can estimate the true value of a business in two ways First, valuers directly apply the ET R developed in this study to current business valuation models The second way is that practitioners still estimate firm value using conventional models with ST R Subsequently, they used the general formula in this study to calculate the value of inflation-related tax distortions independently The true value is equal to the result from the first step minus that from the second step For the third study: This study helps valuers avoid valuation errors when estimating growth rates We also provide a guideline for valuers to apply the modified equations for the two most popular situations regarding earnings growth to ensure consistency in business valuation models 1.3.3 Educational contributions Many universities in the world provide business valuation as a subject (e.g., Harvard Extension School, National University of Singapore (NUS), London Business School (LBS), the Wharton School) or a major (e.g., Curtin University, Lincoln University) Besides, business valuation is also a professional certificate (e.g., Valuer Certificate of RICS, Business Valuation Certificate of ACCA) or a topic area of professional qualifications (e.g., certificate of CFA) More importantly, nowadays, applying and developing valuation models have become increasingly popular in valuation perspectives and other fields of the economy As a result, the educational system needs to change to meet these practices The thesis makes the following educational contributions • The research findings contribute to educational materials in the valuation field so that lecturers, valuers, and learners can use them in teaching and learning • Textbooks of professional certificates (e.g., RICS, CFA, ACCA, AICPA) are usually updated and upgraded with new knowledge, especially new models, every year; hence, the research contributes to these materials 12 Next, we show that the explicit reference to V ET S and V EBC in Eq (3.1) can be eliminated with a bit of ΥTt S ΥBC t St V EBCt BC additional work Towards that end, we define Tt S = VDET = + and = = + T S BC t Dt Φ tΦ κT S Dt κBC Dt T S,D t BC,D t T S,Υ t BC,Υ t The equivalence between κE and other costs of capital shown in Eq (3.1) is generalized by the following expression: Eu κE − t =κ Dt Et TS TS t Φ κEu − κT S + Dt Et BC BC t Φ κEu − κBC + Dt Eu κ − κDp Et (3.2) The formula that shows the relation between κA and κEu is one of the most vital equations in valuation (Cooper and Nyborg, 2008) From Eq (3.2), the equivalence between κA and other costs of capital is Eu κA − ftD t =κ TS TS t Φ κEu − κT S + ftD BC BC t Φ κEu − κBC − ftD ΦT S − ΦBC (3.3) Eqs (3.2) and (3.3) are the consistent valuations between the CoC and APV methods in a world without PIT Precisely, they capture (i) the trade-off between benefits and costs due to debt financing (through ΦT S vs ΦBC ) and (ii) the time-varying debt (through (.),Υ t (.) t ) in 3.2 CoC–APV equivalence with personal income taxes 3.2.1 VETS with personal income taxes Throughout this section, we denote notations considering PIT by a "tilde", (.) We define τ as the equivalent tax rate to capture this tax integration: τ = − (1−τcs )(1−τpe ) (1−τpd ) Hence, V ET S in a world with PIT is given by the following expression: ∞ V ET S t = Dt+i−1 κDp τ i i=1 (1 + κT S ) (3.4) 3.2.2 CoC–APV equivalence with personal income taxes Kolari (2018) argues that distinguishing between gross and net tax shields arising from tax deductions of interest payment is vital to business valuation Hence, the net tax shields of using $D of debt rather than equity at time t + i, T S t+i = Dt+i−1 κDp [(1 − τpd ) − (1 − τcs )(1 − τpe )], can be given by the following expression: e d T S t+i T S t+i