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St. Petersburg University

Graduate School of Management

Master in Corporate Finance

Model of identification of optimal debt structure in corporations

Master's Thesis by the 2^nd year student

Anton Solovev

Research advisor:

Professor, Alexander V. Bukhvalov

St. Petersburg - 2016



Table of contents

Introduction

1. Concept of the capital structure and existing models of capital structure optimization

1.1 Capital structure

1.2 Theoretical review of existing models of capital structure optimization

1.2.1 Classical theories of the optimal capital structure

1.2.2 Models based on approach to equity as a real option

2. Methodology of identification optimal capital structure

2.1 Theoretical model description

2.2 Model development

2.2.1 Model parameters estimation

2.2.2 Optimal leverage identification

3. Model implementation

3.1 Russian crisis 2014-2015 description

3.3 Model implementation

3.3.1 Case of Lukoil compony

3.3.2 Case of Rosneft company

3.3.3 Case of Novatek compony

3.4 Analysis of the results and the managerial implications

Conclusion

List of references



Introduction

Background and actuality of the study. Optimal capital structure - is the capital structure that is chosen in a way, in which the company maximizes its overall value, since the maximizing value for the stakeholders can be called the main goal for every company. But in the same time optimal capital structure also must minimize the overall cost of capital for the company. There are a huge variety of approaches to identification the optimal capital structure. However, most of them have certain limitations and there is no common agreement in the scientific and business society, what model should be generally used. This explains the actuality of the research, since there are plenty of works published every year concerned this topic, which indicates the relevance and importance of the topic for the business and scientific societies, but there is still no commonly accepted model. Further literature review gives an evidence of the primarily theoretical character of the current general models and approaches of optimal debt structure identification. In spite of plurality and variety of this type of models, most of them have a lot of assumptions that are hardly suitable for use in the cases of real companies. Alternatively, plenty of practically oriented works in the volume of interest are devoted to examination of specific situations or markets that limits their potential for more common use.

The models of identifying the optimal capital structure can be divided to the two big groups. The first one, are the models based on the Modigliani-Miller's theory, and they can be called classical models. However, despite the fact that classical optimal structure models are widely used and discussed in academic papers, they often require unrealistic assumptions that make them not practically valuable. Moreover, the models based on Modigliani-Miller's theory do not fully and accurately describe the relationship between required return on equity capital and risk of default, because in the situation of company's default even if shareholders require incredibly high return, they will not get it eventually.

However, this thesis is aimed at building an applicable model that may be analyzed and leads to the certain practical implications. That is why this paper is focused on the other type of optimal capital structure models defining equity as an option. Those types of models predicting the best capital structure using Black-Scholes-Merton differential equation. This type of models seems to have more realistic approach. Nevertheless, these models are really complicate and hard to modify, because mostly pure mathematics approach is used in them. So, there is not really wide range of the works, that implement this models on practice and analyze the gained results. In the following work, one of the models with option-like approach was chosen as a basis model for the further modification and implementation. This also explains actuality and relevance of the study, due to the fact that the application of the option-like models to the real cases is not so well learned as classical models. As the base model was chosen the model presented by Leland in 1994, due to the interesting approach to the debt values, and the fact that from all option-like models it has the best balance of possibility to be modified, and realistic approach.

Managerial implications of the model seem to be very broad. Interest of the managers of the companies to the research is often due to possibility of optimal debt structure determination for their companies. Moreover, this work should be relevant for different kinds of investors by virtue of opportunity to predict the kind of debt of the companies they wish to invest, understand the amount of risk that company holds and it's perspectives to grow in terms of efficient using of financial leverage. As supplementary to the aforementioned applications, kind of model developed may be a useful tool for banks and other similar organizations in the perspective of either clients' risk identification or targeting their services by offering special kinds of debts to companies.

The objectives and research methodology. The research goal can be stated as developing the optimal debt structure identification model. Practical focus of the research goal provides an advantage to this paper as against existing studies in examined field of expertise. Thereby, the research goal of this paper can be attributed to the aspiration of developing the universally applicable model with the capability of implementation in the real cases, which might be suitable for the wide range of real companies.

The subject of the research is the company's capital structure, which shows the balance between only long-term sources of funding, i.e. balance between debt and equity. Whereas the object of the research is the “optimal” capital structure, which can be described as chosen capital structure from all possible ones, which creates the biggest value for the company's stakeholders.

