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This study develops a dynamic model of optimization of the value of the company following the postulates of the theory of Trade-Off. The model includes Bystrom formulation for calculating the cost of bankruptcy in the search the optimal debt. Our model is innovative in several respects: 1) raises calculate the cost of bankruptcy easily. This model is easy to implement in firms 2) calculates the cost of debt endogenously. 3) The calculation is dynamic. We determine the cost of debt and the cost of capital for each unit of additional debt. The proposed model has been applied to the companies that make up the Dow Jones Industrial Average (DJIA) in 2007. We have used consolidated financial data from 1996 to 2006, published by Bloomberg. We have used simplex optimization method to find the debt level that maximizes firm value. Then, we compare the estimated debt with real debt of companies using statistical nonparametric Mann-Whitney. The results indicate that 63% of companies do not show a statistically significant difference between the real and the estimated debt.

If there is one subject that both fascinates and frustrates financial theorists and researchers, it is capital structure. It is the basic question that all financial managers should be able to answer: How much debt and how much equity should my company have? Even although these and other questions have occupied economists for a long time, we are far from that being answered. Most of the empirical literature on firms’ capital structure analyzes this issue within a static framework. Since the seminal papers of Modigliani and Miller (referred to as MM pro- positions from now on) written between 1958 and 1963 [

The resulting research that stemmed from the MM propositions yielded two different schools of thought. The first avenue, which directly followed the MM proposition line, has been attempting to find a relationship between the tax benefit of debt, the cost of bankruptcy and the optimal level of debt versus equity of a firm. This line of thought is commonly known as Trade-Off Theory. The other school of thought put forth by Myers [

There have been hundreds of deterministic models presented since 1958, starting with the leverage gains model presented by Modigliani and Miller [

The objectives of this study are as follows: First, develop a model for calculating the optimal level of debt, which adjust the cost of funds (both debt and equity) for bankruptcy risk that each monetary unit of debt brings to cash flow of the company. Second, apply the model to real data of the companies included in the Dow Jones Industrial Average (DJIA) to check the robustness of it.

This paper is organized as follows. Section 2 provides an overview the most outstanding papers on capital structure including bankruptcy costs in their models. Section 3 describes the dynamic model of capital structure developed. Section 4 describes the methodology and data used. Section 5 shows the findings. The final session discusses the overall conclusion of the study.

Previous literature to estimate the cost of bankruptcy is extensive. One of the best examples is Altman [

Through a complex system of simultaneous equations based on the following formulas:

where

And C^{*} is the optimal coupon for the debt, D^{*} is optimal debt and v^{*} is the optimal value of the company’s assets, while V is the value of the company, r is the risk free rate, σ^{2} is the volatility of the assets of the company, T is the corporate tax rate and α is the costs related to bankruptcy. The solution to the simultaneous equations yields L^{*}, the optimal leverage. Unluckily, Leland’s [

Philosophov and Philosophov [

In this case, the cash flow was discontinuous at the point of bankruptcy:

The value V of the company is a function of a number of factors ƒ and the moment of bankruptcy t_{b}. After bankruptcy, the cash flow suffers a reduction in value represented by the term δ. The net present value of the company is determined by the estimated profit of the company,

where T_{E} is the personal tax rate on equity income, T_{C} is the corporate tax rate, T_{D} is the personal income tax rate on interest income and D is the value of the company’s debt. To improve the formula Hull [

where α represents _{U} is the unleveraged value of the

shareholder’s invested capital. Thus, the side of the equation that represents V_{U} (unleveraged value, right hand side) is generally negative (if _{U} gives us a negative value) while the

side of the equation that represents V_{L}_{ }(leveraged value, left hand side) is generally positive (if

lied by the result would be a positive number).

