Value and capacity of tax shields: An analysis of the slicing approach

Journal of Banking & Finance 35 (2011) 166–173

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Value and capacity of tax shields: An analysis of the slicing approach Howard Qi * School of Business and Economics, Michigan Tech University, Houghton, MI 49931, USA

a r t i c l e

i n f o

Article history: Received 1 October 2009 Accepted 20 July 2010 Available online 24 July 2010 JEL classification: G10 G12 G30

a b s t r a c t Correctly valuing tax shields has been a challenge in corporate valuation. A recent study by Liu (2009) introduces the slicing approach to separating the tax shield into the earned and unearned parts. Liu also shows that the MM results are wrong, and claims that the slicing approach has finally resolved the issue of pricing tax shields, thereby bringing closure to the topic. However, through careful analysis, we refute Liu’s main claims and restore the MM results. There are still open questions and the topic is not completely resolved as claimed in Liu (2009). Ó 2010 Elsevier B.V. All rights reserved.

Keywords: Capital structure Tax shields Discount rate

1. Introduction The value of tax shields from the use of debt is an important source for the value creation process in the corporate world. Accurately pricing the tax shields not only helps to better determine the optimal debt structure and make sound corporate strategic decisions, but it is also a vital component in maximizing firm value. However, it has been a puzzling issue that has existed since the seminal work by Modigliani and Miller (1958, 1963), herein after referred to as ‘‘MM”. As of today, there seems to be some consensus on how to value tax shields under some extremely special and simple circumstances. Once the situation becomes even somewhat realistic, controversies immediately emerge. See, for example, Fernández (2004, 2005, 2006), Cooper and Nyborg (2006, 2007), Ruback (2002), Booth (2002, 2007), Sabal (2005, 2007), Qi (2010), Pereiro (2002), Johnson and Qi (2008), and Graham (2000), among many, regarding how to value tax shields under a variety of circumstances. A recent surge in the debate over how the discount rate should be chosen for tax shields is a good demonstration of the liveliness and importance of the tax shield issue. Among many recent interesting works in this area of research is the study by Liu (2009) which proposes a novel theoretical framework – the slicing approach – to understand the nature of tax shields and their present value. In Liu (2009), not only a new theoretical approach is proposed to help us better understand the issue of the shields * Tel.: +1 906 487 3114; fax: +1 906 487 1863. E-mail address: [email protected] 0378-4266/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jbankfin.2010.07.030

and corporate valuation, but also the seminal results by MM (1958, 1963) are challenged on various fronts. If these are valid, it would be a major advancement in financial economics, which, as Liu claims, would put a finishing closure to this line of research and finally settle the issues in dealing with tax shields and their value. We thoroughly analyze some of the main claims made in Liu (2009) and demonstrate that they are not entirely valid. Specifically, the fundamental risk-return principle is violated, which is not easily seen on the surface. Earned and unearned tax shields, if these concepts are still valid, are mispriced. However, these concepts are cast in serious doubt once we pin down the central issue of what the feasible borrowing rate should be. We make our points in direct connection with a succinct derivation of the MM results. In this way, the issues are revealed more clearly. Not only are we able to discover the subtle errors made in Liu (2009), but we are also able to restore the MM results, and as a side effort show why MM (1963) replace their main results made in their earlier seminal work of MM (1958). We carry out our analysis as rigorously as possible and try to link the issues to their fundamental theoretical underpinnings. After all, we make some important and timely clarifications in the research of tax shield value. The rest of the paper is organized as follows. Section 2 introduces and analyzes the slicing approach. A variety of issues and claims made by Liu (2009) are discussed and shown to be incorrect or have serious flaws. We make our cases in close connection with Section 3 where we provide a succinct derivation of the main MM results. Section 4 concludes.

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2. Analyze the slicing approach Liu (2009) proposes that the present value of the tax shields sD should be considered to be the sum of the earned and unearned value of tax shields (VTS) as follows (Liu’s equation (5))

sD ¼

RD ROI

sD þ

ðROI  RD Þ ROI

sD ¼ Earned VTS þ Unearned VTS;

2.1. The definition of ROI The return on invested capital (ROI or ROIC) is not defined in Liu (2009), or if it is, it is not done clearly. The literature is not without discrepancies regarding this concept. For example, according to Brigham and Ehrhardt (2004, p. 108), ROIC is defined as

EBIT  ð1  sÞ ; Operating Capital

ð2Þ

where operating capital (OC) is defined as [Cash + Accounts Receivable + Inventories] + (Operating) Long-term Assets.2 The return on assets (ROA) is defined as

ROA ¼

Net income available to common stockholders : Total assets

ð3Þ

Obviously, ROI and ROA, defined above, are not the same concept. But according to Brealey et al. (2005, p. 794), ROI and ROA are the same and defined as

ROI ¼ ROA ¼

EBIT  tax ; V

ROI ¼ E



 EBIT  tax : V

ð5Þ

ð1Þ

where RD is the interest rate (i.e., cost of debt), s is the corporate tax rate, D is the debt (a fixed value), and ROI is the expected rate of return on investment. Liu calls the first term on the RHS of (1) the ‘‘earned tax shields,” and the second term the ‘‘unearned tax shields.” Following his argument, ðRD =ROIÞ  sD is the true value of the tax shields, ðROI  RD Þ=ROI  sD is the lost tax shields which could have been realized,1 and sD is the upper limit of the VTS and, he therefore argues, the capacity of the VTS. Eq. (1) is the central result of Liu (2009) and critically essential to his conclusions. Liu claims that the problem of the VTS is solved and settled for good by this slicing approach. However, the way (1) is derived, the underlying meaning and mechanism, the logic, the implication, among other results, are all questionable. Next we point out a few important flaws in Liu’s arguments and derivations.

