If V=Vb, D(V) = (1- б)Vb, (2.5)

As V ?, D(V) C/r, (2.6)

This means, that if business value falls to the bankruptcy triggering asset value, then debt cost the amount that will be given to debt holders, and this amount is part of the Vb except cost of bankruptcy. On the other hand, if the business value is really large, then the debt value is the value of all coupons that will be paid by the firm (since, the firm will never collapse in terms of infinite value of business, the value of debt is just a value of infinite annuity, with risk-free rate, since firm with infinite value of business is risk free). From this conditions, and equation F(V), it can be found that in this case, A₁=0 and, since V^-X 0 as V?, A₀=C/r. Also, because of first condition . All in all with all A defined< we consume that:

It can be said that p here is the present value of 1$ contingent, in case the bankruptcy appears. This formula for debt is developed results of Black and Cox model, but with bankruptcy costs. The taxes here change the value Vb, that will be shown later, so it can be said that tax benefits are included in this model. AS it was already mentioned, debt changes the firms total value in two ways: firstly, it cuts the value due to higher riskiness of the company, but it also increases value due to tax shields. Both this effects are time independent, however, they both depends on the value of unlevered firm. So, both of them: tax benefits and bankruptcy costs can be valued as time independent securities, through the formula F(V).

Firstly, bankruptcy costs will be identified. If the company declares bankruptcy - the value V falls to Vb, then bankruptcy costs accrues, and they can be defined, as amount lost to bankruptcy costs- BC in the following work. However, if the firm did not declare bankruptcy, then BC are equal to zero. In other words, if firm will never collapse ( as it was explained before, it would happen when business value of the firm is infinitely large), the bankruptcy costs will never occur. This gives the following conditions:

If V=Vb, BC(V) = бVb, (2.10)

As V ?, BC(V)0, (2.11)

So, for this conditions, considering equation F(V), bankruptcy costs can be defined as:

What basically means that, present value of bankruptcy costs is the amount lost due to bankruptcy multiplied on the present value of 1$, in case if bankruptcy occurs, and this results corresponds to common sense. Of course, bankruptcy costs are decreasing function of V, since more business value the firm has, less likely it will go bankrupt, and bankruptcy costs will occur.

The value of tax benefits occurs with debt financing can also be defined through equation F(V). Firstly, it is assumed that tax rate that company pays is considered known and have value of ф. This tax is paid until the bankruptcy is declared and is time-independent. The value of tax benefits - TB in the following work - is the same in all periods and depends on the amount of coupon paid on debt ( which is known and constant for all periods) and equals tax rate multiplied by coupon value - ф*C. However, in case of bankruptcy tax benefits are lost and equals to zero. In case of the firm that will not declare bankruptcy at any moment (risk-free firm, with value of business near to infinite) the tax shields will appear at all periods. So from the declared above, boundary conditions are following:

If V=Vb, TB(V) = 0, , (2.14)

As V ?, TB(V)фC/r, , (2.15)

So, solving F(V) for TB(V) under above condition gives:

This formula indicates the value of tax benefits, considering the risk of bankruptcy. Of course, this is rising function of V, since more business value firm has, less likely it will go bankrupt. However, this approach to tax benefits requires an important assumption that firm will fully benefit from tax shields at all time but it is not always true. Firstly, to have this benefits firm must have EBIT higher than zero. Moreover, under some tax codes ( U.S. tax code, for example), to benefit fully from the tax deduction, firm must have EBIT at least as large as coupon payments. So, it should be taken into account and the model should be tested only on those corporations, for which this assumptions are true.

Now it's time to find the overall firms value. Due to the logic of the model, the total firms value consists of three components: value of unlevered firm, or business value, benefits gained from the taxation, due to choice of leverage and bankruptcy costs. Of course, tax benefits and bankruptcy costs are calculated with the respect of bankruptcy probability, and higher is this probability, higher are bankruptcy costs and smaller tax benefits. Bankruptcy probability is considered in Vb, and later in this work it will be shown how, but for now regards it as fixed. So all in all, total value of the firm can be presented as:

Here it can be seen that u(V) is rising function of V, and it make sense, since more value in business there is, more valuable is a company itself. However, it should be take into account, that this equation can be applied only with some logically based assumptions, that coupon is non negative, and there are some taxes and recovery rate that are higher than zero, but less than one. Moreover, it is interesting to mention that in case V=Vb, u(V)= (1-б)Vb, which also corresponds to the common sense, since if the value of the firm falls to Vb ( the bankruptcy is triggered), then all the company is worst is the amount of assets ( except sunk costs) that are left for debt holders.

