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Journal of Business Economics

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19.10.2015 | Original Paper

Reconsidering the appropriate discount rate for tax shield valuation

verfasst von: Marko Volker Krause, Alexander Lahmann

Erschienen in: Journal of Business Economics | Ausgabe 5/2016

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Abstract

This paper aims at identifying the appropriate discount rate for tax shield valuation in a setting where a partial default is possible and either principal or interest payments are prioritized in default. As a general valuation framework we use the stochastic discount factor. We assume a tax framework with corporate taxes, tax-deductible interest payments of the firm, no taxes on the cancellation of debt and no personal taxes. We strictly decompose the payments owed to the debtholders into interest and principal payments and analyze discount rates of those claims for the different priorities. As a result of the single-period analysis we find that the discount rate for tax savings, i.e., the conditional expected return on tax savings, is always equal to the discount rate of debt only for a proportional loss distribution on interest and principal payments. If losses are distributed according to one of the priority assumptions, the discount rate of tax savings behaves different from the discount rate of debt and both discount rates are equal only in very special cases. Furthermore, we derive qualitative statements for the relation between the discount rate of debt and the discount rate of tax savings assuming certain correlations between the stochastic discount factor and the debt repayments. Finally, we show how the prioritization assumptions can be implemented in a multi-period setting. We obtain for the presented set of assumptions a pricing equation equivalent to the one by Miles and Ezzell (J Finance 40:1485–1492, 1985).

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This impartial list only contains articles dealing with the discounted cash flow (DCF) approach. Other articles on the optimal capital structure implicitly consider the effect of default on the tax shield (see e.g. Leland 1994 or Goldstein et al. 2001).

While it is still under discussion whether Miles and Ezzell (1980) considered risky debt (see e.g. Rapp 2006, p. 777, footnote 24), they refer to the term “cost of debt” in their derivations. Koziol (2014, p. 656), uses the cost of debt for a default-risk adjusted WACC approach. With a respective rearranging the cost of debt would then represent a part of the discount rate of the tax shield.

Examples are Molnár and Nyborg (2013) or Koziol (2014).

One notable exception is Arzac and Glosten (2005).

For further expositions and derivations we refer to Cochrane (2005).

We abstain from further assumptions with regard to the economy. However, let us briefly line out some additional remarks: For the case of complete markets, all investors have the same unique SDF, there is perfect risk sharing and the investors’ individual consumption moves in lockstep with aggregate consumption, i.e., moves in lockstep with business cycles. For the case of incomplete markets, there can be many SDFs but there is a unique SDF within the payoff space of tradable assets, the projection of every investor’s SDF onto the payoff space of tradable assets (see for example Cochrane 2005). In this case, one might be faced with imperfect risk sharing and individual consumption might not move in lockstep with aggregate consumption.

Since the risk-free rate is known at time t we use t as subscript as opposed to uncertain returns.

Several authors define discount rates as cost of capital (see Kruschwitz and Löffler 2006; Laitenberger and Löffler 2006 or Cooper and Nyborg 2008). In this literature stream cost of capital are defined as conditional expected one-period returns with additional assumptions such as cost of capital being deterministic (see Kruschwitz and Löffler 2006). We keep the term expected return throughout our analysis and add assumptions when necessary.

To derive Eq. (4) we use the price Eq. (1) and expand it to \(p_t(x_{t+1})=E_t[m_{t+1}]E_t[x_{t+1}]+cov_t(m_{t+1},x_{t+1}).\) Now, we divide by the price \(p_t(x_{t+1}\)), substitue in \(\frac{1}{R^f_t}\) for \(E[m_{t+1}]\) to obtain \(1=\frac{E_t[R_{t+1}]}{R^f_t}+cov_t(m_{t+1},R_{t+1}).\) Multiplying by \(R^f_t\) and rearranging for \(E_t[R_{t+1}]\) we arrive at Eq. (4).

Alternatively, we could have applied the risk-neutral pricing approach, which became famous especially through its application in derivative pricing. If applied correctly and stating the necessary assumptions, i.e., in this study an arbitrage-free (capital) market and additionally a complete market, risk-neutral and SDF pricing yield equivalent results. In this case, the existence of a unique risk-neutral probability measure implies a positive SDF. However, as stated above and more specifically in footnote (FN) 6, our pricing arguments hold without the additional assumption of a complete market.

See Kim et al. (1993, p. 119f.), or Kruschwitz et al. (2005, p. 223f.), for a discussion of these assumptions.

Throughout our analysis we regard single-period debt contracts, i.e., the amount of debt issued at time \(s-1\) is promised to be repaid at s.

