Constant Leverage and Constant Cost of Capital: A Common Knowledge Half-Truth

In this teaching note we show that using the findings of Tham and Velez-Pareja 2002, for finite cash flows, Ke and hence WACC depend on the discount rate that is used to value the tax shield, TS and as expected, Ke and WACC are not constant with Kd as the discount rate for the tax shield, even if the leverage is constant. We illustrate this situation with a simple example. We analyze five methods: DCF using APV, FCF and traditional and general formulation for WACC, present value of CFE plus debt and Capital Cash Flow, CCF. In Tham and Velez-Pareja 2002, they derive a general expression for Ke, the cost of levered equity and for the Weighted Average Cost of Capital (WACC) applied to the Free Cash Flow (FCF) and Capital Cash Flow (CCF). For finite cash flows and perpetuities, the derivation presents the analysis for different levels of risk with respect to discounting the tax shields (TS). Taggart 1991 presents a revision of the set of formulations for the cost of levered Ke and WACC. He introduces the formulation with and without personal taxes and for different level of risk for discounting the TS, including the proposal by Miles and Ezzel 1980. However, Taggart does not include the case of Kd, the cost of debt as the level of risk for the TS and finite cash flows. A typical approach for valuing finite cash flows is to assume that leverage is constant (usually as target leverage) and the Ke and WACC are also assumed to be constant. For cash flows in perpetuity, and with Kd as the discount rate for the tax shield, it is indeed the case that the Ke and WACC applied to the FCF are constant if the leverage is constant. However this does not hold true for finite cash flows. Though it might be convenient to perform calculations under such assumption, it is not in fact always true that Ke and WACC are constant under the constant leverage financing policy. As could be seen from the findings and example of Inselbag and Kaufold (1997), and as a general expression for Ke and WACC derived by Tham and Velez-Pareja (2002) shows, both the cost of levered equity and the Weighted Average Cost of Capital depend on the value of the interest tax shield (VTS), and in the case of finite cash flows valuation they could be changing from period to period if certain choice is made for the rate to discount for the expected tax shields. The teaching note is organized as follows: An Introduction to state the problem; in Section Two we present the generalized formulation for the cost of capital for the finite cash flow valuation, and in particular formulae under the assumption that the discount rate for the tax shield (TS) is Kd. In Section Three we show a simple example. In Section Four we conclude.