The value of firms equity can be presented as:

This equation was gained with the simple algebra. Here it also can be seen that E(V) is rising function of V, since - more business value firm holds, more its equity worth. What is more, it should be stated that according to the previous assumptions, as well as real world situations, E(Vb)= 0, due to the fact that in case of bankruptcy equity holders gain nothing from the company's leftovers. This reflects the “option-like” nature of equity, with Vb as a strike price. However, this approach has some possible agency problems associated with the “asset substitution”. This is a problem, when equity holders become risk neutral, from some point of leverage, since they will gain nothing in the case of bankruptcy anyway, so they chose the most risky projects in order to gain higher returns. This is an interesting question to discuss, however, this problem stands beyond the goal and objectives of this master thesis, so in the current work, it will be assumed that this problem is not really going to take place. Of course, this assumption limits our testing companies in such a way, that the author of the thesis has to be quite sure, that there is no this problem taking place in the researched companies.

As it was shown Vb is a crucial parameter for the model. Vb is the level of asset, which triggers bankruptcy, since company has not enough recourses to meet its obligation, and when it happens, equity value of the company is falling to zero, and debt value to Vb without a fraction lost due to the bankruptcy costs. In order to calculate this amount several important assumptions should be made. Firstly, it is understandable that u(V) will be maximized with the Vb setting as low as possible, under other fixed parameters.

However, it is stopped by being smaller than it possibly can be due to limited value of equity, since due to previously made assumption - firm finances its coupon payment by equity. So, the equity value E(V) must be nonnegative for all V>Vb and eq1uels zero for V=Vb.

So, we assume that Vb is the smallest possible value, for the given E and other parameters. Since, it was proven that E(V), under common environment (taxes and recovery rate from zero to one, coupon non-negative) is rising function in terms of V, the lowest possible value of Vb is when differenced E(V) = 0, or dE/dV = 0. Moreover, due to “option-like” nature of equity its value is equal to zero, when V=Vb. So at the same point, lowest equity value is zero ( due to common sense) and it also happens at the point when V=Vb Thus to calculate Vb, we should fulfill all this requirements. The dE/dV will have the following form:

Thus, now showld be defined a value when dE/dV = 0, and at the same tome V=Vb, so:

Solving this in terms of Vb gives:

Now Vb is defined. As it can be seen, Vb is proportional to coupon. It happens, because the higher level of leverage the company has, the higher risk it takes, and higher return on debt (coupon in this model) it has to pay. Also, Vb decreases with increase in tax rates, since more tax it is required to pay by the firm, more tax shields it gains, so more valuable becomes the debt. Also, the Vb decreases as the risk free rate rises. It corresponds to the common sense, since the the risk of the firm is proportional to the overall risk at the market, and more valuable becomes risk-free activities, more valuable becomes equity of the firm. It is also interesting, that the Vb is independent of the value of unlevered firm. This situation occurs, since the influence value of business on Vb is already counted in the model by the parameter p, so if the V would influence the value of Vb, then the results would be incorrect, since “double counting” will appear. However, as it was already stated, in this model the only risk that is counted in this model, is the internal risk of default. But in the real world there are plenty more risks that appears in the company. In this model, author of the current master thesis decided to include currency risk in this model. Of course, even with currency risk the model will steel be quite an approximation it terms of the risk analysis, but it still be quite relevant if used properly.

So, all in all, this is the theoretical model for unprotected debt. The comparative statistic of it is presented below.

Figure 2.1.1 Comparative statistics of financial variables

In this table, there is the description, of how variables influence each other. Here, R is return on debt, or C/D(V). And R-r is basically a yield-spreads required on debt. This is theoretical analysis, and any firm that have this theoretical model, will act in the following way. So, now it is time to modify a model, firstly to add currency risk and define an optimal level of leverage.

2.2 Model development

In order to achieve fourth and fifth objectives of the research, the model of Leland should be modified in way, that will allow to identify the optimal capital structure of the real company. Before, making any modification to the theoretical model, there should be considered several assumptions that should be taken into account, so the model would give correct results:

The assumptions of the model are following:

· The firms value of the assets, Vt, can be described by a diffusion-type process with stochastic differential equation dVt = m(V, t)dt + уdW

· The stochastic process of V is unaffected by the financial structure of the firm

· All outflows associated with the choice of leverage are financed by selling additional equity and Debt of the firm is unprotected

· The cost of debt can be described as perpetual coupon that is paid to the debt holders once a year, until default

· If firm keeps constant D/E ratio, the C/D(V) ratio also remain constant

· Rate of recovery is unaffected by the financial structure of the firm

· Considering a constant corporate tax rate, the firm obtains tax shields from its debt at a rate C until default

· Bankruptcy occurs when the firm value reaches a threshold Vb, and firm is trying to keep Vb as low, as possible at certain level of leverage

