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Open Access 2020 | OriginalPaper | Chapter

# 4. Personal Income Tax

Authors: Lutz Kruschwitz, Andreas Löffler

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## Abstract

We now shift gears. While in the last chapter we worked on the basis that the firm was taxed, but that the financiers were free from taxes, we now suppose that the financiers have to pay income tax, but that the firm is spared.
We now shift gears. While in the last chapter we worked on the basis that the firm was taxed, but that the financiers were free from taxes, we now suppose that the financiers have to pay income tax, but that the firm is spared.
Certainly not all readers will think that it makes sense in the valuation of firms to take taxes due at the financiers’ level into consideration. The appropriate textbooks at any rate like to leave out income tax all too well.1 If, however, you keep in mind that income tax influences the consumption flow of private investors in every case, then there is hardly any reason why this tax is not taken into consideration in valuation of firms. Somebody who has acquired a firm will have other numbers to enter in on her income tax statements than someone who invests her money on the capital market. Just this fact itself speaks for including the income tax in the valuation of firms. The German profession of certified public accountants, for instance, officially decided in 1997 to advocate the consideration of income tax in valuation of firms.2
The reader can expect a timetable in this chapter very close in structure to the plan of action in the previous chapter. We will shortly recognize that despite clear differences, there exist enough similarities between corporate income tax and personal income tax to justify treating them structurally in the same way. So, surely to the surprise of some readers, we will again speak of levered and unlevered firms. We will use the same symbol for the tax rate and also look into the different firm policies again in order to value the resulting tax advantages of a levered firm versus an unlevered firm.
We will certainly not surprise our readers by the statement that considerable new ground is being broken with the inclusion of personal income tax in theory of valuation. The first work on WACC and APV appeared already half a century ago. In contrast, international literature on valuation of firms has very often ignored income tax on the level of the financiers up to present. We thus have considerably less literature on which to build. So being, this chapter cannot deal with the systematic presentation of available knowledge. We will, much rather, have to compile new results. In doing so, we cannot check to see if we are moving in the right direction by comparing our results with the outcomes of other papers. We thus see the following chapter not as an attempt at presenting already existing knowledge, but much rather as a contribution to the theoretical discussion of income tax and valuation of firms within the DCF theory.

## 4.1 Unlevered and Levered Firms

DCF theory, in essence, continually deals with the question as to how tax shields are appropriately valued. If talk of a tax advantage (or also of a tax disadvantage) is to be economically substantial, a reference point is needed against which the advantage (or disadvantage) can be measured. This reference point concerns a firm which pursues a very definite policy, a firm which we will say is unlevered.

### 4.1.1 “Leverage” Interpreted Anew

Do you remember the beginning of the previous chapter? We had supposed there that the firm has to pay taxes, but the financiers remained free from tax. We had made it clear there that a levered firm is less heavily burdened with taxes than is an unlevered. We had further yet considered that you can immediately understand what an unlevered firm is supposed to be without any further details, but much more detailed information is needed in order to exactly comprehend what a levered firm is.3 We had agreed at the beginning of the previous chapter to refer to the non-debted firm as unlevered and to characterize the indebted firm with the adjective levered.
Reference Firm
We are now dealing with a completely different tax situation. The firms remain free from tax, while the financiers will be enlisted to pay income tax. If you bear in mind that from the basis of income tax, dividends, and interest payments are the object of taxation, then a lot of income tax is due to a firm which has a policy of full distribution. An income taxes savings, in contrast, is expected if the firm only partially distributes the profits obtained to the financiers. Profit retention brings about a tax shield for the investors. It is therefore recommended to use the firm with full distribution as reference firm, since no other distribution activity makes what we are talking about immediately clear. Whoever, in contrast, has to value a firm with partial distribution, must very precisely describe which share of the cash flows is to be withheld in which periods. It is completely clear, in contrast, what is designated by full distribution. Since more income tax has to be paid in the case of full distribution than in the case of reduced distribution, we will speak in the first case of an unlevered firm, and in the second of a levered firm.
Please observe that we use the terms levered differently in the current chapter than in the preceding chapter. Levered now no longer means indebted, but rather partial distribution. The capital structure of the firm does not play any role in this chapter. Leverage in the ongoing chapter relates solely to the question if and how much is retained in the firm. Although this may possibly be somewhat irritating for readers, we purposely chose the terminology in order to be able to make use of structural similarities in the previous chapter.
In the previous chapter a full retention policy was assigned to the unlevered and the levered firm as well. Any other possible retention policies were not object of our discussion. In this chapter we argue similarly. Now both companies, the unlevered (or full distributing) firm as well as the levered (or partially distributing) firm, are self-financed and therefore without debt. The financing policy is not object of our discussion. How financing and retention policy can be linked together will be the topic of the last chapter. Seen in this light, the unlevered firm of the previous and this chapter is identical: both are without debt and both fully distribute their cash flow to the owners.
If we interpret the notation anew in the sense explained here, then it should also apply that a firm with full distribution has the same value as a firm with partial distribution, as long as no taxes are imposed. That agrees with the theorem of Miller and Modigliani (1961) on the irrelevance of dividend policy. This theorem says that it does not matter when a firm distributes its earnings so long as taxes do not play a role.
Notation
We will note the market value of the unlevered firm with $$\widetilde {V}^u_t$$ and the market value of the levered firm with $$\widetilde {V}^l_t$$, respectively. Please note that both firms are not indebted, hence both values coincide with the market value of the equity, respectively. And we will completely analogously use the symbol $$\widetilde {\mathit {FCF}}^u_t$$ for the post-tax free cash flows of the firm with full distribution, while the post-tax free cash flows of the firm with partial distribution are referred to by $$\widetilde {\mathit {FCF}}^l_t$$. Correspondingly, $$\widetilde {\mathit {Tax}}^u_t$$ has to do with the shareholders’ taxes of a firm with full distribution, but $$\widetilde {\mathit {Tax}}^l_t$$ with the shareholders’ income tax of a firm with partial distribution.
Notice another important difference between the previous and present chapters. Previously $$\widetilde {\mathit {FCF}}^l_t$$ described the free cash flow that accrued to all investors (to shareholders as well as to debt holders). Now, there are no debt holders. In contrast with the previous chapter, here the free cash flow exists only as payment flow, which accrues to the shareholders.
Positive Dividends
If we work on the grounds that a firm’s cash flows are dealing with payments from which the dividends to the owners are defrayed, then a further limitation results. If in the last chapter cash flows turned negative, it meant nothing more than that the financiers infused the firm with further equity. If this was no longer possible, we then spoke of default and in more than one section we specifically expanded upon the implications of such a situation. But negative payments do not make any sense in the case of dividends. Thus in the following we will presuppose that the (levered as well as the unlevered) firm’s cash flows are always large enough so that the dividends cannot turn negative.

### 4.1.2 The Unlevered Firm

In this chapter firms with full distribution play the same role as firms without debt in the chapter on corporate taxes. We assume that firms with full distribution are just as seldom the case in economic reality as self-financed firms. Nonetheless, it is important to be able to value them. Just as we maintained in the last chapter that you can only value an indebted firm if you are also capable of valuing a firm that is self-financed, we can now maintain that you can only value a firm with partial distribution if you can find a way of valuing a firm with full distribution.
If the cost of equity and the free cash flows of a firm with full distribution are known, it is very simple to write down a correct valuation equation. To do this we first define the cost of equity.
Definition 4.1 (Cost of Equity)
Cost of equity $$\widetilde {k}^{E,u}_t$$ of an unlevered firm are conditional expected returns
\displaystyle \begin{aligned} \widetilde{k}^{E,u}_t:=\frac{\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_{t+1}+\widetilde{V}^u_{t+1}|\mathcal{F}_t\right]}{\widetilde{V}^u_t}-1. \end{aligned}
Since these cost of equity are formally not different from the cost of equity of an unlevered firm according to Definition 3.​1, the valuation equation for a firm with full distribution naturally results very easily. If we again assume that the cost of equity are deterministic, then the valuation equation looks exactly like the corresponding valuation equation for a fully self-financed firm according to Theorem 3.​1.
Theorem 4.1 (Market Value of the Unlevered Firm)
If the cost of equity of the unlevered firm $$k^{E,u}_t$$ are deterministic, then the value of the firm, which fully distributes its free cash flows, amounts at time t to
\displaystyle \begin{aligned} \widetilde{V}^u_t=\sum_{s=t+1}^T\frac{\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_{s}|\mathcal{F}_t\right]}{\left(1+k^{E,u}_{t}\right) \ldots\left(1+k^{E,u}_{s-1}\right)}\,. \end{aligned}
We do, however, find it important to point out a fact that may take some time getting used to for one or more readers. The cost of equity kE, u are dealing with the cost of equity post- taxes, and not say pre-taxes ! It is just as important for us to ascertain that we avoid every statement about the connection between the pre-tax and the post-tax cost of equity. We much rather see the cost of equity post-taxes as simply given. We will later come back to a likely relation between both quantities in still more detail.4
In the chapter on corporate income tax, it was necessary to be able to fall back upon a premise that we had designated as the assumption of weak auto-regressive cash flows.5 In order to develop valuation equations and also, above all, adjustment equations, we used the Theorem 3.​3 many times in the chapter on corporate tax. We then need a commensurate theorem when dealing with personal income tax. But in order to get the theorem, we first have to introduce the assumption on weak auto-regressive cash flows. Thus:
Assumption 4.1 (Weak Auto-Regressive Cash Flows)
There are real numbers g t such that
\displaystyle \begin{aligned} \operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_{t+1}|\mathcal{F}_t\right]=(1+g_{t})\,\widetilde{\mathit{FCF}}^u_t \end{aligned}
is valid for the unlevered firm’s cash flows.
Since we have already discussed the economic significance of this assumption above, we do not need to go into it again. Before we can begin with proving specific valuation equations, we have to complete a series of preparatory steps. We must first describe the tax which is at the center of this chapter’s discussion in more detail. We must furthermore go into the question as to what happens to the fundamental theorem of asset pricing when we have to take personal income tax into consideration.