The objectives of the paper represent the sequence of actions in place to reach the research goal, specified above:

*to analyze the existing models of optimal debt structure identification;

*to identify factors that can determine capital structure;

*to examine risks that can be mitigated by capital structure;

*develop debt structure management model for optimization of capital structure

*to establish a standard optimal capital structure calculation procedure;

*apply developed model to real firm and analyze the results

Outline of the study. The thesis is divided on the tree chapters. In the first chapter, firstly the concept of the capital structure will be introduced. Then, the existing models of capital structure optimization will be analyzed in order to achieve the first objective of the thesis, and partly achieved second and third objectives.

In the second chapter firstly the base theoretical model will be described, thus the second and the third objectives will be fully achieved. Then the process of the model development will be described and the modifications that were made to the base model. Then, the standard optimal capital structure calculation procedure will be established, so the fourth thesis objectives will also be achieved.

In the third chapter, the real companies, on which the model will be tested will be described, with the analysis of the market conditions, on which these companies operates. So the last objective will be achieved in the third chapter, as well as the goal of the research. Also, main managerial and theoretical contributions of the current research will be presented.



1. Concept of the capital structure and existing models of capital structure optimization

1.1 Capital structure

This paragraph describes advantages and inconveniences of each source of capital; furthermore, conceptions of capital structure and structure of sources of financing are explained; relationship between increase in debt and equity risk and return (financial leverage concept) is analyzed; financial, commercial risks and taxes are described as main factors reflecting the choice of capital structure; moreover, the key capital structure concepts that seek for the optimal structure of capital (maximizing its value) are explained and their pros and cons are given; finally, the key concepts of optimal capital structure are introduced and explained.

Financial management decisions may be divided into two main directions: investment decisions that describe funds invested in real or financial assets, and financial decisions that consist in a choice of sources of current and fixed financial assets. The second direction reflects the conception of capital structure decisions, dedicated to finding a balance between equity and debt. Suggesting the only firm activity is a continuous process of investments projects implementation, it may be stated that financial decisions are all about funding investment projects and choosing the sources of financing. Here occurs the dilemma - a need to choose a proper balance between different sources of financing that distinct in their costs because of tradeoff between risk and return Moreover, finding a proper balance between sources may increase the value of the capital itself.

Financial decisions are divided into short-term and long-term, defining the goal of funding - current or longer-term needs in assets. The key point of long-term financial decisions is the choice of the combination of borrowed capital and equity that would maximize the total value of capital V = E + D, where E - is a market value of equity and D - is a market value of debt. The problem of choosing a proper (or optimal as stated in different sources) combination of debt and equity is widely disputed and requires a closer look that will be described later.

The structure of financing sources is the balance between different short-term and long-term sources. The capital structure shows the balance between only long-term sources of funding, i.e. balance between debt and equity. The main difference between debt and equity is about the risk level and consequent required return of the owner of capital.

Cost of capital

The key condition that influences the choice of using a particular type of resources is its price or costs. This condition is also related to the financial resources. Different components of financial resources have different costs as obtained from different marketplaces: money market, stock market or commodity market. The costs of financial resources are also explained by maintenance price. The relative costs on maintaining different elements of capital is the cost of capital. The cost of capital of a certain resource may be found as shown in the formula1.1.

The cost of capital is strongly related to the return required by the owner of capital. Generally, owners of capital require bigger returns on riskier assets (in terms of the companies' activities) and, therefore, the cost of capital is also larger than in the case of less riskier assets. In the conditions of competitive market, owners of capital may choose and relate the risk and return of different investment assets. The cost of capital is defined as current risk-free return on investments, estimated pace of inflation and risk premium. The owner of capital observes different options of investments on different markets and chooses a certain return on investments in accordance with the risk implied. Making financial decisions, financial manager evaluates the cost of each element of the capital and the total cost of capital as combination of these elements. The total cost of capital of the corporation is often used as a discount rate for the present value of future cash flows. To understand the total cost of capital of a company financial manager use Weighted Average Cost of capital (WACC) showed in the formula 1.2.

where r_debt ? is the debt interest rate; r_equity ? is sharehoders' required rate on return; and r_tax ? is the tax rate on companies' profit.

Evaluation and comparison of different elements of the capital allow to choose the most cheap way of long-term financing, in other words to choose the target structure of capital. The cost of capital differs not only between different elements, but also into different conditions implied on each of the elements. For example, required return and, correspondingly, risk of each element of the capital change over time, that leads to the changes in cost of capital.

Borrowed capital

The main advantage of the borrowed capital is the low costs associated with this source, in comparison with equity. This may be explained by different factors:

- Maintenance costs of the debt are loer than for equity as it does not require registrator services or underwriting services;

- Before tax interest rate on debt is lower that required rate of return for equity as the risk is lower (because in the case of the company's default requirememts of the debt holders will be satisfied first);

- Debt payments make the tax base lower, this effect os called tax shield;

- Debt-holders do not have rights to run the company and it does not imply risk of firing for managers;

Relatively to the equity, borrowed capital has wide range of opportuninties to attract the capital. Moreover, the debt may provide a good potential to financial growth, as it enables growth of profitability and return on equity.