ROICð¼ ROIÞ ¼

Based on the context, let us assume that (4) is closest to Liu’s ROI (and ROA) if not the same. We define the expected ROI (and ROA) by

We note that ROI is fairly unreliable because (1) it uses book values instead of market values for EBIT and firm value V, (2) the variables do not represent cash flows because noncash amounts such as depreciation are involved in EBIT and V, and (3) A hidden assumption is that the current year’s realized ROI is treated as the default proxy for ROI. Therefore, it is not difficult to see the various problems that would make the use of ROI potentially unreliable. 2.2. The criticism of the MM results Any new development in the pricing of tax shields cannot avoid a meaningful discussion of the seminal work by Modigliani and Miller (1958, 1963). Liu explains a few flaws in the MM framework and how his slicing approach resolves these flaws for good (2009). However, we find that the MM results are not accurately cited or correctly understood. It is essential to straighten out these important issues. 2.2.1. VTS capacity and confusion Liu cites the following two equations from MM (1963)4

r f sD ¼ V U þ sD; Eq: ð3Þ of MM ð1963Þ rf r f sD VL ¼ VU þ ; Eq: ð4Þ of MM ð1963Þ VL ¼ VU þ

q

ð7Þ

where q is the after-(corporate) tax asset return,5 or opportunity cost of capital. Liu uses them and Footnote 8 in MM (1963) to argue that Eq. (6) ‘‘gives an upper bound on the value of the tax shields.” Later, Liu (2009) argues that sD represents capacity, and rfsD/q or its variation, for example r f sD=ROI, represents earned tax shields. We feel some clarifications are needed here. In fact, MM (1963) intend to revise equations (11) and (12) of MM (1958) which were proposed without a rigorous proof. The main result from the original MM (1958), where corporate tax is considered, is MM Proposition I, which directly implies that

ð4Þ

where V is the total asset value; and tax is taxes paid by the levered firm, which already includes the reduction of the tax shield. We note that (2) and (4) are not the same unless the firm is all-equity financed and the firm’s assets are 100% operating capital.3 Obviously, these definitions are not unique and can be quite different depending on the context. Nonetheless, we note that these discrepancies are not in fact flaws. However, given their existence and the importance of the role that ROI plays in Liu (2009), a clear definition of ROI would be necessary.

ð6Þ

VL ¼ VU þ

r f sD

q

;

Eq: ð4Þ MM ð1963Þ

ð7Þ

and MM Proposition II is given below

RE ¼ q þ ðq  r f Þ

D ; E

Eq: ð12Þ MM ð1958Þ;

ð8Þ

where RE is the expected rate of return on equity, D is the debt value and E is the equity value. MM (1963) ‘‘correct” these two propositions by replacing them with Eqs. (3) and (12c) of MM (1963) as follows:

MM Proposition I :

V L ¼ V U þ sD;

ð6Þ

1

Liu (2009) does not explain how to earn it and why firms do not want to earn it. These two questions are essentially the same. 2 All these variables are given in the balance sheets. In order to compute OC, one must not count items that are not related to ‘‘operations.” Brigham and Ehrhardt (2004) list two common items that are considered operations-unrelated: firm’s shortterm investment and notes payable. However, whether this classification of operating assets is debatable or not is irrelevant here, because what is important is that at least the definitions must be clear within its own reference. We simply would like to point out that there may exist differences in how the terms are defined. 3 Some textbooks do not count notes payable and firm’s short-term investment as operating capital and make sure they are excluded in calculating some ratios such as ROI; see for example, Brigham and Ehrhardt, 2004. Other textbooks do not make this distinction; for example, Brealey et al., 2005.

MM Proposition II :

RE ¼ q þ ð1  sÞðq  rf Þ

D : E

ð9Þ

4 We note that these two equations are incorrectly cited in Liu (2009), i.e., the cited Eq. (3) of MM (1963) contains an extra term; and the cited Eq. (4) of MM (1963) is mistakenly labeled as Eq. (5) of MM (1963). We thank the anonymous referee for spotting these errors and bringing them to our attention. 5 In MM (1963), qs is used to denote asset return after corporate tax is accounted for. Here for simplicity, we use q as in Liu (2009).