### 4.1.3 Income and Taxes

Economists usually describe a tax type by saying who pays the tax, how the tax base is established, and which tariff is to be applied. Individuals are always subject to tax.
Categories of Income
In most countries of the world, income tax is measured according to an amount, that as a rule is not so easy to calculate since very detailed legal provisions must be observed. The core of this amount is comprised of the sum of the so-called income. For the present we will differentiate between the incomes of the owners and the creditors, although no creditors are present in our model yet.
1.
Owners’ incomes can mean the firm’s achieved earnings or the firm’s dividends. If the firm’s achieved earnings are taxed, and in fact taxed regardless of whether these earnings are distributed or withheld, then it is “accrued income” that forms the tax base. If, in contrast, the cash which the shareholders receive is the object of taxation, then “realized income” is spoken of. Income from shares, which are traded on capital markets, is always included in the second group.

2.
When we speak of the creditors’ incomes, we must think of interest. In many countries interest income and income from dividends are taxed differently.

Redemption of Capital
Owners sometimes receive payments that do not have the character of dividends. Think of the repayment of capital in relation to capital reductions or the liquidation of firms. It is important not to mix such payments up with dividends since they are, as a rule, spared from the burden of income tax.
Earnings Retention
It is possible to retain parts of the distributable cash flows in the firm. We write $$\widetilde {A}_t$$ for these earnings retention amounts. The earnings retention amounts are always non-negative. We proceed on the basis that these amounts are invested by the firm for one period at the (later more precisely described) interest rate of $$\widetilde {r}_{t}$$, to then be distributed to the financiers.
Table 4.1 describes how to get from the pre-tax gross cash flows of a firm to the levered taxable income. With that we get the term given in the last line of Table 4.1 for the financiers’ taxable income. The owners must, however, still pay their taxes from this: the unlevered as well as the levered free cash flows result from the respective taxable income minus the tax payments.
Table 4.1
From pre-tax gross cash flow to income
 Pre-tax gross cash flow $$\widetilde {\mathit {GCF}}_t$$ − Investment expenses $$\widetilde {\mathit {Inv}}_t$$ = Shareholder’s unlevered taxable income $$\widetilde {\mathit {GCF}}_t-\widetilde {\mathit {Inv}}_t$$ − Retained earnings $$\widetilde {A}_t$$ + Reflux from retained earnings $$(1+\widetilde {r}_{t-1})\widetilde {A}_{t-1}$$ = Shareholder’s levered taxable income $$\widetilde {\mathit {GCF}}_t-\widetilde {\mathit {Inv}}_t$$ $$\quad -\widetilde {A}_t+(1+\widetilde {r}_{t-1})\widetilde {A}_{t-1}$$
In this figure the gross cash flows and the investment expenses are identical for the levered and the unlevered firm. The unlevered firm, however, does not have retained earnings.
Yield of the Retained Earnings Amounts
Under what conditions does the firm invest the amount $$\widetilde {A}_t$$? To answer this question we want to recall that managers—independent of distribution policy—should institute every investment project with positive net present value. Investing the amount $$\widetilde {A}_t$$ in operating assets is as impossible as that of the principle repayment of debts or the mark-down of equity. The only remaining possibility is that of investing on the capital market.
Investing on the capital market can now turn out to be riskless, but it can also be risky. And one may suppose that it makes a difference as to which variants we abide by here. If personal income taxes are not (yet) taken into account we will, however, soon recognize that due to the fundamental theorem we do not have to take this into consideration. rf shall denote the risk free interest rate before income tax is deducted. If the amount $$\widetilde {A}_t$$ is invested in riskless assets, then the following simply applies:
\displaystyle \begin{aligned} \widetilde{r}_t=r_f . \end{aligned}
(4.1)
In case of a risky investment by the firm, the following happens: The firm invests the amount $$\widetilde {A}_t$$ and receives back the amount $$(1+\widetilde {r}_t)\widetilde {A}_t$$ one period later. If the capital market is arbitrage free, then the reflux at time t + 1 must be just as great as the cash value on the money investment at time t. We use the fundamental theorem to determine this cash value. But since the investment is carried out by the firm and according to our conditions the firm does not pay taxes, we must apply the fundamental theorem here according to Theorem 2.​2, which is valid in a world where companies do not pay taxes. The following results:
\displaystyle \begin{aligned} \widetilde{A}_t = \frac{\operatorname*{\mathrm{E}}_Q \left[ (1+\widetilde{r}_t)\widetilde{A}_ t |\mathcal{F}_t \right]}{1+r_f}\;. \end{aligned}
With rules 2 and 5, we then get
\displaystyle \begin{aligned} \text{E}_{Q} \left[ \widetilde{r}_t |\mathcal{F}_t \right] = r_f. \end{aligned}
(4.2)
That is a generalization of Eq. (4.1). In the following we will proceed from this relation.
Tax Rates for Interest and Dividends
Just as with the firm tax, the tax rate is again linear. There are therefore neither exemptions nor exemption thresholds. As in the preceding section, the tax rate is certain as well as constant. We already clearly said in the previous chapter that this presents a heavy, but unfortunately necessary assumption. Furthermore, tax rates on dividends and interest differ: the tax rate on dividends will be denoted by τD, the tax rate on interest is τI. Although our firms are self-financed, this difference is of relevance in proving our fundamental theorem which will become clear in a moment.
We need to take the following into consideration: Let us assume that an investor has an amount of G at her disposal. There are two different possibilities to invest this sum securely. Either she could now keep G in his private means in order to acquire an asset of $$G+\left (1-\tau ^I\right )r_f G$$ for the course of one time period or she pays this into her company and invests it there. In this case her assets will at first increase to G + rfG during the next time period. Once distributed to the entrepreneur it is crucial as to how this additional cash flow is taxed. If the interest share of the additional cash flow is treated like a dividend, the investor will get $$G+\left (1-\tau ^D\right )r_f G$$ from the safe investment after income tax. Obviously this amount differs from $$G+\left (1-\tau ^I\right )r_f G$$ which we ascertained beforehand, unless, of course, the tax rate for interest and dividends happens to be identical. It is now quite easy to formulate a possibility for arbitrage from the disparity of the two values (even when considering only self-financed firms). This would make our model superfluous. To avoid this, we only have two possibilities.
1.
We could assume that
\displaystyle \begin{aligned} \tau^I=\tau^D \end{aligned}
applies.6 Unfortunately it can easily empirically be verified that many industrial nations of the world do not currently match up to the identity we presumed. So we will not consider this particular option.

2.
The second possibility consists in defining the assessment basis of the income tax in a different way than in our arbitrage example. For this purpose we need to define a capital market, which takes into account our considerations more precisely. We are assuming a fixed quantity of convertible assets to which we shall refer as basic assets.7 One of these basic assets is risk free, all the others are risky. Additionally, every company and every investor can only dispose of a portfolio of these basic assets.

The formulation of such an assessment basis is thus crucial to the elimination of the arbitrage opportunities. Any investment in the capital market is geared to the portfolio of basic assets. Furthermore, payments from the risk free asset will always be taxed by τI, while payments from the risky asset will be taxed by τD. Such a definition has far reaching consequences. Let us assume an investor chooses to invest risk free not privately but rather into a company. Once the returns of this investment are distributed then the return will not be taxed as a dividend but will be treated like interest. Ultimately it can be traced back to the risk free basic assets. With such a definition of the assessment basis we avoid the arbitrage opportunity mentioned above.
Tax Equation
Our model’s income tax equation for shareholders can now be written principally in the form
\displaystyle \begin{aligned} \widetilde{\mathit{Tax}}^u_t = \tau^D \, \left(\widetilde{\mathit{GCF}}_t-\widetilde{\mathit{Inv}}_t\right) \end{aligned}
for the unlevered firm.
As far as the levered company is concerned, it is not quite as easy to deal with this issue. Much depends on how the amount $$\widetilde {A}_t$$ has been invested: because a risk free financial investment will be taxed differently as opposed to an investment of risky securities. In this chapter we shall assume that the retention investment will be risky. The owners of the levered firm together pay taxes in the amount of8
\displaystyle \begin{aligned}\widetilde{\mathit{Tax}}^l_t = \tau^D \, \left(\widetilde{\mathit{GCF}}_t-\widetilde{\mathit{Inv}}_t -\widetilde{A}_t+(1+\widetilde{r}_{t-1})\widetilde{A}_{t-1} \right) . \end{aligned}
The two firms tax payments differ by one amount, which we, analogously to the chapter on firm tax, want to designate as tax shield.
Tax Shield
If we want to determine the cash flows of the levered firm in t, we do not only have to observe the earnings retention at time t, but also the earnings retention from the previous period. In total the tax shield amounts to
\displaystyle \begin{aligned} \widetilde{\mathit{Tax}}^l_{t}-\widetilde{\mathit{Tax}}^u_{t} = \tau^D \, \left( -\widetilde{A}_t+(1+\widetilde{r}_{t-1})\widetilde{A}_{t-1} \right). \end{aligned}
In order to calculate the difference between the free cash flows of the two firms, we take into consideration that gross cash flows and capital repayments as well as investments all result in identical amounts in both firms. The one and only difference between the two firms is the single fact that one renounces all earnings retention measures, while the other institutes such measures, resulting in a different amount of income tax. We can then entirely concentrate on the retained earnings amounts and tax payments in calculating the difference between the free cash flows of the levered firm and the free cash flows of the unlevered firm,
\displaystyle \begin{aligned} \widetilde{\mathit{FCF}}^l_{t}-\widetilde{\mathit{FCF}}^u_{t} &= \left(\ldots-\widetilde{A}_t+(1+\widetilde{r}_{t-1})\widetilde{A}_{t-1} -\widetilde{\mathit{Tax}}^l_{t}\right)-\left(\ldots-\widetilde{\mathit{Tax}}^u_{t}\right)\\ &= \left(1-\tau^D\right)\left((1+\widetilde{r}_{t-1})\widetilde{A}_{t-1}-\widetilde{A}_t\right). \end{aligned}
With the help of (4.2) and rule 5 we get for the expectation under risk-neutral probability
\displaystyle \begin{aligned} \operatorname*{\mathrm{E}}\nolimits_Q \left[\widetilde{\mathit{FCF}}^l_{t}-\widetilde{\mathit{FCF}}^u_{t}|\mathcal{F}_{t-1}\right] = \left(1-\tau^D\right)\left(1+r_f\right)\widetilde{A}_{t-1}-\left(1-\tau^D\right)\operatorname*{\mathrm{E}}\nolimits_Q \left[\widetilde{A}_t|\mathcal{F}_{t-1}\right]. \end{aligned}
(4.3)
A tax shield then comes about if the firm does away with fully distributing the free cash flows.