In this fashion, Hull [

Let us set up a basic scenario for our model. We will start by looking at a simple company with many shareholders, no debt and one manager. For the time being we will forgo taxes. Under such circumstances, the value of the company will be represented by:

where V is value of the company, WACC is the weighted average cost of capital, Kinv are the funds invested by the company either from its o own funds or from equity issues (we consider all funds in the company as belonging to the shareholders, so we see little difference between money coming from shareholder or money coming from the company’s operations) EBITDA is the earnings before interest payments, taxes, depreciation and amortization and t is the time period. Since we are looking for the sustainability of the investment over time, we include the depreciation (DEP) as a cost as it generates de funds necessary to keep the investment operating indefinitely. Since there is no debt, the WACC and the company’s return on equity would be the same. Thus, the company invests an amount Kinv at time zero in equipment, which allows the company to generate a cash flow

represented by

of the cash flow from the investment will be higher than the funds invested and the company’s value will be positive.

Let us now include debt in the equation to allow this company manager to decide between using his shareholder’s money or financing the investment. For simplicity we will assume there is only one type of debt: unsecured bond or bank debt. From a cash flow perspective to this manager, the company’s value function would look as this:

were AMRT is the amortization of the debt contracted and is described as a function of the original debt at time 0 divided by the number n of payments contracted:_{t}) multiplied by the debt interest rate i. Now Kinv represents the portion of invested funds that comes from equity, be it company funds or new equity issue. Since now there is debt, the WACC

would be represented by the well-known function

of equity, D is the face value of debt, k is the return on common equity and i is the interest rate on debt. In turn, k is represented by the Capital Asset Pricing Model in the formula (

We have found very few research papers that use an interest rate that is determined internally by the model, as common practice is to assume the interest rate as exogenous. The works of Leland [

where variable K is defined as:

where C is the debt coupon, V is the value of the company’s assets, r is the risk free rate, σ^{2} is the volatility in the value of the assets, T is the corporate tax rate and α is the costs associated with bankruptcy. Unluckily, as with most research that follows this particular line of inquiry, the work done is mostly theoretical and not tested on real data. Another problem that this line of inquiry faces is that the variables used by these models are not easily observed or easy to measure.

Merton [

where V_{A} is the company’s market value of assets, D is the total amount of debt, _{A} is the volatility in the value of the assets,

The main problem with Merton’s formula is that the market value of assets of a company is not an easily observable data. Even calculating the market value of assets would be cumbersome. Obtaining the volatility and standard deviation in the market value of assets would require that such a cumbersome calculation be repeated enough times to make the data reliable. This means that most research based on similar formulas, like Leland [

drift term, as it was generally very close to zero and used a one period debt. He then restated σ_{A} as

where σ_{E} is equity volatility and V_{E} is the market value of equity. Cleaning up the formula and replacing

Finally, Bystrom determined that

Bystrom’s approximation to Merton’s formula showed a 99.9% correlation with Merton’s formula, which allows us to use easily obtainable data to calculate the distance to default.

We also used Altman and Kishore [

where L is the _{E} is the volatility of the company’s equity, which is used as a valid approximation to the volatility of the company’s asset return and N is the cumulative normal distribution. This distance to bankruptcy estimation is but an approximation to Merton’s [

small. We will refer to the term

Looking at Equation (1.2), we notice that the debt value is divided into two components: one represented by the present value of all payments of the principal of the debt and a second one that represents the net present value of the interest payments on the outstanding debt. Again, since we are looking at this cash flow from the manager’s perspective, both terms are negative as they detract from the shareholder’s cash flow. Furthermore, since the principal of the debt has been broken up into an n number of payments to be paid in an n number of periods, both components of debt are meaningful only up to the nth period since at time n + 1 all the debt will be repaid and thus both terms will equal zero. At this point in the model, we have yet to include the government’s silent participation in our company’s cash flow. The addition of corporate taxes has two opposing effects. On the one hand it subtracts from the company’s cash flow leaving the inflow as

Since we are including taxes, we also need to revisit the calculation of the return on equity k and the WACC. Recall that we talked earlier of the effect of debt on b and the need of calculating an unleveraged b. The tax ef-

fect on debt will also have an effect on b so that

sents the corporate tax rate, D the debt and E the equity. This will allow us to account for changes in the return on equity k due to increases in debt levels. The calculation of WACC will also have to take into account the lower cost of debt due to the tax effect on interest payments. This would result in a

interest rate i, such that the new equation would be:

WACC and thus increasing the net present value of future cash flows (generating a second order tax shield which is not directly seen in the cash flow per se, but rather through the present value of said cash flows).