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It has been the consensus in academia that (6) and (9), instead of (7) and (8), are the final and standard MM propositions.6 MM (1963) justify these replacements by showing that if (7) was in fact true, then investors could ‘‘secure a more efficient portfolio by switching from relatively overvalued to relatively undervalued firms.” MM recognize that, given their assumptions,7 (7) is not sustainable in equilibrium, and only (6) is self-consistent and holds true within their framework. However, MM (1963) just fall short of completely discrediting (7) (and (8)) by very vaguely claiming in their footnote 9 that (7) ‘‘is perhaps still of some interest if only if as a lower bound,” and the only (guessed) example they give for this claim is the finite horizon of debt. ‘‘Alternative versions of (3) (i.e., (6) here) can be readily developed for cases in which the debt is not assumed to be permanent, but rather to be outstanding only for some specific finite length of time. For reasons of space, we shall not pursue this line of inquiry here beyond observing that the shorter the debt period considered, the closer does the valuation formula approach our original (4) (i.e., (7) here).” In fact, this statement about (7) may deserve further investigation itself,8 but the point here is that MM clearly acknowledge that (6) is the only correct version for a level and perpetual debt of D. In Liu (2009), the citations of MM (1963) and specifically their footnote 8 are used to serve the argument that (6) is for VTS’s upper bound (i.e., capacity) and ‘‘MM confuse the VTS capacity with VTS.” However, this is done without adequately pointing out that MM (1963) recognize sD as the upper bound only in the sense that a lower VTS may exist as the result of the finite debt horizon, not because of anything else. The slicing approach in Liu (2009) that separates ‘‘the VTS capacity” of sD into earned and unearned parts is not related to ‘‘the VTS upper bound” (or capacity) in light of finite debt horizon as we explain above. The two capacities are not in fact the same. In the MM context, that upper bound (sD) must be upheld, otherwise it would not be arbitrage-free; in the Liu (2009) context, the capacity (sD) simply represents the possible maximum VTS value that is usually not and need not be maintained.9 This point must be made clear because not only should the citing be accurate, but most importantly, if one wants to argue against a theory, he or she should either attack the assumption(s), or demonstrate the result(s) that are inconsistent with the assumptions, or both. In Liu (2009), it seems that the MM result (i.e., (6)) is challenged simply by stating that it is not logical to replace VTS with the VTS capacity without showing (1) whether it is inconsistent with the MM assumptions, or (2) which specific MM assumptions should be replaced with more realistic assumptions, thereby resulting in a new framework such as that outlined in Liu (2009). Rigorously speaking, without pinpointing any error in the theory, it is not a convincing and valid argument against a theory to simply disbelieve that the result is logical. 6 However, we note that there could be a common confusion regarding MM Proposition II. Normally (8), instead of (9), is actually more commonly called the MM proposition II; see for example, Brealey et al. (2005). But this is inaccurate. It is true that when corporate tax rate is zero, (9) collapses to (8), but the original Proposition II (i.e., (8)) does consider non-zero corporate tax rate. In other words, both (8) and (9) are obtained when corporate tax is considered and they agree only when s is set to zero. Therefore, (8) and (9) are fundamentally different and shall not be treated as one being another’s simplified version, and the correct MM proposition II should be unambiguously (9). 7 For example, (1) safe stream of debt payments; (2) individuals and firms have the same borrowing and lending rate; (3) perfect market, etc. Later on, we will show that the crux of the tax shields issue is at what rate individuals and firms can borrow and lend. 8 It is not difficult to show that this ‘‘observation” by MM is too casual and may not be correct when the debt horizon is finite. 9 This is why VTS is always separated into earned and unearned parts in Liu (2009).

In summary, Liu’s claim that ‘‘MM confuse the VTS capacity with VTS,” and therefore MM Proposition I is wrong, is invalid in light of the problematic issues discussed above. In Section 3, we show in detail why MM Proposition I is indeed consistent with MM assumptions and the claimed confusion does not exist. 2.2.2. Is VTS independent of interest rate? Another claim made in Liu (2009) is that the MM result (i.e., (6)) is illogical because MM’s VTS is sD, which is not a function of cost of debt r (or rf when the debt is safe). Specifically, Liu argues that ‘‘Unfortunately, MM’s logic of capitalization is inconsistent with the definition of the tax shield because the IRS grants the tax shields on the basis of interest expenses rather than (the value of) debt. By definition, VTS is a function of interest rate no matter whether the income is sure or not as long as q > rf .” This is a misunderstanding of the tax shield RDDs over the time period (t  1, t] and the present value of all future tax shields, VTS, where we use RD to denote the cost of risky or risk-free debt. Once again, this further claim against the validity of MM’s VTS is not furnished with a detailed explanation of what is incorrect in MM’s framework. Rather, some casual proofs are given to support the claim. Next, we illustrate why these proofs are not necessarily valid arguments against MM’s result. First, numerical Cases 1 and 2 in Liu (2009) are used to argue that the VTS given by MM (i.e., MM Proposition I in (6)) cannot be true. In these two cases, EBIT = $10,000, ROI ¼ q ¼ 10%, s = 40%, D = $100,000, and only the borrowing rate on debt RD changes from 2% to 6%, which results in different annual tax payments of $3200 and $1600 (=[EBIT  RDD]  s). Yet, VTS = D  s = 100, 000  40% = 40,00010 remains unchanged for both cases, and Liu thereby argues that ‘‘Obviously, it is illogical for different tax shields to have the same VTS.” However, this reasoning is incomplete in that the present value of future cash flows does not only depend on cash flows, it also depends on the discount rate. Indeed, the annual tax shield increases from $800 (in Case 1 where RD = 2%) to $2400 (in Case 2 where RD = 6%). But a greater future cash flow does not necessarily mean a greater present value. The very reason that RD increases is because its associated cash flows ($2400) are not as attractive as those associated with the lower RD ($800) for some legitimate reasons. This is one of the most fundamental principles in financial economics – riskier (or more inferior) cash flows result in a greater discount rate. Furthermore, given the MM assumption of the same corporate and individual borrowing/lending rate, the proper discount rate is RD, otherwise an arbitrage profit can be created, which is explained in detail in Section 3. Taken together, contrary to the claim in Liu (2009), it would be illogical for Cases 1 and 2 to have different VTS values after RD changes. Finally, the basis of these two cases is questionable as well.11 MM characterize their bonds as offering sure cash flows. A firm that could do so and have a debt ratio of 100% is a firm with entirely risk-free cash flows, which means that the firm’s unlevered equity capitalization rate q must be equal to the risk-free rate rf. Thus, a case with 100% debt ratio, rf = 6% and q = 10% does not exit in the MM world. Second, Liu also uses another interesting and somewhat extreme numerical example (Case 3) to lay out further proof that VTS = Ds cannot be true. The example is a humanitarian loan ($10 Million) with zero interest. He argues that according to MM, VTS = Ds = 10,000,000  40% = $4 Million, which contradicts the fact that this zero-interest humanitarian loan provides no annual tax shields at all. In other words, how can many zeros add up to 10 11

We note that Liu (2009) mistakenly obtains VTS = D  s = $10,000  40% = $4000. We thank the referee for providing this insight.