On the other hand, borrowed capital generates risk of decrease in financial stability and sovency. That is why the main disadvantage of this type of capital is exessive risk for the equity-holders, because in this case shareholders need to suffer additional risks besides commercial - financial risk. As the result, shareholders increase required rate of return as the additional risk compensation. In this way, borrowed capital may be attractive for its cheapness, but it is also associated with additional shareholders' risks.

Another disadvantage of borrowed capital is related to the need of money concentration by the time loans have to be paid off. As the loan repayment term is accurately defined, management of a company has to work out special repayment schemes. Moreover, companies may cover their debts with issuing new capital.

Despite the fact that creditors do not have rights to run a company, in a situation when the share of debt is significant, debt-holders may control some of the company's decisions. In some cases in bank loan agreement may be marked a mandatory share of net income retention that may contradict with the managers' interests. Furthermore, guarantee of pledge are often required. If company's shares serve as collateral, in a case when company is unable to serve the debt, creditors may take control over the company. This situation took place during the crisis of 1998 in Russia. Companies may occur in a more difficult situation if they provide theur currency export earnings as the collateral, because in this case the need of repaying the debt at the same time being unable to issue the new one.

Equity

Generally, equity may be presented as difference between total assets and liabilities, as an accounting measure. However, under equity the market price of issued stock is usually understood.

The main advantage of this source of capital is the level of financial responsibility towards shareholders. Companies are not obliged to make regular interest payments and may redirect their cash flows on the business growth; furthermore, if business is not successful, shareholders are those who take the hit, because creditors are the first who receive contribution.

However, because of higher risk shareholders take they may take control over the managers' decisions. Furthermore, the share issue is costly and requires a lot of time; that is why it is not rational to finance separate projects with new share issues, because equity is a longer-term source of capital that debt.

1.2 Theoretical review of existing models of capital structure optimization

Optimal capital structure

Optimal capital structure - is the capital structure that is chosen in a way, in which the company maximizes its overall value, since the maximizing value for the stakeholders can be called the main goal for every company. But in the same time optimal capital structure also must minimize the overall cost of capital for the company. The problem here, is that when company is taking more debt, thus increase debt to equity ratio, its overall value is increasing, due to benefits of the debt, most important of which is tax shield. However, when the company is increasing its debt levels, the risk of the company is also getting higher, since it have to repay the debts, and at one point it the repayments of debt might become so high, that the company will have to call themselves a bankrupt. Due to this fact, the riskiness of the firm is rising with rising debt to equity ratio, which is also measured by increased overall cost of capital. So, the point here is to find a balance between these two trends is the main question of finding the optimal capital structure.

1.2.1 Classical theories of the optimal capital structure

Modigliani-Miller model

By the end of 1950s the theory of the capital structure had not existed itself. In the year 1958 F. Modigliani and M. Miller published the article “The Cost of Capital, Corporation Finance and the Theory of Investment” [Modigliani, Miller, 1958], where the basics of capital theory structure were laid. The main point in their theory is that the firm value does not depend on the capital structure, this fact is explained by the basic Modigliani-Miller model. Afterwards the authors completed the model taking into account corporate taxation on profits [Modigliani, Miller, 1963]. F. Modigliani and M. Miller defined the impact of tax shield on the firm value; however, they had not a possibility to propose accurate model to be used on practice. The use of the extended model of Modigliani-Miller gave the paradoxical conclusion that the capital structure is optimal when approaching the magnitude of financial leverage to infinity.

For example, there are two companies - A and B. A is fully financed with equity and does not have any debt, and B is financed both with equity and debt. If an investor does not want to take additional risk, he or she would prefer shares of the company A as it is not liable anything to the creditors. Imagine, the investor bought 1% of the A's shares outstanding. That means that the shareholder has rights for 1% of the company's profits. If the shareholder wants to buy the same share of both equity and debt in the company B, that means that he or she invested 1% in B's equity and 1% in B's debt and as a return he or she will get 1% of the debt interest and 1% of the profit after interest payments. This means that in the end this person will get the same 1% of the company's profit. According to the law of one price, in conditions of perfect market two investments having the same return must have the same price [Brealy, Myers, 2008], so the value of unlevered company A will be equal to the value of levered B.

However, this model does not take into consideration the opportunity to reduce tax payment for the firm B by the amount of debt rate (because debt interest is paid before taxes). This effect is called tax shield. As mentioned above, it is a great advantage of the borrowed capital.