### 4.1.4 Fundamental Theorem

In the last chapter we made thorough use of the fundamental theorem of asset pricing. We had already introduced this theorem in the first chapter of this book, since it is of such central importance for the derivation of valuation equations.
The fundamental theorem says that under the condition of an arbitrage free capital market, risk-neutral probabilities Q exist. Risk-neutral expectations can thus be discounted in a world without taxes with the riskless interest rate rf. The correctness of this statement does not change if a corporate income tax is entered into the model. This is because in taxation, which is only affective at the level of the firm, the riskless interest is identical pre- and post-tax. But now we are dealing with taxation at the financiers’ level. And whoever invests riskless money as a financier and is at the same time liable to pay taxes, no longer attains net returns in the amount of rf, but rather a return in the amount of $$r_f\left (1-\tau ^I\right )$$. What comes of the fundamental theorem under these conditions? Do risk-neutral probabilities Q still exist? And if so, how are risk-neutral expectations to be discounted?
For the value of any discretionary portfolio from risky and riskless assets as well the following theorem now applies.
Theorem 4.2 (Fundamental Theorem with Different Taxation of Dividends and Interest)
If the capital market with a personal income tax is free of arbitrage, the conditional probabilities Q can be chosen to the extent that the following result is valid:
\displaystyle \begin{aligned} \widetilde{V}_t=\frac{\operatorname*{\mathrm{E}}_Q\left[\widetilde{\mathit{FCF}}_{t+1}^{\mathit{\text{post-tax}}}+\widetilde{V}_{t+1}|\mathcal{F}_t\right]}{1+r_f\left(1-\tau^I\right)}\, \,. \end{aligned}
The tax rate for returns on interest is found in the denominator even if it includes assets with risky returns. This result is neither trivial nor is it immediately evident. Due to the fact that proofs like this cannot be found in the relevant literature, we nevertheless felt that we had no other option than to publish it here. For further information please refer to the appendix.9 Now we shall continue to develop results analogous to the ones we have proved in the previous two chapters.
Theorem 4.3 (Williams/Gordon-Shapiro Formula)
If the cost of equity are deterministic and the assumption on weak auto-regressive cash flows holds, then for the value of the unlevered firm
\displaystyle \begin{aligned} \widetilde{V}^u_t=\frac{\widetilde{\mathit{FCF}}^u_t} {d^u_t} \end{aligned}
holds for deterministic $$d^u_t$$.
Theorem 4.4 (Equivalence of the Valuation Concepts)
If the cost of equity are deterministic and the assumption on weak auto-regressive cash flows holds, then the following is valid for all times s > t
\displaystyle \begin{aligned} \frac{\operatorname*{\mathrm{E}}\nolimits_Q\left[\widetilde{\mathit{FCF}}^u_{s}|\mathcal{F}_t\right]}{\left(1+r_f\left(1-\tau^I\right)\right)^{s-t}}=\frac{\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_{s}|\mathcal{F}_t\right]}{\left( 1+k^{E,u}_{t}\right)\ldots\left(1+k^{E,u}_{s-1}\right)} . \end{aligned}
In terms of the value of equity of the unlevered firm, cost of equity and discount rates are the same. But the taxed interest rate $$r_f\left (1-\tau ^I\right )$$ now appears in place of the riskless interest rate rf. The cost of equity kE, u we are now dealing with is also a post-tax variable. We do, however, avoid establishing an explicit relation to the levered firm’s pre-tax cost of equity here.
We do not really need to prove the two theorems here for a second time. Instead, we refer our readers to the applicable pages in the chapter on the corporate tax.10

### 4.1.5 Tax Shield and Distribution Policy

In this section we want to characterize the difference in value between an unlevered and a levered firm.
Let us begin with the firm with full distribution. From Theorem 4.1 in relation to (4.3), we immediately get the representation
\displaystyle \begin{aligned} \widetilde{V}^u_t=\frac{\operatorname*{\mathrm{E}}_Q\left[\widetilde{\mathit{FCF}}^u_{t+1}|\mathcal{F}_t\right]}{1+r_f\left(1-\tau^I\right)}+\ldots +\frac{\operatorname*{\mathrm{E}}_Q\left[\widetilde{\mathit{FCF}}^u_{T}|\mathcal{F}_t \right]}{\left(1+r_f\left(1-\tau^I\right)\right)^{T-t}}\;. \end{aligned}
We get the value of a levered firm in the exact same way from
\displaystyle \begin{aligned} \widetilde{V}^l_t=\frac{\operatorname*{\mathrm{E}}_Q\left[\widetilde{\mathit{FCF}}^l_{t+1}|\mathcal{F}_t\right]}{1+r_f\left(1-\tau^I\right)}+\ldots +\frac{\operatorname*{\mathrm{E}}_Q\left[\widetilde{\mathit{FCF}}^l_{T}|\mathcal{F}_t \right]}{\left(1+r_f\left(1-\tau^I\right)\right)^{T-t}}\,.\end{aligned}
Yet, we still have to think about how the free cash flows from the firm with partial distribution differ from those of the unlevered firm. We thereby regard the following principle: The first earnings retention takes place in t. It is economically unsuitable to forgo with distributions at the last time t = T. From that results $$\widetilde {A}_T=0$$. If we compare the value of the levered and unlevered firm, we then get by applying the rules 4 as well as 5
\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \quad \widetilde{V}^l_t=\widetilde{V}^u_t+ \frac{\left(1-\tau^D\right)\operatorname*{\mathrm{E}}_Q\left[\left(1+r_f\right)\widetilde{A}_{t}-\widetilde{A}_{t+1}|\mathcal{F}_t\right]}{1+r_f\left(1-\tau^I\right)}+\ldots+\\ &\displaystyle &\displaystyle +\frac{\left(1-\tau^D\right)\operatorname*{\mathrm{E}}_Q\left[\left(1+r_f\right)\widetilde{A}_{T-2}-\widetilde{A}_{T-1}|\mathcal{F}_t\right]}{1+r_f\left(1-\tau^I\right)^{T-t-1}}+ \frac{\left(1-\tau^D\right)\operatorname*{\mathrm{E}}_Q\left[\left(1+r_f\right)\widetilde{A}_{T-1}|\mathcal{F}_t\right]}{(1+r_f(1-\tau^I))^{T-t}}\;. \end{array} \end{aligned}
After some minimal reshuffling, the following results:
\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \widetilde{V}^l_t=\widetilde{V}^u_t +\frac{\left(1-\tau^D\right)\left(1+r_f\right)\widetilde{A}_t}{1+r_f\left(1-\tau^I\right)}+ \frac{\operatorname*{\mathrm{E}}_Q\left[\frac{\left(1+r_f\right)\left(1-\tau^D\right)}{1+r_f\left(1-\tau^I\right)}\widetilde{A}_{t+1}-\left(1-\tau^D\right)\widetilde{A}_{t+1}| \mathcal{F}_t\right]}{1+r_f\left(1-\tau^I\right)}+\\ &\displaystyle &\displaystyle \qquad \qquad \qquad \qquad \qquad +\ldots+ \frac{\operatorname*{\mathrm{E}}_Q\left[\frac{\left(1+r_f\right)\left(1-\tau^D\right)}{1+r_f\left(1-\tau^I\right)}\widetilde{A}_{T-1}- \left(1-\tau^D\right)\widetilde{A}_{T-1}|\mathcal{F}_t\right]}{\left(1+r_f\left(1-\tau^I\right)\right)^{T-t}}\;. \end{array} \end{aligned}
This brings us to the conclusion
\displaystyle \begin{aligned} &\widetilde{V}^l_t=\widetilde{V}^u_t+\left(1-\tau^D\right)\widetilde{A}_t+\frac{\tau^I\left(1-\tau^D\right)r_f\operatorname*{\mathrm{E}}_Q\left[\widetilde{A}_{t}| \mathcal{F}_t\right]}{1+r_f\left(1-\tau^I\right)}+\\ &\qquad \quad \qquad \qquad \qquad \qquad \qquad \ldots+ \frac{\tau^I \left(1-\tau^D\right)r_f\operatorname*{\mathrm{E}}_Q\left[\widetilde{A}_{T-1}|\mathcal{F}_t\right]}{\left(1+r_f\left(1-\tau^I\right)\right)^{T-t}}. \end{aligned}
(4.4)
This equation shows itself to be the personal income tax pendant to Eq. (3.​11). In place of debt $$\widetilde {D}_t$$, $$\left (1-\tau ^D\right )\widetilde {A}_t$$ simply enters in, that being the amount by which the maximum distribution to the financiers is reduced.
Alternative Retained Earnings Policies
If we think the tax shields through to a consequential end, then we would have to advise every firm to hold off the distribution only so long as possible for tax reasons. There is nothing to object to this recommendation within the framework of our model. We could have also argued accordingly in the last chapter. It was the debt which brought a tax shield there. And it would have then made sense to recommend that the firm allow for the maximum debt ratio permissible. As we did there, we here refrain from such recommendations since we very well know that suitable advice is unreasonable if it is solely based on tax considerations. That is also why we want to master the situation differently here. We take the firm’s distribution policy—just as we did the debt policy in the last chapter—as a given and question how it affects the value of the firm.
In the following we will more clearly analyze five distribution policies. Since distribution and retention are always complementary measures, we can naturally speak of alternative retention policies as well. This involves a phraseology in relation to the subsequent characterizations which can more easily be remembered.
1.
With the autonomous retention, a certain amount is retained each period.