INTRODUCTION
In this document we show that using the findings of Tham and Velez-Pareja (2002), for finite cash flows, Ke (cost of levered equity) and hence WACC (Weighted Average Cost of Capital), depend on the discount rate that is used to value the tax shield (TS), and as expected, Ke and WACC are not constant with Kd (cost of debt) as the discount rate for the tax shield, even if the leverage is constant. We illustrate this situation with a simple example. We analyze five methods: DCF Discounted Cash flows (the Free Cash Flow, FCF) using APV, FCF and traditional and general formulation for WACC, present value of Cash Flow to equity (CFE) plus debt and Capital Cash Flow (CCF).
A typical approach for project or firm valuation which could be found in practice (See for example World Bank (2002), Benninga (1997Benninga ( , 2006, Myers (2000, 2003), Brealey, Myers and Allen (2006), Murrin (1995, 2000) is to discount cash flows expected within the finite time horizon at constant cost of capital, (usually as a target leverage) assuming that target leverage is maintained throughout the life of the project, and thus its cost of levered equity Ke and the WACC are constant. Though it might be convenient to perform calculations under such assumption, it is not in fact always true that Ke and WACC are constant under the constant leverage financing policy. As could be seen from the findings and example of Inselbag and Kaufold (1997), and as a general expression for Ke and WACC derived by Tham and Velez-Pareja (2002), both the cost of levered equity and the Weighted Average Cost of Capital depend on the value of the interest tax shield (V TS ), and in the case of finite cash flows valuation, they could be changing from period to period if certain choice is made for the rate to discount expected tax shields.
The case of variable leverage has been studied elsewhere by Mian and Velez Pareja (2008), Velez-Pareja (2004, Velez-Pareja and Burbano (2006), Velez-Pareja and Tham (2001Tham ( , 2006aTham ( , 2006b, and Velez Pareja (2002, 2004). In these cases, they find complete consistency between all methods and with different assumptions about the discount rate for the tax shields.
Practitioners frequently assume that the risk (and corresponding discount rate, ψ) of the interest tax shield is the cost of debt, Kd. This is done explicitly when, for example, the APV method is applied, or implicitly, if popular formula Ke= Ku + (Ku−Kd)×(1−T)×D/E (Ku, the cost of unlevered equity; T, corporate tax rate; D and E are market values of debt and equity, respectively) is used to estimate the cost of equity capital. As Taggart (1991) and Velez-Pareja (2002, 2004) prove, this formulation is valid only for a fixed (in perpetuity) dollar amount of debt, thus under constant leverage assumption it could be applied only to perpetual cash flows. However, this formula is used by Fernandez (2002), Shapiro (2005) and others even within finite planning horizon and when dollar amount of debt is changing from period to period. Another example of implicit ψ = Kd assumption is applying Hamada's formulation to unlever and relever betas. Initially developed by Hamada (1972) for flat perpetuity and risk free debt, his formula is persistently used in conjunction with discounting at constant WACC under constant leverage assumption, 1 potentially producing significant valuation errors as can be seen from the comprehensive example analyzed by Mian and Velez-Pareja (2008). Velez-Pareja and Tham ( , 2006aTham ( , 2006b repeatedly show that, if assumptions and formulae are mismatched, inconsistencies arise when calculating value with different methods. So analysts should be very careful dealing with finite cash flows. To obtain correct and consistent valuation results one should specify assumption for the risk of the tax shield (ψ) first, and from that assumption choose the proper formulation for Ke and WACC.
Different values one proposes for ψ, the risk or discount rate for the TS might be questioned based on the particular debt policy and underlying expectations for the cash flow profile. However, when selecting the assumption or approach for ψ, we have to be consistent in the use of the formulation for the cost of capital. If under the constant leverage financing policy the risk ψ of the interest tax shield is assumed 2 to be equal to Kd, then Ke and WACC could not be assumed constant. Put it another way, ψ = Kd and constant Ke and WACC are incompatible assumptions within the constant leverage set up.
To illustrate the scenario of non constant cost of capital with constant leverage 3 we present a simple example, and analyze five DCF methods: The rest of the document is organized as follows: in Section Two, we present the generalized formulation for the cost of capital for the finite cash flow valuation, and in particular formulae under the assumption that the discount rate for the tax shield (TS) is Kd. In Section Three we show a simple example. In Section Four we conclude. Taggart (1991) presents a revision of the set of formulations for the cost of levered Ke and WACC for perpetuities and finite cash flows. He introduces the formulation with and 1 Here we can mention very different texts from practitioners and academics: Pratt, Reilly and Schweihs (2000), Abrams (2001), Damodaran (2002), 2 This assumption is by itself debatable 3 Which for example could be achieved through debt rebalancing at the end of every period to keep constant its percentage of the estimated project value) 4 See Velez-Pareja (2002, 2004). without personal taxes and for different level of risk for discounting the TS, including the Miles and Ezzell (1980). However, Taggart does not include the case of ψ = Kd for finite cash flows. Inselbag and Kaufold (1997) include the formulation of Ke and WACC for the case of Kd, the cost of debt as the level of risk for the TS and finite cash flows, but neither Taggart (1991) nor Inselbag and Kaufold (1997) show the formulation for the cost of capital appropriate to discount capital cash flow (CCF) under ψ = Kd and finite cash flows scenario. Tham and Velez-Pareja (2002) present a derivation of the general expression for Ke, the cost of levered equity for different levels of ψ corresponding to the risk of the tax shields, and resulting formulations for the general WACC, which should be applied to discount the Free Cash Flow (FCF) and Capital Cash Flow (CCF) both for finite time horizon valuation and for perpetuities.