In 1963, Modigliani and Miller published a second work dedicated to the capital structure, which entered into the original model of corporate taxes. Taking into account corporate profits taxation, it was shown that the share price is directly related to the use of debt financing: the higher the proportion of borrowed capital, the higher the share price. According to the revised theory of Modigliani-Miller, businesses should be funded only with borrowed capital, as it provides the highest stock prices.

Modigliani and Miller made two propositions related to the relationship between value of levered and unlevered firm, and to the relationship between required rate of return on equity and capital structure. According to Modigliani and Miller, value of levered firm (V_L) is equal to the value of unlevered firm (V_U) adding the gain from the tax shield effect (product of the corporate tax rate r_tax, and value of debt D) as shown in the formula 1.3:

According to the other conclusion made in the paper required rate of return on equity of the levered firm (r_L) is composed with the return on equity of unlevered form (r_u) and a certain kind of risk premium for the debt presence in the company corrected to the tax shield positive impact (formula 1.4):

Where

R_L - required rate of return on equity of the levered firm,

R_U - required rate of return on equity of the unlevered firm,

R_tax - corporate tax rate,

D - value of debt,

E -value of equity,

R_D - Rate of return of debt,

E - is equity of the firm.

However, Modigliani-Miller's model has a little implication on real business, because of unrealistic assumptions made: absence of transaction costs, tax rate does not depend on the size of debt and is fixed forever, the debt is also permanent, individuals and corporations borrow at the same rate.

Various research have tried to modify the theory of Modigliani-Miller who, in order to explain the actual situation, neglected many of the original terms of the theory. It was found that some of the conditions have no significant effect on the result. However, with the introduction of a model of such a factor, as additional financial costs due to poor capital structure, the picture changes dramatically. For example, economy on the tax payments enhances the value of the enterprise with increasing share of borrowed capital, but at some point the value of the company starts to decrease with the further increase in debt capital, as savings on tax payments are offset by rising costs on a riskier capital structure.

Hamada's Equation and its modifications

The first part of the paper has the form of the search for an optimal debt ratio of the company's capital. This search is substantially based on the Hamada's Equation [Hamada, 1972]. R. Hamada proposed the following formula for calculating в_L by combination of the Modigliani-Miller model with CAMP and taking taxes into account:

where:

в_L- Beta levered,

в_U - Beta unlevered,

T- Corporate tax rate,

ц - Debt to equity ratio.

In consequence of quantity of CAMP and Modigliani-Miller models assumptions the equation, worked out by R. Hamada was acknowledged advantageous, but too theoretical. Due to restrictions originally present in the model its implementation to the real companies can lead to serious errors in the results, which was confirmed by various studies. It caused several attempts of upgrading Hamada's Equation usability for real-world examples by the reduction of the restrictions' quantity.

Particularly, survey figured that using the original Hamada's formula leads to the situation when EBIT of the company varies with the level of leverage. Regarding this problem, Conine developed the following modified formula by adding «the beta of debt» almost immediately after the release of the original article:

where:

в_L- Beta levered,

в_U - Beta unlevered,

T- Corporate tax rate,

ц - Debt to equity ratio,

R_D- Rate of return of debt,

R_PM - Market risk premium,

R_F - Risk free rate.

Conine's modified formula has the «beta of debt» as a conceptual foundation while this concept has been criticized in several researches [Conine, 1985]. The important drawback of this model is coming from the impossibility of offering such a WACC or VL that would fit the definition of an optimal capital structure. Another problem is attributed to noncompliance with the financial principle, according to which the company's value should fall on the some level of leverage as the efficiency of the tax shield sooner or later will be lower than the required rate of return of the amount of risk On the contrary, the value of the company with increasing leverage grows infinitely as specified in Conine's model.

In an effort to handle described noncompliance R. D. Cohen proposed his own modification of Hamada's Equation [Cohen, 2007]. The central message of this modification is reprising the debt with the following formula:

where:

D* - Adjusted debt,

R_D - Return on debt,

D - Debt value of the firm,

R_F - Risk free rate.

Alternatively to previous formula the approach of Conine can be used as a part of Hamada's Equation for finding the optimal capital structure, which is expressed by the sequent set of formulas:

where:

V_U* - Value of unlevered firm,

E - Value of equity,

D* - Adjusted debt,

T - Corporate tax rate,

в_L- Beta levered,

в_U - Beta unlevered,

ц? - Adjusted leverage,

R_D- Rate of return of debt,

R_F - Risk free rate,

D - Debt value of the firm,

E - Value of equity.

This type of modification seems to be one of the most relevant in the terms of its appliance to the real companies. Such method of Hamada's Equation usability elevation lets to gain better results and improve the accuracy of the model.