What we now have is a value based function that can be maximized to find the optimal debt level for the com- pany, but that also adjusts the cost of funds (both equity and debt) to account for the added risk that debt brings to the cash flows of the company.

As we mentioned earlier, ours is a two-step model where the determination of the necessary investment, Kinv, is determined prior to the value maximization process of determining how much debt and how much equity will be used to finance the investment. Therefore, the EBITDA of the company, as well as the depreciation, is calculated and set prior to the maximization process. The final step is to look for the optimal debt level required by the company to finance its operations. We define the optimal debt level as that which maximizes the net present value of the company, as per Formula (1.4). The increase in the net present value of the company when maximizing the debt level comes from the increase in the value of tax shields as more debt is added, versus the added cost of said debt. As such, the increase in debt will increase the probability of bankruptcy to a point when the addition of further debt will increase the cost of debt over and above the benefit of the marginal tax shield.

So we see that the relationship between debt and company value is an inverted U shaped curve, where the higher interest rate resulting from the increase in the probability of bankruptcy of additional debt is lower than the positive effect of a lower WACC and the

The parameters of our maximization procedure are the following:

1. The amount of the investment needed by the company, Kinv, is exogenous to the maximization process of the capital structure. That is to say that the level of investment is not conditioned by the source of the funds.

2. Equity, E, has to be positive and at least 1. This has to do with the social construct of companies in the North American market, where the creation of a company requires that equity be supplied (debt is not a requirement to create or incorporate a company in the US).

3. Debt, D and Equity, E, as well as the value of the company, V, must be positive.

4. The probability of bankruptcy should not exceed 5%.

The fourth parameter requires some explanation. Usually, the model should be able to calculate interest rates from bankruptcy probability levels of 0% to 100%. However, if we look at the formula, at 100% of probability of bankruptcy, the formula turns into 1R and therefore the cost of the debt represented by the probability of bankruptcy is equal to the expected loss in value of the debt at the moment of bankruptcy as determined by Alt- man and Suresh [

Nevertheless, the use of Altman and Suresh’s data to determine the interest rate sets a limit to the probability of bankruptcy that our interest rate model can predict.

If we look at the classification of debt as Standard and Poor’s, we see that debt is classified from AAA which SandP itself describes as “Extremely strong capacity to fulfill its financial obligations”, all the way to D “In default of its financial obligations”.

As such, any debt rated below C would be encompassed in Altman and Suresh’s [

To test our model, we used it to try to determine the optimal debt level (short and long term straight debt) of the companies that made up the Dow Jones Industrial Average (DJIA) in 2007 and compare it to their real debt between the years 1996 and 2006. However, since this is an EBITDA based model we had to eliminate from the sample all financial companies (this takes out of the sample, American Express, J. P. Morgan Chase, Citigroup and AIG) leaving 26 companies on the sample group. Furthermore, for the sake of our study, General Motors was also eliminated as it had a negative shareholder’s equity in 2006 and negative EBITDA in 2005, leaving the sample at 25 companies. We decided to use the period from 1996 to 2006 because if we started in 1995 we would have had to eliminate also Microsoft and Honeywell (because both the β and the σ_{E} are calculated from data from the previous 10 years and both companies did not list back in 1985). We used consolidated year end 1996 to 2006 numbers as reported in Bloomberg.

