Among the theories of capital structure, the two main ones are the Brusov–Filatova–Orekhova (BFO) theory and the Modigliani–Miller (MM) theory, which is the eternal limit of the BFO theory. Both of them consider the case of constant income, although in practice, a company’s income is, of course, variable. The generalization of these two theories of capital structure for the case of variable income is very important, since it allows them to expand their applicability in practice. Recently, we have generalized the case of variable income the Modigliani-Miller theory [
1], and here we have generalized for the first time the case of variable income of the Brusov–Filatova–Orekhova theory. This generalization significantly expands the applicability of the modern capital structure theory, which is valid for companies of any age, and in practice, for investments, corporate finance, business valuation, banking, ratings, etc.
1.2. Before the Modigliani and Miller Work
In the traditional approach, the WACC and the value of the company, V, depend on the level of leverage, L (and, therefore, on the capital structure). Debt is always cheaper than equity, since the former has less risk, and because in the event of bankruptcy, the claims of creditors are satisfied earlier than the claims of shareholders.
Thus, an increase in the share of cheaper borrowed capital of the total capital structure to the limit that does not violate financial stability and does not increase the risk of bankruptcy leads to a decrease in the WACC and an increase in the value of the company, V.
Further increase of debt financing could lead to financial stability violation and an increase in the bankruptcy risk. WACC increases and the company value, V, decreases. The competition of advantages of debt financing and its shortcomings at low and at high leverage levels forms the optimal capital structure where WACC is minimal and the company value, V, is maximum. These traditional approach results have been used in the trade-off theory.
1.3. Modigliani–Miller Theory
1.3.1. Modigliani–Miller Theory without Taxes
Modigliani and Miller (MM) [
2], under a lot of assumptions, including the absence of corporate and individual taxes, the perpetuity of all companies and all cash flows, etc., obtained results that are completely different from the results of the traditional approach: capital structure does not affect capital cost and company value.
Under the above restrictions, Modigliani and Miller have shown that without taxes, the company value,
V, is equal
Here, EBIT is Earnings Before Interest and Taxes, k0 is discount rate, and V0 stands for the unlevered company value.
From (1) it is easy to obtain
WACC:
k0 is the cost of equity for a company without borrowed funds, and for a company with borrowed capital, k0 is the cost of equity with a zero level of borrowed funds (L = 0).
From (1) and the expression for
WACCOne can obtain the cost of equity,
keHere, WACC is weighted average cost of capital; L is leverage level; D is debt capital value; S is equity capital value; kd and wd are the cost and share of the company’s debt capital; and ke and we are the equity capital cost and share. From (4), it follows that the equity cost increases linearly with the leverage level.
1.3.2. Modigliani–Miller Theory with Taxes
Taking into account income tax, in 1963, Modigliani and Miller [
3,
4] obtained the following result for the value of the levered company,
V,
Here, V0 stands for the unlevered company value, t is the tax on income, and D is debt value.
From (5), it is easy to derive the expression for the
WACCFrom (6), one can obtain the formula for equity cost,
ke, within the Modigliani–Miller theory with taxes
Formula (7) differs from Formula (4) (MM without taxes) by the factor (1 − t), which is called the tax corrector. This is less than one, so the tilt of the ke(L) curve decreases with taxes.
1.4. Unification of Capital Asset Pricing Model (CAPM) with Modigliani–Miller Model
Unification of Capital Asset Pricing Model (CAPM) with the Modigliani–Miller model was done in 1969 [
5]. Hamada derived the below formula for the cost of equity of a levered company, which included the financial and business risks of a company:
where
bU is the
β–coefficient of the company of the same group of business risk, that the company under consideration, but with
L = 0. The Formula (8) consists of three terms: risk-free income ability
kF, compensating shareholders a time value of their money, business risk premium
, and financial risk premium
.
For a financially independent company, the financial risk is equal to zero (the third term disappears), and its owners will only receive the business risk premium.
Miller Model
In [
6], Miller has accounted for corporate and individual taxes to obtain the following formula for unlevered company value,
VU,
Here, TC—tax rate on corporate income, TS—the tax rate on incomes of an individual investor from his ownership through corporation stocks.
1.5. Brusov–Filatova–Orekhova (BFO) Theory
The perpetuity of all company cash flow and of a company’s lifetime was one of the main restrictions of the Modigliani–Miller (MM) theory, which has been lifted up in 2008 by Brusov–Filatova–Orekhova [
7,
8]. They generalized the MM theory for the case of the company of any age,
n, and derived the following Brusov–Filatova–Orekhova formula for the
WACCTo obtain the Modigliani–Miller formula for the WACC from (10), one should substitute .
It was shown [
7,
8] that a number of innovative effects, discovered in the BFO theory, are absent in the MM theory [
2,
3,
4].
Some main existing principles of financial management spanning many decades have been destroyed by the BFO theory; among them is the keystone of optimal capital structure formation—trade-off theory, and the bankruptcy of this theory has been proven within the BFO theory [
7,
8].
1.6. Alternate WACC Formula
An alternate formula for the
WACC has been suggested [
9,
10,
11,
12]. It has the form below (Equation (18) in [
9])
Here, k0, kd, and kTS are the returns on the financially independent company, the debt, and the tax shield, respectively, t is the corporate tax rate, and is the debt share.
Although Equation (11) is quite general, additional conditions are needed for practical applicability. When the
WACC remains constant over time, the value of a leveraged company can be found by discounting the unleveraged free cash flows using the
WACC. In this case, specific formulas can be found in textbook [
11].