Hamada's equation modifications or some pieces of them can be used for the purposes of calculation of the optimal ratio between equity and borrowed capital of the company. Significance of this idea is proved out by the fact that further attempts of modifying the Hamada's formula are undertaken by researchers even today. For example, modification of the formula, proposed by Conine, forms the basis for another model of Munshi [Munshi, 2014], who avoids the restriction about the company's possibility of borrowing at a risk-free rate and takes into account the increase in interest rates on the debt, depending on the ratio of debt and equity by empirically determined formula б = 1 + D, where б is added to the R_D.

Consequently, it is possible to state the necessity occurrence of existing methodology modification due to huge number of assumptions that should be made to implement the model while most of them are not holding in the real company cases. Since this implementing some propositions for the optimal debt structure identification model, based on Hamada's Equation modifications or some other branches of examined field of expertise is one of objectives of this master thesis, which is clear justification of the relevance of the each reviewed paper for my research.

Trade-off theory

Thade-off theory also states that for each company there is a target capital structure that may be obtained. According to this model, optimal capital structure may be found by analyzing costs and gain of every additional dollar in debt. The gain may occur because of the tax deductibility or increasing free cash flow. The debt costs are related to the probability of bankruptcy and potential agency conflicts between equity holders and debt-holders. In the situation when the capital structure (or financial leverage) is optimal, the last dollar's gains offset costs. As stated in the theory, the same things happen to the firm's dividends. Companies select such an amount of dividends that maximizes its value.

Pecking order theory

Pecking order theory suggests that there is not target capital structure and financial managers choose capital sources in the order: internal capital, borrowed capital and equity as “the last resort” [Myers, Majluf, 1984]. That happens because of the information asymmetry occurring between firm's management and its potential investors, that is why, according to the theory, companies first prefer use of the internal capital which do not imply costs related to the information asymmetry than the short-term debt. After the internal capital in the firm is depleted, it would apply for short-term debt rather than longer-term, and long-term debt to the new shares issue. Myers and Majluf stated that the information asymmetry may be overcame if companies use their plowback instead of issuing new shares.

In other words, retained earnings, they do not provide the problem of the adverse selection that occurs because of information asymmetry. On the other hand, equity is dependent on the problem of adverse selection, whereas debt is a subject of an insignificant impact of such a problem. From the point of view of external investor, equity is riskier than debt, because both have adverse selection problem, but the risk premium is higher in the case of equity. That is why external investor require higher return on equity capital, rather than on borrowed capital. Fama and French empirically proved that less levered firms are more profitable, according the pecking order theory [Fama, French, 2002].

According to this theory, company will not issue equity being in normal conditions, and in the need of financing deficits, the debt will be issued.

Summing up, despite the fact that classical optimal structure models are widely used and discussed in academic papers, they often require unrealistic assumptions that make them not practically valuable. Moreover, the models based on Modigliani-Miller's theory do not fully and accurately describe the relationship between required return on equity capital and risk of default, because in the situation of company's default even if shareholders require incredibly high return, they will not get it eventually. However, this thesis is aimed at building an applicable model that may be analyzed and leads to the certain practical implications. That is why this paper is focused on the other type of optimal capital structure models defining equity as an option. This type of models, first, seems to be more realistic that enable us to make practical conclusions.

1.2.2 Models based on approach to equity as a real option

Besides classical models using weighted average cost of capital and method of discounted cash flows, there is the second type of models predicting the best capital structure using Black-Scholes-Merton differential equation.

Merton model

In 1973, Black and Scholes [Black, Scholes, 1973] and afterwards in 1974 Merton proposed simple model [Merton, 1974], that related credit risk to the firm's capital structure. Firstly, Black-Sholes model was used as an instrument for options evaluation, but namely R. Merton first applied options theory to the problem of debt evaluation in a case of default possibility.

As known from Finance, value of assets is equal to the sum of values of debt and equity. Furthermore, it is not a secret that debt-holders first satisfy their interests in a company in the case of default, and only thereafter stockholders receive their money. In this way, shareholders' capital is the residual value of firm. Consequently, the value of shareholders' capital may be negative if debt value exceeds assets value. If the value of shareholders' capital is under zero, shareholders may take shares off their hands without any costs.

In other words, shareholders may use do not exercise call-option and leave the firm to the creditors. Taking into account that the value of assets is less that the value of debt, the creditors requirements will not be satisfied fully that means the company's default. If the firm value exceeds the value of debt, its shareholders a sort of redeem the firm for the amount of money equal to the company's debt. Otherwise, they leave the firm and the “redeem agreement” is not realized that means call option is not exercised.

