2020 | OriginalPaper | Buchkapitel Open Access

# 5. Corporate and Personal Income Tax

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Stochastic Discounted Cash Flow

In the last chapter of this book we would like to look into the question of how to evaluate a firm if taxes are raised both at the investor and at the company level. Attentive readers of Chaps.
2 and
3 might expect a discussion on different strategies of financing policy (Chap.
2) and of dividend policy (Chap.
3). But, we are not going to do this. We will also, however, not dwell on the innumerable variations of possible combinations in this context, rather we will restrict ourselves to a manageable example. At the end, we will only discuss how to proceed in the cases of further possible financing and dividend policies.

## 5.1 Assumptions

In this chapter we will define an unlevered company as a company, which fulfills the following two conditions. First, it has to be entirely self-financed, and second, it has to distribute all free cash flow to the shareholders. If even one of the attributes does not apply, then we will switch to discussing a levered company. Like in Chaps.
2 and
3 we will refer to the symbols
u and
l to distinguish either one case or the other.

Anzeige

In what follows we will begin by characterizing a tax system, which will be used in our example.

Corporate Income Tax

The corporate income tax will possess the attributes we mentioned in Chap.
2: The tax is measured by the company’s profit. The tax scale will be linear and the tax rate, which we will designate with
τ

^{C}is independent on time. Furthermore, taking out loans at time t − 1 provides a tax advantage. This advantage will be equal to the product of tax scale and interest, \(\tau ^C\,\widetilde {\,I\,}_t\).
Personal Income Tax

The personal income tax will follow the same model as in Chap.
3. On an assessment basis we are dealing with dividends and interest. Both will be taxed differently, the dividends with a tax rate of
τ

^{D}and the interest with a rate of τ^{I}. Such a strongly diverging treatment of income categories exists in several industrial nations. Both of the tax scales are linear and the tax rate is independent on time. In case the company does not pay out its entire free cash flow a tax advantage in the amount of \(\widetilde {A}_t\) will result due to the retention.
Interaction of Both Taxes

As we excluded one of the two tax types in Chaps.
2 and
3 we did not have to deal with the question of how to pattern an interaction of both types. It is obvious that here is a problem we have to discuss. For this purpose we shall put ourselves into the position of a shareholder whose earnings consist only of dividends. However, in order to be able to distribute dividends, the company in which the shareholder is involved, has to first yield profits. These profits are liable to corporate taxes. Consequently, the distribution to the shareholder will therefore already have been subject to initial taxes. Naturally the company can only distribute that amount of money that remains after corporate taxes. If the shareholder now has to pay personal income tax as well, the tax authorities will essentially have access to the dividend once again.

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The legislator will have two ways to deal with the situation. Either he accepts that the income tax, which the company has already paid (entirely or in part), is considered just as a first installment, in which case we can talk about indirect relief,

^{1}or the legislator does not provide for such an allowance. This last option would imply a double taxation of dividends.^{2}Occasionally political entities try to combine both concepts. As a general rule this is achieved with a milder dose of taxation of the dividends, especially when compared to the taxation applied to interest.^{3}
In our examples we will assume the classical system with a two-tier rate of taxation for dividends and interest. Further comments as to the question of how to proceed in the case of different indirect relief policies or financial systems are discussed in Cooper and Nyborg (
2004), Husmann et al. (
2006), and Lally (
2000).

## 5.2 Identification and Evaluation of Tax Advantages

Gross and Free Cash Flows

Our intention consists in evaluating the tax advantages of the levered company, especially compared with those of the unlevered company. Again this can be achieved in two steps as in the preceding chapters. In the first step we will identify the tax advantages in
t. In a second step we will evaluate these future tax advantages at the point in time of evaluation. To identify the amount of the tax advantages accurately it is advisable to examine Fig.
5.1.

