Mechanism of formation of the company optimal capital structure, different from suggested by trade off theory

Abstract Under condition of proved by us insolvency of well-known classical trade off theory it becomes important to identify mechanisms for forming the optimal capital structure of a company. This paper presents one of the real such mechanisms based on the decrease of debt cost with leverage, which is determined by growth of debt volume. This mechanism is absent in perpetuity Modigliani–Miller theory, even in modified version, developed by us, and exists within more general modern theory of capital cost and capital structure by Brusov–Filatova–Orekhova, or BFO theory..


Introduction
Analyzing the validity of well-known trade off theory (Brennan & Schwartz, 1978;Leland, 1994), we have investigated the problem of existing of company optimal capital structure within Modigliani-Miller theory (Modigliani & Miller, 1958, 1963, 1966, modified by us by removing the suggestion about risk-free debt capital (MMM theory), as well as within modern theory of capital cost and capital structure by Brusov-Filatova-Orekhova (BFO theory) applicable to companies with arbitrary lifetime and investment projects of arbitrary duration Brusov, Filatova, Eskindarov, et al., 2011;Brusov, Filatova, & Orekhova, 2013b;Filatova, Orehova, & Brusova, 2008). Within both theories (MMM and BFO), the absence of the optimal capital ABOUT THE AUTHORS Peter Brusov has a Master of Science in Physics and Maths from the Rostov State University, and a PhD from the Leningrad branch of Math Institute. He is now a professor in the Department of Applied Math, Financial University. He is currently interested in corporate finance, investment and capital structure. Tatiana Filatova graduated from the Moscow Finance Institute (now Finance University) and then gained her PhD from the same institute in 1978. She is now the Dean of Financial Management at the Finance University. Her interests include financial management, corporate finance, investment and capital structure. Natalia Orekhova has a Master of Science in Physics and a PhD from the Rostov State University. She is now a professor and Head of Financial and Economical Technology at the Institute of Business, Management and Low, Rostov-on-Don. She is also an expert in corporate finance & investment.

PUBLIC INTEREST STATEMENT
Establishing the optimal capital structure of a company is important and cannot be explained by trade off theory at all. This paper develops an alternative mechanism based on the decrease of debt cost with leverage, which is determined by growth of debt volume. This mechanism is absent within any version of Modigliani-Miller theory, but exists within a more general modern theory of capital cost and capital structure called Brusov-Filatova-Orekhova theory (BFO theory). structure has been proved under the modeling of financial distress and danger of bankruptcy by increase in debt cost. This proves the insolvency of the classical trade off theory (Brusov, Filatova, & Orekhova, 2013a), which is based on the following suggestions (Brennan & Schwartz, 1978;Leland, 1994): At low leverage levels, the advantages of using of debt financing (which is cheaper than equity one) are connected to the fact that the weighted average cost of capital (WACC) decreases with leverage and consequently the company capitalization grows. Starting from some leverage level, as financial distress appears and grows, bankruptcy risk grows as well. With an increase of WACC, the decrease of company capitalization starts. The leverage level, at which profits of debt capital using are approximately equal to the bankruptcy cost, determines the company optimal capital structure.
As our investigations show (Brusov et al., 2013a), within both theories, growth of WACC and consequent decreases of company capitalization are absent. We have given the explanation of for such phenomena: for leverage levels above some value L*, the equity cost decreases with leverage, providing continuously (at all leverage levels) reduction of WACC.
Our conclusion is as follows: the mechanism of formation of the company optimal capital structure suggested in the trade off theory about 40 years ago turns out to be insolvent (Brusov et al., 2013a). The continuous and unlimited reduction of WACC and consequent unlimited growth of company capitalization with leverage seems to contradict the existing experience.
In studying the problems associated with existing concepts of company optimal capital structure, we investigate the influence of debt cost on equity cost and on WACC. We have discovered the presence of correlations between debt cost and equity cost, which could give an another mechanism of formation of the company optimal capital structure (different from the trade off one) at leverage levels which are far enough from the "critical" levels, at which financial distress appears and the bankruptcy risk increases. The discovery of such a mechanism is the main purpose of this paper.
Suggested mechanism of formation of the company optimal capital structure is based on the decrease of debt cost, which (in some range of leverage levels) is determined by increasing the debt volume.