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nonzero? Therefore Liu concludes that MM Proposition I must be incorrect. However, this argument is quite problematic in that, technically speaking, it creates a singularity situation where the discount rate is zero, and no meaningful result can be obtained under a 0/0 situation (see the following section for a discussion of another singularity case in Liu’s argument); rather, the final result of VTS = 0 should be quickly reached, since it is already given that the loan does not bring in any annual tax shields. In fact, this puzzle may be resolved more easily by recognizing that the MM framework uses market values.12 We now take an asymptotic approach to better illustrate why this humanitarian loan example does not pose any valid challenge to the MM framework. Suppose a humanitarian loan of $10 Million requires the firm to pay rH = 1% annual interest, risk-free, in perpetuity and the market risk-free rate is 5%. This implies that such a loan would have a market value of $2 Million. Since it is the market value that applies to the MM framework, MM’s VTS would be only $0.4 Million (Ds = $2 Million  40% = $0.4 Million) instead of $4 Million as claimed in Liu (2009). Now, it is easy to see why VTS decreases to zero when this logic is advanced by letting rH asymptotically approach zero. Once again, the puzzle is spurious and the MM results are restored. In summary, the confusion in Liu’s arguments comes from some misunderstandings about how VTS = Ds is derived, and the requirement of market value in particular. The casual and ‘‘intuitive” argument of what ‘‘it logically should be” ignores the true robustness of the MM framework as we have illustrated. In Section 3, we show in detail why VTS = Ds (with nonzero borrowing rate, RD > 0) must be true given MM’s assumptions. 2.2.3. The singularity case – 100% leverage Another claim made in Liu (2009) is that the MM result (i.e., (6)) is illogical because it leads to a paradox when the leverage approaches 100%. Liu argues that ‘‘If the MM VTS were true, all-debt financing (V = D) with risk-free rate could waive tax payment (first half), . . . , However, a firm of all-debt financing with risk-free rate produces the highest taxable income and has to pay the highest tax, higher than with any other interest rate (second half) . . . ” Thus, there appears to be a contradiction in MM’s VTS. But this argument is problematic in that it is built upon the well-known singularity within the MM framework, because once E/D ? 0, the risk in equity approaches infinity, hence the requirement for infinite returns. It is not convincing to use the singularity case to discredit the MM results because the MM results are not intended for extremely high leverages. This is because when leverage approaches 100%, an infinitesimal fluctuation in the cash flow could lead to bankruptcy. One of the key assumptions of the MM framework is ‘‘no bankruptcy.” Thus, the singularity case (i.e., E/D ? 0) is implicitly ruled out in the MM settings. However, even with the original MM assumptions, it is still possible to show that the above contradiction is not a real one. It is a technical issue rather than a theoretical truism. Notice that the first half, where tax payment is waived, is equivalent to assuming that every cent of cash flow must go to the debtholders without any surplus, as any amount exceeding the debt interest payment would create a tax liability. The second half, where the firm pays the highest tax with a risk-free rate, assumes that there exists a sure surplus after debt payment is fulfilled. Therefore, there is an implicit and critical difference in the assumptions that must be addressed beforehand; otherwise we can play it into different arguments as desired. This confusion is caused by the same problem 12 We thank the anonymous referee for making this important point and helping us with the following asymptotic argument which makes the humanitarian–loan puzzle easily resolved.

undermining the basis of the previous two cases where we demonstrate that ‘‘a case with 100% debt ratio, rf = 6% and q = 10% does not exit in the MM world.” Another way to see through this mind-boggling singularity dilemma is to realize that the very bottom of the paradox is how the cash flows are distributed (a fixed amount or 100%) and whether fluctuations in EBITt are allowed by the assumptions. If EBITt is constant for sure, then ‘‘100% debt financing with fixed debt interest” is feasible, which is essentially all-equity financing under a different name. Whether the government acknowledges this as a ‘‘legal name change” is a different and interesting issue.13 If EBITt is a risky cash flow with possible fluctuations, then ‘‘100% debt financing with fixed debt interest” is simply not feasible and is therefore ruled out by the assumptions due to an embedded inconsistency with the assumptions. This is because one must answer two questions: (1) where does the excess profit go after debtholders get their fixed interest payment, and (2) who is going to cover the shortfall if EBITt < the fixed debt interest? The singularity paradox originates from the question of whether we can satisfactorily settle these two questions a priori within our system of assumptions because it has to do with the feasibility of the cash flow allocation arrangement. Allowing fluctuating EBITt to co-exist a priori with the feasibility of having fixed debt interest with all-debt financing within our assumption framework is simply a wishful way of saying ‘‘let’s evaporate the risk by calling investors of an all-equity financed firm debtholders.” Indeed, if we ask a question inconsistent with our assumptions, we set ourselves into a logical trap.14 In summary, the MM Propositions are not discredited by this singularity paradox. More information may be needed once any singularity situation is entered. Indeed, the two contradicting situations (i.e., the first half and the second half of the cited paradox put forth by Liu) are not in fact contradictory at all in the sense that they are never simultaneously true, because they are based on two mutually exclusive conditions regarding how the cash flows are (feasibly) distributed (a fixed amount or 100% of EBITt which may fluctuate). 2.2.4. Slicing approach and violation of the risk-return principle Now that Liu’s inferences from paradoxical special cases have been debunked, the central result of (1), i.e., Eq. (5) in Liu (2009), deserves some technical analysis in its own right. We note that technically we can always have

sD ¼

RD

x

sD þ

ðx  RD Þ

x

sD;