2.
With the cash flow based earnings retention, a certain amount of the free cash flows is deducted each period.

3.
With the dividend based earnings retention, the amounts to be deducted are so chosen that in the first n periods it comes to a distribution of a fixed dividend.

4.
With the market value based earnings retention, the deducted amounts are so chosen that the relation of earnings retention and equity value remains deterministic.

Another potential dividend policy of relevance can be of consequence if we consider disbursement stoppages. In many countries statutory provisions allow that, at the very most, the earnings are distributed and nothing more. This applies as well even if the free cash flow should exceed this amount. Unfortunately, these disbursement stoppages are very difficult to manage. Therefore we will refrain from discussing them.

### 4.1.6 Example (Continued)

We also want to use the data from the two examples in the previous chapter in terms of the personal tax. We assume that the tax rates on dividend and interest will coincide and be denoted by τ. Since the expositions in this chapter almost completely formally correspond to those of the last, we could simply repeat the previous chapter’s calculations here once again. we would just have to interpret the respective variables differently: with $$\widetilde {\mathit {FCF}}^u_t$$ as cash flows post income tax, kE, u as taxed cost of equity and τ as income tax rate, the calculations take shape for the finite as well as infinite example, but formally just as in the Sects. 2.​2.​3 and 3.​1.​3.
In the infinite case, we would—despite the formal agreement with the previous chapter’s concept—indeed fumble into a trap. Analogously to Sect. 3.​1.​3, we could determine the risk-neutral probabilities Q1(d) and Q1(u) for a certain time period. Now the outcomes no longer agree with the values calculated for the infinite case in Sect. 3.​1.​3, because the fundamental theorems for the case of the corporate income tax and the personal income tax differ from each other. In the first case, the riskless interest rf is to be calculated with, and in the other case it is the riskless tax interest rf(1 − τ). It now appears that with the data constellation we have chosen, the probabilities would be negative and that correlates to an arbitrage opportunity!11
We will therefore suppose a cost of equity rate of kE, u = 15 % in the following. It can be seen that with such a cost of equity rate the arbitrage opportunity vanishes. With cost of equity in the amount of kE, u = 15 %, the value of the unlevered firm in the infinite example amounts to
\displaystyle \begin{aligned} \text{V}^u_0&= \frac{\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_1\right]}{1+k^{E,u}}+ \frac{\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_2\right]}{(1+k^{E,u})^2}+\frac{\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_3\right]}{(1+k^{E,u})^3}\\ &=\frac{100}{1.15}+\frac{110}{1.15^2}+\frac{121}{1.15^3}\approx 249.692. \end{aligned}
The values of $$\widetilde {V}^u_1$$ and $$\widetilde {V}^u_2$$ have to be determined anew.
We can take the other original numerical values for the infinite example without restriction.

### 4.1.7 Problems

1.
In Sect. 3.​1.​3 we were able to evaluate the risk-neutral probabilities Q1(d) and Q1(u) for the finite example.12 Show that
\displaystyle \begin{aligned}Q_1(u) \approx -0.125, \qquad Q_1(d) \approx 1.125 \end{aligned}
if a personal income tax with τ = 50 % is present.
Verify that for kE, u = 15 % this arbitrage opportunity vanishes and determine Q1(u) and Q1(d). Determine Q2(dd), Q2(du), Q2(ud), and Q2(uu).

2.
Prove that the tax shield $$\widetilde {V}^l_t-\widetilde {V}^u_t$$ in the case of the personal income tax satisfies
\displaystyle \begin{aligned} \widetilde{V}^l_{t}-\widetilde{V}^u_{t}= \frac{\operatorname*{\mathrm{E}}_Q\left[\widetilde{V}^l_{t+1}-\widetilde{V}^u_{t+1}|\mathcal{F}_t\right]}{1+r_f(1-\tau^I)} +(1-\tau^D)\frac{\operatorname*{\mathrm{E}}_Q\left[(1+r_f)\widetilde{A}_t-\widetilde{A}_{t+1}|\mathcal{F}_t\right]}{1+r_f(1-\tau^I)}\,. \end{aligned}

3.
Similar to problem 4 show that the main valuation Eq. (4.4) can be written as
\displaystyle \begin{aligned} \widetilde{V}^l_t =\widetilde{V}^u_t+\frac{1-\tau^D}{1-\tau^I}\widetilde{A}_t+\frac{\tau^I\left(1-\tau^D\right)}{1-\tau^I}\sum_{s=t+1}^T \frac{\operatorname*{\mathrm{E}}_Q\left[\widetilde{A}_{s}-\widetilde{A}_{s-1}|\mathcal{F}_t\right]}{(1+r_f\left(1-\tau^I\right))^{s-t}}. \end{aligned}