Absence of suggested mechanism of formation of the company optimal capital structure in modified Modigliani-Miller theory (MMM theory)
Let us show first that in Modigliani-Miller theory (Modigliani & Miller, 1958, 1963 modified by us by removing the suggestion about risk-free debt capital, suggested that a mechanism of formation of the company optimal capital structure is impossible. Consider the case of arbitrary dependence of debt cost on leverage f (L).
Suppose, that debt cost k d is described by the following function Here f (L) is arbitrary (growing or decreasing) function of leverage level L. We are interested in leverage levels L > L 0 , because at L < L 0 the standard Modigliani-Miller theory works, and WACC is decreased with leverage while an equity cost grows linearly with leverage (1) Here k e is an equity cost; k 0 is an equity cost of financially independent company; k d is debt cost; t is tax on profit rate; WACC is a weighted average cost of capital.
In this case for WACC one has Substituting Equations 1 and 3 into Equation 4, one has finally for WACC, One can see, that WACC, does not depend on f (L). Moreover, it is described by the same expression (Equation 2), as in case of riskless debt capital. Note that obtained result is consistent with the conclusions of Rubinstein (1973) and Stiglitz (1969), that cost of company within Modigliani-Miller theory is not changed upon introduction of debt riskiness. In our approximation, as well as at Hsia (1981) debt cost is not already constant.
For derivative from WACC, on leverage level one has We have proved the following theorem: In modified Modigliani-Miller theory (allowing riskiness debt capital) under arbitrary change of debt cost with leverage (growing, as well as decrease) weighted average cost of capital, WACC always fall down with leverage. This means the absence of the company optimal capital structure and proves insolvency of well-known classical trade off in its original formulation.

Formation of the company optimal capital structure within Brusov-Filatova-Orekhova (BFO theory)
The situation is different in modern theory of capital cost and capital structure by Brusov-Filatova-Orekhova (BFO theory). As it will be shown below, decrease of debt cost with leverage leads to formation of minimum in dependence of WACC on leverage at moderate leverage levels far from the "critical" levels, at which financial distress appear and the bankruptcy risk increases. Existence of such minimum leads to appearance of maximum in capitalization of the company. So, we suggest a new mechanism of formation of the company optimal capital structure, different from one, suggested by (already insolvent) trade off theory.
Before studying the problem within BFO theory (Brusov et al., 2013a(Brusov et al., , 2013b let us consider oneyear companies, which have been studied by Myers (2001). This case is the particular case of more general BFO theory.

For WACC of one-year company one has
Here w d is the debt ratio.
The debt cost k d still has the following form Thus, WACC, at leverage levels L > L 0 is equal to and, obviously, depends on the form of f (L). Thus, the difference of the simplest case of one-year companies from perpetual ones, which, as we have shown above, is independent of the form of f (L), becomes obvious.
We will not analyze here one-year companies in detail, but instead we will go now to the analysis of companies with arbitrary lifetime, described by BFO theory (Brusov et al., 2013a(Brusov et al., , 2013b. Let us consider a few dependences of debt cost on leverage f (L). We consider decrease of debt cost at exponential rate as well as at quadratic rate with weak fall α = .01 as well as with strong fall α = .1 of debt cost.

Decrease of debt cost at exponential rate
We have the following parameters and the debt cost has the form Calculation of the WACC, will be done, using the BFO formula By the function "Matching parameter" in Excel we will find the WACC values.
Then, using obtained values of WACC, we will find the cost of equity values k e by the equation :   Equation 12 is the definition of the WACC, for the case of existing of taxing.
The application of BFO Equation 11 is very wide: authors have applied it in corporate finance, in investments, in taxing, in business valuation, in banking and some other areas (Brusov et al., 2013b;Brusov, Filatova, Orehova, Brusov, & Brusova, 2011a, 2011b. Using this Equation 11, one can study the dependence of the WACC, as well as the equity cost k e on leverage level L, on tax on profit rate t, on lifetime of the company n.
The case α = .01. Let us consider first the case α = .01.
We will study below the dependence of debt cost k d , equity cost k e and WACC, on leverage level L in case of k d exponential decrease (Figure 1 and Table 1).
Let us valuate the optimum position L* and its depth ΔWACC, using obtained results (see Table 3).

The quadratic decrease of debt cost k d with leverage
Let us consider the quadratic decrease of debt cost k d with leverage.
We will study below the dependence of debt cost k d , equity cost k e and WACC, on leverage level L in case of k d quadratic decrease.
We use the same parameters, as above    with the following dependence of debt cost k d on leverage

Three-year companies
For three-year companies we get the following results (Figures 10 and 11, Table 4).

Ten-year companies
For ten-year companies we get the following results (Figures 12 and 13, Table 5).