By excluding an explicit modelling of personal taxes we operate - from a tax perspective - in a simplified framework, which is predominant in the corporate finance literature stream dealing with the tax benefits of debt financing (firm valuation literature see e.g. Rapp 2006; Koziol 2014; capital structure theory Titman and Tsyplakov 2007 or Hackbarth and Mauer 2012). Nevertheless, we discuss at the respective points of our analysis the consequences.

At this point, disregarding the effects of default, we already can outline one consequence of not considering personal taxes. While interest payments on the corporate level help to avoid corporate income tax, (1) interest income on the personal level has to be taxed, implying a decrease of the tax savings from perspective of the personal level and (2) payments to equityholders are taxed on the personal level as well. Here, it is relevant to note that interest payments reduce the payments to equityholders, which results in lower personal equity taxes. See for a detailed analysis of the US e.g. Graham (2003).

Note that \(PP_{t+1}\) and \(PP^{net}_{t+1}\) coincide in the single-period analysis.

The perspective is here from the moment in \(t+1\) when the cash flow has just been paid out.

Partial default has been analyzed by Molnár and Nyborg (2013) in a binomial model where the loss in default is ex ante specified. In comparison, we regard a model with more than two states and without relying on the assumption of a deterministic loss.

We use the subscript t for the period from t to \(t+1\) because the promised yield is already known, i.e., determined at t.

Bankruptcy costs could easily be implemented as a factor reducing the firm value in case of default (see e.g. Koziol 2014).

Note that in the single-period case principal and net principal payments are the same, because in \(t+1\) the firm ceases to exist.

Blaufus and Hundsdoerfer (2008) discuss the impact upon the paid taxes of whether the tax authority allows for a tax deductibility of interest payments or not.

Kruschwitz et al. (2005, p. 228), proceed on an equivalent assumption in combination with a strict prioritization of interest payments. With their assumption of a minimum cash flow for at least paying the taxes this results—even in default— in a constant tax deductibility of the contractually fixed interest payments. In Rapp (2006, specifically p. 777), the therein stated assumptions imply that the firm’s contractually fixed interest payments are always tax-deductible.

Cooper and Nyborg (2008) elaborate that one of the differences with the highest impact on tax shield valuation between Sick (1990) and Miles and Ezzell (1980) is the different assumption with respect to the tax treatment of a COD. This difference has been further analyzed by Blaufus and Hundsdoerfer (2008) by including personal taxes. They show that the main difference between the aforementioned approaches as well as the ones by Kruschwitz et al. (2005) on the one hand and on the other by Homburg et al. (2004) and Rapp (2006) result from the general tax treatment of a default, i.e., the tax deductibility of interest payments, the taxation of a possible COD and the taxation of a possible debt write-down on the level of the debtholder.

As stated above, assuming no tax on COD is a valid assumption for mapping the case of a reorganization (see e.g. Miller 1991 or Cooper and Nyborg 2008). Therefore, at first glance, it might not be reasonable to combine a single period model with the assumption of no taxes on a COD. In a single period setting, the firm is simply not able to reorganize. However, we have decided to state this assumption with regard to the more general setting of the multi-period analysis in Sect. 4 which aims at mapping the valuation of a firm that uses in default the opportunity to reorganize. Notice that no tax on a possible COD is an implicit standard assumption in most works on optimal capital structure (e.g. Leland 1994), even though most works dealing with optimal capital structure do not consider the case of a reorganization.

We consider a strict prioritization of the payments to the tax authority (equivalently Kruschwitz et al. 2005). Thereby, the firm cannot default on its tax payments. With our set of assumptions, we always compare the unlevered free cash flows with the debt repayments where taxes have been already deducted.

Typically, \(Tax^L_{t+1}\) is defined in terms of the equity cash flows (\(ECF_{t+1}\)), with our set of assumptions, by \(Tax^L_{t+1}=(ECF_{t+1}+PP^{net}_{t+1})\frac{\tau }{1-\tau },\) where \(ECF_{t+1}=(EBIT_{t+1}-Int_{t+1})(1-\tau )-PP^{net}_{t+1}.\)

Assuming a taxation of a COD implies a kind of irrelevance of default on the value of tax payments. This tax effect has already been shown by Sick (1990) and Kruschwitz et al. (2005). Blaufus and Hundsdoerfer (2008, p. 173), show the effects of such a tax system and draw upon the taxation effect of debt write-downs on the creditor level. Combining the case of tax deductibility of interest payments on the corporate level with a taxation of a COD and the tax deductibility of debt write-downs on the creditor level results in a tax system which does not distort the effects of a default (credit default neutral tax system). For the assumptions of Homburg et al. (2004) and Rapp (2006), as already discussed in FN 23, Blaufus and Hundsdoerfer (2008) analyze how the tax treatment of debt write-downs on the creditor level has to be constructed in order to imply a credit default neutral tax system.