Now, if the assumptions holds, the parameters of the model can be estimated. It is important to state that al, the parameters and proxies are found for the certain time moment. In this master's thesis, there were four time moments taken: first January 2010, first January 2013, first January 2014, first January 2015. This particular time moment were chosen, because author of the thesis decided that it would be interesting to test the model in different economic situations. As it is known, 2014 and 2015 were crisis years for Russia, and especially large oil companies, because of two factors: low prices on oil, and huge fall of rubble currency. The first factor is important due to quite obvious situation, that low oil prices damage the oil companies. The second factor, however, is important because large Russian companies have a huge debt in Eurobonds. So, combination of this two factors damaged this type of companies quite a lot, and that's why those years were taken. I chose 2010 and 2013 for camperecent pre-crisis years, so the difference in optimal leverage structure suggested by the developed model will be seen clearly. However, the algorithm used is the same, despite the time moment taken. Moreover, in every moment in time there is an assumption that main proxies will stay the same for infinite time, for simplicity and accurateness of the model in terms of mathematics. In the future studies, and model development, it is recommended to change the model, so it can show the time-dependence and change with the changing environment, but this task is far beyond the field of master thesis competence, so for now the model will be calculated as it was described previously.

The parameters that are proxied are following:

• Risk free rate r;

• Tax rate ф;

• Total firm value u(V) ;

• Equity value E(V);

• Debt value D(V) ;

• Monthly volatility of equity returns у_m;

• Monthly average of equity returns м _m.

The proxies were chosen with respect to the previous studies, and common sense.

Risk-free rate of return reflects the level of income that could be received by investors without incurring the risks associated with the investment. It is believed that the risk free rate is the same for all investors, at least in one country and currency. However, there is no absolutely no risk object of investment, so as the risk-free rate is usually is used the yield on government securities of the country, in which the company operates. Debt securities are usually issued to certain terms. Because the model is mainly conducted on the basis of the infinite operation of the company (or until bankruptcy) as the risk-free rate should be selected state bonds with the largest remaining term to maturity. Due to this reasons, as a risk free rate the long term rate on the Federal Loan Obligations (OFZ) in the certain date were taken. Those are the state Russian bonds, issued by government of Russia and we take the longest maturity period possible.

Tax rate influences the all the model equations quite a lot, since, the tax benefits dependent on the tax rates as well as the level on assets that triggers bankruptcy. The problem here, is that the companies that are analyzed and on which the model is tested has earnings not only in Russia, but abroad as well. So, in order to simplify the model, and eliminate the need to reflect all the possible taxes and different tax rates in tax codes of different countries it was chosen to use an effective tax rate. This rate is calculated by dividing tax paid, taken from the balance sheet, on profit before tax. This approach was used by many researchers [Patel and Pereira, 2005], who were testing similar models, and the results that were gained in their works were quite reasonable, so the same approach will be used in the current master's thesis.

As a total firm value at a certain point in time market capitalization was taken. This is also a common approach [Ericsson and Reneby, 1998], and seems to reflect the real situation quite well. The market capitalization of the company - a financial indicator, which determines the market value of the company, based on the current value of the shares on the stock market. Calculate the company's market capitalization by multiplying the number of shares issued in circulation at their cost, the current stock market. In this particular case, in order to have more unbiased results, as a share price is taken the average share price through the year before the analyzed date. Foe example for 01.01.2010, the price for the share will be average price from 01.01.2009 to 01.01.2010. This allows reflecting average company value trough a year, when it is analyzed.

As a proxy for the debt value, the balance value of debt was taken. The balance value of debt is short term debt plus long term debt. The debt was adjusted in the way that most of the non-currency debt was subtracted from the debt value. However, in terms of simplicity and more analytical model testing all left debt was assumed to be euro-bond debt. So, to have more accurate results, the companies that were taken, have huge value of its debt in Eurobonds. The approach to debt value as a balance debt value, was already used in the previous studies, that were testing the similar models [Teixeira, 2007]. So, this approach can be called unbiased and accurate.

Since value of the firm is sum of debt value and equity value, the value of equity for a certain time period was calculated as E(V)= u(V)- D(V). Because u(V) and D(V) was described previously, we consider equity value of the firm as a known parameter.

Now it is needed find the volatility of the firm's equity. It is needed to find the volatility of unlevered firm, with the help of Ito's lemma. We make an assumption, that the volatility of early returns of the total firm is the same as volatility of equity returns of the firm. This can be proven by the fact that as it was mentioned above, equity is a simple linear function of the total firm value, since E(V)= u(V)-D(V), and debt remains constant. The volatility of the u(V) can be described as volatility of firm's shares. However, this assumption also makes a boarder to the number of companies that can be analyzed with proposed model, because the shares of the analyzed firms should have high liquidity. Another problem associated with the volatility of firms equity is that it should correspond to chosen time periods- in our cases its 1 year. But the year volatility for certain year cannot be found straight from the share prices. So, in order to find it, we will calculate it on the basis of monthly volatility for the analyzed year and the monthly average for the same time period. So, as a proxies for this numbers the monthly volatility and monthly average of the share prices returns will be taken, with the respect to the time period - for example, for the date 01.01.2010 the will be taken monthly numbers from 01.01.2009 to 01.01.2010.