T S t+i = Dt+i−1 κDp [τcs (1 − τpe ) + τpe ] − Dt+i−1 κDp τpd , e (3.5) d where T S t+i is gross tax shields available to the levered shareholders, T S t+i is PIT paid by debt holders, the e d e d net tax shields are the difference between T S t+i and T S t+i (i.e., N etT S = T S t+i − T S t+i ) VETS with integration between CIT and PIT at time t can be rewritten as d e T S t+i T S t+i ∞ V ET S t = i=1 Dt+i−1 κDp [τcs (1 − τpe ) + τpe ] (1 + κT S,e (1 − τpe )) e i ∞ − i=1 Dt+i−1 κDp τpd (1 + κT S,d (1 − τpd )) i (3.6) d V ET S t V ET S t Similarly, the value of expected bankruptcy costs with integration between CIT and PIT at time t can be 13 rewritten as [I] [D] BC t+i ∞ V EBC t = ρt i=1 BC t+i ∞ Dt+i−1 φ(1 − τpe ) κBC,e (1 (1 + − τpe )) i Dt+i−1 ψ(1 − τpe ) + ρt i=1 (1 + κBC,e (1 − τpe )) [I] i (3.7) [D] V EBC t V EBC t Eq.(3.7) parallels Eq.(2.9), but considers the CIT–PIT integration rather than only CIT We introduce some additional notations: ΥTt S,e , ΥTt S,d , ΥBC,e , ΦT S,e , ΦT S,d , and ΦBC,e t Eqs (3.6) and (3.7) can be rewritten as the following system of equations: Dt ΦT S,e Dt ΦT S,d + ΥTt S,e ΦT S,e − + ΥTt S,d ΦT S,d , T S,e κ κT S,d V ET S t = (3.8) e V ET S t d V ET S t and Dt ΦBC,e ΦBC,e + ΥBC,e t κBC,e V EBC t = Towards that end, we define T S,e t e = V ET S t Dt ΦT S,e = κT S,e + ΥTt S,e , Dt T S,e,D t BC,e t = V EBC t Dt ΦBC,e = + κBC,e BC,e,D t ΥBC,e t Dt T S,d t (3.9) d = V ET S t Dt ΦT S,d = κT S,d + T S,d,D t T S,e,Υ t ΥTt S,d , and Dt T S,d,Υ t BC,e,Υ t With time-varying debt and the APV method considering both V ET S and V EBC in a world with the CIT–PIT integration, as well as V ET S following two different types of discount rates due to PIT consideration, the equivalence between κE and other costs of capital is generalized by the following expression: Eu κE − t =κ Dt + Et Dt Et T S,e T S,e Φ t BC,e BC,e Φ t Eu κ κEu − κT S,e + BC,e −κ Dt Et T S,d T S,d Φ t κEu − κT S,d (3.10) Dt Eu + κ − κDp Et From Eq (3.10), the equivalence between κA and other costs of capital is [after some algebra in a way that similar to Eq (3.3)] Eu κA − ftD t =κ + ftD T S,e T S,e Φ t BC,e BC,e Φ t κEu − κT S,e + ftD T S,d T S,d Φ t κEu − κT S,d κEu − κBC,e − ftD ΦT S,e − ΦT S,d − ΦBC,e (3.11) Eqs (3.10) and (3.11) are the consistent valuation between the CoC and APV methods at the highest level of the generalization Precisely, they capture (i) the trade-off between benefits and costs due to debt financing (through ΦT S,e vs ΦBC,e and ΦT S,d vs ΦBC,e ), (ii) the trade-off between CIT and PIT (through τcs vs τpe , τpd and the after-PIT discount rates, i.e., κEu , κDp , κT S,e , κT S,d , and κBC,e ), (iii) the time-varying debt (through (.),Υ in (.) ), and (iv) the distinguishing between gross and net tax shields (through ΦT S,e vs ΦT S,d ) 3.3 Comparison to standard formulas Table 3.1 compares the consistent valuation developed in this study to standard equivalence 14 3.4 Concluding remarks This study presents a general framework for a consistent valuation between the CoC and APV methods, under the stochastically variable debt Based on the development of the APV method in recent years, the general scenarios of the CoC–APV equivalence were derived This study is vital, because the APV method has significantly changed while the latest versions of the CoC–APV equivalence have only captured the older scenarios of the APV The study’s main contribution is the CoC–APV equivalence at the highest level of the generalization Specifically, formulas allow for both PIT (i.e., the trade-off between CIT and PIT) and bankruptcy costs (i.e., the trade-off between tax benefits and bankruptcy costs) Thus, there are two groups of scenarios provided in the study The first is the CoC–APV equivalence with the APV