Discussion of results
Thus, we have considered the impact of reducing of the cost of debt k d with increases of debt volume. The article deals with two cases: quadratic and an exponential dependence of cost of debt k d on leverage. We have considered other dependences as well, giving similar results.
It is shown that in considered cases, the equity capital cost of firm correlates with the debt cost that leads to the emergence of an optimal capital structure of companies. Cause of the emergence of an optimal structure is that the speed of increase of equity cost k e of the firm begins to grow, starting from some leverage level L* that not only compensates of the fall in cost of debt k d with leverage, but it has also led to an increase in WACC with leverage, starting from some leverage level. This leverage level determines the optimal capital structure of the company.
It is found that, in all examined cases (quadratic as well as exponential one fall of debt cost) in the case of weak drops in debt cost with leverage (α = .01), the optimal capital structure of the company is formed at bigger leverage values than the beginning of the fall (in our case L * ∝ 2L 0 ), and in the case of a stronger fall of k d (α = .1) the optimal capital structure of the company is formed directly near the start point of the fall of k d (L * ≈ L 0 ).
It turns out that the depth of optimum (and, accordingly, the achieved in optimum company capitalization) is bigger at weak drops of debt cost with leverage (α = .01), that is due to the more longterm fall in this case of the WACC with leverage L.

Conclusion
(1) The Modigliani-Miller theory in its classical version in principle does not consider risky debt funds, therefore, within this theory it is not possible to investigate the current problem.
(2) In the modified (by us) theory of Modigliani-Miller with the modeling of riskiness of debt funds by dependence of their cost of leverage level, as shown in this article, at arbitrary change of debt cost with leverage (the growing as well as the fall) the WACC always decreases with leverage, that demonstrates the absence of the optimal capital structure and proves insolvency of well-known classical trade off theory in its original formulation as well as the inability to implement the mechanism of formation of an optimal capital structure proposed in this article.  (3) Within modern theory of capital cost and capital structure by Brusov-Filatova-Orekhova (BFO theory) it is shown that decrease of debt cost with leverage leads to formation of minimum in dependence of the WACC on leverage at moderate leverage levels far from critical ones, at which financial distress appear and the bankruptcy risk increases. Existence of minimum in dependence of the WACC on leverage leads to maximum in company capitalization.
Thus, a new mechanism of formation of the company optimal capital structure, different from one, suggested by trade off theory (now insolvent) and which is based on the decrease of debt cost with leverage, has been developed by us in this paper ( Figure 14).
The cause of optimum formation is as follows: decrease of debt cost with leverage leads to more significant grows of equity cost, which is not compensated by the fall of the debt cost and WACC starts to increase with leverage at some (moderate) leverage level. On the other hand, the increase of debt cost with leverage at higher leverage level leads, as we have shown before (Brusov et al., 2013a), to the fall of WACC with leverage.
Thus within BFO theory under suggestion of decrease of debt cost at moderate leverage levels and of its increase at high leverage levels WACC first decreases with leverage, then, going through minimum, starts to grow and finally fall continuously (under growing or constant debt cost).
Let us discuss the continuous reduction of WACC with leverage at high leverage levels in more detail. Note that at high leverage levels (contrary to the case at moderate leverage levels, where debt cost can decrease via the growth of debt volume), debt cost increases because financial distressed appear and the bankruptcy risk increases. As it has been proved by us in our previous paper (Brusov et al., 2013a), where the insolvency of well-known classical trade off theory has been demonstrated, any type of growth of debt cost leads to continuously fall of WACC with leverage.
Note, that in classical Modigliani-Miller theory WACC always falls with leverage via expression where WACC falls from k 0 to k 0 (1 − t), when L → ∞.
As we have shown in current paper above in modified Modigliani-Miller theory (allowing riskiness debt capital) under arbitrary change of debt cost with leverage (growing, as well as decrease) WACC always fall down with leverage.
Obtained by us within Brusov-Filatova-Orekhova theory conclusions concerning the existence of optimal capital structure do not depend qualitatively on velocity of debt cost fall. Only optimum depth and its position (but not its existence) depend on the particular form of dependence of debt cost on leverage (mainly on velocity of debt cost fall and significantly less on the particular form of function f (L)). WACC = k 0 (1 − w d t) Figure 14. Mechanism of formation of the company's optimal capital structure, different from one, suggested by trade off theory. Decrease of debt cost with increase credit volume in leverage range from L 0 up to L 1 leads to formation of optimum in dependence WACC (L) at L = L opt .
Results of this paper (under the condition of proved by us the insolvency of the classical trade off theory) allow companies to manage the debt volume in order to form a real optimal capital structure, where the WACC has minimum and consequently the company capitalization has a maximum. Suggested mechanism is real, because it is based on real situation in business: a credit rate depends on debt volume-banks give a lower credit rate for higher debt volume.