2.2.1 Model parameters estimation

Now, when all endogenously determined proxies were explained, the estimated parameters of the model can be analyzed. The process of estimating the parameters of the model will consist of the two steps. Firstly, the parameters without any currency risk would be estimated. Secondly, this parameters will be adjusted in order to include the currency risk of debt. This approach is required due to technical limitations, since most of the equations in the model cannot be solved analytically, and requires numerical solving in order to gain results.

First estimated parameter will be yearly volatility of the returns of equity. This parameter will be the same, regarding of the currency risk, since the volatility of the company's equity is determined with the help of company shares volatility, where it is believed, the currency risk associated with the company is already included. The early volatility of the company's equity returns can be found from the following equation:

Where, is squared yearly volatility of company's equity. Since, all the parameters in this equation are already defined, we simply compute yearly equity returns volatility.

It is important to state that, as it was described before, the model uses very specific approach to the debt of the company, as a perpetual coupon. It is known, that in real cases, this scenario is hardly ever a case. So, in order to implement the model, the first thing that should be done, is adjusting the existing debt of the company, to the “theoretical” debt, that is used in the model. It can be done by calculating the theoretical coupon, that the firm should pay for the existing amount of debt. Since, we have the today's value of debt - D(V)- it can be used to define the coupon, that is used in the model. However, the theoretical model does not include the currency risk of debt, so the identified coupon should be adjusted on some amount to be more accurate interpretation of the companies required return on debt. So, basically the algorithm of identifying the parametes will be: calculating the coupon without currency risk, then adjusting it to the currency risk, and then, identifying all the other model parameters.

To complete the first step of the model parameters identification process, there is a need to identify following parameters:

• V_naive - value of business without currency risk

• у_naive - volatility of business without currency risk

• C_naive - perpetual coupon without currency risk

• б_naive - rate of recovery without currency risk

Now, to gain a number of equations that can be solved simultaneously, one of this parameters should be specified. In this case, we assume that we will identify the parameter б_naive, because of the two reasons. Firstly, the change of this parameter in all possible ways - from 0 to 1, influence model the least, unlike all the other parameters. Basically, it only changes the value of debt, and as it was found by the author of the thesis, the difference between two critical values of this parameter - zero and one, only gives 10% difference in terms of value of debt. In other words D(б=0) 0.9* D(б=1), so as it can be seen, there is no huge difference. Second reason, is that in most of the works⁷, where similar models are analyzed⁹, б is most likely to be determined as 0.5. What is more, the studies concerning the recovery rate often give the same results. Several studies in the literature report that, bondholders' recovery rate varies with the type of debt. For example, [Altman, 1991] finds that, during the period 1985-1991, the average recovery rate for a sample of defaulted bond issues was: 0.605 for secured debt, 0.523 for senior debt, 0.307 for senior subordinated debt, 0.28 for cash-pay subordinated debt, 0.195 for non cash-pay subordinated debt. Given this evidence, previous studies: [(Longstaff and Schwartz, 1995), (Delianedis and Geske, 1999), (Leland, 2002), (Huang and Huang, 2003)] assume an average recovery rate of 51.31%. So the assumption of recovery rate being 50% is quite reasonable, and, what is more, this value still will be modified into the more precise value in the second step of the models parameters estimation.

In order to gain the third equation of parameters estimation, Ito's lemma should be used. According to the lemma, if one of the processes is a function of the other process, which is the standard Brownian motion process, then the dependent process is also the standard Brownian process, with volatility dependent on the volatility of the second process. In other words, if:

And

Then

And

In our case, value of unlevered firm - V - was defined as a standard Brownian process with constant volatility, and equity was defined as a function of unlevered firm - E(V) - so E(V) is also a standard Brownian process with constant volatility. So, due to this important facts we can define volatility of unlevered firm as a function of volatility as equity, which was already found earlier. This basically means that:

Now, since values ф, r, and , as well as E() and D( were already defined, we have a situation with three equations and three unknowns, which can be presented in the following form:

capital structure crisis lukoil

Where:

So, now we can define C_naive. When it is done, we can proceed to the next step of identifying model parameters. For proper identification of the model parameters the currency risk should be included in the model. To achieve it, we should make changes to the cost of debt of the companies. However, it should be stated, what fraction of the coupon should be adjusted. Since, the large Russian oil companies gain some profits in other currency then rubble, it should be stated, that some part of the currency risk of debt is covered with this profits. In order of simplicity, it will be assumed, that since the debt of the companies is in dollars, all non-rubble profits is also in dollars. This assumption is quite reasonable, since companies themselves state that most of their not rubble profits are dollar profits [Lukoil annual report, 2015]. The other connected assumption will be that the company covers the dollar debt by dollar profits on the same fraction as its dollar to rubble profits. From some point of view this assumption might not hold on practice, since the company is covering all its currency debts with currency profits, however, in this case, the same amount of rubble profit is lost. So, we will say, that due to simplicity this assumption holds, and the company covers only part of its currency debt with currency profits. This fraction will be defined as:

Now, we will include currency risk in the coupon. For doing so, we assume, that every year company should hedge the currency risk obligatory. For doing so, the company pays certain amount of coupon, for cover the risk of significant change of the coupon payments, due to currency differences. As a significant amount 2, 5% of the coupon payments will be taken. So, basically, the company guarantees, that on the next years, the amount of the coupon payment will not rise more than 102.5% from the coupon. But, as it was mentioned, some of the currency risk is already covered by dollar profits. So, basically, the changeable part of the coupon can grow more than 2.5%, the real amount of growth of can be calculated by the estimation:

To guarantee that, the company buys a call option, with the strike price

The time to maturity of this option is one year. The current price of underlining asset is C_naive. One more important assumptions associated with this type of including “costs of currency risk” is that the yearly volatility of the returns of the RUB/ USD pair will be the same for all following years. Of course, this type of assumption is not really the case in the real market situation, but from the point of view of the company from the certain year, it makes sense. However, this assumption was taken in order to simplify the model, and as further model development, I would recommend to revise it. Still, from the theoretical point of view, it has its use, so it will be kept like this. The yearly volatility of RUB/USD returns, was taken from the monthly volatility and average, by the same equation as was described before for the equity volatility returns. Now, the price of this “theoretical” call options is calculated, with the use of Black-Scholes option pricing model:

Where:

• -- the value of estimated coupon

• N(d) -- normal distribution function

• Strike -- strike price of option;

• r -- risk free rate

• t -- time in years, in this case its one year

• у_usd - early volatility of RUB/USD returns.

After defining the price of an option, we add it to the coupon. This represented the currency risk in a fact, that for more risky debt (due to higher currency risk), firm has to pay higher return on debt. Also, as well as the coupon, firm has to finance this additional spending with issuing more equity. What is more, it is assumed, that if firm is unable to pay this additional price of debt, it declares default. This assumption is true, since the call option here represents not the real hedging case, but the additional costs of currency risk, so it means, that if firm cannot afford to do hedging, the required costs of debt had risen too high due to currency change, and now firm is unable to meet its obligations. So now the perpetual coupon is:

After identifying the coupon with included currency risk, the other parameters of the model should be estimated. Again, with the help of Ito's Lemma and identified ф, r, and , as well as E(V) and D(V) the system of equations can be build:

With:

So here, from three equations with three unknowns, the parameters V, у and б can be identified. However, unlike the previous system, this system of equations might not have strong-form numerical solution. Moreover, it can be stated, that there are quite few situations, where the numerical solution can be found, so, the situation when the parameters of the model can be identified with solving this system, it, can serve additional proof that for the analyzed company most of the assumptions that were previously made are true.

On the other hand, since the model is only a reflection of a real life, some approximations are possible. So, if there is no numerical solution for this model that can be found, following procedure of identification model parameters will take place. In order to solve the system, firstly it is needed to identify estimation errors of the model:

After this, we can use the least residuals sum squares, in order to gain the best possible solution in terms of un biasness So, with minimizing the sum:

And if there is a solution with all the residuals less than 2.5 %² and their sum is also less than 2.5%, it will be assumed the found solution is close enough to be considered as the unbiased estimation. So, technically, identifying the parameters estimation is the task of function minimization, which can be done in Excel solver. The number of 2.5%² was taken as an 5% analog. Since 5% is common trust interval in the statistics, here it was also taken as a border condition. So, all in all, if a numerical solution for this problem is found, we assume that the parameters estimation are true, and, what is more, the assumptions necessary for the model holds in the particular case, since there is quite small amount of conditions, that lead to possibility of numerical solution.

2.2.2 Optimal leverage identification

After all the unknown parameters of the model were identified, the optimal leverage structure of the companies can be defined. The thing here is, that with rising the debt value, the required coupon value will also rise. What is more, with the rising coupon values, the level of assets, when the bankruptcy is triggered is also rising, so from the certain point of leverage, the values of leverage the value of debt is falling, as well as value of the firm. So basically, the value of the optimal leverage structure is identifying by the following procedure. Firstly, the connection between leverage fraction and required coupon should be found. To found this function, it is needed to state that leverage fraction (L) can be defined as:

Since, it was already mentioned that value of unlevered firm (V) does not depend on value of leverage, as well as the other values, like r, б, у², ф, it could be said that the parameters that are changing with the value of leverage is coupon, and so debt value, and the equity value. So the fraction can be presented as:

So:

or

After simplifying and adjustment the model can be presented as:

Where everything except coupon (C) and value of leverage (L) are constants, so this model is a function of coupon from leverage level - C(L), but due to complication of the model it cannot be presented in the classical form. However, this model can be solved simultaneously.