method that considers bankruptcy costs under an environment without PIT The second is the consistent valuation among them, in the presence of PITs Following Kolari (2018) analysis, we distinguish between gross and net tax shields to capture the generalization for tax-shield formulas when PITs are introduced For the value of the expected costs of bankruptcy, we employ the risk probability of default and the present value of bankruptcy costs (see Damodaran, 2012), to ensure that it is possible for valuers to practically apply our formulas The study also shows that the indirect bankruptcy costs due to the excess promised yield, should not follow the idea of tax benefits while distinguishing between gross and net excess rates Thus, it is possible to collapse the general models in this study into standard formulas This argues that the CoC–APV equivalence developed in this study, is consistent with previous classical equivalences when adopting their respective assumptions The consistent formulas between the CoC and APV methods developed in this study show (i) the trade-off between benefits and costs due to debt financing through ΦT S,e vs ΦBC,e and ΦT S,d vs ΦBC,e ; (ii) the trade-off between CIT and PIT through τcs vs τpe , τpd , as well as the after-PIT discount rates, i.e., κEu , κDp , κT S,e , κT S,d , and κBC,e ; (iii) the time-varying debt (through Υ(.) and gross and net tax shields through Φ T S,e vs Φ T S,d (.),Υ in (.) ; and (iv) the distinguishing between Therefore, registered valuers can understand a general picture about the consistent valuation when they apply the CoC and APV methods in business valuation Besides, the assumptions in the valuation report will be also fully identified and presented to meet the valuation standards when one or more of the items in the consistent equation can be impossible in practice 15 Chapter Macroeconomic contexts and business valuation models: Inflation-related tax distortions 4.1 Inflation-related tax distortions The relation between EBT A and EBT T is EBT A = EBT T + β + α (4.1) where EBT A and EBT T are the reported accounting income from using financial accounting principles and the taxable income from using tax accounting principles, respectively; α and β are permanent differences and temporary differences between financial accounting and tax accounting principles, respectively This study considers GAAP ETR as τce is more appropriate because our study focuses completely on analyzing valuation models rather than providing empirical evidence Then τce is τce EAT A EBT T τcs + βτcs ατcs =1− = = τcs − A EBT EBT A EBT A =τ cs ⇒ τce > τcs < τ cs We use the superscripts and FV NB and RB α = 0, ∀β α < 0, ∀β (4.2) α > 0, ∀β to indicate the nominal and real tax bases for (.)T , respectively; HC indicate the historical-cost- and fair-value-based accounting for (.)A , respectively Let η = be the measurement bases based on current value accounting (i.e., fair value), and η = be the measurement bases based on historical cost accounting Let ϑ = be the nominal tax basis code and ϑ = be the real tax basis code The following general ET R formula, which is valid for fair-value- and historical-cost-based accounting under nominal and real tax basis, is τce = τcs + (1 − η)(1 − ϑ) ατcs ατcs − ηϑ , EBT A,HC EBT A,F V τ IT D where τ IT D denotes inflation-related tax distortions, which is transformed and shown in tax rates 16 (4.3) 4.2 Tax distortions and firm value: No personal income taxes Denote inflation-related tax distortions in F CF F by IT D F CF F in the case of tax distortions, E[F CF F ], is t t (1 + gιEBIT )(1 − τcs )(1 − RIRFt ) − EBIT0 E[F CF Ft ] = EBIT0 ι=1 (1 + gιEBIT )(1 − RIRFt )τtIT D (4.4) ι=1 F CF Ft IT Dt kt is the percentage of IT Dt to F CF Ft , and