2.2.1. Optional valuation of debt and the issues associated with getting into debt

According to Merton (1974), the fluctuations of the value of a company, over time, are seen as a stochastic process according to the following differential equation:

[2.1]

Where:

• – α is the instant expected profit rate of the company by unit of time. If C is positive, it represents the total revenue by unit of time paid by the company to its shareholders and bondholders (e.g. dividends, interest charges). If C is negative, it represents the money received by a company for new financing;
• – σ² is the instantaneous variance of the return for the company by unit of time;
• – dz follows a standard Wiener process.

Moreover, F is the economic value of the debt and D is the nominal value of the debt, that is, the amount that the company has promised to repay to its bondholders according to a precise schedule. In the case where the repayment of D is not made, the bondholders take over the company and shareholders receive nothing. If there is no coupon, the stochastic differential equation applied to D gives:

[2.2]

Let us call:

• – F(V,τ) the economic value of the debt when the remaining time to maturity is τ. Thus, F(V,0) = min (V,D);
• – f(V,τ) the economic value of equity when the remaining time to maturity is τ. Thus, f(V,0) = max(0;V-D) and: f(V,τ)= V.Φ(d₁) – De^-rt.Φ(d₂).

Since: F = V – f, we have:

[2.3]

Let us consider:

[2.4]

Thus:

[2.5]

This formula allows us to express the exercise price of the risky debt. In this context, we call the return at maturity R. Then:

[2.6]

Consequently:

[2.7]

[2.8]

[2.9]

In the end, the spread is equal to:

[2.10]

Thus, the consideration of an economic value of net debt (substituting the amount of accounting net debt) in the process of company valuation necessarily impacts the systematic risk of equity and the profitability that shareholders demand from the company.

2.2.2. Combination of CAPM and the options model: the systematic risk of equity and the rate of return required by shareholders

Galaï and Masulis (1976)¹⁸ combine the capital asset pricing model (CAPM) with that of options in order to valuate company equity. The synthesis of these models leads to an adjustment of systematic equity risk and to a distinction between different situations impacting the financial structure of companies. The authors recall the CAPM formula:

[2.11]

with:

• – : instantaneous return expected for a company share i;
• – : instantaneous return expected from the market;
• – β_i: instantaneous volatility coefficient for the share with respect to the market, also called the systematic risk of the share;
• – r_f: risk-free interest rate;
• – : – r_f: market risk premium.

And:

[2.12]

Thus, the present value of the company J can be found using the following expression:

[2.13]

where:

• – T: liquidation date of the company;
• – : final expected value of the company;
• – : economic value of company shares at maturity;
• – : economic value at maturity of shares from companies that make up the market;
• – R_f: instantaneous risk-free market rate;
• – : risk unit;
• – γ: market value by risk unit, defined by: ;
• – : expected market return in discrete time.

By using the Black–Scholes (1973) formula consisting of the valuation of a European call, Galaï and Masulis use the partial derivatives equation for the call formula in order to demonstrate the existing relationships between the different parameters¹⁹:

[2.14]

Additionally, Galaï and Masulis retain the following Merton relations (1974)²⁰:

[2.15]

with:

• – S: economic value of equity;
• – V: economic value of assets;
• – σ²: instantaneous variance of asset returns;
• – C: nominal value of debt;
• – D: economic value of debt;
• – r_f: risk-free interest rate;
• – T: probable lifetime of a company.

The value of the share is a growth function of the value of company assets, the risk-free interest rate, the variance of returns of company assets and the maturity. On the contrary, it is a decay function of the nominal value of the debt.