So, in order to find optimal leverage fraction, we define the needed coupon for every leverage fraction, which represents the growing cost of debt for the company, since with higher leverage means higher risk of default, so the debt holders will require higher return on debt in order to compensate the higher risks. The leverage value will be changed with the step of 0.1%, from L_0..100, and for every L_n the corresponding coupon C_0..100 will be defined through the way that previous equation will be true. After completing this procedure, there will be defined 1000 coupons, and each of them will correspond to the certain leverage fraction. Then, for each coupon will be calculated the value of the debt and the value of equity. Now, since

The maximum possible variable of u(C) will be found. The coupon that corresponds to this value, will be the “optimal” coupon for the firm, so after this the leverage value that corresponds to this coupon should be defined, so this leverage value can also be defined as optimal. Here, the optimal level of leverage, is a level of leverage, at which the company's value is maximized. Unlike other models of identifying optimal leverage structure, this model does not have a goal to minimize possible cost of capital, or risk of the company, since the value of risk as well as cost of the debt is already included in the calculations of overall company's value.

So, as it can be seen, the objective of this paper of developing a model of identification of optimal capital structure was completed. However, in order to gain better understanding of the model the analysis should be done, of how changes in different variables effects the choice of optimal leverage structure. It could give a clue, why the model suggests certain level of leverage for the one companies, and other one for other companies. Moreover, most of the changes should be explained from the logical point of view, in order to prove model's consistency. The most important part of this analysis is the analysis of changes in the optimal leverage, connected with different variables, since it can explain the whole logic of identifying optimal leverage structure for different companies.

Theoretically, the model performs in a following way:

Table 2.2.1 Comparative statistics of financial variables at the optimal Leverage ratio

Variable	Shape	у2	r	б	ф	Call (Cnaive, 1)
Coupon ( C)	Linear in V	< 0	> 0	< 0	> 0	< 0
Value of Debt (D)		< 0	> 0	< 0	> 0	< 0
Value of Equity (E)		> 0	< 0	>0	< 0	> 0
Asset value, that triggers bunkruptcy (Vb)		<0	> 0	< 0	> 0	< 0
Leverage (L)	Invariant to V	<0	> 0	< 0	> 0	< 0
Yield of debt (R-r)		> 0	< 0	< 0	>0	< 0

In the following table it is shown, how different variables change the nature of optimal leverage fraction. To begin with, it should be explained, that the yield of debt here, is the difference between the required return on debt for the optimal leverage structure and the risk free rate. Of course, the changes described in the model are calculated with the assumptions, that all other factors, except the analyzed ones are staying the same.

The first changes are connected with the increasing of the firm's business volatility- the volatility of unlevered firm. If the volatility of firm increasing it means higher risks of the firm, so the optimal leverage structure will be decreasing, and the yield of the debt will grow higher, due to higher risks. As the yields are going up, the optimal coupon is going down, since, as it was explained earlier, the coupon payments are representation of required return on debt. The explanation of the falling optimal bankruptcy triggering asset value (Vb), is also quite understandable, due to the fact that higher possible changes in value of unlevered firm leads to the higher risks of it reaching the Vb, so in order to compensate it the optimal Vb value should decrease.

Increasing in the risk free rate, leads to the higher optimal leverage structure. This situation, that is surprising from one point of view, can be explained with the fact that the basic assumption of this changes is that every other parameter, except risk free rate remains the same. From the logical point of view, it means, that even though, the overall market becomes more risky, the company was able to keep the same amount of risks and returns, which means that from the point of view of overall market the company become less risky. So, now it is easy explained why in this case the model is suggesting to increase the level of leverage. The other thins here, is that the yields of the debt is decreasing, so the debt for the company is becoming cheaper, than it was before, so it also explains why in case of the rising risk-free rate the level of company's leverage should increase. On the other hand, in the real cases, rising of the risk free rate leads to the rising the risk of the company itself, since changed economic situation, so the previous results should be described with the caution.

The rising rate of recovery is causing quite obvious consequences. Since, the higher level of bankruptcy costs the debt holders are experiencing more risks, connected with the default, because in the case of default they will return less of their money. This situation leads to the higher required return on debt, to cover these additional possible losses, so the optimal leverage structure should be reduced, due to the higher cost of debt. So, for the firm with the higher losses of assets in case of bankruptcy the optimal level of debt will be lower, comparing with the firm, who will have lower б under the same circumstances.

Higher taxes, quite obviously means higher optimal leverage fraction, because of higher tax benefits from debt. So, if the taxes are rising and nothing else is changing, it is recommended to the firm to take more debt, to reduce the taxes paid.