we have kt = τtIT D 1−τcs and IT Dt = F CF Ft × kt The true value of the unlevered firm can be rewritten as ∞ E[VtAu ] = i=1 F CF Ft+i (1 + i κEu ) ∞ − kt+i F CF Ft+i i i=1 VtAu (1 + κEu ) (4.5) VtAu,IT D Considering inflation-related tax distortions, the bias in Υ(.) (i.e., Υ(.),IT D ) is ∞ (.),IT D Υt D ft+i−1 = t+i−1 (kj ReInvFj ) j=t+1 , (1 + κ(.) )i i=2 (4.6) The true values of V ET S and V EBC are the following system of equations: D BC E[V ET St ] = V ET St − ΥTt S,IT D ΦT S and E[V EBCt ] = V EBCt − ΥBC,IT Φ t V ET StIT D V EBCtIT D Hence, the true intrinsic value of the levered firm is E[VtA ] = VtAu + V ET St − V EBCt − (VtAu,IT D + V ET StIT D − V EBCtIT D ), (4.7) V IT Dt VtA where the second term on the right hand side is the value of inflation-related tax distortions at time t, V IT Dt This value can be rewritten as D V IT Dt = VtAu,IT D + ΥIT ΦT S − ΦBC t (4.8) 4.3 Tax distortions and firm value with personal income taxes Similar to Eqs (4.4), (4.5), and (4.6): t (1 + gιEBIT )(1 − τpe )(1 − RIRF t )τtIT D , E[F CF F t ] = F CF F t − EBIT0 (4.9) ι=1 IT D t ∞ E[VtAu ] = i=1 F CF F t+i (1 + i κEu ) ∞ − i=1 kt+i F CF F t+i (1 + κEu ) VtAu,IT D VtAu 17 i , (4.10) and ∞ (.),IT D Υt D ft+i−1 = t+i−1 (kj ReInvF j ) j=t+1 , i + κ(.) i=2 (4.11) The true value of the V ET S and V EBC is the following system of equations: D BC,e E[V ET S t ] = V ET S t − ΥTt S,e,IT D ΦT S,e − ΥTt S,d,IT D ΦT S,d and E[V EBC t ] = V EBC t − ΥBC,e,IT Φ t e,IT D V ET S t IT D V EBC t d,IT D −V ET S t Hence, the true intrinsic value of the levered firm in an environment with both CIT and PIT is e,IT D E[VtA ] = VtAu + V ET S t − V EBC t − VtAu,IT D + V ET S t VtA d,IT D − V ET S t IT D − V EBC t (4.12) V IT D t The second term on the right-hand side of Eq (4.12)—the value of inflation-related tax distortions at time t in the presence of PIT, V IT Dt —can be rewritten as D BC,e V IT Dt = VtAu,IT D + ΥTt S,e,IT D ΦT S,e − ΥTt S,d,IT D ΦT S,d − ΥBC,e,IT Φ t e,IT D d,IT D V ET S t V ET S t (4.13) IT D V EBC t 4.4 Concluding remarks We developed a general formula for effective tax rates in the presence of inflation In doing so, we provide a new indicator—the percentage of tax rates due to inflation-related tax distortion (i.e., τ IT D )—to show the difference between ETR (i.e., τce ) and STR (i.e., τcs ) The current CIT expense occupies just one line item on the statement of profit or loss and other comprehensive income; however, it is a unique line item following tax rules without financial reporting rules Thus, what makes the formula useful is its ease of use, and it analyzes tax distortions through tax rates rather than independent components in cash flows to avoid confusion between tax and financial accounting treatments The developed ET R formula helps advance valuers’ understanding of the distinction between tax and accounting purposes in business valuation In addition, it allows integration between different tax bases and the measurement bases of accounting The general formula can be employed for four cases: (i) historical-cost-based accounting under nominal tax basis, (ii) fair-value-based accounting under a nominal tax basis, (iii) historicalcost-based accounting under a real tax basis, and (iv) fair-value-based accounting under a real tax basis The value of inflation-related tax distortions is the difference between the true value estimated when valuers employ ET R and firm value in which the tax rate is ST R We have shown these distortions in F CF F and firm value through τ IT D and the k factor for two scenarios, with and without personal income tax, if enterprise value is correctly estimated, compared to valuers ignoring the ET R Thus, our analysis provides a general formula for estimating the value of