Following a stochastic logic and the hypotheses of a market in equilibrium by Galaï and Masulis (1976), the return on equity can be defined by the following expression:

[2.16]

with:

[2.17]

By dividing ∆S by S, we get, when t tends towards 0:

[2.18]

Now, according to formula [2.12],

[2.19]

Keeping in mind that, according to the Black–Scholes model (1973), S_V = N(d₁)²¹, Galaï and Masulis combine the CAPM with the options model:

[2.20]

By replacing S with the call formula from Black and Scholes, we have:

[2.21]

Now, 0 ≤ S = V(d₁) – Ce^{− rf.T} N(d₂), which implies that:

[2.22]

The systematic risk of equity is greater than or equal to that of the company, for β_V > 0. If the latter is stable, then equation [2.20] indicates that the systematic risk of equity is not. Thus, K, the vector for the parameters V, C, r_f, σ² and T:

[2.23]

and:

[2.24]

[2.25]

The existing relationships between β_S or β_D and the parameters of the vector K are²²:

[2.26]

[2.27]

The systematic risk of equity for a company is a growth function of the nominal value of debt and a decay function of the value of company assets, the risk-free interest rate, the variance of return on company assets and the maturity.

Next, Galaï and Masulis (1976) continue their study on the profitability rate required by shareholders. The instantaneous return on company assets is equal to that of its debt and equity weighted with their relative importance in the company accounts. Thus:

[2.28]

[2.29]

By replacing by [2.11], β_S by [2.20] and recalling that we get:

[2.30]

According to Rubinstein (1973)²³:

• – represents the expected profitability rate of company assets;
• – represents the financial risk of an indebted company taken on by the shareholders.

The equation [2.30] can also be written as:

[2.31]

Now, represents the return on investment for shareholders, which includes the financial risk. From this:

[2.32]

This result underscores the elements that contribute to a higher profitability rate, required by shareholders because of the debt. By using equation [2.11] and the results obtained in [2.26] concerning β_s, we have:

[2.33]

In this context, Galaï and Masulis (1976) determined that it would be interesting to focus on different financial situations impacting the structure of capital and consequently the value of the company.

2.2.3. Situations that impact financial structure

In order to analyze the potential transfers of wealth from one security to another, Galaï and Masulis (1976) consider two companies, A and B, and define and then annotate the following variables.

Variables	A	B
Market value of company assets
Market value of company assets at maturity
Market value of company shares
Market value of company debt
Systematic risk of company assets
Systematic risk of company shares
Variance of return on company assets
Return on company assets
Return on company shares
Nominal value of debt	CA	CB

In order to study the impact of acquisitions and divestments, Galaï and Masulis consider the following hypotheses:

[2.34]

[2.35]

[2.57]

[2.36]

Following [2.13], [2.35] and [2.36], the authors affirm that . The two companies have the same systematic risk but different variances. If , then and . Indeed, the value of the option is a growth function of the variance of its underlying securities, all other things being equal. Now, equity can be assimilated to a call. It is therefore possible to directly apply the relation .

Two companies in possession of the same nominal value of debts and the same company value but with different variances will have different financial structures. The market value of will be greater for the company with the weaker variance. In the example, If a company makes an acquisition or an investment that modifies the variance of return on its assets, then the economic value of the debt and equity will be impacted.

In order to study the impact of the variation of the size of a company, Galaï and Masulis consider the following hypothesis:

[2.37]

which implies that:

[2.38]

Following equation [2.13], we have:

[2.39]

Hypothesis [2.37] also implies that company returns have a perfectly positive correlation, such that:

[2.40]

[2.41]

If we consider that:

[2.42]

Then, according to equation [2.17], which corresponds to the Black– Scholes formula, we remark that and that . Therefore:

[2.43]

If the market value of assets and the nominal value of debts of two companies are proportional, then the economic value of their equity and their debt will also be proportional. If the value of a company rises by , the latter can produce debt until This amounts to increasing the debt and the equity in proportion to the increase in the size of the company. In the contrary case, the company faces the dissolution of its securities (debt or equity).