ð10Þ

where x – 0. Therefore, deriving (1) is technically trivial. But calling the first term the present value of (earned) tax shields after the substitution of x ¼ ROI raises a fundamental technical question of whether the risk-return principle is violated. To see this problem, it is necessary to separate the cash flow of ROI  Ds into two streams – RDDs and ðROI  RD ÞDs. The first term, RDDs, is the tax shield over one specific time period, and ðROI  RD ÞDs, following Liu’s logic, is a cash flow that could have been generated but is not, hence it is termed unearned. In order to earn this unearned portion, the firm may pay off the existing debt and refinance with a new debt that is worth exactly the same D but with higher interest payments, i.e., ROI > RD . The question 13 By ‘‘acknowledge,” we refer to the tax treatment. It is interesting to view this problem in light of money laundering. For example, a similar (unethical and perhaps illegal as well) trick of tax evasion is to donate money to charities (who do not pay taxes) and then have them ‘‘invest” it back. Debt financing may be disguised as lease financing (and the IRS tries to sniff out non-genuine leases). The case of Enron using SPEs to hide off-balance sheet financing was overlooked. 14 To help better understanding this issue, we pose two more questions. First, what is the difference between an all-equity financed firm and an all-debt financed firm? Second, who is the owner of an all-debt financed firm?

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does not stop here – the question remains of what kind of debt this is. Since RD P r f where rf is the risk-free rate, we must have ROI > r f , the newly refinanced debt would be risky, and riskier (to debtholders) than the existing debt to justify ROI > RD . No firm would offer higher interest payments on identical debt terms. Thus, if the new debt calls for an annual interest payment of ROI  Dð> RD DÞ, there must be some additional flexibility given to the firm (i.e., equityholders) in the new equity-debt contract. There are an infinite number of ways to reallocate the total risk between equity and debt. For example, some feasible features, including callability, or terms similar to preferred stock, etc., would make the debt less attractive to debtholders. The heart of the problem is that there must be added flexibility on the side of equity and lost flexibility on the side of debtholders. This is why the new equity-debt contract can have ROI > RD . Ceteris paribus, the change in interest rate must come from an underlying transfer of flexibility in the contract covenants.15 Many studies in this line of research seem to neglect this fundamental issue as if one could simply assume any interest rate at will without the need to be consistent with its consequences and implications. Observing the necessary risk-and-return correspondence, some variations must be allowed in the new debt interest payment.16 Now, the closest situation imaginable that is similar to Liu’s setup is to have the same expected cash flows rather than an absolute constant, i.e., E½r D t D  ROI  D, where r D t D is the actual new debt interest payment over the time period of ðt  1; t, and E½RD t D  RD D, where RD t is the actual interest payment on the existing debt. Once again, these modifications are minimally necessary to reconcile with a debt interest rate greater than the risk-free rate. Therefore, the present values of the tax shields generated by these two kinds of debt can be determined by discounting them at the appropriate discount rate, say, x1 and x2 for the existing debt and the new debt, respectively. Then, following Liu (2009), we can find the earned tax shield and the former unearned tax shield as follows, 1 X RD t Ds PV 1 ¼ Earned tax shield ¼ t t¼1 ð1 þ x1 Þ 1 X ðr D t  RD t ÞDs PV 2 ¼ Unearned tax shield ¼ ð1 þ x2 Þt t¼1

ð11Þ

1 X E½RD t Ds RD Ds RD Ds – t ¼ x1 ROI t¼1 ð1 þ x1 Þ 1 X E½ðr D t  RD t ÞDs ðROI  RD ÞDs ðROI  RD ÞDs : ¼ – PV 2 ¼ x2 ð1 þ x2 Þt ROI t¼1

PV 1 ¼

ð12Þ Comparing (12) with (1), it is not difficult to observe that, in general, neither the earned nor the unearned tax shields, assuming these concepts are still valid, would be equal to those given by Liu (2009). As is now clear, the error in Liu’s determination of the present value of the earned and unearned VTS comes from the violation of the underlying risk-return principle in financial economics. Of course, assume other relevant factors are unchanged for simplicity. We note that a financial asset’s risk and return may be influenced by a variety of factors such as human psychology and January effect. A recent study by Moller and Zilca (2008) offers a detailed account of the evolution of the January effect. There is even debate that weather may play a role as well; see for example, Chang et al., 2008; Jacobsen and Marquering, 2009; Kamstra et al., 2009. However, in the current context, we restrict the concept of risk to sheer return variations that are represented by standard deviation in the traditional mean-variance framework. 16

s  ROI  D RD

¼ sD þ

RD ROI

þ

ROI  RD

!

ROI

ROI  RD sD; RD

Eq: ð10Þ of Liu ð2009Þ

ð13Þ

D ÞsD D ÞsD and claims that ðROIR and ðROIR imply ðROI  RD ÞsD is ‘‘doubleRD ROI counted with different capitalization rates.” We note that Eq. (13) per se is perfectly mathematically correct. However, we fail to see the meaning of it. We only make a few comments.