For assets that are uncorrelated with the marginal utility growth of consumption of an investor (the SDF) the investor expects to earn a rate equal to the risk-free rate. Such an asset does not offer risk-reduction services and does not insure against bad states, i.e., low consumption, because on average it does not pay more in such a state. On average, it also does not pay more in good states, i.e., states of high consumption, so that the volatility of the consumption stream is not increased. See for an extensive treatment (Magill and Quinzii 2002, p. 160).

If we would have additionally assumed the spanning assumption to hold, it is possible to perfectly duplicate the single payoffs of principal and interest payments on the capital market. See for a discussion of the spanning assumption e.g. Kruschwitz and Löffler (2006). However, we have abstained from making further assumptions with respect to the economy (see FN 6 and 10). As mentioned in Cochrane (2005) the basic pricing equation can also represent the private evaluation of non-traded assets.

Proportional means pro rata according to promised interest and principal payments.

For returns on principal payments we write \(R^{PP}_{t+1}=\frac{PP_{t+1}}{p_t(PP_{t+1})}=\frac{D_t - L_{t+1}\frac{1}{(1+r^c_t)}}{p_t(D_t - L_{t+1}\frac{1}{(1+r^c_t)})},\) multiply numerator and denominator by \(1+r^c_t\) and arrive at \(R^{PP}_{t+1}=\frac{D_t(1+r^c_t) - L_{t+1}}{p_t(D_t(1+r^c_t) - L_{t+1})}=R^D_{t+1}.\)

For example Cooper and Nyborg (2008) conduct their analysis based on a full loss of principal and interest in default. However, without questioning their results, they use the promised yield as discount rate for the tax savings.

Notice that principal payments only suffer losses when interest payments are already equal to zero.

We do not transform the gross return into net returns, i.e., \(R-1\) into r, because in the following we will regard the 1 as a strike of a call option. We continue to show an important consequence using this logic.

For a full derivation see Coval and Shumway (2001).

\(E_t[R^D_{t+1}]=\frac{E_t[Int_{t+1}]+E_t[PP_{t+1}]}{D_t}=E_t[R^{Int}_{t+1}]\frac{p_t(Int_{t+1})}{D_t}+E_t[R^{PP}_{t+1}]\frac{p_t(PP_{t+1})}{D_t}.\)

Coval and Shumway (2001) assume no restrictions regarding the differentiability of the expected return with respect to K. We have discrete returns such that with increasing K at \(K=R^D_{t+1}(\omega )\) the function is not smooth, i.e., it is not differentiable (\(\max (R^D_{t+1}(\omega )-K,0)=0\) in this state). But the expected return is still continuous in K along the range of \(R^D_{t+1},\) i.e., there are no jumps, and therefore the expected return will not suddenly decrease or increase. When we know that before and after such a point the derivative is positive, expected returns always increase with K.

The EBIT increases linearly from 5 to 30 from state 1 to state 100. We assume that the investor’s consumption increases linearly from state 1 to 100, so that we can replace the \(\omega\) indicating states with consumption \(C_{t+1},\) i.e., \(C_{t+1}(\omega =1)=1, C_{t+1}(\omega =2)=2, ..., C_{t+1}(\omega =100)=100.\) We assume an SDF of the form \(m_{t+1}=\beta \frac{u'(C_{t+1})}{u'(C_t)},\) where \(\beta =\frac{1}{R^f_t}\) and \(u'(\cdot )\) is the first derivative of a utility function \(u(\cdot )\) (see Cochrane 2005 for a basic treatment). We use the power utility \(u(C_{t+1})=\frac{C_{t+1}^{1-\gamma }}{1-\gamma },\) where the first derivative is \(u'(C_{t+1})=C_{t+1}^{-\gamma }.\) Thus, the SDF is \(m_{t+1}=\beta \left( \frac{C_{t+1}}{C_t}\right) ^{-\gamma }.\) The constant \(\gamma\) is the coefficient of risk-aversion and must be positive for risk-averse investors. We set \(\gamma =0.25\) and \(C_t=33.686.\) With numeric values the SDF is \(m_{t+1}=\frac{1}{1.05}\left( \frac{C_{t+1}}{33.686}\right) ^{-0.25}.\) The SDF is a convex function over states and consumption with values from \(m_{t+1}(\omega =1)=2.294\) to \(m_{t+1}(\omega =100)=0.726.\)

See for example Leland (1994), Goldstein et al. (2001) or Koziol (2014), who discuss this in more detail. Kruschwitz et al. (2005, p. 228), and Rapp (2006, p. 778), build their analysis on the equivalent assumption.

