On the value and determinants of the interest tax shields

Abstract

We use a dynamic model of the firm to ascertain both the value and the determinants of the debt tax shields. For a representative U.S. firm, we find that the value of the interest tax shields represents less than 5 % of firm value, and it varies considerably across U.S. industries. Our results also show that this component of firm value behaves counter-cyclically over the business cycle. Finally, besides the interest rate on debt and the corporate income tax rate, we find that the curvature of the production function is one of the main determinants of the tax advantage of debt.

Appendices

Appendix 1: proofs

The proof of Proposition 1 requires an intermediate result that we present next.

Lemma 2

Restricting debt to be risk-free, the maximum level of book leverage in each period is given by

$$\begin{aligned} \ell ^{*}=\frac{1-\left( f+\delta \right) \left( 1-\tau \right) }{ 1+r_{B}\left( 1-\tau \right) }. \end{aligned}$$

(25)

Proof

We say debt is risk-free if, in every period, the following inequality is true for all $z^{\prime }$

$$\begin{aligned} \left( z^{\prime }K^{\prime ^{\alpha }}-fK^{\prime }-\delta K^{\prime }-r_{B}\ell K^{\prime }\right) \left( 1-\tau \right) +K^{\prime }-\ell K^{\prime }\ge 0. \end{aligned}$$

(26)

That is, risk-free debt implies that next-period, after-shock book value of equity must be weakly positive for all $z^{\prime }$.¹⁴ In other words, net profits, $\left( z^{\prime }K^{\prime ^{\alpha }}-fK^{\prime }-\delta K^{\prime }-r_{B}\ell K^{\prime }\right) \left( 1-\tau \right)$, plus the value of assets, $K^{\prime }$, must be sufficient to cover debt, $\ell K^{\prime }$.

Given that the worst-case scenario is $z^{\prime }=0$, the maximum book leverage ratio consistent with risk-free debt, $\ell ^{*}$, satisfies

$$\begin{aligned} \left( -fK^{\prime }-\delta K^{\prime }-r_{B}\ell ^{*}K^{\prime }\right) \left( 1-\tau \right) +K^{\prime }-\ell ^{*}K^{\prime }=0. \end{aligned}$$

(27)

Working on the previous expression, we can derive the maximum level of book leverage as

$$\begin{aligned} \ell ^{*}=\frac{1-\left( f+\delta \right) \left( 1-\tau \right) }{1+r_{B}\left( 1-\tau \right) } \end{aligned}$$

(28)

which completes the proof.¹⁵ $\square$

Proof of Proposition 1

The maximization in Eq. (4) requires a normalization of growing variables that keeps the expectation of the payoff function in the future periods bounded. This normalization is equivalent to the one used to find the solution of the canonical Gordon Growth Model. Let vector $\widetilde{X}_{t}=\left\{ \widetilde{K}_{t},\widetilde{B}_{t}, \widetilde{N}_{t},\widetilde{L}_{t},\widetilde{S}_{t}\right\}$ contain the growing variables of the model. We then transform vector $\widetilde{X}_{t}$ in the following way: $X_{t}=\widetilde{X}_{t}/(1+g)^{t}$. Using the normalized variables and modifying the payoff function accordingly, the market value of equity can be expressed as

$$\begin{aligned} S_{0}\left( K_{0},B_{0},z_{0}\right) =\max _{\left\{ K_{t+1},B_{t+1}\right\} _{t=0}^{\infty }}E_{0}\sum \limits _{t=0}^{\infty }\frac{\left( 1+g\right) ^{t}}{\prod \nolimits _{j=0}^{t}\left( 1+r_{S_{j}}\right) }L_{t} \end{aligned}$$

(29)

subject to keeping debt risk-free. Because we use the Adjusted Present Value method of firm valuation, we solve the problem of the firm in Eq. (29) in three steps. First, we determine the value of the unlevered firm, $S_{u_{0}}\left( K_{0},z_{0}\right)$. Second, we solve for optimal debt and compute the present value of the financing side effects. Finally, we obtain the value of the levered firm in Eq. (29).