In order to study the impact of fusions, Galaï and Masulis consider the following hypothesis:

[2.44]

Now, according to the previous case:

[2.45]

[2.46]

[2.47]

[2.48]

Moreover:

[2.49]

[2.50]

Hypotheses [2.48] and [2.49] imply that:

[2.51]

Hypotheses [2.44], [2.47] and [2.50] lead to:

[2.52]

The study of the first case, results [2.51] and [2.52] and equation [2.17], which corresponds to the Black–Scholes formula, gives us:

[2.53]

The market value of debt for company G is higher than the sum of the market values of A and B’s debts. On the other hand, the market value of G’s equity is lower than the sum of the market values of A and B’s equity. The position of shareholders will deteriorate if the merger of companies A and B leads to a drop in the variance of asset returns for the new company G. As for the situation of G’s creditors, it is improved since the risk of bankruptcy has gone down.

In order to study the impact of splits, Galaï and Masulis consider the following hypothesis:

[2.54]

which implies that:

[2.55]

Company G is made of two economically independent divisions, A and B. At t = 0, company G separates from division B, thus G is composed only of division A. Thus:

[2.56]

After the split, the creditors of company A (creditors of company G after the split) see their situation deteriorate since the assets that were backing the debt have diminished. The company that does not protect itself when financial decisions are being made can worsen the situation for its creditors.

Consequently, agency conflicts between the different interested parties of the company are provoked by the financial structure itself, by the strategies that are attached to it, as well as by the presence of investment options that give the company value a potential surplus.

2.3.1. The interaction of financing decisions

Myers (1977) shows that the production of loans is inversely proportional to the portion of company value that is attributable to real options. Thus, even in the presence of favorable fiscal measures that aim to incite company debt, the optimal strategy encourages them to find a compromise between these fiscal advantages and the costs of this optimal strategy which need to be absorbed in order to envisage any future investment. The Myers article is, in many ways, in the continuum of the one by Jensen and Meckling (1976)³⁰.

These authors are in fact interested in agency costs and the optimal structure of capital. Myers, for his part, considers a sub-optimal investment policy to be an agency cost for a company. And Jensen and Meckling, as well as Myers, identify and analyze the costs that are traditionally considered imperfections of the market³¹. In this context, Mello and Parsons (1992)³² recall the Modigliani–Miller principle (1963)³³, namely that in the absence of agency costs tied to debt, the value of an indebted company is equal to that of the debt-free company plus the fiscal deductibility of interest charges.

Then, Mello and Parsons overlap with Myers’ work (1977) by taking an interest in agency costs of company debt in the mining sector. They note that as the size of the debt increases, the agency costs grow such that in situations where the debt is large, they can surpass the value of fiscal advantages tied to the fiscal deductibility of interest charges. In this case, the value of the company in debt can turn out to be less than that of the one that is not, insofar as we have to subtract agency costs from the company value adjusted for the fiscal advantage of interest charges.

Chava and Roberts (2008)³⁴ performed an empirical study on more than 37,700 loan contracts collected from more than 6,700 companies between 1994 and 2005. Their objective is to study the mechanisms that tie investment to financing, that is, the friction caused by different financing methods that influence the choice of investments. The authors study these agency problems, namely the impact when the covenants associated with a company’s investment debt are not followed.

In this context, they concentrate on the violation of restrictive financial clauses imposed on companies by creditors, such as the conservation of certain ratios or certain values. These violations, which the authors call “technical faults,” caused by the fact that companies do not honor their commitments to pay off borrowed capital or interests, encourage the breakdown of financial alliances. In principle, the creditors then use their control rights and threaten to accelerate the schedule for repaying the credit. And these commitments are in fact omnipresent in financial contracts.

For example, public debt can exist alongside private equity. It is in these terms that the potential for violation exists and, moreover, as Gopalakrishnan and Parkash affirm (1995)³⁵, the non-respect of covenants arises in situations that are completely devoid of anger or any financial distress and only rarely leads to payment defaults or an acceleration of the debt repayment plan.

Thus, according to Chava and Roberts, the potential impact of violations of the covenants is not limited to a small fraction of companies confronted with unique situations. The authors study the impact of the transfer of control rights over the company investment. The optional logic suggests that the shareholders keep their rights of control as long as the investment value remains over a certain alliance threshold. In the inverse case, the rights to control can be granted to creditors, who then influence the company’s investment policies. They can then increase interest rate, shorten the deadline for repayment, which reduces the free cash flow of the company, or become involved in subsequent investment decisions.

