## 3.1 Unlevered Firms

### 3.1.1 Valuation Equation

### 3.1.2 Weak Auto-Regressive Cash Flows

^{2}The reader may rest assured that without recourse to this assumption, development of correct adjustment formulas is doomed to fail.

_{t}> −1 such that

_{t}is greater than − 100% in order to prevent cash flows from having oscillating signs which would be rather unrealistic. It is not necessary to assume that g

_{t}is positive, so shrinking cash flows are not excluded with Assumption 3.1.

_{t}is a deterministic amount that is already known in t = 0. What does our assumption on weak auto-regressive cash flows imply for the increments ε

_{t+1}? It will imply that these increments have expectation zero and are uncorrelated to each other (sometimes called “white noise” although this in fact refers to independent increments).

^{3}

_{s}, ε

_{t}] disappears. We already know that the noise terms’ expectations are zero. And so from the covariance it immediately follows:

_{t+1}as well as their uncorrelation results from the Assumption 3.1. You may get the impression that the reverse is true as well. This is not the case. For reasons of order, we have to ascertain that it is the condition

_{t+1}have an expectation of zero, are distributed identically and are mutually independent.

^{4}Compare this to the equation above, where f needs only to be linear. If the equation does not hold for any linear function, then there may well be any other nonlinear functional relation f. Independent random variables have always correlation zero, but uncorrelated random variables are—apart from normally distributed random variables—not independent from each other. This again makes clear that in the assumption on weak auto-regressive cash flows we are dealing with a weaker formulation. We are not insinuating any random walk with regard to the cash flows within the framework of our theory, but are instead working on the basis of the less demanding uncorrelated growth.

_{t}guarantee that the cash flows turn out to be weak auto-regressive once again. By inserting Assumption 3.1 in the definition we get

^{5}Based on these considerations we will restrict ourselves on additive error terms from now on.

^{6}But that does not mean anything else than that an evaluator working with estimated expectations of cash flows can operate upon the basis that—so to speak behind the scenes—there is always a state space that corresponds to the Assumption 3.1. All in all, that is why we hold the assumption on weak auto-regressive cash flows to be practically acceptable.

^{7}

^{8}

_{t}we understand that number, which allows the price of the cash flow \(\widetilde {\mathit {FCF}}^u_{t+1}\) at time t to be determined. According to our statements up to present, the discount rate κ

_{t}shall serve as an instrument to value the single cash flow \(\widetilde {\mathit {FCF}}^u_{t+1}\), or using the subjective probabilities we must have

_{t}as well as with κ

_{t+1}in just such a way that

_{t}is serving the valuation of the cash flow \(\widetilde {\mathit {FCF}}^u_{t+1}\) or the cash flow \(\widetilde {\mathit {FCF}}^u_{t+2}\). And we can also no longer assume that the discount rate κ

_{t}from Eq. (3.3) agrees with κ

_{t}from Eq. (3.4).

^{9}We will thus suggest a definition for the discount rates that takes the cash flows to be valued into consideration and which requires a somewhat more complicated notation.

^{E, u}prove to be appropriate candidates for discount rates of the unlevered cash flows. In answering this question we fall back upon the assumption that the cash flows of the unlevered firm are weak auto-regressive. Cost of capital does not then just only prove itself as appropriate discount rates. It has, much further, the pleasant characteristic that it is independent of the particular cash flow to be valued \(\widetilde {\mathit {FCF}}^u_s\). This characteristic will later prove itself to be very helpful.

^{10}

^{11}Similarly, e.g., the UK Insolvency Act initiates bankruptcy proceedings if a firm either does not have enough assets to cover its debts (i.e., the value of assets is less than the amount of the liabilities), or it is unable to pay its debts as they fall due.

^{12}However, the mere existence of negative net cash flows does not automatically imply such a run of events. We should therefore speak of the danger of illiquidity that arises if net cash flows turn out to be negative or \(\widetilde {\mathit {FCF}}^u_t(\omega )<0\).