Consequently, assuming the assets value to be independent of firm's capital structure, the equity value may be explained as a call option on the company's assets:

Where

Call option_short - price of the call option on the company's assets in short position.

D_risk - Value of the firm's debt,

V_firm - Value of the firm.

Or in terms of Black-Sholes model:

Where

E_t - equity value at the time t,

V_t - value of the company's assets at the time t,

N(d₁) and N(d₂) - standard normal distribution functions,

T - maturity of the debt,

F - exercise price of the option.

So, if the value of assets (Vt) is higher than the value of debt (F) at the maturity (T), debt-holders will receive F and shareholders V_T - F. In the opposite situation when the face value of debt is higher than the assets value, the company announces default and control over it moves from shareholders to bondholders. This model assume that default will occur at the maturity, where N(-d₂) is the risk-neutral probability of default.

Black-Cox model

In 1976 F. Black and J. Cox proposed the model [Black, Cox, 1976] which assumes that default may happen at any time before the maturity of the debt. As in the previous case, default occurs when the assets value (value of the company) decreases, but in the Black-Cox model this decrease is not defined with the value of debt, but with the specific level that is not constant.

The authors make an assumption that the company observed made only one issue of bonds that do not imply coupon payments (discount bonds). Black and Cox also assume the trigger level (the level of the firm value when default will occur) to be:

Where

C - safety covenant at the present moment,

C_1t - safety covenant at the moment t,

г - discount rate,

T - debt maturity.

As the time of exercise is undetermined, the authors suggest the continuous time analysis using exponential form of discount rate.

Then the time of default may be defined as follow:

Where

ф - time of default,

t - present time,

V_S -value of the firm.

K - value of the covenant C₁.

Then the probability that default occurs before the time ф under conditions of risk-neutrality may be expressed as follow:

Where

ф, T, V, K, and C1t have been defined earlier,

N - standard normal distribution functions,

r -interest rate,

a - dividend rate,

у² - instantaneous variance of the return on the firm.

The main difference between Black-Cox and Merton model is that the first one accepts the situation when the default may occur earlier that the debt maturity (option expiration time). In this case, we consider European option.

In order to employ this model, authors make several assumptions:

- “Every individual acts as if he can buy or sell as much of any security as he wishes without affecting the market price;

- There exists a riskless asset paying a known constant interest rate r;

- Individuals may take short positions in any security, including the riskless asset, and receive the proceeds of the sale. Restitution is required for payouts made to securities held short;

- Trading takes place continuously;

- There are no taxes, indivisibilities, bankruptcy costs, transaction costs, or agency costs;

- The value of the firm follows a diffusion process with instantaneous variance proportional to the square of the value” [Black, Cox, 1976];

Leland-Toft model

Leland and Toft proposed an extended variation of Black and Cox model in 1996 [Leland, Toft, 1996] that provided the choice of both maturity of the debt and its amount. According to this model, default is defined mostly by endogenous factors, rather than by some external factors or cash flows problems. Leland-Toft model presents wider range of possible optimal capital structures than Leland's previous article published in 1994 [Leland, 1994]. Furthermore, this model enables to find not only optimal amount of debt, but also optimal maturity of the debt, credit spreads, default rates and reduction in the assets value. It is also stated by the authors that predictive power of the model is close to the reality that was checked on historical data. Moreover, this model make difference between short-term and long-term debt, as the first one do not provide such advantages due to the tax shield effect. That is why, according to Leland and Toft, gain from the debt tax shield must be contrasted by the risk of assets transfer that lie on the bond-holders' shoulders. Leland-Toft model is presented in formulas 1.15-1.17,

V_b - endogenous bankruptcy trigger, when tax deductibility is lost,

r - risk-free interest rate,

a - Bankruptcy costs,

у - Assets risk or volatility of the firms' assets,

T - Debt maturity,

N - Standard normal distribution function,

C - Coupon payment,

д - Payments to shareholders.

The main benefit of the model is that it predicts more accurate and interesting results that may be analyzed and applied on practice. As an improvement of the [Leland, 1994] model Leland and Toft proposed optimal capital structure taking into account debt principal value, but not only coupons (as done before). That seems to be more realistic scenario of default as the main problem that causes the company's bankruptcy is inability to pay of the principal value, but not coupons payments. However, the scope of the master thesis do not allow to use such a complex model requiring many resources. That is why it was decide to use Leland model described in [Leland, 1994], as it is less resources requiring, but also more realistic than other models using equity as an options.