(1) The reason MM use RD(=rf) to discount their tax shields is because their debt and tax shields are fixed in the MM settings. (2) MM have never claimed that RD(=rf) is the only appropriate discount rate for the tax shields regardless of the nature of the debt. As we will show in Section 3, this discount rate depends on what borrowing/lending rate is available to the individual and the company. (3) On the right-hand-side of (13), using RD to discount cash flows whose risks do not match RD will produce a number but it is not a meaningful present value. Therefore, the so-called double counting problem with the MM approach is not a valid notion. Readers can check that it disappears as long we use the right discount rate that matches the risk in the cash flows. Liu also introduces three kinds of VTS values, defined as follows,

8 > < VTSWACC ¼ ðRD sDÞ=WACC VTSROI ¼ ðRD sDÞ=ROI : > : VTSROI ¼ ðR s DÞ=ROI D WACC

ð14Þ

There is some confusion regarding these definitions. First, once again, the risk-return principle seems to be unheeded. It must be explained why the same cash flows can be arbitrarily assigned with different discount rates. Liu (2009) fails to do so. 3. Deriving the MM formula and discussing its foundations

Since RD t D and ðr D t  RD t ÞDs will have different risks, these two cash flows correspond to different discount rates, that is x1 – x2, and the expected present values can be written as

15

2.2.5. Double counting of VTSMM and other minor comments Liu (2009) also derives the following equation using his ‘‘slicing approach on the basis of RD: MM,”

To facilitate a rigorous discussion, it is necessary to show how the MM results are obtained in a fairly succinct way. We assume that the market is arbitrage-free,17 without bankruptcy, the cost of debt to the firm is RD, the firm’s business profit is EBITt in year t, debt is D, equity is E, unlevered firm value is VU and levered is VL, and the corporate tax rate is s. If one pays E today for the levered firm’s equity, then in the tth period she or he will be entitled to

½EBIT t  RD D  ð1  sÞ ¼ fEBIT t  ð1  sÞg  fRD D  ð1  sÞg: ð15Þ We can replicate the above cash flow with a portfolio of the unlevered firm and a personal loan. The strategy involves buying the unlevered firm for VU and creating a personal loan that requires an annual interest payment of {RDD  (1  s)}. Table 1 lists the relevant cash flows associated with such a strategy. In order to perfectly match the cash flow from Strategy #1 in all periods of time, investors must create a personal loan that would 17 This is in some sense equivalent to the law of one price, a basic underlying assumption made in most asset pricing frameworks. Akram et al. (2009) provide evidence in support of this assumption in general but also document violations in international financial markets with durations high enough to make it worthwhile searching for one-way arbitrage opportunities in order to minimize borrowing costs.

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flows from buying the unlevered firm VU. Suppose the lending rate is RPersonal,Lending, following the previous logic, a feasible range for the value of tax shields may in fact be established,

Table 1 Two strategies offer identical future cash flows.

Strategy #1: Buy levered firm’s equity Strategy #2: Buy unlevered firm Borrow a loan (i.e., negative payment)

Today you pay

Collect in period t

E (=VL  D)

EBITt(1  s)  RDD(1  s)

VU

EBITt(1  s)

L = ?

RDD(1  s)

require a payment of RDD(1  s). The remaining question is over the present value of this loan, L. It must be determined how much in cash investors can get today by agreeing (legally binding) to make future interest payments of RDD(1  s). This is the most crucial and potentially controversial component. The answer to this question is that it depends on the lowest possible interest rate RPersonal that is available to the investors making the offer of future cash payments. We note that these future payments need not be constant or risk-free as long as the risk (or fluctuations in value) does not trigger bankruptcy, hence the nobankruptcy assumption is not violated.18 Suppose investors can finally take out such a loan with an interest rate of RPersonal, for simplicity assuming perpetuity of the expected cash flows,19 then this loan must be worth

L¼

RD  D  ð1  sÞ : RPersonal

ð16Þ

Because this loan is a cash inflow at t = 0, in terms of ‘‘pay,” it should be recognized as a negative value. Since these two strategies now offer identical future cash flows in all states of the world, i.e., this match is perfect and independent of probability, they must be worth the same today. Therefore we can establish the following equation,

V U þ ðLÞ ¼ E ¼ V L  D:

ð17Þ

Rearranging (17) and substituting (16) for L, we have

V L ¼ V U þ ½D  RD D  ð1  sÞ=RPersonal ;

ð18Þ

and the VTS is then defined as

  RD  ð1  sÞ : VTS ¼ V L  V U ¼ D 1  RPersonal

ð19Þ

However, the above arbitrage-free replicating argument is carried out in an ideal situation. In reality, the homemade portfolio (Strategy #2) cannot be better than the market-available product of Strategy #1, otherwise the levered firm as well as its equity E would be easily replaced by Strategy #2. This implies that