inflation-related tax distortions independently As a result, valuers can estimate the true value of a business in two ways First, valuers directly apply the ET R developed in this study to current business valuation models The second way is that practitioners still estimate firm value using conventional models with ST R Subsequently, they used the general formula in this study to calculate the value of inflation-related tax distortions independently The true value is equal to the result from the first step minus that from the second step In the context of a numerical example, we illustrate that the valuation errors due to the lack of consideration of tax distortions are exactly equal to the value of inflation-related tax distortions estimated from our formula 18 Chapter Earnings growth rates in business valuation models: Impossible quaternity 5.1 Formulas for earnings growth rates An alternative approach to the estimation of earnings growth rates is based on the fundamental characteristics of earnings of a firm Under the assumption that capital is reinvested at the end of the year: gtE = RIRt−1 × ROICt + ROICt ROICt−1 (5.1) 5.2 Analysis ReInvt Et (1−τcs ) t (1−τcs ) Cap—ROICt = ECap t−1 RIR is defined as the ratio of reinvestment, ReInv, to earnings after taxes, E(1 − τcs )—RIRt = ROIC is measured as the ratio of earnings after taxes, E(1 − τcs ), to invested capital, (for simplicity, we assume that ReInv is reinvested at the end of the year) gtE can be rewritten as follows: gtE = Et (1 − τcs ) Et (1 − τcs ) Capt−2 ReInvt−1 × −1 , + × Et−1 (1 − τcs ) Capt−1 Capt−1 Et−1 (1 − τcs ) (5.2) Et (1 − τcs ) ReInvt−1 + Capt−2 × − Et−1 (1 − τcs ) Capt−1 (5.3) or gtE = Eq (5.1) is still correct in practical if practitioners assume that capital is reinvested at the end of the year For RIR, both the numerator (i.e., ReInv) and the denominator (i.e., E(1 − τcs )) capture transactions over the entire year In contrast, for ROIC, the numerator (i.e., E(1 − τcs )) is the entire year’s data, whereas the denominator (i.e., Cap) captures only a snapshot in time To handle this drawback in ROIC estimation, valuers usually apply the idea of the financial aspect by considering the average capital to reflect the period of the year Therefore, we rewrite ROIC without the assumption of capital being reinvested at the end of the year: ROICt = Et (1 − τcs ) Et (1 − τcs ) Et (1 − τcs ) = = , µt Capt−1 + νt Capt µt Capt−1 + νt (Capt−1 + ReInvt ) Capt−1 + νt ReInvt (5.4) where µt is the number of days from January 01 to the reinvested day in year t divided by 365 in year t, νt is the number of days from the reinvested day in year t to 31 December divided by 365 in year t The condition for µt and νt is µt + νt = Equations for RIR and ROIC above are always correct, regardless of whether capital is assumed to be invested at the end of the year, if RIR and ROIC are estimated and analyzed independently Eq (5.4) implies 19 that (i) νt (%) of reinvestment in year t (i.e., νt ReInvt ) contributes to the growth rate in year t and (ii) the rest (i.e., (1 − νt )ReInvt or µt ReInvt ) pushes earnings in year t + because it will be used to estimate ROICt+1 As a result, we have gtE = Et (1 − τcs ) Et (1 − τcs ) Capt−2 + νt−1 ReInvt−1 ReInvt−1 × + × − (5.5) Et−1 (1 − τcs ) Capt−1 + νt ReInvt Capt−1 + νt ReInvt Et−1 (1 − τcs ) Applying Eq (5.5) for estimation of growth rate in year t + 1, we have E gt+1 = ReInvt Et+1 (1 − τcs ) Et+1 (1 − τcs ) Capt−1 + νt ReInvt × + × −1 Et (1 − τcs ) Capt + νt+1 ReInvt+1 Capt + νt+1 ReInvt+1 Et (1 − τcs ) (5.6) E To avoid the overlap in gt+1 , the reinvestment in RIR in the first term on the left-hand side of Eq (5.6) must not include νt (%) of reinvestment in year t (i.e., νt ReInvt ) because νt ReInvt is used