## 4.2 Excursus: Cost of Equity and Tax Rate

Statement of the Problem
In this section we will go into the question which until now has been left out. The question goes: How do a firm’s cost of equity change when the income tax rate changes? And further: What can we say about the value of a fully self-financed firm as function of the income tax rate? In all our expositions up to now, we kept the tax rates constant. We did not want to clarify how a firm’s market value behaved with a different tax rate. In this section it will be seen why we constantly avoided this question.
Whoever wants to investigate the relation between tax rate and value of the firm has to grapple with two problems. On the one hand, she has to analyze the influence of the tax rate on the cash flow, and on the other hand, the influence of the tax rate on the cost of equity. The first problem mentioned is not considered particularly difficult in the literature, which is why it generally does not get much attention. The question as to how the cost of equity react to changes in the tax rate, in contrast, deserves much more consideration. We will limit ourselves to the case of the eternal return in our subsequent analysis. The reasons for doing so will soon be clear. To simplify matters, we are operating in this section on the basis that both income types are burdened by income tax in the same way. The tax rate will be denoted by τ.
Influence on Cash Flows
We presuppose expected cash flows remain equal. We further assume that no changes of the drawn capital and no investments result during the entire duration. Thus the free cash flow is dealing with an amount that can be easily calculated as product of the gross cash flow and the term 1 − τ,
\displaystyle \begin{aligned} \widetilde{\mathit{FCF}}^u_t=\widetilde{\mathit{GCF}}_t(1-\tau). \end{aligned}
(4.5)
Influence on Cost of Equity
In connection to a work by Johansson (1969), a declaration on the functional relation between the cost of equity (post-tax) and the tax rate is often made in the literature,13 which can be formulated as follows: the post-tax cost of equity kE, u is dependent linearly on tax rate τ; there is then a number kE, so that
\displaystyle \begin{aligned} k^{E,u}=k^E(1-\tau) \end{aligned}
(4.6)
is valid. kE is also referred to as “pre-tax cost of equity,” although this interpretation is not necessary for our argument. We ourselves have always very wisely forgone asserting such a relation.
Equation (4.5) now leads in connection to Eq. (4.6) to the familiar outcome of the income tax being completely redundant under the conditions assumed here. It simply cancels from the valuation equation,14
\displaystyle \begin{aligned} \widetilde{V}_t=\frac{\widetilde{\mathit{FCF}}^u_t}{k^{E,u}}=\frac{\widetilde{\mathit{GCF}}_t(1-\tau)}{k^E(1-\tau)} = \frac{\widetilde{\mathit{GCF}}_t}{k^E}. \end{aligned}
(4.7)
But our discussion concerns a deeper matter. We want to critically analyze the linear relation (4.6).
Stochastic Structure of the Cash Flows
There is a strong relation between the cost of equity and the risk-neutral probability measure Q.
We always stressed that earlier as well. The relation which we want to call attention to here once again is particularly well expressed in Theorems 3.​3 and 4.4. If, as now in Eq. (4.6), a statement on the dependence of the cost of equity on the tax rate is made, then that also implies a relation between risk-neutral probabilities and the tax rate.
We want to show with help from an example that this relation can lead to a dramatic problem. In order to more exactly characterize the stochastic structure of the future gross cash flows, we use our example of an infinitely long living firm. We now assume that the gross cash flows $$\widetilde {\mathit {GCF}}_t$$ of the firm evolve according to Fig. 2.​3.
In order to show that the free cash flows as modelled are in fact weak auto-regressive, we have to make it clear how the conditioned expectation $$\operatorname *{\mathrm {E}}\left [\cdot |\mathcal {F}_t\right ]$$ is calculated. At time t the cash flow $$\widetilde {\mathit {GCF}}_t$$ is already known, and that is why the uncertainty can only relate the subsequent movement u or d. We thus have in connection to rule 2
\displaystyle \begin{aligned} \operatorname*{\mathrm{E}}\left[(1-\tau)\widetilde{\mathit{GCF}}_{t+1}|\mathcal{F}_t\right]&=(1-\tau)\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{GCF}}_{t+1}|\mathcal{F}_t\right] \\ &= (1-\tau)\, P(u) \, u\, \widetilde{\mathit{GCF}}_t + (1-\tau)\, P(d)\, d\, \widetilde{\mathit{GCF}}_t \\ {} &= \Big( \underbrace{P(u) u + P(d) d}_{:= 1+g}\Big) (1-\tau)\widetilde{\mathit{GCF}}_t . \end{aligned}
(4.8)
Because of (4.5) that is exactly the assumption of weak auto-regressive cash flows.
An Arbitrage Opportunity
Now we do not only suppose the existence of one, but also of two firms. Both should be without debt and pursue a policy of full distribution. For the parameters u, d in the first firm
\displaystyle \begin{aligned} P(u) u+P(d)d=1 \qquad \Longrightarrow\qquad g=0 \end{aligned}
should be valid. The value of the firm at time t is denoted by $$\widetilde {V}_t$$. For the sake of simplicity, we designate the cost of equity when taxes are neglected with k; they should remain constant in time.
The second firm should also possess gross cash flows with the stochastic structure as in Fig. 2.​3. If the cash flows grow in the first firm (that is, move up), then they also grow in the second firm. If they fall in the first firm, then they also sink in the second. It can thus be determined that the cash flows of the two firms are perfectly correlated. We will denote the cash flows of the second firm with $$\widetilde {\mathit {GCF}}^{\prime }_t$$. The factors u′, d′ are different from those of the first firm, but
\displaystyle \begin{aligned} P(u)u'+P(d)d'=1\qquad \Longrightarrow\qquad g'=0 \end{aligned}
should again be valid. Because of this connection, the gross cash flows do not point to any expected growth in either case. The second firm’s cost of equity rate when taxes are neglected is k′ and the firm’s value in t is denoted with $$\widetilde {V}_t^{\prime }$$.
The investor can continue selling or acquiring riskless bonds, which at time t have the value Bt. When taxes are left out, the bonds promise a return in the amount of rf, the riskless interest rate.
We use an idea reasonably well known in the literature, that of the so-called pricing by duplication. This way we will be able to uncover a relationship between the value of the two firms, $$\widetilde {V}_t$$ and $$\widetilde {V}_t^{\prime }$$, and so too between the cost of equity k and k′. This relationship is based on the idea that a portfolio can be put together from shares from the first company and the riskless bond, the cash flows of which do not differ from the payments with which an owner from the second firm can plan.
For that purpose we make up a portfolio which at time t includes exactly nB riskless bonds and nV shares of the first firm. We choose the numbers nB and nV in such a way that, independently from the state which manifests at time t + 1, the equation
\displaystyle \begin{aligned} n_B B_t \left(1+r_f(1-\tau)\right) + n_V \left(\widetilde{\mathit{GCF}}_{t+1}(1-\tau)+\widetilde{V}_{t+1}\right) = \widetilde{\mathit{GCF}}^{\prime}_{t+1}(1-\tau)+\widetilde{V}^{\prime}_{t+1} \end{aligned}
is satisfied. With the help of (4.7), applied to both firms, the equation can be simplified to
\displaystyle \begin{aligned} n_B B_t \left(1+r_f(1-\tau)\right) + n_V \left(1+k_{t+1}(1-\tau)\right)\widetilde{V}_{t+1} = \left(1+k^{\prime}_{t+1}(1-\tau)\right)\widetilde{V}^{\prime}_{t+1}. \end{aligned}
In the period following from the end of time t, there are exactly two possible directions (up or down) along the cash flow path in the binomial model. That is why the above condition can be resolved in a system of two equations, which must be simultaneously satisfied: in the case of an up movement
\displaystyle \begin{aligned} \left(1+r_f(1-\tau)\right)n_B B_t + u\left(1+k(1-\tau)\right)n_V \widetilde{V}_{t} = u'\left(1+k'(1-\tau)\right)\widetilde{V}^{\prime}_{t} \end{aligned}
must be valid, while in the case of a down movement
\displaystyle \begin{aligned} \left(1+r_f(1-\tau)\right)n_B B_t + d\left(1+k(1-\tau)\right)n_V \widetilde{V}_{t} = d'\left(1+k'(1-\tau)\right)\widetilde{V}^{\prime}_{t} \end{aligned}
must be satisfied. Both equations make up a linear system, which can be unequivocally resolved according to variables nB and nV,
\displaystyle \begin{aligned} n_B & := \frac{\widetilde{V}^{\prime}_t}{B_t}\,\frac{(u-u')(1+k'(1-\tau))}{u(1+r_f(1-\tau))}\\ n_V & :=\frac{\widetilde{V}^{\prime}_t}{\widetilde{V}_t}\,\frac{u'(1+k'(1-\tau))}{u(1+k(1-\tau))} . \end{aligned}
All variables are uncertain. They depend upon the firm value in t.
Since the portfolio ex constructione at time t + 1 generates the same payments as the second firm, it has to have the same price under the arbitrage free conditions as this has,
\displaystyle \begin{aligned} n_B B_t + n_V \widetilde{V}_t = \widetilde{V}^{\prime}_t. \end{aligned}
(4.9)
If we employ the solutions for nB and nV in the solution at hand, then we get a valuation equation for the second firm. In so doing $$\widetilde {V}^{\prime }_t$$ cancels out and we get a functional relation between the cost of equity k of the first firm and k′ of the second. It reads as follows:
\displaystyle \begin{aligned} \frac{u-u'}{1+r_f(1-\tau)}+\frac{u'}{1+k(1-\tau)} =\frac{u}{1+k'(1-\tau)}\;. \end{aligned}
(4.10)
A possible economic interpretation of this equation could consist of the cost of equity 1 + k′(1 − τ) being established as harmonic mean of the cost of equity 1 + rf(1 − τ) and 1 + k(1 − τ), whereby this harmonic mean is weighted with the parameters of the up and down movement.
The following is decisive for this equation: it must be valid for all conceivable tax rates τ. But that does not work. In addition to the trivial solution τ = 1 the value τ = 0 will yield a relation between k and k′. Hence, τ = 0 and τ = 1 solve the above equation already. But, a simple rearrangement shows that (4.10) is a quadratic equation in τ that cannot have more than two solutions! The equation cannot be satisfied for a single further τ.
That is a violation of the no arbitrage principle, a principle we always uphold. If namely, the cost of equity do not fulfill the given relation (4.10), then that means nothing more than that the relation of the firm values (4.9) is also not valid—and a free lunch can easily be construed from there. Depending upon whether
\displaystyle \begin{aligned} n_B B_t + n_V \widetilde{V}_t > \widetilde{V}^{\prime}_t \qquad \text{or}\qquad n_B B_t + n_V \widetilde{V}_t < \widetilde{V}^{\prime}_t \end{aligned}
is valid, you must either go short or long with the shares of the second firm and cover this transaction with the bond and the shares of the first firm.
We had posed the question as to what connection existed between cost of equity and the tax rate. Until now, this question was always left out of our considerations. In order to answer the question, we fall back upon a concept which is very popular in applied work. This concept produces a simple linear relation between the cost of equity and the tax rate. We could show that the unlimited application of the appropriate equation results in an arbitrage opportunity. We are thus left with the following realization: whoever wants to know how cost of equity react to the changes in the tax rate, may not rely on Eq. (4.6). The DCF theory simply does not give any answer here. And that is exactly why we have until now deliberately avoided the question.

### 4.2.1 Problems

The following problems are devoted to the understanding of the arbitrage opportunity revealed in this section.
1.
One particular feature of our tax system in that section was that only dividends were taxed. Assume now that also capital gains are taxed. In particular, we assume that the capital gains (even if they are not realized!) also add to the tax base, i.e., instead of (4.5) we assume
\displaystyle \begin{aligned} \widetilde{\mathit{FCF}}^u_t=\widetilde{\mathit{GCF}}_t-\tau\Big(\widetilde{\mathit{GCF}}_t+\underbrace{\widetilde{V}^u_{t}-\widetilde{V}^u_{t-1}}_{\text{unrealized capital gain}}\Big)\;. \end{aligned}
Such a tax system is also called neutral tax system or taxation of economic rent. Show that if the value of the assets remain unchanged by the tax rate the cost of equity have to satisfy
\displaystyle \begin{aligned} k^{\text{post-tax}}=k^{\text{pre-tax}}(1-\tau). \end{aligned}
Remark
It can be shown that if this system is free of arbitrage pre-tax it will remain free of arbitrage post-tax.

2.
This problem derives a possible relation between company value and tax rate without violating the arbitrage principle.
Assume that the risk-neutral probability measure Q does not change with the tax rate τ and that the cash flows form a perpetual rent (no growth). We further assume that the company has constant pre-tax cost of equity kpre-tax. Using Theorem 4.2, derive an equation for post-tax value of the unlevered firm that explicitly contains kpre-tax.
Hint: The heart of the solution is the precise definition of a pre-tax cost of equity. Make certain that
\displaystyle \begin{aligned} \frac{\operatorname*{\mathrm{E}}_Q\left[\widetilde{\mathit{GCF}}_s|\mathcal{F}_t\right]}{\left(1+r_f\right)^{s-t}}=\frac{\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{GCF}}_s| \mathcal{F}_t\right]}{(1+k^{\text{pre-tax}})^{s-t}} \end{aligned}
is a good choice.

3.
The (pre-tax) gross cash flows from two companies follow the binomial tree as in Fig. 2.​3 and let FCF0 = 100, g = 0, rf = 5 %, k = 15 %, u = 10 %, and u′ = 20 %.
a.
Consider the first company having cost of equity k. Determine the risk-neutral probabilities Q1(u) and Q1(d).

b.
Use the arbitrage argument above to determine k′.

c.
Assume that for the first company the post-tax cost of equity are given as (1 − τ)k. Determine the risk-neutral probabilities depending on τ.

d.
Now assume that for the second company the post-tax cost of equity are given as (1 − τ)k′. Calculate again the risk-neutral probabilities depending on τ. Does the result coincide with c)?

Hint: You might consult the finite example from Sect. 3.​1.​3 on how to calculate Q1(u) and Q1(d).

## 4.3 Retention Policies

In the following we will analyze alternative forms of the earnings retention policies. With one exception, it essentially deals with such strategies in which the scale of the earnings withholding is uncertain.