2. Methodology of identification optimal capital structure

In this chapter, in order to complete fourth and fifth objectives of the thesis, firstly the model developed by Hayne Leland will be described, and after this, the modifications of it will be described. As it was already stated, the model invented by Hayne Leland was chosen, on the one hand due to its' positive sides, and close to reality approach, and in the other hand, because of its simplicity. Of course, for more realistic results it would be better to use modification of this model that have been done by Leland himself in 1994 [Leland, 1994], to include time to maturity and the principal of debt. However, the process of solving that model and having a strict - numbers result seems to be the task that lays far-beyond the master thesis requirements. So, in order to achieve thesis objectives four and five this chapter was divided by two parts: firstly, the base model will be described, and in the seconded part necessary modifications will be described, and the standard algorithm, that should be implemented for the real company in order to obtain results in terms of optimal leverage structure will be described.

2.1 Theoretical model description

To begin with, it is important to state that corporate capital structure and value of debt are interlinked variables. The debt value, because it depends on yield spreads, also depend of level of leverage, because higher leverage level, means more risk in the company, and higher risk means higher yields. The model developed by Leland in his article is further development of the model of Brennan - Schwartz [Brennan, Schwartz, 1978], that provided one of the first quantitative examination of optimal leverage. In their model Brennan and Schwartz made the assumption that unlevered firm value (the value of business) follows a diffusion process with constant level of volatility. However, in Brennan Schwartz model there were several limitations, that were eliminated in the Lelands' model.

First and most important one is that the model was based on the numerical approach, so, it could not give any general solutions for value of the risky debt and optimal choice of leverage. Of course numerical examples are quite useful, but they cannot be applied generally to the possible companies.

Second one is that the model of Brennan - Schwartz is built around certain case, where the bankruptcy of the company triggers only when the business value falls to the principle value of debt. Nevertheless, the described situation is not really common, or even, really uncommon in the real cases of the companies. The described situation is commonly fined only in the cases of the short-term debt contracts, but is rare in long term debt cases, which was described by the future researches [Smith and Warner, 1979]. In current Masters' thesis, the model is developed for the long-term debt cases, so the alternative bankruptcy triggering conditions, described in the Leland's model are more suitable.

Third limitation of the Brennan- Schwartz model is the maturity dates, which also are quite short. Even in their model they take long-term debt, with big time to maturity as an example, it was already discussed before, that the conditions implied bi Brennan and Swartz are mostly true for short term debt, so again, for this Master thesis model implemented by Leland is more suitable. All the limitations of the Brennan and Schwartz model were eliminated in the Leland's model, as it will be shown further,

Leland's article considered two possible options of bankruptcy determinants: first is when bankruptcy is triggered endogenously - when firm has insufficient equity capital to meet it current debt obligations. The second when it was triggered similar to the Brannan Schwartz model conditions, and in the Leland article it is called - the case of protected debt. However, due to more general approach, the model of the current Master's thesis will concentrate only on the first case of the Leland's model that is called unprotected debt.

In the Leland's model there is important approximation that corporate securities depend on the underlying firm value, but, are time-independent. The author of the article states, that despite the fact that debt securities in most cases have a specified maturity date and, therefore have time-dependent cash flows and values, the time independence assumption can be justified. In some cases very long time horizons, are as good as infinite, since after a certain moment, the value of the future payments are nearly zero in the present moment. And very long time horizons for fixed obligations are not new, either in theory or in practice. They were used in [Modigliani Miller, 1958] model had assumptions of infinite maturity debt, [Black and Cox, 1976] took infinite maturity debt in their model and in real life as well some examples can be found. For instance, Bank of England issued Consols - bonds with a fixed coupon with infinite maturity, as well as preferred stock pays dividend without any time limit. The other explanation given by the author if following: “time-independent environment is when, at each moment, the debt matures but is rolled over at a fixed interest rate (or fixed premium to a reference risk-free rate) unless terminated because of failure to meet a minimum value, such as a positive net-worth covenant.”. Basically, it means, that the debt policy of the company is that they take some debt to cover previous debt, so the leverage level is keeping the same trough time. This policy can be used by the firms, so because of this two explained reasons in this Master's thesis, as well as in the Leland's model the time of debt maturity considered to be infinite and assumption can be considered as realistic.

This time independents assumptions allows to establish closed-form solutions for optimal capital structure, and the Leland's article itself is a further development of the results gained by following researchers [Merton, 1974], [Black and Cox, 1976] with taxes and bankruptcy costs. The article itself has following research questions to cover:

· How do yields spreads depend on corporate debt, firm leverage, taxes and risk free rate

· What is influenced by the level of leverage (bankruptcy costs, tax shields, etc.)