Chava and Roberts’ study also shows that the transfer of control rights following a violation of the covenants, on average, leads to a fall in a company’s investment activities by approximately 1% per quarter, since creditors intervene in order to prevent investments judged to be inefficient and also to punish the people responsible for the bad policy.

However, the authors point out a heterogeneity at the very heart of this phenomenon. Divestment is in fact more notable in cases where information and agency problems are larger than in cases where the friction from asymmetrical access to information are less severe. The authors even show that in cases where information and agency problems are less severe, the decrease in investment expenses is almost non-existent. It can occur in situations where the current creditors are long term and believe their client to benefit from a good reputation and where the loan was made before a bank federation (rather than only one credit institution).

For example, in line with the conclusions of Diamond (1989)³⁶, the authors note that on average, after a covenant violation by a company without any long-term moneylenders, the reduction in investments is 1.7% with respect to the value of shares while the companies with long-term moneylenders, on the basis of the same calculation, observe a negligible reduction in investments of 0.2%. Likewise, by referring to the works of Bolton and Scharfstein (1996)³⁷ which suggest an attenuation in the moral vagaries of borrowers, in the case of a covenant violation, companies under contract with a sole moneylender see a greater decline in investments (2.3%) than those who benefit from a bank federation (0.3%).

Morellec, Nikolov and Schürhoff (2008)³⁸ build a dynamic model of the structure of capital in order to determine if agency conflicts can explain the financial decisions of companies. The model contains variables such as tax on individuals or the costs of refinancing. Their analysis shows that the private profits linked to control lead managers to create less debt so they can rebalance the capital structure, which is often sub-optimal for shareholders. By using data on financing choices and by making predictions for different levels of debt, the three researchers observe that on average, 1.02% of agency costs on the value of equity is sufficient to resolve the issue of debt and justifies the chronological series of debt ratios observed. They also observe that agency costs vary between companies and that governance mechanisms significantly affect the value of control and the financing decisions of companies.

Galaï and Wiener (2013)³⁹ underscore the fact that the recent financial crisis made apparent the harmful effect of policies of excessive dividends leading companies to have financial difficulties. As Black says (1996)⁴⁰, “the more we look at the image of the dividend, the more it looks like a puzzle whose pieces simply don’t go together”. Benartzi, Michaeli and Thaler (1997)⁴¹ find a very weak relationship between the variations of dividend payments and the content of information on the future beneficiaries of the company. Indeed, the increases (decreases) in dividends are correlated with the increases (decreases) in year-end and previous year results but do not explain any expectations on future profits.

Galaï and Wiener thus try to define the qualitative and quantitative restrictions on the dividend payment policy by taking any conflicts with creditors into account. The distribution of dividends, from the perspective of creditors, is similar to a reduction in company value and, consequently, increases the value of the put, the latter corresponding to the credit risk value. In other words, the forecasts for dividend payments over the lifetime of a bond loan can explain, at least in part, the overvaluation of bonds and the underestimation of their premium.

This is why DeAngelo et al. (1992)⁴² observe that companies smooth out the payment of their dividends and try to maintain a stable distribution ratio. Moreover, it turns out that empirical studies before Galaï and Wiener’s article on the spread of credit risk show that the latter are bigger than those anticipated by the classic optional model. Jones, Mason and Rosenfeld (1985)⁴³ consider the spread using the Merton model and observe that it is less than the one observed in the data. Elton et al. (2001)⁴⁴ as well as Huang and Huang (2003)⁴⁵ attempt to explain this surplus of premiums with unobservable variables such as the effects of tax rates or liquidity.