In the Modigliani-Miller theory [
3], the debt value
D is constant.
V0 is also constant, as the expected after-tax cash-flow of the financially independent company is fixed. By assumption,
kTS =
kd and the tax shield value is
TS =
tD. Therefore, the company value
V is a constant and the alternate
WACC Formula (11) simplifies the MM formula:
The “classical” MM theory, suggesting that the returns on the debt
kd and the tax shield
kTS are equals (both these values have debt nature), is much more reasonable, so this is why in [
1], we modify the “classical” MM theory, namely.
1.7. Trade-Off Theory
In the study of the problem of optimal capital structure of the company during many decades, the cornerstone was the world-famous trade-off theory. It is still widely used now for decisions on capital structure. In [
13], the relative importance of different factors in capital structure decisions of publicly traded American companies has been studied. The most important factors to explain leverage level are: median industry leverage (+ effect on leverage), log of assets (+), market-to-book assets ratio (−), inflation (+), tangibility (+), and incomes (−). It was noted that companies that pay dividends tend to have lower leverage levels. The related effects have been found under considering book leverage. Authors found empirical data consistent with some trade-off theory versions.
In [
14], authors compare the applicability of the trade-off theory and pecking order theory for small and medium-sized companies’ decisions about capital structure. It was found that the most lucrative and oldest companies have smaller leverage levels, which confirms the forecasts of the pecking order theory. Larger companies have a higher level of leverage, which is consistent with the predictions of the trade-off theory and pecking order theory. It is concluded that the trade-off theory and pecking order theory for small and medium-sized companies are not mutually exclusive when explaining capital structure decisions.
However, in 2013, Brusov et al. [
7,
8] proved the inconsistency of the trade-off theory in the framework of the BFO theory they created. It is shown that the assumption of risky debt financing does not lead to an increase in the
WACC, which still decreases with increasing leverage. Thus, there is no minimum depending on the level of
WACC leverage and no maximum depending on the value of the company from the level of leverage. Therefore, in the world-famous theory of trade-offs, there is no optimal capital structure. Brusov et al., in 2013 [
7,
8], having analyzed the equity cost dependence on the level of leverage under the assumption that debt capital is risky, gave an explanation for this fact.
The Modigliani–Miller theory proved that tax shields provided substantial gains to the company. In [
15], the theoretical study of tax shields was continued. It was noted that companies may have tax deductibles other than debt. Such non-debt tax shields are investment tax credits, depreciation, and net-loss carry forwards. In [
16], the tax effects suggested in [
15] have been tested. In contrast to the prediction in [
15], it was shown that debt is positively related to non-debt tax shields, as measured by investment tax credits and depreciation. The results of [
14] do not provide support for an effect on debt ratios arising from nondebt tax shields. In [
17], it was pointed out that a positive relationship between such proxies for non-debt tax shield and debt may result if a company invests heavily and borrows to invest. Any substitution effects between debt and non-debt tax shields could be suppressed by a mechanical positive relation of this type.
The Brusov–Filatova–Orekhova (BFO) theory methodology and results are well known in the literature (for example, see references [
18,
19,
20,
21,
22,
23,
24,
25,
26,
27,
28]). Papers [
22,
23,
24,
28] use the BFO theory in practice. The impact on capital structure decisions of the overconfidence of finance managers of family-run businesses in India has been studied in [
23]. The study concludes that manager decisions about capital structure could be explained by measurable managerial characteristics. In [
24], the correlation between capital structure and company risk was studied using datasets from Pakistani companies. It was shown that the role of capital structure and risk valuation is vital for the increase in the wealth of shareholders and the sustainable growth of companies. In [
25], the adjusted present value method, the free cash flow (FCF) method, the flow-to-equity method and the relationships between these methods have been considered. The authors used a stationary FCF method and the Miles and Ezzell method instead the Modigliani–Miller method to derive DCF valuation formulas for annuities. In [
26], the influence of internal and external corporate governance mechanisms on the financial performance of banks in the MENA region is studied. It was shown that the corporate governance had positive effects on the financial indicators of banks. The energy companies capital costs by including an investor and market risk approach have been evaluated in [
27]. The
WACC intra-industry analysis of the companies has been done. The connection of capitalization and income ability in the BRICS banking sector has been examined in [
28] under the signaling theory, the bankruptcy cost theory, the agency theory, the pecking order theory, the Modigliani and Miller theory, and the general theory of the cost of capital and capital structure—the Brusov–Filatova–Orekhova (BFO) theory. Over the past two years, the theory of capital structure has received a new impetus. A large-scale modification of both main theories of the capital structure, BFO and MM, has been carried out and continues in order to better take into account the conditions for the real functioning of companies, such as variable income, advance income tax payments, frequent income tax payments, their combinations, etc. [
1,
29,
30]. A study of different aspects of emerging markets was carried out in [
31,
32,
33,
34,
35,
36,
37,
38]. In [
34], the impact of intellectual capital on firm performance within a modified and extended VAIC model has been studied.
In the near future, the authors plan to publish a large review, which will examine in detail the problems of capital structure.
1.8. Materials and Methods
We combine analytical and numerical methods. First, we derive formulas for the company’s leveraged value, V; the leverage less value of the company, V0; the tax shield TS, and finally the WACC in the case of variable income.
Then, using Microsoft Excel, we study the dependences of the following values: weighted average cost of capital, WACC, discount rate, WACC–g, company value, V, and the cost of equity, ke, on the level of leverage, L, at different values of the growth rate, g. We have created a large database for sets of cost of equity k0 and cost of debt kd, which is available upon request.