### 3.1.3 Example (Continued)

^{13}

_{f}= 10% and consider a particular time, for example, t = 3. Due to Theorem 3.3, we have

^{E, u}= 20%. Then the value of the unlevered firm is using rule 4 (remember g = 0)

_{f}= 10%. Due to Theorem 3.3, we have at any time t

### 3.1.4 Problems

_{t}and the dividend-price ratio \(d^u_t\). Do the same for the capital gains ratio

^{E, u}are deterministic and constant. The firm is infinitely living (T →∞). Assume that the cash flows of the unlevered firm are weak auto-regressive as in Assumption 3.1 for deterministic and constant g with − 1 < g < k

^{E, u}.

^{E, u}?

_{t}is a random variable satisfying

_{t}is uncorrelated to \(\widetilde {\mathit {FCF}}^u_t\). Hence, this random variable is white noise and has no price at t − 1. Several problems will be devoted to this special case.

^{E, u}< 100%.

^{14}Furthermore, the discount rate \(\kappa ^{r\to s}_t\) shall depend neither on r nor on s

_{f}= κ

_{t}.

## 3.2 Basics About Levered Firms

^{15}In addition, we will work out in which the taxation of levered firms is different from the unlevered firm. These differences in taxation influence the value of the firm. And the degree of influence on value is dependent upon the type of financing policy the managers of the firm to be valued are operating under. In connection to the fundamental representation of this relation, we will analyze how numerous conceivable forms of financing policy effect value of firms and derive each appropriate valuation equation.

### 3.2.1 Equity and Debt

_{t}. Interest paid at time t + 1 is \(\widetilde {\,I\,}_{t+1}\).

### 3.2.2 Earnings and Taxes

### 3.2.3 Financing Policies

_{f}as well as the tax rate τ are certain according to the requirements.

^{16}particularly the valuation statements coming out of Theorem 2.2. We thus know that the value of the unlevered firm can be established with equation

^{17}

### 3.2.4 Debt and Transversality (Again)

### 3.2.5 Default

^{18}In case of default, the amount, which the company amortizes at the time t + 1, will not coincide with the repayment sum to which it is legally obligated. \(\widetilde {D}_t\) shall be the credit which has been raised at time t and \(\widetilde {D}_{t+1}\) the corresponding amount a year later. Consequently, the difference between \(\widetilde {D}_t\) and \(\widetilde {D}_{t+1}\) accounts for the amount which the company needs to pay back to the creditor (or, if this amount is negative, has to be raised). In the following we will assume that the company pays back the amount \(\widetilde {R}_{t+1}\) which can be at the most as high as \(\widetilde {D}_t - \widetilde {D}_{t+1}\), hence

^{19}Authors who address valuation problems seem to assume that it does not matter which insolvency trigger is used—a view we challenge although in Proposition 3.4 we have exactly shown that. While default has been intensively investigated in prior research, until now the relationship between these two triggers has not been subject to a detailed analysis. However, for investors and financiers (e.g., in context of insolvency risk forecast) it is important to understand, whether these triggers are substitutes to each other, or whether one trigger is stricter than the other in the sense that one default criterion is met earlier.

^{20}

_{f})

^{s−t}and adding up over all t results in

_{f}with the creditor. Since we will later consider default it might be that in some (possibly very uncertain) states of the world the payments for interest and redemption lie below the riskless rate and therefore the firm demands a higher interest rate in the remaining states. Analogous to the cost of equity this requires a definition of cost of debt. Someone who invests \(\widetilde {D}_t\) today is entitled to payments amounting to \(\widetilde {D}_t+\widetilde {\,I\,}_{t+1}\) less remission of debts. Due to a remission of debts of \(\widetilde {D}_t-\widetilde {D}_{t+1}-\widetilde {R}_{t+1}\), we obtain the following definition.

^{21}cost of debt will not be used itself to determine the value of firms and hence need not to be deterministic.

### 3.2.6 Example (Finite Case Continued)

_{t−1}. Consequently, in such a case they would be acting irrationally if they were to grant a credit of a dimension like that anyway.

_{2}. Due to Eq. (3.19) and D

_{3}= 0, then the following must also apply,

### 3.2.7 Problems

## 3.3 Autonomous Financing

### 3.3.1 Adjusted Present Value (APV)

^{22}

_{0}as the product of the debt ratio l

_{0}and market value of the levered firm can be written,

^{23}

### 3.3.2 Example (Continued)

^{24}

_{2}= 50, we get at ω = dd a cost of debt of

_{2}, then for the interval D

_{2}≤ 70, the relation depicted in Fig. 3.8 is gotten (see the Problem 5 below).