for the estimation of growth rate in year t (cf Eq (5.5)), and must include νt+1 (%) of reinvestment in year t + (i.e., νt+1 ReInvt+1 ) Thus, E the reinvestment in RIR to estimate gt+1 is ReInvt − νt ReInvt + νt+1 ReInvt+1 (or µt ReInvt + νt+1 ReInvt+1 ) instead of ReInvt , as shown in Eq (5.6) Figure 5.1: Correct reinvestment contributes to earnings growth 5.3 Correction 5.3.1 Correct formulas First, if valuers apply 100% of reinvestment this year to estimate the growth rate in the next year, they must assume that total reinvestment is at the end of this year This leads to νt = 0% for all t Eqs (5.5) and (5.6) are generally modified as1 ReInvt ReInvt−1 Capt−2 Et (1 − τcs ) Et (1 − τcs ) + × × − gtE = Et−1 (1 − τcs ) Capt−2 + ReInvt−1 Capt−2 + ReInvt−1 Et−1 (1 − τcs ) Capt−1 ReInvt Capt−1 gtE,RIR (5.7) ReInvt gtE, ROIC Second, if valuers argue that the assumption in the first scenario is inappropriate, Eqs (5.5) and (5.6) are Because gtE = ReInvt−1 Et (1 − τcs ) Et (1 − τcs ) Capt−2 × + × −1 Et−1 (1 − τcs ) Capt−1 Capt−1 Et−1 (1 − τcs ) 20 generally modified as2 ReInvt µt−1 ReInvt−1 + νt ReInvt Et (1 − τcs ) gtE = × Et−1 (1 − τcs ) Capt−2 + νt−1 ReInvt−1 + µt−1 ReInvt−1 + νt ReInvt Capt−1 ReInvt (5.8) gtE,RIR + Et (1 − τcs ) Capt−2 + νt−1 ReInvt−1 × −1 Capt−1 + νt ReInvt Et−1 (1 − τcs ) gtE, ROIC Similar to the first case, gtE in Eq (5.8) also includes two fundamental determinants, gtE,RIR and gtE, ROIC ReInvt is a unique difference in the nature of Eq (5.8) compared to Eq (5.7) In this situation, ReInvt = µt−1 ReInvt−1 + νt ReInvt is the reinvestment for gtE from the valuation aspect with ReInvt and ReInvt−1 , which is known as the accounting perspective µt−1 % (or (1 − νt−1 )%) of ReInvt−1 is the component that has E not yet been estimated for gt−1 and is forwarded to year t The difference in ReInvt leads to a difference in Capt−1 and Capt in Eq (5.8) compared to Eq (5.7) Specifically, Capt−1 is Capt−2 + νt−1 ReInvt−1 instead of only Capt−2 In addition, the term Capt−2 + νt−1 ReInvt−1 + µt−1 ReInvt−1 (i.e., Capt−1 + µt−1 ReInvt−1 ) is also known as Capt−1 from the accounting aspect 5.3.2 Impossible quaternity We find the principle of impossible quaternity for estimating earnings growth Theoretically, all formulas of Et (1 − τcs ) − Hence, it is impossible for a business fundamental growth are the results obtained from gtE = Et−1 (1 − τcs ) valuation to have an available-expected growth rate, a fixed change in ROIC, independence in the timing of reinvestment, and a fixed level of actual reinvestment We call this fact the "impossible quaternity" illustrated in Panel B of Figure 5.2 Valuers must select two adjacent sides of the rhombus, giving up the institutional features at the opposite corner Figure 5.2: Impossible quaternity of earnings growth rates 5.3.3 Comparison between incorrect growth and actual growth Figure 5.3 shows errors between growth rates estimated from Eq (5.5) (or Eq (5.6)) and correct growth rates calculated from Eq (5.7) and Eq (5.8) This figure also shows the negative relationship between incorrect Because gtE = µt−1 ReInvt−1 + νt ReInvt Et (1 − τcs ) × Et−1 (1 − τcs ) Capt−1 + νt ReInvt + Et (1 − τcs ) Capt−2 + νt−1 ReInvt−1 × −1 Capt−1 + νt ReInvt Et−1 (1 − τcs ) 21 growth and different levels of νt 5.4 Simulation analysis Panel A of Figure 5.4 presents the relationship between earnings growth rates and different levels of ν in year t for different levels of µ in year t − for the target firm Similarly, Panel B shows the association between the growth rate bias and different levels of ν in year t 5.5 A practitioners guide To avoid valuation errors, as mentioned in Section 5.2, Eq (5.1) can be rewritten as gtE = RIRt × ROICt + ROICt ROICt−1 (5.9) where ()t is defined as