The Galaï and Wiener model is based on the conflicts of interest that exist between shareholders and bondholders. By establishing a stable policy of dividend distribution, shareholders certainly get value (expected) from the bondholders, but this stability facilitates surveillance by the bondholders. Indeed, Kalay (1980⁴⁶, 1982⁴⁷) emphasizes that the restrictive clauses imposed on shareholders by bondholders push them to withhold dividends they would be authorized to distribute. The latter remain well within the imposed constraints. Thus, the bondholders can feel more secure with this accounting policy using covenants, projections and acceptable levels of probability of default. If a profitable company does not pay dividends (and does not develop a program for buying back shares), the value of the debt improves because of the decrease in probability of bankruptcy. If a dividend distribution program is expected without being realized, there is a transfer of equity wealth to the holders of the debt. Thus, the dividends are also a way of getting value from shareholders. A well-known and stable dividend payment policy, however, reduces the cost of the debt because the risk of credit is itself stabilized, making it more predictable.

In this context, Galaï and Wiener propose the introduction of a dividend distribution policy in the Merton model. They suppose that dividends are paid at a fixed, continuous rate. In this way, the shareholders of an indebted company keep a call over the company’s assets in addition to having the right to obtain dividends. The dividends that are paid out reduce the value of the company, V, at a rate of δ and thus the put of creditors (which reflects the risk of credit affected by the fluctuation in dividends). By writing the value of the put over company V as P^EX, with an exercise price of F, T maturity and a risk-free rate of r, we have:

[2.57]

with:

[2.58]

[2.59]

We can note that:

[2.60]

The value of the put rises when the dividend distribution rate rises. Let us write B^EX, the price of the bond when the dividend is paid. Thus:

[2.61]

[2.62]

[2.63]

Using Boness (1964)⁴⁸ and Galaï (1978)⁴⁹ as a starting point, the probability of default PD when the bond loan is repaid, considering the dividend policy, is:

[2.64]

with:

[2.65]

Campbell and Taksler (2003)⁵⁰ observe that between 1995 and 1999, the American stock and bond markets evolved differently. That is, the price of shares rose sharply while the performance of bonds was mediocre. In fact, if the return on company bonds has risen, those on Treasury bonds will too, provoking stagnation in the spread. The authors explore numerous explanations:

• – insofar as the shareholders predict better profits, they anticipate, optimistically, more revenues while the bondholders see neither more nor less than the anticipated coupon;
• – the companies which resort to bond loans do not correspond to those that dominate the stock indices;
• – a rise in the liquidity premium on company bonds with respect to Treasury bonds lowers the value of bonds without affecting the value of shares;
• – the returns on newly issued company bonds can vary because of the change in characteristics. For example, a rise in the value of issuing impacts the drop in the price of newly issued bonds and the rise in their return without impacting the older bonds;
• – given the expectation of future profits, the volatility of assets impacts bonds because of a rise in the probability of bankruptcy. To this end, Campbell and Taksler note that the more the volatility rises, the more the cost of the debt grows.

To mitigate agency conflicts, protection clauses and information costs figure in the contracts involving financing decisions. These are therefore two parameters to take into account in the valuation of a company using real options.

2.3.2. Accounting for information costs and protection clauses

Bellalah (2000)⁵¹ seeks to optimize the value of companies by using the theory of options, first according to a model that conforms to those by Black and Scholes (1973) and Galaï and Masulis (1976), then in a context that accounts for the effects of information costs with reference to the structure of capital, as envisaged by Bellalah and Jacquillat (1995)⁵² and Bellalah (1999)⁵³.

The protection clauses preserve the interests of creditors. The general clauses allow for agreements between the different interested parties tied to financing and control operations. They reduce the influence of shareholders and directors in matters of financing and investment policy in order to limit transfers of wealth. The classic clauses that obligate the borrowing company to provide an exhaustive report of its activity fall within the scope of agency theory. The objective is to minimize conflicts of interest between different types of creditors. When special clauses are written, the company must use the loaned amounts to make profitable investments.

Additionally, because of the existence of different types of debts, a hierarchy is established. The senior debt, of the highest importance, is generally subject to securities on the assets of the borrower. The junior debt, of lesser importance, is repaid after the senior debt and is situated between equities and classic loans. The subordination clauses accompanied by special clauses, on their part, preserve the interests of each creditor.