### 4.3.1 Autonomous Retention

The free cash flow of the unlevered firm can be either fully or partially distributed. We will now examine the most simple form of a retention policy. It is distinguishable in that the firm, on principle, annually holds back a certain amount of the maximally distributable cash flows.
Definition 4.2 (Autonomous Retention)
A firm is following autonomous retention policy if the retention is deterministic.
The value of a firm which follows this policy is easy to calculate. We employ Definition 4.2 in Eq. (4.4) and get
Theorem 4.5 (Autonomous Retention)
In the case of an autonomous retention, the following is valid:
\displaystyle \begin{aligned} \widetilde{V}^l_t=\widetilde{V}^u_t+\left(1-\tau^D\right)A_t+\frac{\tau^I\left(1-\tau^D\right)r_f A_{t}}{1+r_f\left(1-\tau^I\right)}+\ldots+ \frac{\tau^I \left(1-\tau^D\right)r_f A_{T-1}}{\left(1+r_f\left(1-\tau^I\right)\right)^{T-t}} . \end{aligned}
Eternally Living Firm with Autonomous Retention
If the firm pursues the autonomous retention in such a way so that At does not change in time, the last statement is simplified. We furthermore suppose at the same time that the firm lives infinitely. Notice that the value of the tax shield in Theorem 4.6 does not depend from the tax rate if τD = τI.
Theorem 4.6 (Constant Retention)
The firm lives on forever. For constant A, the following applies:
\displaystyle \begin{aligned} \widetilde{V}^l_t= \widetilde{V}^u_t+\frac{1-\tau^D}{1-\tau^I}A. \end{aligned}
The proof of this statement is straightforward.
Example
Let us turn to our finite example. The amounts of retention with the levered firm are exactly
\displaystyle \begin{aligned} A_0=10,\quad A_1=20,\quad A_2=0\,. \end{aligned}
With that we get
\displaystyle \begin{aligned} V^l_0 & =V^u_0+(1-\tau)A_0+\frac{\tau(1-\tau)r_f A_{0}}{1+r_f(1-\tau)}+ \frac{\tau(1-\tau)r_f A_{1}}{(1+r_f(1-\tau))^2}\\ & =249.692+(1-0.5)\cdot 10 +\frac{0.5\cdot(1-0.5)\cdot0.1 \cdot 10}{1+0.1\cdot(1-0.5)}+ \frac{0.5\cdot(1-0.5)\cdot 0.1 \cdot 20}{(1+0.1\cdot(1-0.5))^2}\\ &\approx 255.383 \end{aligned}
for the value of the levered firm.
In order to establish the value of the levered firm in the infinite example, assuming A = 10 and using the statement from Theorem 4.6 we get
\displaystyle \begin{aligned} V^l_0 = V^u_0+A=510\,, \end{aligned}
which is independent from the tax rate as mentioned earlier.

### 4.3.2 Retention Based on Cash Flow

The next policy concerns the case where a fraction of the distributable cash flow is retained.
Definition 4.3 (Retention Based on Cash Flows)
A firm is following an earnings retention policy based on cash flows if the retention is a determinate multiple of the free cash flow of the unlevered firm,
\displaystyle \begin{aligned} \widetilde{A}_t=\alpha_t\,\widetilde{\mathit{FCF}}^u_t. \end{aligned}
αt is a number greater than zero.
The value of a firm which follows this policy is easy to calculate. We employ Definition 4.3 in Eq. (4.4) and get
\displaystyle \begin{aligned} \widetilde{V}^l_t &=\widetilde{V}^u_t+\left(1-\tau^D\right)\widetilde{A}_t+\frac{\operatorname*{\mathrm{E}}_Q\left[\tau^Ir_f\left(1-\tau^D\right)\widetilde{A}_{t}| \mathcal{F}_t\right]}{1+r_f\left(1-\tau^I\right)}+\ldots+ \frac{\operatorname*{\mathrm{E}}_Q\left[\tau^Ir_f\left(1-\tau^D\right)\widetilde{A}_{T-1}|\mathcal{F}_t\right]}{\left(1+r_f\left(1-\tau^I\right)\right)^{T-t}} \\ &=\widetilde{V}^u_t+\left(1-\tau^D\right)\alpha_t\widetilde{\mathit{FCF}}^u_t+\frac{\tau^I r_f\left(1-\tau^D\right)\alpha_t\widetilde{\mathit{FCF}}^u_t}{1+r_f\left(1-\tau^I\right)}+\\ &\qquad + \frac{\tau^I r_f\left(1-\tau^D\right)}{1+r_f\left(1-\tau^I\right)} \left( \frac{\operatorname*{\mathrm{E}}_Q\left[\alpha_{t+1}\widetilde{\mathit{FCF}}^u_{t+1}|\mathcal{F}_t\right]}{1+r_f\left(1-\tau^I\right)}+\ldots+ \frac{\operatorname*{\mathrm{E}}_Q\left[\alpha_{T-1}\widetilde{\mathit{FCF}}^u_{T-1}|\mathcal{F}_t\right]}{\left(1+r_f\left(1-\tau^I\right)\right)^{T-t-1}}\right). \end{aligned}
All we need to do now is to use Theorem 4.4, and the following statement is already proven.
Theorem 4.7 (Cash Flow-Retention)
In the case of a retention based on cash flows, the following is valid:
\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \widetilde{V}^l_t=\widetilde{V}^u_t+\frac{\left(1+r_f\right)\left(1-\tau^D\right)\alpha_t\widetilde{\mathit{FCF}}^u_t}{1+r_f\left(1-\tau^I\right)}+\\ &\displaystyle &\displaystyle \ +\frac{\tau^Ir_f\left(1-\tau^D\right)}{1+r_f\left(1-\tau^I\right)} \left( \frac{\operatorname*{\mathrm{E}}\left[\alpha_{t+1}\widetilde{\mathit{FCF}}^u_{t+1}|\mathcal{F}_t\right]}{1+k^{E,u}}+\ldots+ \frac{\operatorname*{\mathrm{E}}\left[\alpha_{T-1}\widetilde{\mathit{FCF}}^u_{T-1}|\mathcal{F}_t\right]}{(1+k^{E,u})^{T-t}}\right). \end{array} \end{aligned}
Eternally Living Firm with Constant Retention
If the firm pursues the retention based on cash flow in such a way so that the share percentage α > 0 does not change in time, the last statement is simplified. True, we only arrive at this simplification if we suppose at the same time that the firm lives infinitely. Otherwise it is so that αT = 0 is necessarily valid, and then a constant retention rate can no longer be spoken of. The subsequent statement applies under these conditions.
Theorem 4.8 (Constant Retention Rate)
The firm lives on forever. For constant α, the following applies:
\displaystyle \begin{aligned} \widetilde{V}^l_t= \left(1+\frac{\tau^Ir_f\left(1-\tau^D\right)\alpha}{1+r_f\left(1-\tau^I\right)}\right)\widetilde{V}^u_t+ \frac{\left(1+r_f\right)\left(1-\tau^D\right)\alpha}{1+r_f\left(1-\tau^I\right)}\,\widetilde{\mathit{FCF}}^u_t\,. \end{aligned}
The proof of this statement is simple. We only need to employ the constant retention rate,
\displaystyle \begin{aligned} \widetilde{V}^l_t&=\widetilde{V}^u_t+\frac{\left(1+r_f\right)\left(1-\tau^D\right)\alpha\widetilde{\mathit{FCF}}^u_t}{1+r_f\left(1-\tau^I\right)}+ \frac{\tau^Ir_f\left(1-\tau^D\right)\alpha}{1+r_f\left(1-\tau^I\right)} \sum_{s=t+1}^\infty\frac{\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_{s}|\mathcal{F}_t\right]}{(1+k^{E,u})^{s-t}}\\ &=\widetilde{V}^u_t+\frac{\left(1+r_f\right)\left(1-\tau^D\right)\alpha\widetilde{\mathit{FCF}}^u_t}{1+r_f\left(1-\tau^I\right)}+\frac{\tau^Ir_f\left(1-\tau^D\right)\alpha}{1+r_f\left(1-\tau^I\right)} \widetilde{V}^u_t\,. \end{aligned}
And that was it.
Example
Let us turn again to our finite example. The retention coefficients with the levered firm are exactly
\displaystyle \begin{aligned} \alpha_0=0\,\%, \quad \alpha_1=10\,\%\,,\quad \alpha_2=20\,\%\,. \end{aligned}
With that we get
\displaystyle \begin{aligned} V^l_0 &= V^u_0+ \frac{\left(1+{r_f}\right)(1-{\tau})\alpha_0 FCF^u_0}{1+{r_f}(1-{\tau})}+\frac{{\tau}{r_f}(1-{\tau})}{1+{r_f} (1-{\tau})}\left(\alpha_1\frac{\operatorname*{\mathrm{E}}\left[{\widetilde{\mathit{FCF}}^u}_1\right]}{ 1+{k^{E,u}}} \right. \\ & \qquad \left. + \alpha_2\frac{\operatorname*{\mathrm{E}}\left[{\widetilde{\mathit{FCF}}^u}_2\right]}{(1+{k^{E,u}})^2}\right)\\ &= 249.692+0+\frac{0.5\cdot 0.1\cdot(1-0.5)}{1+0.1 \cdot (1-0.5)} \cdot \left(0.1 \cdot \frac{100}{1+0.15}+0.2 \cdot \frac{110}{(1+0.15)^2} \right) \\ &\approx 250.295 \end{aligned}
for the value of the levered firm.
In order to establish the value of the levered firm in the infinite example, we use the statement from Theorem 4.8 and get
\displaystyle \begin{aligned} V^l_0 & = \left(1+\frac{\tau r_f(1-\tau)}{1+r_f(1-\tau)}\,\alpha\right)V^u_0+ \frac{\left(1+r_f\right)(1-\tau)\alpha}{1+r_f(1-\tau)}\,FCF^u_0\\ &=\left(1+\frac{0.5\cdot 0.1\cdot (1-0.5)}{1+0.1\cdot(1-0.5)}\cdot0.5\right)\cdot500 + \frac{(1+0.1)\cdot(1-0.5)\cdot0.5}{1+0.1\cdot(1-0.5)}\cdot 100\\&\approx 558.333\,. \end{aligned}