_{t}= 100. Debt is not threatened by default. As already indicated in the Modigliani–Miller formula the value of the levered firm is given by

### 3.3.3 Problems

_{f}≠ k

^{E, u}.

_{2}≤ 70.

## 3.4 Financing Based on Market Values

^{25}If we solve the debt ratio’s definition equation (3.5) according to \(\widetilde {D}_t\), then with a deterministic ratio we get

_{f}. It is uncertain from today’s perspective as to how much tax the firm proportionately financed by debt will save. Even with no default the discounting from t to 0 cannot be done with r

_{f}. Today we cannot know how large the tax advantages from debt are.

### 3.4.1 Flow to Equity (FTE)

^{27}We can thus spare our readers it here.

### 3.4.2 Total Cash Flow (TCF)

### 3.4.3 Weighted Average Cost of Capital (WACC)

_{ t }are deterministic, then the value at time t of the firm financed by debt comes to

### 3.4.4 Miles–Ezzell- and Modigliani–Miller Adjustments

^{28}Following this direction, we get this relation for the value of the levered firm

^{29}If we use this theorem, we get

_{t}represents the debt-equity ratio in the sense of Eq. (3.6). The cost of capital of the levered firm can be determined with this formula, if the cost of capital of the unlevered firm, the cost of debt, the income tax rate as well as the aspired leverage ratio and the riskless interest rate are known. If the above equation is solved according to \(k^{E,u}_t\), the leverage ratio, and income tax rate are known, it calls for converting the cost of capital of a levered firm into the cost of capital of the unlevered firm. The condition is that the levered firm follows a financing policy based on market values.

^{ E,u }) are deterministic, then the firm is financed based on market values.

^{30}Not one of these authors thought of an interpretation of these quantities as expected returns. This is why we are far away from blaming Modigliani and Miller for making an error.

### 3.4.5 Over-Indebtedness and Illiquidity with Financing Based on Market Values

### 3.4.6 Example (Continued)

_{2}= 0 applies.

^{31}

_{0}as well as the owing interests r

_{f}D

_{0}cannot be paid completely. A possible solution would be an extension for payment. Assuming that the creditors accept an extension for capital deficit of 2.23 until t = 2, a default would be prevented. But such a step disagrees with our assumptions: We assumed that the investors pursue an exogenous leverage policy even if it is not optimal. In our case, it is the described extension for credit which disagrees with the assumption. This example shows exactly the limitations of our model. If (and only if) the leverage policy is given, the default risk has no influence at all on the value of the company.

### 3.4.7 Problems

^{E, u}. We do not require that the company maintains a deterministic leverage ratio \(\widetilde {l}\) and will show with this problem that the other DCF methods of this chapter are not applicable.

^{E, u}> r

_{f}the Miles–Ezzell-WACC from (3.24) is always larger than k

^{E, u}(1 − τl

_{0}) if l

_{0}= l is the leverage ratio of the firm.

_{M}is the return on the market portfolio and β

_{t}the so-called beta factor. Let \(\beta ^{WACC}_t\) be the beta factor of WACC

_{t}and \(\beta ^{E,u}_t\) of \(k^{E,u}_t\). Write down a beta form of the Miles–Ezzell formula.

## 3.5 Financing Based on Book Values

### 3.5.1 Assumptions

^{32}

^{33}

^{34}But admittedly, we are not able to develop valuation equations without such relationships.

### 3.5.2 Full Distribution Policy

^{35}

### 3.5.3 Replacement Investments

^{36}We also want to deal here with this case concerning the debt which continuously stays the same.

### 3.5.4 Investment Policy Based on Cash Flows

^{37}And Definition 3.13 sees to it that the investment policy is independent from the leverage.

^{38}The equation named in the current theorem is only formulated for time t = 0, and is nevertheless anything but pleasant to read. It can be generalized with considerable technical effort so that a result for \(\widetilde {V}^l_t\) can be obtained. This representation certainly does not give any new insights. That is why we forego presenting it here.