The market value of equity for the unlevered firm can be expressed as

$$\begin{aligned} S_{u_{0}}\left( K_{0},z_{0}\right) =\max _{\left\{ K_{t+1}\right\} _{t=0}^{\infty }}E_{0}\sum \limits _{t=0}^{\infty }\left( \frac{1+g}{1+r_{A}}\right) ^{t}L_{u_{t}} \end{aligned}$$

(30)

where $L_{u_{t}}=N_{u_{t}}-\left( K_{t+1}-K_{t}\right)$ and $N_{u_{t}}=\left( z_{t}K_{t}^{\alpha }-fK_{t}-\delta K_{t}\right) \left( 1-\tau \right)$. We let normalized variables with primes indicate values in the next period and normalized variables with no primes denote current values. Then, the Bellman equation for the problem of the firm in Eq. (30) is given by

$$\begin{aligned} S_{u}\left( K,z\right) =\underset{K^{\prime }}{\max }\left\{ \left( zK^{\alpha }-fK-\delta K\right) \left( 1-\tau \right) -\left( 1+g\right) K^{\prime }+K+\frac{\left( 1+g\right) }{\left( 1+r_{A}\right) }E\left[ S_{u}\left( K^{\prime },z^{\prime }\right) |z\right] \right\} . \end{aligned}$$

(31)

We use the guess and verify method as the proof strategy. Thus, we start by guessing that the solution is given by

$$\begin{aligned} S_{u}\left( K,z\right) =\left( zK^{\alpha }-fK-\delta K\right) \left( 1-\tau \right) +K+M\left( z\right) P_{u}^{*} \end{aligned}$$

(32)

where

$$\begin{aligned} M\left( z\right)= & {} e^{-\frac{1}{2}\sigma ^{2}\frac{\alpha }{\left( 1-\alpha \right) ^{2}}}\sum _{n=1}^{\infty }\left\{ \left( \frac{1+g}{1+r_{A}}\right) ^{n}\left( c^{\frac{1-\rho ^{n}}{1-\rho }}z^{\rho ^{n}}e^{\frac{1}{2}\sigma ^{2}\frac{\left( 1-\rho ^{2n}\right) }{\left( 1-\rho ^{2}\right) }\frac{1}{\left( 1-\alpha \right) }}\right) ^{\frac{1}{1-\alpha }}\right\} , \end{aligned}$$

(33)

$$\begin{aligned} P_{u}^{*}= & {} \left( W^{*^{\alpha }}-fW^{*}-\delta W^{*}\right) \left( 1-\tau \right) -r_{A}W^{*}, \end{aligned}$$

(34)

$$\begin{aligned} W^{*}= & {} \left( \frac{\alpha }{\frac{r_{A}}{1-\tau }+f+\delta }\right) ^{\frac{1}{1-\alpha }}. \end{aligned}$$

(35)

We obtained this initial guess as the solution of Eq. (31) by the backward induction method.

We now verify our guess. To this end, let us write

$$\begin{aligned} S_{u}\left( K,z\right) =\underset{K^{\prime }}{\max }\left\{ F\left( K^{\prime },K,z\right) \right\} \end{aligned}$$

(36)

with F defined as the objective function in Eq. (31).

The FOC for this problem is

$$\begin{aligned} \partial F\left( K^{\prime },K,z\right) /\partial K^{\prime }=-\left( 1+g\right) +\frac{\left( 1+g\right) }{\left( 1+r_{A}\right) }\left[ \left( E \left[ z^{\prime }|z\right] \alpha K^{*\alpha -1}-f-\delta \right) \left( 1-\tau \right) +1\right] =0 \end{aligned}$$

(37)

and optimal capital turns out to be

$$\begin{aligned} K^{*}=E\left[ z^{\prime }|z\right] ^{\frac{1}{1-\alpha }}W^{*} \end{aligned}$$

(38)

where $W^{*}$ is as in Eq. (35).