The increasing in the price of “obligatory hedging” is basically the interpretation of the rising currency risks, associated with the debt. It can happen due to the decrease of the currency profits, or the increase in the RUB/USD volatility, or both. In the case of the huge Russian companies, this value significantly increased, due to both mentioned reasons, since the oil prices went down, as well as rubble currency. This increase the value of the theoretical call option, that was implemented in the model as a value of currency risk of debt. If this call price is going up, the cost of debt is rising, since this price is included in the coupon payments, so the optimal level of leverage should be decreased, because the risk of bankruptcy is rising.

All in all, analysis of the model performance shown that every change in the model can be explained from the logical point of view. Some of the analyzed patterns will be used as an explanation for the observed results in the next chapter. So, now, when the model is build and described, it should be tested on the real cases, in order to analyze its performance, make conclusion about its accuracy, and fulfilling the last objective of the thesis.

To summarize the develop model, there algorithm of its' implementation to the real company was designed. In order to implement the developed model, for every tested company the following algorithm should be used:

1. Determine the proxies of the model for the tested time-moment

2. Estimate parameters without currency risk

3. Fond the cost of hedging for the tested time-moment

4. Adjust coupon, in order to take into account the currency risk associated with debt

5. Estimate parameters with the adjusted coupon

6. If there is no closed form solution, use the minimum sum of squared residuals method

7. If there is a solution gained with the minimum sum of squared residuals method define coupon values at all values of leverage

8. Find a coupon that maximizes total value of the firm

9. Determine leverage value for this coupon

10. Analyze observed results, determine all the meaningful values for the found solution

So, in the last chapter, this algorithm will be implemented to different companies in the different time moments.



3. Model implementation

3.1 Russian crisis 2014-2015 description

One objective of the model is to show differences between Russian companies' cases in two different conditions. The first part is relatively stable economic situation that can be described by reliable growth in GDP. To identify utility of the designed model, comparison of cases in conditions of stable economy with cases of crisis will be implemented. Degree of differences between these two options is presented in Table 3.1.1.

Table 3.1.1 Features of Russian economy in different time periods, [Gregory 2015]

Period	A. 2008/2009 crisis	B. 2009/2011 recovery	C. Pre-crisis Q3 2014 - Q1 2015
GDP growth	-7.8%	4.9%	0.6%
Oil price	$116 to $42	$42 to $112	$103 to $53
Foreign borrowing (net change in foreign debt)	Falls $85 bn	Rises $63 bn	Falls $100 bn
Reserves of central bank	Falls $222 bn	Rises $123 bn	Falls $161 bn

However, crisis shows that international reserves of Russian Federation that consist of various types of foreign assets has significant drop [Mchugh, 2015]. This evidence represents that coming out of recession requires essential funds. Nevertheless, it is hard to admit positive changes in economic situation since 2014.

The most considerable impact on economy was done by implementation of US and European sanctions that restricted access for several Russian companies to international financial resources. Reason for sanctions was “annexation” of the Crimea and supposed support of separatists in eastern Ukraine. Consequently, shortage of sufficient funds and investors outflow because of high uncertainty in the market contributed to fall of the ruble exchange rate.

This is another significant problem that worsened conditions for the Russian companies. Because of devaluation of ruble more than 50%, overall costs of operations increased, especially, in cases where companies use imported material for production or service [Hobson, 2015]. Number of organizations had bank loans and issued debts denominated in foreign currencies. Those who do not have earning in foreign currencies faced problem of debts' return.

Additional effect created drop in oil prices as Russia has one of the biggest volume of oil export in the world. The prices of oils dropped dramatically during seven months - from June to December. The price was changing from 100 US dollars per barrel to the level below 50 US dollars [Bowler, 2015]. The main reason of these circumstances is decline in demand all over the world while producers increased volumes. Almost the half of federal budget of Russian Federation consists of oil and gas sales. For the last two decades, national specialization of Russia was in extraction and export of natural resources. Without strong diversification in different industries except natural resources production, it was almost impossible to minimize effect of crisis on economy. Even during preparation of budget for 2014, government used oil price equaled to $100 per barrel. Therefore, execution of suggested budget was complicated because of prices' fall twice. Implemented solution that was aimed to fulfil shortage in budget was to increase production and export of oil.



Figure 3.1.1 Russian GDP growth, oil prices, foreign borrowings and reserve fund over 2007-2016, [Gregory, 2015]

Figure 3.1.1 shows even during 2007-2012 high volatility in these indices existed. However, general tendency from crisis 2008-2009 represented gradual increase in oil prices and reserved funds that indicates high revenue of the Russian Federation budget. Moreover, GDP was rather stable and companies were able to borrow money in foreign markets to cover internal shortage of funds. Period from the second quarter of 2013 is crucial moment for reserve funds as access to foreign loans were denied for several government-related companies. For the following years list of companies expended and the first quarter of 2016 showed the lowest level of foreign borrowings. However, from this period government intended on convergence with China that allowed to cover shortage of funds partly. In addition, price of oil dropped significantly that collapsed Russian GDP. The only source of significant resources was reserve fund that was used by the government to support key industries.