  RD  ð1  sÞ : VTS P D 1  RPersonal

     RD  ð1  sÞ RD  ð1  sÞ ; D 1 : VTS 2 D 1  RPersonal RPersonal;lending

In other words, VTS is not determined precisely. To pin down VTS and arrive at the MM result of VTSMM = Ds, it is easy to see from the above analysis that a necessary condition is RPersonal,Lending = RPersonal = RD.20 Indeed, a crucial assumption made in the MM framework is that individuals can borrow and lend at the same rate as the firm does, i.e., personal and corporate borrowing/lending rates are identical. As long as this assumption is valid, the present value of the tax shields is indeed VTSMM = Ds. This result is very robust. It is now clear that the MM result of VTS = Ds hinges on the assumption that investors can borrow and lend at the same rate as the firm’s cost of debt RD. Therefore, the heart of the problem really comes down to a simple yet fundamental topic: any new theory that refutes VTSMM = Ds or intends to improve the MM results must face and explain very clearly how the discount (or borrowing) rates are handled within its assumed environment. So far, there are basically two mainstream methods of settling the discount rates. The first assumes that debt D is fixed and perpetual, along with fixed corporate tax rate s and cost of debt RD, and the individuals and the firm can borrow or lend at the same rate of RD. The second deals with the situation where the firm keeps a constant leverage ratio by rebalancing its capital structure instantaneously, and RPersonal would be different from RD. These choices of the discount rate above are in fact based on the argument that the tax shield RDDs is in strict proportion to debt D. Therefore, if the fluctuation (or risk) in D per se requires a compensation of RD, then RDDs would have identical fluctuation (or risk) that warrants the same compensation of RD as well. When the leverage ratio is kept constant, there will be additional risk coming from future uncertain debt policies, to the extent that D as well as RDDs would have the same fluctuation (or risk) as that of the firm’s risky assets. If the assets’ risk warrants a compensation of q for the investors, then the annual tax shield RDDs warrants the same discount rate q as well. In essence, identical risk must correspond to identical compensation, i.e., the discount rate, regardless of the asset type – equity, debt, derivatives, or other asset. Therefore, assuming perpetual debt and full deductibility of tax shields, we have the following two scenarios. (1) For fixed perpetual debt,

MM Proposition I : V L ¼ V U þ sD

ð20Þ

Also notice that there exists a similar procedure but in the opposite direction – buying levered firm’s equity E and lending out a perpetual loan with an annual coupon of RDD(1  s) can match the cash

MM Proposition II : RE ¼ q þ ð1  sÞðq  RD Þ

Here we distinguish between bankruptcy risk from mild fluctuations in debt value due to market news or the nature of the debt contract. The reason we do not want to restrict to risk-free debt with constant coupons (as in the original MM framework) is because on the one hand, cash flow matching in the current replicating portfolio analysis does not require constant risk-free cash flows; all it requires is perfect matching. On the other hand, we want to allow for a range of possible discount rate RPersonal for the debt cash flows rather than being limited to the risk-free rate rf. There are many types of debt that can be risky (in terms of changing debt value or interest payments) but without causing bankruptcy, such as floating-rate perpetual debt, or renegotiable mortgages. 19 This is in line with the traditional studies in this line of research, such as MM (1958, 1963), Ruback (2002), Fernández (2004, 2005, 2006), Cooper and Nyborg (2006, 2007), Harris and Pringle (1985), Sabal (2005, 2007), etc.

ð6Þ D E

ð22Þ

(2) For firms with constant leverage,

VL ¼ VU þ 18

ð21Þ

RD D s

ð23Þ

q

RE ¼ q þ ðq  RD Þ

D E

ð24Þ

In light of (19)–(21), and matching the value of VTS, one can finally see the connection between the validity of the above four key relations and the implicit assumptions regarding the personal borrowing/lending rate. Next, we comment on these key results. 20 We thank the anonymous referee for pointing out this feasible range of VTS due to the possible difference between borrowing and lending rates at the personal level.

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First, we note that (22) is slightly different from the original MM Proposition II in that the debt need not be risk-free, which is a minor issue. Applying the capital asset pricing model (CAPM) to (22) and assuming the debt is safe would immediately lead to Hamada’s (1972) beta formula. The above results are frequently cited in academia with some minor modifications for certain special cases.21 Comparing (23) and (24) with (7) and (8) demonstrates a substantial similarity in form, however, the underlying assumptions and their meanings are very different to the extent that they should be considered theoretically different in all senses. Specifically, (23) and (24) relax the risk-free debt restriction; (8) is incorrect and MM replaced it with (9) (which is a special case of (22) in this paper) in their (1963) paper because they noticed an arbitrage opportunity implied by their original (7) and (8). In light of this study, it is easy in hindsight to see the origin of the arbitrage opportunity in their (1958) paper. Note that in MM (1958) there is no difference between the individual and corporate borrowing rates RD (=r in their notation). This implies RPersonal = RD in our derivation of the MM results with the portfolio-replicating strategy explained earlier. Therefore, an arbitrage-free requirement would immediately rule out their (1958) results (with corporate taxes) and arrive at their (1963) results of (6) and (22). When the personal borrowing rate is different, as in the case where leverage is continuously rebalanced to be constant, we get the different results seen in (23) and (24). The fundamental mechanism underlying these results is the same. Finally we note that the choice of the discount rate, RPersonal (or RPersonal,Lending), in determining VTS in (19) is largely based on arguments for certain special cases, i.e., fixed debt and constant leverage ratio. So far, there has been no rigorous theoretical framework for how to handle VTS, especially for more complicated cases. For example, when growth is introduced in assets and debt, controversies quickly emerge even for the simple case of a constant leverage ratio. We refrain from exploring these further subtleties in this research. For interested readers, we refer to studies such as Fernández (2004, 2005, 2006), Qi (2010), Cooper and Nyborg (2006, 2007), Ruback (2002), and Booth (2002, 2007), among many others.

4. Conclusions Correctly determining the value of tax shields is an ongoing effort ever since the seminal work by Modigliani and Miller more than half a century ago. One central issue is the discount rate; another is how to deal with growth in debt and firm value, which is related to the first issue. The debate has produced many intellectual efforts that have forged a deeper understanding of the issue. While substantial advances have been made, there are still many puzzles and open questions. One recent study by Liu (2009) introduces a novel theoretical framework which separates the tax shield into two parts – the earned and unearned tax shield, and calls it the ‘‘slicing approach.” Liu also shows that the MM results are wrong on many fronts. For example, besides many ‘‘illogical errors,” the MM results have a so-called ‘‘double counting problem.” These are serious charges. If valid, they would shake a major cornerstone of financial economics, as well as shed new light on one of the most puzzling issues in academic research. Indeed, Liu (2009) claims 21 For example, Miles and Ezzell (1980) point out that the tax shield for t = 1 is known because it depends on how much debt is borrowed at t = 0, the discount rate for the first tax shield should change to rf independent of capital structure choices. However, this argument is true only when we can absolutely rule out bankruptcy before or at t = 1. As a matter of fact, basically no one can forecast bankruptcy with 100% certainty in reality. Here we just want to give an example of an attempt to advance this line of research and do not intend to get into any details.