The test assumes that the book value of debt and the book value of equity represent the capital raised by the company to meet their investment needs. We understand that some capital structure prediction models use market value of both equity and debt. Welch [

More important, changes in the market value of debt or equity have no impact on the company’s cash flow. For the company to benefit from the increase in market values it would need to issue either debt or equity, which would be reflected in the book value of the company, bringing us back to the book value of equity and debt as the proper measure for our purposes. The investment financed with either debt or equity (or a mixture of both) generates an EBITDA and a depreciation which are constant over time. The assumption behind this is that all capital was invested in productive assets, operating at capacity and no new capital will be raised until the next year (any new capital raised would warrant a new calculation). Thus, we can calculate

rate. Thus,

sary to reduce the workload to run our test from thousands of calculations to hundreds of calculations.

The amortization of debt and the interest payments are calculated over the weighted average length of the company’s debt as reported in their 2006 10-K and then brought to present value using the company’s WACC. The average 1996 to 2006 tax rate paid by the company is used as corporate Tax (T).

Each company’s

The equity volatility (σ_{E}) is calculated as the standard deviation of the monthly log normal returns of each stock. Ten years of previous data are used to determine both the ^{1} for the period 1990 to 2006, which is 10.2%. The percentage of the company’s value lost at bankruptcy (R) is calculated as [1-bond recovery rate] which, in turn, is taken from

There are a number of concessions we have had to make either due to lack of information or in order to facilitate the performance of our test. For the purpose of determining a recovery rate from Altman and Suresh [

The results of the Mann-Whitney tests show that, at a 99% confidence interval, 63% of the companies tested had no statistically significant difference between the median target debt calculated by our model and the median real debt for the company as reported in Bloomberg (See

Company | W-Statistic | P-Value | P > 0.01 | Company | W-Statistic | P-Value | P > 0.01 |
---|---|---|---|---|---|---|---|

Alcoa Inc. | 150.0 | 0.1310 | Equal | McDonalds Cp | 82.0 | 0.0039 | Not Equal |

Boeing Co. | 167.0 | 0.0086 | Not equal | 3M Company | 66.0 | 0.0001 | Not Equal |

Caterpillar Inc. | 187.0 | 0.0010 | Not equal | Altria Group Inc. | 132.0 | 0.7427 | Equal |

Du Pont | 122.0 | 0.7928 | Equal | Merck Co Inc. | 104.0 | 0.1486 | Equal |

Walt Disney-Disney Co. | 108.0 | 0.2372 | Equal | Microsoft Cp. | No Significant | No Significant | |

Gen Electric Co. | 143.0 | 0.2934 | Equal | Pfizer Inc. | 111.0 | 0.3246 | Equal |

Home Depot Inc. | 68.0 | 0.0001 | Not equal | Procter Gamble Co. | 128.0 | 0.9476 | Equal |

Honeywell Intl Inc. | 142.0 | 0.3246 | Equal | AT and T Inc. | 121.0 | 0.7427 | Equal |

Hewlett Packard Co. | 166.0 | 0.0104 | Equal | United Tech. | 101.0 | 0.1007 | Equal |

Intl Business Mach | 187.0 | 0.0001 | Not equal | Verizon Commun. | 159.0 | 0.0356 | Equal |

Intel Cp. | 68.0 | 0.0001 | Not equal | Wal Mart Stores | 129.0 | 0.8955 | Equal |

Johnson and Johnson Dc. | 66.0 | 0.0001 | Not equal | Exxon Mobil Cp. | 66.0 | 0.0001 | Not Equal |

Coca Cola Co. | 95.0 | 0.0418 | Equal |

From our results, we can draw the following conclusions:

1. We created a simple model that can be used by any financial officer to determine the optimal level of debt of a company. The model used Bystrom’s [

2. Our model finally does away with one of the main criticisms of Trade-Off theory. In the words of Stuart Myers: “The tradeoff theory cannot account for the correlation between high profitability and low debt ratios” (Myers [

3. The strong difference between the debt level calculated by our model and the actual level of debt that some of the companies in our sample had, like 3M, Johnson and Exxon, seems to confirm the results of Fischer et al. [