the indicators to estimate gtE Scenario A: Valuers directly estimate earnings growth rates Valuers then apply Eq (5.10), rewritten formulas from Eq (5.7) or Eq (5.8), respectively We have ReInvt ReInvt µ ReInv ReInv ROICt ROICt t−1 t−1 t−1 + νt ReInvt × ROICt + = × ROICt + gtE = ROICt−1 Et−1 (1 − τcs ) ROICt−1 Et−1 (1 − τcs ) gtE, ROIC gtE, gtE,RIR ROIC gtE,RIR (5.10) The whole procedure implies that valuers employ the (B)-(C) sides of the rhombus in the “impossible quaternity." Scenario B: Clients provide future earnings and related information Scenario A uses reinvestment rates and changes in return on invested capital—two growth fundamentals—to calculate earnings growth rates In contrast, scenario B adopts the expected growth provided by clients to 22 estimate the reinvestment rates Although earnings growth rates are available, valuers still employ earnings growth equations to ensure consistency between reinvestment rates and growth rates Thus, we can restate Eqs (5.7) and (5.8) as follows: ReInvt ReInvt−1 Et (1 − τcs ) Et (1 − τcs ) + , × − gtE = × Et−1 (1 − τcs ) Capt−2 + ReInvt−1 Capt−2 + ReInvt−1 ROICt−1 Capt−1 ReInvt Capt−1 gtE,RIR (5.11) ReInvt gtE, ROIC and ReInvt µ ReInv Et (1 − τcs ) Et (1 − τcs ) t−1 t−1 + νt ReInvt −1 gtE = × × + Et−1 (1 − τcs ) Capt−1 + νt ReInvt Capt−1 + νt ReInvt ROICt−1 (5.12) Capt gtE, ROIC gtE,RIR The whole procedure in scenario B implies that valuers employ the (C)-(D) sides of the rhombus in the “impossible quaternity.” 5.6 Concluding remarks This study clarifies the estimation of earnings growth rates, the primary indicator of business valuation under the DCF technique Standard textbooks present three basic growth calculation methods for any firm, including historical growth, growth from analysts’ estimations, and fundamental growth Expected earnings depend naturally on reinvestment (i.e., RIR) and firm performance through changes in the return on invested capital (i.e., ROIC) All versions above are still correct in estimating growth rates However, errors in growth rates continue to persist if the valuers cross-reference across different ways in calculating RIR and ROIC Specifically, both the numerator and denominator for RIR capture transactions over the entire year Thus, valuers can use reinvestment and earnings in year t to estimate RIR in year t In contrast, the numerator for ROIC is the entire year’s data, while the denominator captures only a snapshot in time Hence, valuers tend to average invested capital to resolve this problem These approaches are correct, but they only correct if and only if the analysis of RIR and ROIC is independent A different reinvestment in RIR and ROIC from these two approaches leads to growth errors if these approaches are used simultaneously in growth estimation formulas We use a numerical example to illustrate that the standard formulas found in textbooks for earnings growth rates not yield correct growth, even if the proper estimation of RIR and ROIC is used We also derived the correct formula for earnings growth rates under two situations The first is that reinvestment is at the end of each year The second is that capital can be reinvested at any time The correct formula for estimating growth rates implies that when capital is reinvested, it is also the primary determinant in addition to reinvestment rates and changes in return on invested capital In addition, we provided guidelines for application in business valuation under the two scenarios First, we modify the correct formulas under the scenario that the valuers must directly forecast earnings Second, we provide alternative procedures when clients provide future earnings Most importantly, our results imply the principle called the “impossible quaternity” in earnings