· How bankruptcy risks depend on level debt and leverage

So, as it can be seen the research questions of the Leland's article is quite similar to research questions of the current Master's thesis. However, in the Leland's article, there is only theoretical description of the model. In my Master's thesis, I will modify the Leland's model to show the optimal leverage level calculation procedure.

Next assumption if the Leland's model is that the face value of debt remains constant in time. However, in the model developed in current Master's thesis that assumption is modified, so only the leverage level should keep constant which seems to be more realistic assumption, since there are quite a lot of companies that have constant leverage policy. Anyway, it seems to be quite realistic assumptions and H. Leland proves it in the last section of his work, when it is shown that issuing additional debt will hurt debt holders, however, this part of work will not be described, since this topic relays beyond the topic of current Master's thesis. All in all, this is general description of the roots of the model and its preresearches, and now it is time to describe the model itself.

Description of a model with time -independent security values

To begin with, it should be stated, that business value - the value of unlevered firm, follows the diffusion process with constant volatility of rate of return and W as a standard Brownian motion:

As it can be seen, this value is the value of firm's assets, and does not depend on choice of leverage. This approach to business value is quite common in the financial science, so the author of the current Master's thesis, believes that this assumption about firms unlevered value is quite accurate, and can be applied for the cases of the real companies. Another important assumptions, that any cash outflows connected with debt payments, which depend on the choice of leverage, are financed with issuing additional equity. This assumption also seems quite reasonable, since - firstly it is necessary to fulfill the general theoretic statement, that in the “perfect world” without transaction costs and taxes the value of the firm does not depend on its financial structure, and secondly, as Leland states: “this is consistent with the bond covenants that restrict firms from selling assets”. The same assumption is made on the Brennan and Schwartz model and other research papers, so it seems to be quite reasonable and applicable for the real company's cases. However, this assumption is quite important, when we will test the model on the real companies, since, according to this assumption, the company must take their debt not to finance its' current activities, but to invest in some fixed assets or similar projects. This assumption is one of the reasons, that for the testing companies were chosen huge oil companies, that uses their debt not to finance its current activities (like resellers, for example), but to invest in huge long-term projects. So, following Modigliani Miller, black and Cox and Merton, it is assumed that some riskless asset exits, and it pays rate r as an interest. To proceed with, it is assumed, that firm constantly pays some non negative coupon - C - every period in time, until default. This coupon is return on debt, which is paid by the firm to the debt holders, presented in some value. Basically this coupon is return on debt, required by debt holders, multiplied on the total value of debt. F(V, t) is value of such a claim, which depends on value of business and time. It was proved by Black and Cox that any assets value, must satisfy this partial differential equation, if firm finances the cost of coupon by issuing additional equity:

here, coupon outflows should be paid at maturity, and they should be paid until bankruptcy accrues. In general, closed form solutions cannot be found for this equation. Nevertheless, if we state securities time independence, due to the assumptions made above, F_t(V, t) equals to zero, so this equation becomes ordinary differential equation:



And it has a solution:

And now A₀, A₁ and A₂ can be determined with different boundary conditions. So, it was proved, that all time-independent claims, that satisfies previously made assumptions, have this functional form. After this, it is time to determine specific values, based on this equation in order to answer stated research objectives. Also, to simplify the following formulas X will be used instead of 2r/у² (X=-2r/у²).

AS it was already stated, firm pays constant coupon for the value of debt. This value depends on value of unlevered firm and coupon. So the debt value can be described as a function D(V, C).The coupon, however can be suppressed from this function since it is required rate of return on debt multiplied by debt, and it depends on debt, so the debt can be described as D(V). Also, it should be considered that the Vb is the level of asset value, when the bankruptcy occurs. We assumed, that if Vb is reaching V, than the firm is declared as bankrupt, since it cannot meet its' debt obligations. This value will be determined later, and for now we consider it as some fixed value with the condition that if V?Vb the bankruptcy is triggered. What is also important is that, if firm declares bankruptcy, then equity holders are left with nothing, since every asset that are left from the firm are going to the debt holders, due to law restrictions. However, there are some losses of assets, associated with the firms bankruptcy. The value that will be lost is named б, and it's a fraction of the value that is lost and it can change from 0?б?1. It should be stated, that for every firm that value is unique, but there are some common numbers for all the firms, and according to researches, they are near 0.5 So, in the case of bankruptcy shareholders will gain nothing, and debt holders will gain (1-б)*Vb, or in other words, debt holders will gain all left value of firm (since bankruptcy accrues only when V?Vb), except bankruptcy costs. Also, the value б can be defined as a recovery rate. So, with fixed (for now ) Vb, and knowable б, debt value D(V) can be determent, since debt value is a form of equation F(V), we can found it, identifying A₀, A₁ and A₂. The boundary conditions are following:

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