### 4.3.3 Retention Based on Dividends

Valuation Equation
We now want to look at a retention policy in which a prescribed dividend (pre-tax) is distributed for a period of n years. At the end of this period, the firm should change over to a policy of full distribution. We will expect in the following that all decisions about the firm’s investment policy have already been made. But then the firm cannot pay out any dividends if the free cash flow is not large enough. It is thus recommended requiring that the free cash flow is higher or at worst the same as the planned dividend. If the two amounts do not agree, then the difference is withheld. The reflux from this difference is then available as additional distribution potential at time t + 1.
Definition 4.4 (Retention Based on Dividends)
A firm is following a retention policy based on dividends if a prescribed certain pre-tax dividend should be payed in the first $$n\widetilde {l} T$$ periods,
\displaystyle \begin{aligned} \widetilde{A}_t=\left( \frac{1}{1-\tau^D}\widetilde{\mathit{FCF}}^u_t+(1+r_{t-1})\widetilde{A}_ {t-1}-\ Div_t\right)^+\, \end{aligned}
for all t = 1,…,n.
Readers who still remember details from Sects. 3.​6.​1 and 3.​6.​2 of the previous chapter will hardly be surprised by the assertion that a general valuation equation is only possible through the inclusion of derivatives which we believe are not always traded in the market. Such complications can only be avoided if we introduce an additional assumption.
Assumption 4.2 (Non-negative Retention)
At no time does the prescribed dividend policy lead to the retention amount being negative,
\displaystyle \begin{aligned} \frac{1}{1-\tau^D}\widetilde{\mathit{FCF}}^u_t \ge Div_t \qquad \forall t \le n. \end{aligned}
The maximum function in Definition 4.4 is superfluous under this simplified assumption, and we can prove that the following statement is valid.
Theorem 4.9 (Retention Based on Dividends)
The following applies in the case of a retention based on dividends, if the dividend is not greater than the free cash flow of the unlevered firm for all times before T,
\displaystyle \begin{aligned} \begin{array}{lrr}\displaystyle \widetilde{V}^l_u=\widetilde{V}^u_t+\tau^I\left(1-\tau^D\right)\left(\frac{1+r_f}{1+r_f\left(1-\tau^I\right)}\right)^{n-t+1} \widetilde{A}_t+ \\+\tau^Ir_f\sum_{v=t+1}^n\left(\frac{\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_v|\mathcal{F}_t\right]}{(1+k^{E,u})^{v-t}}- \frac{\left(1-\tau^D\right)Div_v}{\left(1+r_f\left(1-\tau^I\right)\right)^{v-t}} \right) \cdot\left(1 +\left(\frac{1+r_f}{1+r_f\left(1-\tau^I\right)}\right)^{n+1-v} \right) . \end{array} \end{aligned}
The proof is found in the appendix. In our opinion it does not make sense to generalize the above statement to the case of an infinitely long living firm with an eternally constant dividend (n →), since we otherwise fall into conflict with the assumption of transversality. We already pointed this out in a similar context in Sect. 3.​6.​2.
Example
Let us work on the basis that the firm will pay the following pre-tax dividend:
\displaystyle \begin{aligned} A_0=0,\quad Div_1=Div_2=40. \end{aligned}
If the tax rate amounts to 50% , Assumption 4.2 is satisfied. Under these conditions we get the following for the value of the levered firm:
\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle V^l_0=V^u_0+ \tau r_f\left(\frac{\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_1\right]}{1+k^{E,u}}- \frac{(1-\tau)Div_1}{1+(1-\tau)r_f} \right) \,\left(1 +\left(\frac{1+r_f}{1+(1-\tau)r_f}\right)^{2} \right)+\\ &\displaystyle &\displaystyle \ \qquad \qquad +\tau r_f\left(\frac{\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_2\right]}{\left(1+k^{E,u}\right)^{2}} - \frac{(1-\tau)Div_2}{(1+(1-\tau)r_f)^{2}} \right) \,\left(1 +\frac{1+r_f}{1+(1-\tau)r_f}\right) \end{array} \end{aligned}
and from that
\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle V^l_0=249.692 +\\ &\displaystyle &\displaystyle \quad +0.1\cdot 0.5\cdot\left(\frac{100}{1+0.15}- \frac{(1-0.5)\cdot 40}{1+(1-0.5)\cdot0.1} \right) \cdot\left(1 +\left(\frac{1+0.1}{1+(1-0.5)\cdot0.1}\right)^{2} \right)+\\ &\displaystyle &\displaystyle \quad \quad + 0.1\cdot 0.5\cdot\left(\frac{110}{(1+0.15)^{2}}- \frac{(1-0.5)\cdot40}{(1+(1-0.5)\cdot0.1)^{2}} \right) \cdot\left(1 +\frac{1+0.1}{1+(1-0.5)\cdot0.1}\right). \end{array} \end{aligned}
That results in
\displaystyle \begin{aligned} V^l_0\approx 263.472 \end{aligned}
for the levered firm.