Finally, the market value of equity for the unlevered firm becomes

$$\begin{aligned} \begin{array}{ll} S_{u}(K,z) &{}=\left( zK^{\alpha }-fK-\delta K\right) ( 1-\tau ) -( 1+g) K^{*}+K \\ &{}\quad +\,\frac{\left( 1+g\right) }{\left( 1+r_{A}\right) }\left[ \left( E\left[ z^{\prime }|z\right] K^{*\alpha }-fK^{*}-\delta K^{*}\right) \left( 1-\tau \right) +K^{*}\right. \\ &{}\quad +\,\left. E\left[ M\left( z^{\prime }\right) |z\right] P_{u}^{*}\right] \\ &{}=\left( zK^{\alpha }-fK-\delta K\right) \left( 1-\tau \right) +K-\left( 1+g\right) E\left[ z^{\prime }|z\right] ^{\frac{1}{1-\alpha }}W^{*} \\ &{}\quad +\,\frac{\left( 1+g\right) }{\left( 1+r_{A}\right) }\left\{ E\left[ z^{\prime }|z\right] ^{\frac{1}{1-\alpha }}\left[ \left( W^{*^{\alpha }}-fW^{*}-\delta W^{*}\right) \left( 1-\tau \right) +W^{*}\right] \right. \\ &{}\quad +\,\left. E\left[ M\left( z^{\prime }\right) |z\right] P_{u}^{*}\right\} \\ &{}=\left( zK^{\alpha }-fK-\delta K\right) \left( 1-\tau \right) +K \\ &{}\quad +\,\frac{\left( 1+g\right) }{\left( 1+r_{A}\right) }\left( e^{-\frac{1}{2}\sigma ^{2}\frac{\alpha }{\left( 1-\alpha \right) ^{2}}}E\left[ z^{\prime \frac{1}{1-\alpha }}|z\right] +E\left[ M\left( z^{\prime }\right) |z\right] \right) P_{u}^{*} \\ &{}=\left( zK^{\alpha }-fK-\delta K\right) \left( 1-\tau \right) +K+M\left( z\right) P_{u}^{*} \end{array} \end{aligned}$$

(39)

which is equivalent to our initial guess in Eq. (32).

Next, we obtain optimal debt. In each period, the firm solves the following problem

$$\begin{aligned} B^{*}=\max _{B^{\prime }}\left\{ B^{\prime }-\frac{1}{\left( 1+r_{B}\right) }B^{\prime }\left[ 1+r_{B}\left( 1-\tau \right) \right] \right\} \end{aligned}$$

(40)

subject to the restriction of risk-free debt. Because $\tau >0$, the firm increases debt as much as possible (as long as it remains risk-free) to maximize the tax benefits of debt. Then, optimal debt is $B^{*}=\ell ^{*}K^{*}$ where

$$\begin{aligned} \ell ^{*}=\frac{1-\left( f+\delta \right) \left( 1-\tau \right) }{ 1+r_{B}\left( 1-\tau \right) }, \end{aligned}$$

(41)

as shown in Lemma 2. The present value of the financing side effects turns out to be

$$\begin{aligned} \begin{array}{l} Q\left( z\right) =\left( \frac{1+g}{1+r_{A}}\right) \left\{ \left( \frac{ 1+r_{A}}{1+r_{B}}\right) r_{B}\tau B^{*}+E\left[ Q\left( z^{\prime }\right) |z\right] \right\} \\\qquad\; =M\left( z\right) \left( \frac{1+r_{A}}{1+r_{B}}\right) r_{B}\tau \ell ^{*}W^{*} \end{array} \end{aligned}$$

(42)

where $M\left( z\right)$ is as in Eq. (33). Under this financial policy, the amount of debt and interest payments will vary with the future asset cash flows (i.e., they depend on future firm performance). Then, because future interest tax shields will have a level of risk in line with that of the firm cash flows, we use the cost of capital, $r_{A}$, as the discount rate.

The third step consists in obtaining the market value of equity for the levered firm. If we assume the firm used debt B in the previous period, and now has to pay interest $r_{B}B\left( 1-\tau \right)$, then the stock price for the levered firm is

$$\begin{aligned} \begin{array}{ll} S\left( K,B,z\right) &{}=S_{u}\left( K,z\right) +M\left( z\right) \left( \frac{ 1+r_{A}}{1+r_{B}}\right) r_{B}\tau \ell ^{*}W^{*}-B-r_{B}B\left( 1-\tau \right) \\ \quad &{}=\left( zK^{\alpha }-fK-\delta K-r_{B}B\right) \left( 1-\tau \right) +K-B+M\left( z\right) P^{*} \end{array} \end{aligned}$$