GENERAL FORMULATION FOR KE AND WACC
The general formulation for Ke is, Where Ku is the unlevered cost of equity, ψ is the risk (discount rate) of the TS, D is market value of debt, E is market value of equity and V TS is the market value of TS; i is the period of analysis.
The general formulation for WACC FCF is, where TS is tax savings, V L is the market value of the levered firm and the other variables were defined above.
Following the path of the classic WACC derivation, we can easily show that general expression for the classic WACC is and we obtain traditional formula The general formulation for the WACC CCF is This formulation is valid for finite cash flows or perpetuities.
The WACC for the FCF simplifies to When taxes are paid when accrued and there is enough EBIT to earn the TS, then WACC FCF is The value of the TS is The WACC for the FCF simplifies to And the WACC for the CCF simplifies to Observe that in the case of ψ = Ku, Ke does not depend on TS and does not depend on the value of TS. Instead, when ψ = Kd, Ke depends on TS and the value of TS. On the other hand, when ψ = Ku, WACC depends on TS and it will be constant when taxes are paid when accrued and there is enough EBIT to earn the TS. Instead, when ψ = Kd, WACC depends on TS and the value of TS.
From these formulations we can conclude that for finite cash flows leverage and cost of capital are constant when: In the Appendix, the reader will find the complete information and the financial statements.
Assume a project (or the firm) with the following information: 1. Some input data.
2. Income Statement, Cash Budget and Balance Sheet.
3. Cash flows derived from the financial statements.
Year   With the change in working capital we can construct the FCF using the indirect method. We show the different cash flows in the next table.  Using the APV and assuming that the discount rate for the TS is Kd: Using the DCF, the traditional WACC, and assuming that the discount rate for the TS is Kd: Observe that Ke and WACC are not constant. This occurs because the Ke is a function of the value of TS.
Using the DCF, the general WACC, and assuming that the discount rate for the TS is Kd: Using the CFE and assuming that the discount rate for the TS is Kd: And finally, using the CCF with the WACC CCF , and assuming that the discount rate for the TS is Kd:  Velez-Pareja (2002, 2005), and Velez-Pareja and Tham (2006aTham ( , 2006b) have shown that when using Ku as the risk for the TS and some conditions regarding the payment of taxes, the existence of enough EBIT to earn the TS and the source of the TS, the cost of capital is constant. 6 Observe that the value calculated assuming ψ equal to Kd is higher than the value when we assume that ψ equal to Ku. A question arises here: is it reasonable to think that, changing the financing policy from constant leverage to predetermined debt schedule (non constant leverage), the firm will increase its value?
We leave the answer to this question for another work.  The new debt schedule is shown in next table. This means that management has to adjust debt from the beginning in order to achieve the target leverage. The difference in debt level is as follows.

SUMMARY AND CONCLUDING REMARKS
We have shown that a constant leverage does not grant that the cost of capital is constant when the risk of the TS is Kd. Moreover, in order to achieve a proper valuation of finite cash flows with a constant leverage when the risk of TS is Kd, we have to use some formulations that differ from the traditional used by practitioners and textbooks. In other words, assuming constant leverage is not a sufficient condition to have constant cost of capital. We need to make explicit assumptions on the risk for the TS and use formulation for the cost of capital that is consistent with the assumed risk of the tax shield.
Using the proper formulation in this scenario, we obtain full consistency in the calculation of value. This means that there are no advantages of one method over another. All of them give the same value (when properly done) and all of them (even the APV) require iterations when the risk of the TS is Kd.
In short, we can conclude that for finite cash flows leverage and cost of capital are constant when: 1. There is enough EBIT to fully earn the TS.
2. Taxes are paid when accrued.
3. The risk of TS is Ku.

Tax rate T, is constant.
5. Interest rate on debt is equal to the (market) cost of debt, Kd.
In addition, we have to be aware that performing cash flow valuation with constant Ke and WACC under constant leverage assumption implies that particular formulations must be used for the estimation of Ke. Since the possibility of constant leverage and constant cost of capital scenario arises only when ψ = Ku, analysts should use formula to calculate the cost of levered equity directly, and formula for unlevering and levering the beta in case they use the CAPM. Here βu and βLev are the unlevered and levered β's and D t-1 and E t-1 are the market values of debt and equity. In the case of the risk for the TS equal to Ku, we can observe the equations for Ke, WACC for the FCF and the CCF, as follows: The general formulation for Ke is, When the risk of the TS is Ku, the third term of the RHS of the equation vanishes and Ke depends only on Ku, Kd and leverage (constants). Hence, Ke is constant.
The general formulation for WAC-C FCF is, When the risk of the TS is Ku, the third term in the previous equation vanishes and the second term is T×Kd×D% and hence WACC FCF depends only on leverage which is constant. Hence, WACC is constant.
In this case, if Ke is constant then WACC FCF is constant.
The general formulation for the WACC CCF is When the risk of the TS is Ku, the second term of the RHS of the equation vanishes and WACC CCF = Ku which is a constant.