growth rates of the DCF framework It is impossible for a business valuation to have an available-expected growth rate, a fixed change in ROIC, independence in the timing of reinvestment, and a fixed level of actual reinvestment This study is vital because it contributes to business valuation in both literature and practice 23 Chapter General Conclusion The first paper worked with the consistent valuation between the CoC and APV methods The equivalent formulas developed in this study show (i) the trade-off between benefits and costs due to debt financing through ΦT S,e vs ΦBC,e and ΦT S,d vs ΦBC,e ; (ii) the trade-off between CIT and PIT through τcs vs τpe , τpd , as well as the after-PIT discount rates, i.e., κEu , κDp , κT S,e , κT S,d , and κBC,e ; (iii) the time-varying debt (through Υ(.) and (.),Υ in (.) ; and (iv) the distinguishing between gross and net tax shields through ΦT S,e vs ΦT S,d For the tax benefits aspect, the equivalence also captures the difference in the point of view of tax shields between stockholders and debt holders when PITs are introduced Finally, from a tax costs perspective, we use the risk probability of default and the present value of bankruptcy costs to estimate the value of expected costs because there is a large body of literature in calculating the risk probability of default and the present value of bankruptcy costs The second paper focuses on the tax perspective of business valuation models by analyzing tax distortions due to inflation We developed a general formula for ETR to capture inflation-related tax distortion by providing a new indicator—the percentage of tax rates due to inflation-related tax distortion (i.e., τ IT D )—to show the difference between ETR (i.e., τce ) and STR (i.e., τcs ) Tax distortions due to inflation (if available) exist only in CIT or CIT rates rather than other components of cash flows Therefore, we focus on ET R to avoid the confusion between tax and accounting perspectives when analyzing tax distortions because other components follow accounting rules The value of inflation-related tax distortions is the difference between the true value estimated when valuers employ ET R and the firm value in which the tax rate is ST R This study has shown these distortions in F CF F and firm value through τ IT D and the k factor for two scenarios, with and without personal income taxes, if enterprise value is correctly estimated, compared to valuers ignoring the ET R We also provide a general equation to estimate the value of inflation-related tax distortions independently Finally, this study presents the guideline for application to estimate the true value of a business The third paper focuses on the accounting perspective of business valuation models by clarifying growth rates of earnings The study shows that the current standard formula for growth rates is still correct, and the estimation of fundamental growth determinants is also valid if they are calculated independently However, this can lead to inconsistencies when all of them are used simultaneously in growth estimation formulas We illustrate that cross-reference in the calculation between RIR and ROIC leads to valuation errors The results show a negative linear relationship between the timing of reinvestment and the incorrect growth rate Specifically, when capital this year is reinvested closer to the end of the year, the annual growth rate due to miscalculations will decrease Besides, we also clarify the formulas of earnings growth rates to avoid mistakes under two situations: reinvestment is at the end of each year and reinvestment is at any time of each year Therefore, we provide a way to apply our model in practice for 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