Other bankruptcy codes as for example in Germany distinguish between overextension and illiquidity as well. In the UK the Supreme Court clarified in a decision published \(9{\rm th}\) of May 2013 UKSC 28 that the cash flow and balance sheet test are both applicable for testing insolvency.

See for an equivalent definition of illiquidity (Kruschwitz et al. 2005). Additionally, note that this condition can be rearranged to \(ECF_t<0\). This implies that the firm defaults as soon as the cash flows distributed to equity holders turn negative, i.e., the equityholders have to inject cash into the firm in order to prevent a default.

See for a discussion on this issue (Kim et al. 1993).

Notice that putting \(FCF^U_{s+1}+V^L_{s+1}\) on the axes leads to the same graph as in the single-period case.

In Eq. (38)\(V^L_{s+1}\) is also divided by \((1-\tau )\) because within the range of zero and \(r^c_s D_s\) interest payments are \(Int_{s+1}=FCF^L_{s+1} + V^L_{s+1}=FCF^U_{s+1} + \tau Int_{s+1} + V^L_{s+1}\). Rearranging this for interest payments leads to \(Int_{s+1}=\frac{1}{1-\tau }(FCF^U_{s+1}+V^L_{s+1})\).

Notice that with i.i.d. \(R^U\) we can write \(E_t[(R^U)^{s-t}]=E_t[R^U]^{s-t}\).

To see that we write \(p_{s}(TS_{s+1})=c p_{s}(CF^{R^U}_{s+1})\), where c is a constant, and take prices \(p_{t}(\cdot )\) for \(t<s\) to obtain \(p_{t}(p_{s}(TS_{s+1}))=c p_{t}(CF^{R^U}_{s+1})\), which is \(p_{t}(TS_{s+1})=c p_{t}(CF^{R^U}_{s+1})\). Since \(\frac{cp_{s}(CF^{R^U}_{s+1})}{cp_{t}(CF^{R^U}_{s+1})}=(R^U)^{s-t}\), where the constants cancel out, \(\frac{p_{s}(TS_{s+1})}{p_{t}(TS_{s+1})}=(R^U)^{s-t}\) must hold as well.

Notice that returns are also ratios.

The converse is always true: When interest payments experience losses, the loss ratio must not bind, i.e., must not be zero.

Take conditional expectations \(E_{u}[.]\) of both sides of Eq. (67).

The \(E_{s}[FCF^U_{s+1}]\) terms cancel out.

Miles and Ezzell (1985) base their analysis on the CAPM. In case the covariance of the market return and \(\epsilon\) is constant over time, the SDF is a linear combination of the market return.

Using Eq. (1) we are able to state \(p_{s}(\epsilon _{s+1})=E_{s}[m_{s+1} \epsilon _{s+1}]=E_{s}[m_{s+1}]E_{s}[\epsilon _{s+1}]+cov_{s}(m_{s+1}, \epsilon _{s+1})\) and observe that with constant \(E_{s}[m_{s+1}]\), \(E_{s}[\epsilon _{s+1}]\) and \(cov_{s}(m_{s+1}, \epsilon _{s+1})\) the price of \(\epsilon\) is constant over time.

They refer to the traditional Sharpe-Lintner-Black CAPM.

A sequence of a random variable \({X_s}\), for \(s=t, t+1, \ldots, T\), is i.i.d. if any term in the sequence has the same cumulative probability distribution function, i.e. \(F_{X_s}(x)=F_{X_v}(x)\) for all realisations x and all \(s, v \in [t, t+1, \ldots ,T]\), and if the joint cumulative distribution function of the sequence equals the product of all single cumulative distribution functions, i.e. \(F_X(x_t, x_{t+1}, \ldots, x_T)=F_{X_t}(x_t)F_{X_{t+1}}(x_{t+1}) \ldots F_{X_T}(x_T)\).

For a further discussion of discount rates in a multi-period CAPM setting see Fama (1977).

Notice that \(R^f=E[m]\).

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Titel: Reconsidering the appropriate discount rate for tax shield valuation
verfasst von: Marko Volker Krause
Alexander Lahmann
Publikationsdatum: 19.10.2015
Verlag: Springer Berlin Heidelberg
Erschienen in: Journal of Business Economics / Ausgabe 5/2016
Print ISSN: 0044-2372
Elektronische ISSN: 1861-8928
DOI: https://doi.org/10.1007/s11573-015-0782-4

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