that his approach has finally resolved the issue of the present value of the tax shields, thereby bringing closure to the topic. However, we closely examine this slicing approach and discovered a few troubling inconsistencies and logical errors. We explain how a few of the errors in Liu (2009) occur. Specifically, we start with a discussion of the concept of ROI since it plays an important role in Liu’s theory. Then we carry out an in-depth analysis of a variety of concepts and claims made by Liu, including his criticism of the MM results. Our investigation shows that the new theory is incorrect, the main reason being the violation of the most fundament risk-return principle. To further reveal this violation and expound our points, we provide a succinct derivation of the MM results, which more clearly narrow down the issue. Using this study as a vehicle, we also help clarify the famous MM Propositions I and II, including a correction made by MM themselves. In summary, we refute the main claims made in the newlyproposed slicing approach; we explain why this approach is incorrect and why the criticisms of the MM results are not valid. Our study highlights the main challenges in the VTS research. There are still open questions concerning the valuation of tax shields and the topic is far from being completely resolved as claimed in Liu (2009). Acknowledgements I am very grateful to an anonymous referee and the Editor Ike Mathur for very helpful comments and guidance. I also thank Ling Zhang, Julia Qi and participants of the SBE Research Seminar at Michigan Tech for useful comments. All remaining errors in this paper are my own. References Akram, Q.F., Rime, D., Sarno, L., 2009. Does the law of one price hold in international financial market? Evidence from tick data. Journal of Banking and Finance 33, 1741–1754. Booth, L., 2002. Finding value where none exists: Pitfalls in using adjusted present value. Journal of Applied Corporate Finance 15, 8–17. Booth, L., 2007. Capital cash flows, APV and valuation. European Financial Management 13, 29–48. Brealey, R.A., Myers, S.C., Allen, F., 2005. Principles of Corporate Finance, eighth ed. New York, McGraw-Hill/Irwin. Brigham, E.F., Ehrhardt, M.C., 2004. Financial Management: Theory and Practice, 11th ed. South-Western College Publisher, Boston, Massachusetts. Chang, S.-C., Chen, S.-S., Chou, R.K., Lin, Y.-H., 2008. Weather and intraday patterns in stock returns and trading activity. Journal of Banking and Finance 32, 1754– 1766. Cooper, I.A., Nyborg, K.G., 2006. The value of tax shields is equal to the present value of tax shields. Journal of Financial Economics 81, 215–225. Cooper, I.A., Nyborg, K.G., 2007. Valuing the debt tax shield. Journal of Applied Corporate Finance 19, 50–59. Fernández, P., 2004. The value of tax shields is not equal to the present value of tax shields. Journal of Financial Economics 73, 145–165. Fernández, P., 2005. Reply to ‘Comment on the Value of Tax Shields is Not Equal to the Present Value of Tax Shields’. IESE Business School working paper, University of Navarra. Fernández, P., 2006. The Correct Value of Tax Shields: An Analysis of 23 Theories. IESE Business School Working Paper No. 628, University of Navarra. Graham, J.R., 2000. How big are the tax benefits of debt. Journal of Finance 5, 1901– 1942. Hamada, R., 1972. The effect of the firm’s capital structure on the systematic risk of common stocks. Journal of Finance 27, 435–452. Harris, R.S., Pringle, J.J., 1985. Risk-adjusted discount rates extensions from the average risk case. Journal of Financial Research 8, 237–244. Jacobsen, B., Marquering, W., 2009. Is it weather? Response. Journal of Banking and Finance 33, 583–587. Johnson, D., Qi, H., 2008. WACC misunderstandings. Journal of Academy of Finance 6 (1), 32–40. Kamstra, M.J., Kramer, L., Levi, M.D., 2009. Is it the weather? Comment. Journal of Banking and Finance 33, 578–582. Liu, Y.-C., 2009. The slicing approach to valuing tax shields. Journal of Banking and Finance 33, 1069–1078. Miles, J., Ezzell, J.R., 1980. The weighted average cost of capital, perfect capital markets, and project life: A clarification. Journal of Financial and Quantitative Analysis 15, 719–730.

H. Qi / Journal of Banking & Finance 35 (2011) 166–173 Modigliani, F., Miller, M.H., 1958. The cost of capital, corporation finance and the theory of investment. American Economic Review 48, 261–297. Modigliani, F., Miller, M.H., 1963. Corporate income taxes and the cost of capital. American Economic Review 53, 433–443. Moller, N., Zilca, S., 2008. The evolution of the January effect. Journal of Banking and Finance 32, 447–457. Pereiro, L.E., 2002. Valuation of Companies in Emerging Markets. A Practical Approach. John Wiley & Sons, New Jersey.

173

Qi, H., 2010. Valuation methodologies and emerging markets. Journal of Business Valuation and Economic Loss Analysis 5 (1), 14–31. Ruback, S.R., 2002. Capital cash flows: A simple approach to valuing risky cash flows. Financial Management 31 (2), 5–30. Sabal, J., 2005. WACC or APV?: The Case of Emerging Markets. Unpublished Working Paper, ESADE Business School. Sabal, J., 2007. WACC or APV? Journal of Business Valuation and Economic Loss Analysis 2 (2), 1–17.