_{t}and \( \underline {l}_t\) remain constant. We further suppose that the firm exists infinitely long. In contrast to Theorem 3.7, we do not, however, assume that the cash flows have a constant, or constantly growing, expectation. We can then, nevertheless, substantiate the outcome—which at first seems surprising and is by no means obvious—that a valuation formula, which is very similar to the Modigliani–Miller equation, is valid.

^{39}In comparison to the original equation from Modigliani and Miller, two terms appear, which are easy to handle mathematically with all the technics in use today. But since it is, nevertheless, not readily understandable, we want to simplify it somewhat. For low interest rates, it appears that

^{40}With that a preliminary estimate of the order of magnitude of this term is easily possible.

^{E, u}is known. We see two ways of obtaining this information if there is no reference firm available that is actually free of debt.

^{E, u}of a levered reference firm, the cost of capital being sought after can be determined with the help of an iteration.

^{41}The fact that we cannot simply solve the valuation equation according to k

^{E, u}, would then be a cosmetic blemish at best.

### 3.5.5 Example (Continued)

### 3.5.6 Problems

_{t+1}are independent and normally distributed with expectation zero and variance one. Cost of capital k

^{E, u}are constant, the firm follows an investment policy based in cash flows. There were no investments before t = 0 and there will be no increases in subscribed capital in the future. Furthermore, α does not depend on t.

^{42}Write down a simple formula for the book value \(\widetilde { \underline {V}}_t\). (You might have to look at the proofs…) How is the book value distributed?

_{f}= 5%, k

^{E, u}= 15%, n = 4, \( \underline {l}=0.7\), D

_{0}= 500, α = 50%, and τ = 34% and write down both values. Is it fair to evaluate a company financed by book values with WACC?

## 3.6 Other Financing Policies

### 3.6.1 Financing Based on Cash Flows

^{43}

^{44}

^{45}

^{46}

### 3.6.2 Financing Based on Dividends

^{47}

^{ th }period. The expectation of the cash flows of the unlevered firm grows with the constant rate g. The market value of a levered firm is then established from

^{48}

^{T}then moves towards zero.

### 3.6.3 Financing Based on Debt-Cash Flow Ratio

^{49}

### 3.6.4 Comparing Alternative Forms of Financing

_{0}, D

_{1}, …. A valuation equation, which is possible under this assumption and delivers the correct value of the firm is the APV equation. If in contrast financing is based on market values, the evaluator knows the firm’s future debt ratios l

_{0}, l

_{1}, … measured in market values. A valuation equation that results in the correct value of the firm under this condition is the WACC formula. In the case of financing based on book values, the future debt ratios \( \underline {l}_0, \underline {l}_1,\ldots \) measured in book values are known to the evaluator. Which valuation equation is applied under this condition is dependent upon whether the firm follows a policy of full distribution, only takes on replacement investments or conditions its investments upon attained cash flows. With financing based on cash flows, the firm reduces its amount of debt—for a limited time—by a certain proportion of its free cash flows. Special valuation equations can be given that bear just this sort of financing calculation. Financing based on dividends are distinguished in that a firm pays constant dividends over a longer period of time.

^{50}From an academic viewpoint, such valuation formulas may be interesting, but they are not practically relevant. An autonomous evaluator supposes certain future amounts of debt. What sense is there then of the fiction of not knowing the amounts of debt (otherwise only the APV formula would be needed in order to value), but instead falling back upon the expected debt ratios in order to enter them into a WACC formula? It is likewise so for the opposite case. If an investor supposes future certain debt ratios, why should she fabricate not knowing them, and then instead access the expected amount of debt in order to put them into an APV formula? Whoever wants to get from A to B can either take the direct path or take the long way round. Economists normally avoid such long ways. We find it even more strange at the least to propagate such round about ways.

^{51}certain payments are always worth more than uncertain payments from the point of view of risk-averse investors. It is thus completely plausible if one supposes that both assumptions do not necessarily lead to identical values of firms. This is not always adequately stressed in the literature. WACC and APV result in—at least with economic procedures—thoroughly different values of firms.

### 3.6.5 Problems

^{E, u}= 20%, any up- or down-movement has a probability 0.5. The riskless interest rate is r

_{f}= 10%, the tax rate is τ = 34%. The levered company has debt D

_{0}= 0 at time t = 0.