3.2 Investigated companies' profiles

3.2.1 Lukoil

Lukoil is Russian petroleum company and the second largest after Gazprom on revenue volumes in Russia. The interesting fact that previously Lukoil was the largest organization of Russian oil industry on production volumes till 2007, when Rosneft replaced it.

The company were established in 1991 and combined several Siberian organizations. In 1992 it was transformed in open joint stock company by president order. Since 1996 Lukoil began to finance operations with a help of ADR and was able to begin construction of own tanker fleet for maritime transportation. In addition, company did range of acquisitions of large oil companies in 1999.

Beginning of 2000s were marked by the first international acquisition of the company with purchase of Getty Petroleum Marketing Inc. The crucial moment of Lukoil's history was sale of last government owned shares of the company to ConocoPhillips. The valuable part of strategy was not only acquisitions strategy, but also creation of joint enterprises with such companies as Gazprom in 2007 and ERG in 2008. Cooperation with international companies was utilization of common advantages for Lukoil. For example, organization participated with Statoil in tender for exploration of Iranian oil fields and won it.

General concentration of Lukoil is petroleum industry and related activities: exploration, extraction, refinement, transportation and sales of petroleum and gas products. However, Lukoil has diversified in related businesses like electrical power by acquiring numerous number power plants all around Russian. Moreover, except B2B segment Lukoil developed network of gas station and sales petrol to end customers.



3.2.2 Rosneft

Rosneft is current leader of Russian oil market and estimated as the largest traded on the stock exchange oil company in the world. However, it is government ruled company as government is the owner of big stake in Rosneft's equity. Main concentration of the company is exploration, extraction, refinement and sales of petroleum products. The geography of companies' operation covers major regions of Russia. Core stream of sales is export.

Rosneft was founded in 1993 on the base of another enterprise - Rosneftegaz, which was the core petroleum organization in Soviet Union. The interesting fact is that for the purpose of avoiding monopolization ten subsidiaries of the company were excluded and established as independent organizations. However, over time several of them were consolidated again in a form of acquisition. The owner of Rosneft, which is strategic enterprise even nowadays, was government. Nevertheless, in 1995 solution was accepted to change legal form of organization to open joint stock company.

Last years of 1990s can be characterized as reorganization of the company with previously ineffective management and low outcome of production assets. The next stage were connected with intention of Rosneft's privatization. Assets of large oil company attracted different big representatives of the industry. One possible and the most probable buyer was Sibneft company; nevertheless, transaction didn't take place because alliance of Russian oil companies made own proposal. This competition led to cancellation of privatization.

The beginning of 2000s were connected with management activities directed to consolidation of the organization's assets, decrease of leverage appeared during Russian crisis of 1998 and purchasing of licenses for oil deposits. During this decade organization acquired several big Russian oil companies and expanded geography of extraction. Moreover, Rosneft began to acquire foreign entities and to strengthen relationships with large international organizations. In 2001 there was proposal of merger with Gazprom, which, however, were not accepted by the representatives of the companies.

One of the most significant event that is connected to acquisition policy of Rosneft was in 2013. This year Rosneft acquired 50% of joint venture TNK-BP, which made Rosneft the largest public oil company in the world [Rapoza, 2013]. According to annual financial reports for 2015, the last year acquisitions include the following events:

· Rosneft acquired AET Raffineriebeteiligungsgesellschaft mbH by purchasing 66.67%.

· Acquisition of 100% ownerships were utilized for LLC Trican Well Service (TWS), Petrol Market Company, Novokuibyshevsk Petrochemical Company, Orenburg Drilling Company, Bishkek Oil Company.

· Rosneft acquired 8 enterprises of Venezuelan Weatherford International plc.

In 2016 Rosneft capitalization exceeded capitalization of Gazprom that gives opportunity to call Rosneft the most valuable company of Russia [Bierman, 2016].

Figure 3.2.1. Rosneft (blue) and Gazprom (white) capitalization, (Bloomberg)

3.2.3 Novatek

Novatek OFSC is Russian public company that has shares that are traded in Moscow and London stock exchanges. The company is the representative of gas industry and the second large producer of natural gas in Russian according to production volumes. The interesting fact is that Novatek is 6^th company in the world in volume production, whereas its costs on exploration and development of deposits are one of the lowest among gas producers. According to annual report for 2015, the organization has share of 20% in gas supply of Russian market.

Main concentration of the company is exploration, extraction, conversion, transportation and sales of natural gas. The core facilities of Novatek are situated in Yamalo-Nenets Autonomous District. All produced gas is sold in Russian market. Therefore, company tends to enter international markets by creation of new sales channels.

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