The relationships that unite the shareholders, quasi-private equities and bonds, define the content of the clauses. The theory of options thus allows us to analyze them and evaluate the structure of liabilities.

A company with a value V subscribes to a bond loan of a face value of E and maturity T. The nominal of the debt can either correspond to a first-order debt P or a second-order debt Q. At the maturity of the debt:

• – if V > E, the bonds are repaid at the amount of E and the value of shares S is equal to the residual value of the company V – E;
• – if V < E, the company goes bankrupt. The creditors are partially reimbursed and the shareholders get nothing.

The value of the option therefore becomes all the more important as the value of the company at maturity rises. Furthermore, investment in risky projects raises the value of equity⁵⁴. When the bondholders are informed of a company’s intention to proceed with a bond loan and modify its risk, protection clauses are included in the subscription contract.

We thus discover the benefit of the reorganization clauses in that they allow bondholders to begin proceedings for legal redress when the company does not respond to its obligations such as, for example, the payment of capital or interests. And creditors can begin bankruptcy proceedings when the value of the company reaches a critical level. Moreover, subordination clauses favor the hierarchization of the repayment of debt. In this context, a junior bond is repaid after a senior bond. At maturity, the value of assets for a company that has issued first-order bonds P and second-order bonds Q is:

The restriction clauses, which concern limits on creditors imposed over the dividend payment policy and on the charging of interest, can also protect bondholders against new issuing. The point is to include a clause that prevents the company from issuing new bonds without the agreement of second-order bondholders. Any new issuance of junior debt is in fact advantageous for first-order bondholders but unfavorable for second-order creditors.

The evaluation models thus relate to equity, debt and clauses present in debt subscription contracts.

The Bellalah and Jacquillat model (1995)⁵⁵ takes two new parameters into account: information costs for the share and information costs for the underlying asset. Thus, a call on the funds belonging to a company is equal to:

[2.66]

with:

[2.67]

[2.68]

• – E: nominal value of the debt;
• – λ_V: information cost relative to the economic asset of the company;
• – λ_S: information cost relative to the value of shares;
• – T: lifetime of a company;
• – S: present value of equity capital;
• – r: risk-free interest rate;
• – σ²: instantaneous variance on variations in the market value of the company.

Bellalah and Jacquillat (1995) studied variations⁵⁶ of equity capital value with respect to the parameters V, E, r, σ², T, λ_V and λ_S. And because D = V – S, we get:

[2.69]

[2.70]

The results concerning the variables V, E, r, σ² and T are identical to those obtained by Galaï and Masulis⁵⁷. The value of equity is a growth function of the company value, interest rate, risk and probable lifetime of the company. It is a decay function, however, of the nominal value of debt. The last two inequalities show that the value of equity is a decay function of information costs in the sense that the derivatives are negative. In other words, a rise in information costs lowers the value of equity. Thus:

[2.71]

Bellalah (2016)⁵⁸ refers more generally to the “shadow costs”, which the models for budgeting investments must include. The latter are irrecuperable and are made up of two elements. Following the Merton model (1987)⁵⁹ on the balance of capital markets in the presence of incomplete information, we first find the information costs tied to the imperfect knowledge of the market. It his article, Merton indicates that the effect of incomplete information on the price of equilibrium means applying an additional discount rate. In this context, the equilibrium of the expected return Rs on the share S is the following:

[2.72]

Where:

• – Rm: expected rate of return on the market portfolio;
• – R: risk-free rate;
• – βs: beta of the share s;
• – λs: “shadow cost” tied to share s;
• – λm: mean weighted “shadow cost” of incomplete information of all the securities in the market portfolio.

Let us note that in the presence of entirely complete information, the model becomes that of Sharpe (1964)⁶⁰.

