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.
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.
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.
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:
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.
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.
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.
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.
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
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
$$\displaystyle \begin{aligned} k^{\text{post-tax}}=k^{\text{pre-tax}}(1-\tau). \end{aligned}$$
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.
The value of a firm which follows this policy is easy to calculate. We employ Definition
4.2 in Eq. (
4.4) and get
4.3.2 Retention Based on Cash Flow
The next policy concerns the case where a fraction of the distributable cash flow is retained.
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.
4.3.3 Retention Based on Dividends
The maximum function in Definition
4.4 is superfluous under this simplified assumption, and we can prove that the following statement is valid.
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.
4.3.4 Retention Based on Market Value
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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