×

It specifies how to get from the gross cash flow to the free cash flow of the levered company. According to our assumption, the gross cash flow of the levered company involves the same amount as the gross cash flow of the unlevered company. The same applies to the investment expenses. Notice that (as in Chap.
3)
\(\widetilde {\mathit {FCF}}^l_t\) in Fig.
5.1 does not represent the entire payments of the investors but rather only the payments to the shareholders.

Tax Shields

Let us now have a look at the distributions of the levered company. We intend to compare these payments with those of an unlevered company. The unlevered company is able to pay the following amount to the shareholders:
Due to the indebtedness of the levered company the payments to the shareholders will diminish. In addition, a part of the cash flow is withheld. The following equation for the payments to the shareholders of the levered company can be derived from Fig.
5.1:
Here we can find two tax shields, a corporate income tax shield and a personal income tax shield. In our example we want to focus on a case in which both the indebtedness and the retention are constant over a specified period of time. Hence the following always applies:
Now, let us address the tax shields.

$$\displaystyle \begin{aligned} \widetilde{\mathit{FCF}}^u_t=\widetilde{\mathit{GCF}}_t-\widetilde{\mathit{Tax}}^{C, u}_t-\widetilde{\mathit{Inv}}_t-\widetilde{\mathit{Tax}}^{P, u}_t\,.\end{aligned} $$

(5.1)

$$\displaystyle \begin{aligned} &\widetilde{\mathit{FCF}}^l_t=\widetilde{\mathit{FCF}}^u_t-\widetilde{\,I\,}_t+\widetilde{D}_t-\widetilde{D}_{t-1}-\widetilde{A}_t+(1+\widetilde{r}_{t-1})\widetilde{A}_{t-1}\\ &\qquad \qquad \qquad \quad \qquad \qquad \qquad +\widetilde{\mathit{Tax}}^{C, u}_t-\widetilde{\mathit{Tax}}^{C, l}_t+\widetilde{\mathit{Tax}}^{P, u}_t-\widetilde{\mathit{Tax}}^{P, l}_t\,.\end{aligned} $$

(5.2)

$$\displaystyle \begin{aligned} \widetilde{D}_t=D, \quad \widetilde{A}_t=A\,.\end{aligned} $$

(5.3)

The earnings before taxes determine the corporate income tax. Consequently, analogously to Chap.
2, the following applies:
The personal income tax follows the classical system and is geared to distribution. If we take a closer look at the basic assets, we can see that the shareholder’s tax base of the levered company decreases by the amount of the interest payments and increases by the amount of the proceeds from the retention. Looking at the creditors, however, the tax advantages based on borrowing need to be added. But these are partly with risk and partly without risk. Thus the appropriate tax rate has to be used as follows:
Finally, both equations add up to the entire tax shield of the levered company.

$$\displaystyle \begin{aligned} \widetilde{\mathit{Tax}}^{C, l}_t&= \tau^C \widetilde{EBT}^l_t\\ &= \tau^C \left(\widetilde{EBT}^u_t-r_f D+\widetilde{r}_{t-1}A\right)\\ &=\widetilde{\mathit{Tax}}^{C, u}_t-\tau^Cr_f D+\tau^C \widetilde{r}_{t-1}A.\end{aligned} $$

$$\displaystyle \begin{aligned} \widetilde{\mathit{Tax}}^{P, l}_t&=\widetilde{\mathit{Tax}}^{P, u}_t- \tau^Ir_f D +\tau^D \widetilde{r}_{t-1}A +\tau^I\tau^Cr_f D-\tau^D\tau^C \widetilde{r}_{t-1}A\\ &=\widetilde{\mathit{Tax}}^{P, u}_t- \tau^I\left(1-\tau^C\right)r_f D +\tau^D\left(1-\tau^C\right) \widetilde{r}_{t-1}A\;. \end{aligned} $$