Next, Bellalah indicates that we must consider additional costs caused by the constraint of short sales. In this context, Wu et al. (1996)⁶¹ extend the Merton model (1987) to account for the restrictions on sales and the heterogeneous expectations of investors, that is, by considering two types of “shadow costs”: λk corresponds to the constraint of information, and γk corresponds to the constraint of short-term sales. They thus develop a generalized model for the equilibrium of financial assets:

[2.73]

The Black–Cox model (1976)⁶², which evaluates reorganization and subordination clauses, has the objective of measuring their effects on the value of the financial securities of a company. They imagine a company that has signed up to a zero-risk bond loan paying shareholders a dividend δV. The percentage of dividends to pay is thus δ. In this context, the value of the debt B is the following:

[2.74]

where:

• – B: value of the debt;
• – V: value of the company;
• – σ²: instantaneous risk of company returns;
• – T: lifetime of the company.

The partial derivatives equation must satisfy the value of company assets. The following two conditions must be considered in order to resolve the equation:

[2.75]

[2.76]

The first condition shows the value of bonds at the maturity date. The term Ce^-γ(T-t) is the present value of coupons to pay bondholders. It also corresponds to the critical value of the company, synonymous with the triggering of bankruptcy proceedings. Likewise, the value of equity S is calculated using the following equation:

[2.77]

The two conditions to resolve the above equation are as follows:

[2.78]

[2.79]

With P as the value of senior debt, the first condition corresponds to the value of equity at maturity. The second condition proves that the bankruptcy proceedings are triggered once the value of the security is canceled. The value of bonds with this kind of protection clause is thus:

[2.80]

with:

[2.81]

[2.82]

[2.83]

[2.84]

[2.85]

[2.86]

[2.87]

[2.88]

[2.89]

[2.90]

[2.91]

According to Black and Cox (1976), by designating B(V, t, P, δ Pe^−r(T−t)), the value given by equation [2.80] for a bond paying P euros and including a protection clause (δPe^−r(T−t)), the value of a junior bond is:

[2.92]

[2.93]

[2.94]

Black and Cox thus show that protection clauses significantly augment the value of the debt modifying the financial structure. These clauses impose a “floor” value onto bonds and equity. Thus, shareholders lose control of a company the moment the company value tends towards a specified threshold for reorganization. It is moreover in the interest of bondholders to trigger the bankruptcy proceedings when circumstances allow, without waiting for the loan maturity date. Nonetheless, the bankruptcy proceedings are prohibited the moment the value of the company rises, that is, when a maximization strategy for the value is initiated.

2.3.3. Bankruptcy costs, getting into permanent debt and optimizing the debt ratio

Brennan and Schwartz (1978)⁶³ concentrate on the optimal structure of capital by accounting for the interest rate and bankruptcy costs. They suppose that the company value of a debt-free company, U, follows the geometric Brownian motion below:

[2.95]

where dz is a standard Wiener process. The value of the indebted company, V, is a function of the value of the debt-free company (the two companies having the same assets), and of the time to maturity of the debt, t. In other words:

[2.96]

Thus, the partial derivatives equation is:

[2.97]

At maturity T of the debt, we have:

[2.98]

[2.99]

where C(U) corresponds to the bankruptcy costs if the company is at risk of entering these proceedings. Furthermore, by supposing that t^- and t⁺ respectively indicate the moments before and after dividend payments d:

[2.100]

If the payment of coupons, iD, is included and τ is the tax rate on companies:

[2.101]

where i.(1-τ).D corresponds to the rise in capital required to restore the company value of the indebted company after the payment of coupons. After development and simplification with i.D, we have:

[2.102]

If the dividends and the coupon are paid the same day:

[2.103]

Finally, taking bankruptcy costs C(U) into account:

[2.104]

[2.105]

The latter two formulas correspond to the necessary conditions for resolving the partial derivatives equation mentioned previously. But such an equation does not have a determinable solution. This is why Brennan and Schwartz use numerical techniques to determine optimal debt.