### 4.3.4 Retention Based on Market Value

Valuation Equation
A retention policy could take on the following form: the managers withhold a certain share of the value of the levered firm every year. However, much more value that firm has, that much more is retained. Hiding behind that could be the strategy of internally financing the growth of the firm with invariable intensity.
Definition 4.5 (Retention-Value Ratio)
The firm is following a retention policy based on market values if the relation of retention and value of the firm
\displaystyle \begin{aligned} \widetilde{l}_t=\frac{\widetilde{A}_t}{\widetilde{V}^l_t} \end{aligned}
is deterministic.
We again use the symbol for leverage here, since, as we shall soon see, an equation equivalent to the WACC approach can be proven with the help of this quotient. This justifies the designation of this retention policy.
Theorem 4.10 (Retention Based on Market Values)
In the case of a retention policy based on market values, the following is valid for the levered firm that bears no debt
\displaystyle \begin{aligned} \widetilde{V}^l_t=\sum_{s=t+1}^T \frac{\operatorname*{\mathrm{E}}\left[\prod_{h=t+1}^{s-1}\left(1-\left(1-\tau^D\right)l_{h}\right)\widetilde{\mathit{FCF}}^u_{s}|\mathcal{F}_t\right]}{\prod_{h=t}^{s-1} \left( 1+k_h\right)}\;,\end{aligned}
whereby
\displaystyle \begin{aligned} 1+k_h=(1+k^{E,u})\left(1-\frac{\left(1+r_f\right)\left(1-\tau^D\right)}{1+r_f\left(1-\tau^I\right)} l_{h}\right)\;. \end{aligned}
The last equation looks a bit like the Miles–Ezzell formula.15 In order to prove Theorem 4.10, we notate Eq. (4.4) for $$\widetilde {V}^l_{t+1}$$,
\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \widetilde{V}^l_{t+1}-\widetilde{V}^u_{t+1}=\left(1-\tau^D\right)\widetilde{A}_{t+1}+\frac{\operatorname*{\mathrm{E}}_Q\left[\tau^Ir_f\left(1-\tau^D\right)\widetilde{A}_{t+1}| \mathcal{F}_{t+1}\right]}{1+r_f\left(1-\tau^I\right)}+\ldots\\ &\displaystyle &\displaystyle \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad +\frac{\operatorname*{\mathrm{E}}_Q\left[\tau^Ir_f\left(1-\tau^D\right)\widetilde{A}_{T-1}|\mathcal{F}_{t+1}\right]}{\left(1+r_f\left(1-\tau^I\right)\right)^{T-t-1}}\;, \end{array} \end{aligned}
out of which with the rule for the iterated expectation
\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \frac{\operatorname*{\mathrm{E}}_Q\left[\widetilde{V}^l_{t+1}-\widetilde{V}^u_{t+1}|\mathcal{F}_t\right]}{1+r_f\left(1-\tau^I\right)}=\frac{\operatorname*{\mathrm{E}}_Q\left[\left(1-\tau^D\right)\widetilde{A}_{t+1}|\mathcal{F}_t\right]}{1+r_f\left(1-\tau^I\right)}+\\ &\displaystyle &\displaystyle \qquad +\frac{\operatorname*{\mathrm{E}}_Q\left[\tau^Ir_f\left(1-\tau^D\right)\widetilde{A}_{t+1}|\mathcal{F}_{t}\right]} {\left(1+r_f\left(1-\tau^I\right)\right)^2}+\ldots+ \frac{\operatorname*{\mathrm{E}}_Q\left[\tau^Ir_f\left(1-\tau^D\right)\widetilde{A}_{T-1}|\mathcal{F}_{t}\right]}{\left(1+r_f\left(1-\tau^I\right)\right)^{T-t}} \end{array} \end{aligned}
results. If we now add the expression $$\frac { \operatorname *{\mathrm {E}}_Q\left [\tau ^Ir_f\left (1-\tau ^D\right )\widetilde {A}_{t}-\left (1-\tau ^D\right )\widetilde {A}_{t+1}| \mathcal {F}_{t}\right ]}{1+r_f\left (1-\tau ^I\right )}+\left (1-\tau ^D\right )\widetilde {A}_t$$ to both sides, then we can again under application of (4.4) bring the above equation into the form
\displaystyle \begin{aligned} \frac{\operatorname*{\mathrm{E}}_Q\left[\widetilde{V}^l_{t+1}-\widetilde{V}^u_{t+1}+\left(1-\tau^D\right)\left(1+r_f\right)\widetilde{A}_{t}-\left(1-\tau^D\right)\widetilde{A}_{t+1}| \mathcal{F}_t\right]}{1+r_f\left(1-\tau^I\right)}= \widetilde{V}^l_t-\widetilde{V}^u_t. \end{aligned}
With the help of the fundamental Theorem 4.2, we can eliminate $$\widetilde {V}^u_t$$ and $$\widetilde {V}^u_{t+1}$$ from the last equation for the unlevered firm and get
\displaystyle \begin{aligned} \widetilde{V}^l_t=\frac{\operatorname*{\mathrm{E}}_Q\left[\widetilde{V}^l_{t+1}+\widetilde{\mathit{FCF}}^u_{t+1}-\left(1-\tau^D\right)\widetilde{A}_{t+1}| \mathcal{F}_t\right]}{1+r_f\left(1-\tau^I\right)}+ \frac{\left(1-\tau^D\right)\left(1+r_f\right)\widetilde{A}_{t}}{1+r_f\left(1-\tau^I\right)}\;. \end{aligned}
Since the retention-value ratio is deterministic, we have
\displaystyle \begin{aligned} \widetilde{V}^l_t-\frac{\left(1-\tau^D\right)\left(1+r_f\right)l_t\widetilde{V}^l_{t}}{1+r_f\left(1-\tau^I\right)}= \frac{\operatorname*{\mathrm{E}}_Q\left[\widetilde{V}^l_{t+1}+\widetilde{\mathit{FCF}}^u_{t+1}-\left(1-\tau^D\right)l_{t+1}\widetilde{V}^l_{t+1}| \mathcal{F}_t\right]}{1+r_f\left(1-\tau^I\right)}\;. \end{aligned}
We now divide by the factor $$\left (1-\frac {\left (1-\tau ^D\right )\left (1+r_f\right )}{1+r_f\left (1-\tau ^I\right )}l_t\right )$$ and get the recursion equation
\displaystyle \begin{aligned} \widetilde{V}^l_t=\frac{\operatorname*{\mathrm{E}}_Q\left[\left(1-\left(1-\tau^D\right)l_{t+1}\right)\widetilde{V}^l_{t+1}+\widetilde{\mathit{FCF}}^u_{t+1}|\mathcal{F}_t\right]} {1+r_f\left(1-\tau^I\right)-\left(1-\tau^D\right)\left(1+r_f\right) l_{t}}\;. \end{aligned}
If we apply this repeatedly, the following then results:16
\displaystyle \begin{aligned} \widetilde{V}^l_t=\sum_{s=t+1}^T\frac{ \operatorname*{\mathrm{E}}_Q\left[\prod_{h=t+1}^{s-1}\left(1-\left(1-\tau^D\right)l_{h}\right)\widetilde{\mathit{FCF}}^u_{s}|\mathcal{F} _t\right]}{\prod_{h=t}^{s-1} \left( 1+r_f\left(1-\tau^I\right)-\left(1-\tau^D\right)\left(1+r_f\right) l_{h}\right)}\;. \end{aligned}
With the help of statement Theorem 4.4, the following ensues:
\displaystyle \begin{aligned} \widetilde{V}^l_t=\sum_{s=t+1}^T \frac{\operatorname*{\mathrm{E}}\left[\prod_{h=t+1}^{s-1}\left(1-\left(1-\tau^D\right)l_{h}\right)\widetilde{\mathit{FCF}}^u_{s}|\mathcal{F}_t\right]}{\prod_{h=t}^{s-1} \left( (1+k^{E,u}_h)(1-\frac{\left(1+r_f\right)\left(1-\tau^D\right)}{1+r_f\left(1-\tau^I\right)} l_{h})\right)}\;. \end{aligned}
That is the assertion of the statement.
Example
We assume a retention-value ratio constant in time of
\displaystyle \begin{aligned} l_0=l_1=l_2=10\%\,. \end{aligned}
The following then results for the cost of equity:
\displaystyle \begin{aligned} k_t&=(1+k^{E,u}_t)\left(1-\frac{\left(1+r_f\right)(1-\tau)}{1+r_f(1-\tau)} l_{h}\right)-1\\ &=(1+0.15)\cdot\left(1-\frac{(1+0.1)\cdot(1-0.5)}{1+0.1\cdot(1-0.5)}\cdot 0.1\right)-1\approx 8.976\%\,. \end{aligned}
With that we get the value of the levered firm with
\displaystyle \begin{aligned} V^l_0&=\frac{\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_{1}\right ]}{1+k} +\frac{(1-(1-\tau)l_1)\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_{2}\right]}{( 1+k)^2}+\frac{(1-(1-\tau)l_1)(1-(1-\tau)l_2)\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_{3}\right]}{( 1+k)^3}\\ &\approx\frac{100}{1+0.08976}+\frac{(1-(1-0.5)\cdot0.1)110}{( 1+0.08976)^2}+\frac{(1-(1-0.5)\cdot0.1)^2\cdot121}{( 1+0.08976)^3}\\ &\approx 264.137\,. \end{aligned}
In the infinite example, we again use
\displaystyle \begin{aligned} l_s=10\,\% \end{aligned}
resulting in
\displaystyle \begin{aligned} k_s&=(1+k^{E,u}_s)\left(1-\frac{\left(1+r_f\right)(1-\tau)}{1+r_f(1-\tau)} l_{s}\right)-1\\ &=(1+0.2)\left(1-\frac{(1+0.1)\cdot(1-0.5)}{1+0.1\cdot(1-0.5)} 0.1\right)-1\approx 0.13714\,. \end{aligned}
We thus get
\displaystyle \begin{aligned} V^l_0&=\sum_{t=1}^\infty \frac{\operatorname*{\mathrm{E}}\left[(1-(1-\tau)l)^{t-1}\widetilde{\mathit{FCF}}^u_t\right]}{(1+k)^t}\\ &=\frac{1}{1-(1-\tau)l}\frac{\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_1\right]}{\frac{1+k}{1-(1-\tau)l}-1}\\ &=\frac{\operatorname*{\mathrm{E}}\left[\widetilde{\mathit{FCF}}^u_1\right]}{k+(1-\tau)l}\\ &=\frac{100}{0.13714+(1-0.5)\cdot 0.1}=534.351\,. \end{aligned}

### 4.3.5 Problems

1.
Look at the example of the unlevered firm in this section. Assume that funds in the firm are invested risky. Let A0 = 0. What are the highest possible retentions at times t = 1, 2? What value does the firm have if it institutes these retentions?

2.
Consider the case where retention is invested in riskless assets. Write down the valuation equation similar to (4.4). Determine the value of the levered and eternally living company if the firm follows an autonomous retention with constant At.

## 4.4 Further Literature

The literature on personal income tax is few and far between. Miller and Modigliani (1961) dividends story is a predecessor of our handling of tax shield and distribution policy. Miller (1977) investigated an equilibrium model where a corporate and a personal income tax are present. The papers by Sick (1990), Taggart Jr. (1991), and Rashid and Amoaku-Adu (1995) considered personal income taxes in a valuation setup. The relation between tax and arbitrage (and in particular a fundamental theorem with income taxes) is also developed in a paper by Jensen (2009). Lally (2000) develops the DCF valuation implications with personal and corporate income tax where an imputation system is applicable. Fernández (2004) discusses the effect of different retention policies on the firm value.
The so-called “lock-in-effect” designates a situation where the owner of a company retains parts of the distributable cash flows due to personal income tax savings. If the tax rate differs across individuals people with high tax brackets hold assets with low dividends and people with low tax brackets hold assets with high dividend yields. This “clientele effect” is widely discussed in the literature.
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Footnotes
1
The income tax is typically not gone into in detail. In the first edition of our book we were at least able to point to some textbooks that handled typically the case of an eternal annuity. While preparing the second edition it turned out that even these cases have been abolished in the literature. It seems that the inclusion of personal taxes fell out of favor.

2
See Institut der Wirtschaftsprüfer in Deutschland (2000).

3
See Sect. 3.​1.

4
See Sect. 4.2.

5
In regards see Assumption 3.​1.

6
This assumption is found in Miller (1977) for instance.

7
The use of basic assets is in keeping with common procedure in the financial mathematical literature, see Shreve (2004, section 1.2).

8
Whereas if the retention is invested without risk, as you might expect, the equation will be formulated slightly differently. “Retention for a risk free investment” will only indicate that we are dealing with a portfolio with more risk free securities. As a result, the assessment basis for the income, which is taxed at a rate of τI, will correspondingly decrease. In the period that follows, it increases again by virtue of the repayment of the retention and the returns on interest. Thus we finally arrive at the following equation:
\displaystyle \begin{aligned}\widetilde{\mathit{Tax}}^l_t = \tau^D \, \left(\widetilde{\mathit{GCF}}_t-\widetilde{\mathit{Inv}}_t \right) +\tau^I\,\left(-\widetilde{A}_t+\left(1+r_f\right)\widetilde{A}_{t-1} \right). \end{aligned}

9
See Sect. 6.​7.

10
See Sect. 6.​1.

11
For details see problem 1.

12
See Sect. 3.​1.​3.

13
Johansson addressed the question, indeed in more scope, as to if the formula notated below is in fact applicable. He clearly pointed out in response that a series of conditions must be satisfied here. One typically finds this relation in examples and not in a chapter on theory, as, for example, in Hillier et al. (2008, Example 14.5, page 520) or in texts dealing with practical issues as Institut der Wirtschaftsprüfer in Deutschland (2008, section 4.4.2.5) (in German).

14
There was a heated discussion in the German literature about this relation, which has found a way to enter into the documents of the Institute of Certified Public Accountants, the German association of CPAs (“Institut der Wirtschaftsprüfer” or IdW). You read, for instance, in Institut der Wirtschaftsprüfer in Deutschland (2013, II. Band, Teil A, Rz. 195): “…1983 it was still assumed that a number of cases could forgo the (explicit) inclusion of the investor’s tax burden, since it has no affect upon the firm’s value.” The Institute of Certified Public Accountants, however, gave up this position in 1997. Since then an income tax rate of 35% has been used, if no other income tax rate can be identified, see here Institut der Wirtschaftsprüfer in Deutschland (2000) as well as Institut der Wirtschaftsprüfer in Deutschland (2013, II.Band, Teil A, Rz. 117).

15
See Theorem 3.​14.

16
By definition, $$\prod _{s=t+1}^{t}x_s=1$$.

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