^{4}For this purpose we will concentrate on Eq. ( 5.2) and will take advantage of the fact that indebtedness and distribution remain unchanged over time,
$$\displaystyle \begin{aligned} \widetilde{\mathit{FCF}}^l_t&=\widetilde{\mathit{FCF}}^u_t-r_f D+\widetilde{r}_{t-1}A+\widetilde{\mathit{Tax}}^{C,u}-\widetilde{\mathit{Tax}}^{C,l} +\widetilde{\mathit{Tax}}^{P,u}-\widetilde{\mathit{Tax}}^{P,l}\\ &=\widetilde{\mathit{FCF}}^u_t-\left(1-\tau^I\right)\left(1-\tau^C\right)r_f D+ \left(1-\tau^D\right)\left(1-\tau^C\right) \widetilde{r}_{t-1}A\\ \operatorname*{\mathrm{E}}\!_Q\left[\widetilde{\mathit{FCF}}^l_t|\mathcal{F}_{t-1}\right]&=\operatorname*{\mathrm{E}}\!_Q\left[\widetilde{\mathit{FCF}}^u_t| \mathcal{F}_{t-1}\right]+\left(1-\tau^D\right)\left(1-\tau^C\right)r_f A \\ & \qquad -\left(1-\tau^I\right)\left(1-\tau^C\right)r_f D\;. \end{aligned} $$

The difference of both equations is now easy to determine. All we need to do is to make use of the fundamental theorem (
4.2), which applies to the levered as well as to the unlevered companies.

Again we have to take into account an important detail that we have already mentioned in the previous chapter. We designated the cash flow of the unlevered company, thus this company that conducts a full dividend policy, with
\(\widetilde {\mathit {FCF}}^u_t\). We already pointed out that this merely concerns the cash flow, which accrues to the owners. Hence, the payments (interest and/or amortization) for the creditors have not been included up to now. In order to determine not only the value of equity but moreover the value of the entire company, we need to take into consideration the payments to all investors. It would not be correct to focus just on
\(\widetilde {\mathit {FCF}}^u_t\).

Here we technically have two possibilities: On the one hand we could concentrate on the income of the owner and subsequently add
\(\widetilde {D}_t\). On the other hand we could calculate the payments to all financiers. This can be obtained by adding the interest payments to those accrued from amortization and to the
\(\widetilde {\mathit {FCF}}^u_t\). And then we subtract the personal income tax, which the creditors have to pay. Both methods lead to the same result. This is due to the fundamental theory of asset pricing. Correspondingly this applies to the levered company as well.

According to this, the difference between the values of the companies concerned is equal to the sum of the discounted tax shields plus the market value of the debt,
This describes a more general view of the findings of the precedent chapters (Modigliani–Miller-Theorem
3.7 and Theorem
4.6).

$$\displaystyle \begin{aligned} \widetilde{V}^l_t&=\widetilde{V}^u_t+D+\sum_{s=t+1}^\infty \frac{\operatorname*{\mathrm{E}}_Q\left[\left(1-\tau^D\right)\left(1-\tau^C\right)r_f A-\left(1-\tau^I\right)\left(1-\tau^C\right)r_f D |\mathcal{F}_{t}\right]}{\left(1+r_f\left(1-\tau^I\right)\right)^{s-t}} \\ &= \widetilde{V}^u_t+D+\sum_{s=t+1}^\infty \frac{\left(1-\tau^D\right)\left(1-\tau^C\right) }{\left(1+r_f\left(1-\tau^I\right)\right)^{s-t}}\,r_f A -\sum_{s=t+1}^\infty \frac{\left(1-\tau^I\right)\left(1-\tau^C\right) }{\left(1+r_f\left(1-\tau^I\right)\right)^{s-t}} \,r_f D\\ &= \widetilde{V}^u_t+D+\frac{\left(1-\tau^D\right)\left(1-\tau^C\right) }{r_f\left(1-\tau^I\right)}\,r_f A -\frac{\left(1-\tau^I\right)\left(1-\tau^C\right) }{r_f\left(1-\tau^I\right)} \,r_f D\\ &=\widetilde{V}^u_t+\frac{\left(1-\tau^D\right)\left(1-\tau^C\right)}{1-\tau^I}A+\tau^CD\,. \end{aligned} $$

(5.4)

## 5.3 Conclusion

In the previous paragraph we evaluated a company assuming that taxes are raised at the company’s level as well as at the investor’s level. We could formulate a simple valuation equation under the special conditions concerning the lifespan of the company as well as its debt and dividend policies. The question remains how to proceed if the debt and dividend policies seriously deviate from the conditions which we had originally assumed.