(43)

where variable $P^{*}$ takes the form

$$\begin{aligned} P^{*}=\left( W^{*^{\alpha }}-fW^{*}-\delta W^{*}\right) \left( 1-\tau \right) -r_{A}W^{*}+\left( \frac{1+r_{A}}{1+r_{B}}\right) r_{B}\tau \ell ^{*}W^{*}. \end{aligned}$$

(44)

The last part of the proof consists in transforming normalized variables back into growing variables. For this step, we return to the initial notation with growing variables, where next-period assets are $\widetilde{K}_{t+1}$ and current-period assets are $\widetilde{K}_{t}$. Then, the required transformation is: $\widetilde{X}_{t}=(1+g)^{t}X_{t}$, where vector $X_{t}=\left\{ K_{t},B_{t},N_{t},L_{t},S_{t}\right\}$ contains the normalized variables of the model. Finally, the optimal decisions of the firm with growing variables are given by

$$\begin{aligned} \widetilde{K}_{t+1}^{*}\left( z_{t}\right) =\left( 1+g\right) ^{t+1}E \left[ z_{t+1}|z_{t}\right] ^{\frac{1}{1-\alpha }}W^{*}\text { and }\widetilde{B}_{t+1}^{*}\left( z_{t}\right) =\ell ^{*}\widetilde{K} _{t+1}^{*}\left( z_{t}\right) \end{aligned}$$

(45)

and the growing market value of equity is

$$\begin{aligned} \widetilde{S}_{t}\left( \widetilde{K}_{t},\widetilde{B}_{t},z_{t}\right) = \left[ \left( 1+g\right) ^{t\left( 1-\alpha \right) }z_{t}\widetilde{K} _{t}^{\alpha }-f\widetilde{K}_{t}-\delta \widetilde{K}_{t}-r_{B}\widetilde{B} _{t}\right] \left( 1-\tau \right) +\widetilde{K}_{t}-\widetilde{B}_{t}+ \widetilde{M}_{t}\left( z_{t}\right) P^{*} \end{aligned}$$

(46)

as shown in Proposition 1. $\square$

Appendix 2: calibration of model parameters

We need to find parameter values for $c,\rho ,\sigma ,\alpha ,f,\delta ,\tau ,r_{B},r_{A},$ and g for a representative firm in the U.S. and for each SIC industry at the division level. We do the calibration of model parameters using Compustat annual data. We exclude financial firms (i.e., SIC codes between 6000 and 6999) as well as public administration organizations (i.e., SIC codes greater than 9000). The sample covers the period 1990–2015 and includes a total of 58,596 firm-year observations. For the representative firm in the U.S. we use all firms in the sample while for the representative firm in each industry we use only the firms in the corresponding industry.

In order to obtain parameter $\delta$, we average the ratio Depreciation and Amortization (DP)/Assets $-$ Total (AT) for all firm-years. We follow the same procedure to get $\tau$ as the fraction Income Taxes $-$ Total (TXT)/Pretax Income (PI). We trim these ratios at the lower and upper one-percentiles to reduce the effect of outliers and errors in the data. We calibrate parameter f so that optimal book leverage, $\ell ^{*}$, matches the median value of book leverage observed in the sample.

Following Moyen (2004), we obtain parameters $\rho ,\sigma ,$ and $\alpha$ using the firm’s autoregressive profit shock process of Eq. (1) and the gross profits function in Eq. (2), $\widetilde{G}_{t}=\left( 1+g\right) ^{t\left( 1-\alpha \right) }z_{t}\widetilde{K}_{t}^{\alpha }$. The data we use with these equations are Gross Profit (GP) and Assets $-$ Total (AT). We first log-linearize the gross profits function and obtain parameter $\alpha$ by doing an OLS regression.¹⁶ We then use the error term from that regression with the firm’s autoregressive profit shock process to obtain parameters $\rho$ and $\sigma$.

Given that we are interested in evaluating representative firms, we normalize parameter c to 1.¹⁷ We assume that the market cost of debt $r_{B}$ is 0.06, which is consistent with the empirical evidence on the cost of debt (e.g., Kaplan and Stein 1990; Cooper and Davydenko 2007). We derive $r_{A}$ using CAPM with the corresponding (unlevered) industry betas estimated by Fama and French (1997) and assuming an expected market return $\left( r_{M}\right)$ of 0.08. Finally, we obtain g for each industry following Jorgenson and Stiroh (2000).