^{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}

## 4.1 Unlevered and Levered Firms

### 4.1.1 “Leverage” Interpreted Anew

^{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

^{E, 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}

^{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:

_{ t }such that

### 4.1.3 Income and Taxes

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 | \(\widetilde {\mathit {GCF}}_t-\widetilde {\mathit {Inv}}_t\) |

\(\quad -\widetilde {A}_t+(1+\widetilde {r}_{t-1})\widetilde {A}_{t-1}\) |

_{f}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:

^{D}, the tax rate on interest is τ

^{I}. Although our firms are self-financed, this difference is of relevance in proving our fundamental theorem which will become clear in a moment.

_{f}G 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.

^{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.

^{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.

^{I}, while payments from the risky asset will be taxed by τ

^{D}. 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.

^{8}

### 4.1.4 Fundamental Theorem

_{f}. 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 r

_{f}, 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?

^{9}Now we shall continue to develop results analogous to the ones we have proved in the previous two chapters.

_{f}. The cost of equity k

^{E, 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.

^{10}

### 4.1.5 Tax Shield and Distribution Policy

### 4.1.6 Example (Continued)

^{E, 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.

_{1}(d) and Q

_{1}(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 r

_{f}is to be calculated with, and in the other case it is the riskless tax interest r

_{f}(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}

^{E, 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 k

^{E, u}= 15 %, the value of the unlevered firm in the infinite example amounts to

### 4.1.7 Problems

_{1}(d) and Q

_{1}(u) for the finite example.

^{12}Show that

^{E, u}= 15 % this arbitrage opportunity vanishes and determine Q

_{1}(u) and Q

_{1}(d). Determine Q

_{2}(dd), Q

_{2}(du), Q

_{2}(ud), and Q

_{2}(uu).

## 4.2 Excursus: Cost of Equity and Tax Rate

^{13}which can be formulated as follows: the post-tax cost of equity k

^{E, u}is dependent linearly on tax rate τ; there is then a number k

^{E}, so that

^{E}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.

^{14}

_{t}. When taxes are left out, the bonds promise a return in the amount of r

_{f}, the riskless interest rate.

_{B}riskless bonds and n

_{V}shares of the first firm. We choose the numbers n

_{B}and n

_{V}in such a way that, independently from the state which manifests at time t + 1, the equation

_{B}and n

_{V},

_{B}and n

_{V}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:

_{f}(1 − τ) and 1 + k(1 − τ), whereby this harmonic mean is weighted with the parameters of the up and down movement.

### 4.2.1 Problems

^{pre-tax}. Using Theorem 4.2, derive an equation for post-tax value of the unlevered firm that explicitly contains k

^{pre-tax}.

_{0}= 100, g = 0, r

_{f}= 5 %, k = 15 %, u = 10 %, and u′ = 20 %.

_{1}(u) and Q

_{1}(d).

_{1}(u) and Q

_{1}(d).

## 4.3 Retention Policies

### 4.3.1 Autonomous Retention

_{t}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 τ

^{D}= τ

^{I}.

### 4.3.2 Retention Based on Cash Flow

_{t}is a number greater than zero.

_{T}= 0 is necessarily valid, and then a constant retention rate can no longer be spoken of. The subsequent statement applies under these conditions.

### 4.3.3 Retention Based on Dividends

### 4.3.4 Retention Based on Market Value

^{15}In order to prove Theorem 4.10, we notate Eq. (4.4) for \(\widetilde {V}^l_{t+1}\),

^{16}

### 4.3.5 Problems

_{0}= 0. What are the highest possible retentions at times t = 1, 2? What value does the firm have if it institutes these retentions?

_{t}.