For this purpose we will analyze the Eq. (
5.4). It permits us (according to the special assumptions which we have made) to make a statement of appraisal on the difference in value between the levered and unlevered company. Given different assumptions concerning the debt and dividend policies, one can also easily modify the equation. One only has to identify the tax advantages, which are associated with specific debt and dividend policies to determine the corresponding equation. The expected values in such an equation would be valid for the future amounts of debt
\(\widetilde {D}_t\) and the future amounts of retention
\(\widetilde {A}_t\). But in any case they have to be calculated with the risk neutral probability measure
Q. And since the person who, in practice, has to evaluate a company is usually unaware of this measure we would have an elegant but, nevertheless, useless valuation equation.

In a first step in acquiring a valuation equation (while taking into account the subjective probability measure) it is important to formulate a linear correlation between the future amounts of debt
\(\widetilde {D}_t\) and the future amounts of retention
\(\widetilde {A}_t\) as well as the cash flow of the unlevered company
\(\widetilde {\mathit {FCF}}^u_t\).
In this case we need to assume that current amount of debt and current amount of retention are given as known factors. The parameter
x

^{5}If we cope with this task fairly successfully the valuation equation can be expressed by the following formula:
$$\displaystyle \begin{aligned} V^l_0=V^u_0+x_0D_0+x^{\prime}_0A_0+\frac{x_1\operatorname*{\mathrm{E}}_Q\left[\widetilde{\mathit{FCF}}^u_1\right]}{\left(1+r_f\left(1-\tau^I\right)\right)^2}+\cdots+ \frac{x_{T-1}\operatorname*{\mathrm{E}}_Q\left[\widetilde{\mathit{FCF}}^u_{T-1}\right]}{\left(1+r_f\left(1-\tau^I\right)\right)^T}. \end{aligned}$$

_{t}are the expressions of any deterministic variables, which describe the linear correlation between the amounts of debt and the amounts of retention on the one hand, and the cash flow of the unlevered firm on the other hand. Unfortunately we cannot characterize them more precisely in our final general view. With this we will have retraced the evaluation of the tax shield to an ascertainment of expected values of future cash flow under Q.
Finally, we need to eliminate the risk neutral probability measure. For this we avail ourselves in a second step of the assumption that the free cash flow of the unlevered company is going to be weak auto-regressive. Under these conditions the cost of capital of the unlevered company imply suitable discount rates. Finally, we can ascertain realistic valuation equations, which rely just on known variables (subjectively expected cash flow, cost of capital of the unlevered company, tax rates, and interest rates).

Consequently, if we proceed in the described manner, we will be able to evaluate companies operating with other financing and dividend policies. Initially we always have to identify the ensuing tax advantages. Subsequently the future amounts of debt
\(\widetilde {D}_t\) and the future amounts of retention
\(\widetilde {A}_t\) need to be linked to the future cash flow of the unlevered company with a linear correlation. Afterwards we can evaluate the expected values of this cash flow in that the cost of capital of the unlevered company can also be used as discount rates.

But the limits of our approach are quite in evidence. Every time we fail to construct a linear correlation the concept collapses. Such situations are easy to imagine. One just has to think of cases in which the future investments of a company follow a stochastic process which is independent of the cash flow and in which the manager commit themselves to trade the investments exclusively on the equity. In such a case it would not be possible to depict the future amounts of debt with the cash flows in a linear correlation. And our approach would not reap the desired result in which the tax shield is determined solely by the cost of capital and the subjectively expected values of future cash flows.

## 5.4 Problem(s)

1.

Evaluate the value of the levered company in the infinite example using all our assumptions (from autonomous debt and retention) so far.

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