The Leland model (1994)⁶⁴ integrates taxes and bankruptcy costs when managers maximize the economic value of their assets and seek the optimal debt ratio as well as the possible level for bankruptcy. It supposes that capital is made up of perpetual shares and bonds paying out a coupon notated as C. The author considers two bankruptcy explanations:

• – when the company no longer faces its obligations with respect to the repayment of its debts. In other words, bankruptcy is decided endogenously (the debt is not protected by any clause);
• – when the protection clauses are included in the issuing contracts and trigger bankruptcy for a critical value of the company. Bankruptcy is decided exogenously: the clauses impose a “floor” value on the debt and equity.

In a previous article, Leland (1985)⁶⁵ already considered an optional model in continuous time, like that of Black–Scholes, containing transaction costs in order to balance a portfolio using enough cash to cover the option values. The Leland model (1994) uses the endogenous stochastic volatility function, which depends on all of the structural variables. By calling the simple Brownian motion W and the volatility of assets σ, the value V of the company follows:

[2.106]

Leland resolves the equation with the partial derivatives of Brennan and Schwarts (1978) by supposing that the company has infinite debt. The explanation comes from the fact that companies perpetually renew their debts. Just as in the Merton article (1973), F is the economic value of the debt, V is the company value and C is the payment of the coupon, by unit of time when the company is solvent:

[2.107]

Therefore, if there is no interdependence with time:

[2.108]

Since the derivative only concerns V, the equation with partial derivatives can be simplified:

[2.109]

The solution to this partial derivatives equation requires the consideration of two scenarios:

• – Exclusion of C. Thus:

[2.110]

This refers to the partial derivatives equation by Dixit and Pindyck (1994)⁶⁶, namely:

[2.111]

with δ = 0. In this case, the solutions of the characteristic equations are β₁ and β₂:

[2.112]

[2.113]

[2.114]

[2.115]

and:

[2.116]

[2.117]

The solution to the equation is consequently:

[2.118]

• – Inclusion of C. The general solution to the partial derivatives equation is thus:

[2.119]

The constants (i.e. A₀, A₁ and A₂) are determined by the bond conditions. Let us say that α is the fraction of value that would be lost due to bankruptcy costs, leaving the debtors with the value (1–α).V_B and the shareholders without anything at all, with V_B the level of company value at the moment when the bankruptcy is announced. Thus, the value of the debt, D(V), is such that:

[2.120]

[2.121]

But if V → +∞, V^−x = 0, then condition [2.121] requires that:

[2.122]

Furthermore, taking condition [2.120] into account:

[2.123]

Thus:

[2.124]

and:

[2.125]

where:

• – C: debt coupon;
• – r: risk-free interest rate;
• – α: bankruptcy costs;
• – V: asset value;
• – V_b: asset value relative to the bankruptcy.

Considering the costs of bankruptcy BC:

[2.126]

[2.127]

and if V → +∞, V^−x = 0, then condition [2.127] requires that:

[2.128]

Moreover, taking condition [2.126] into account:

[2.129]

Therefore:

[2.130]

and:

[2.131]

Considering the tax rate on companies, TB:

[2.132]

[2.133]

and if V → +∞, V^−x = 0, then condition [2.133] requires that:

[2.134]

Furthermore, taking condition [2.132] into account:

[2.135]

Thus:

[2.136]

and:

[2.137]

Finally, taking bankruptcy costs and company tax rates τ into account, the company value EV is:

[2.138]

and the value of equity E(V) = EV – D(V). Thus:

[2.139]

The absence of arbitration opportunities allows Leland (1994) to present the following formula for the evaluation of an indebted company’s shares:

[2.140]

[2.141]

In addition, the expression for the company value proves that the value of the asset is maximized by the lowest possible V_B, which allows for the maximization of equity value, E(V), such that:

[2.142]

[2.143]

For V = V_B:

[2.144]

[2.145]

[2.146]

Consequently, when bankruptcy is exogenous, V_B is proportional to C, independent of V and α. Moreover, if r, σ or τ rises, V_B decreases; finally, if C rises, V_B also decreases. Thus, by replacing the value of V_B obtained using [2.146] within [2.125], [2.139] and [2.141], we get:

[2.147]

[2.148]

[2.149]

with:

[2.150]

[2.151]

[2.152]

Let us consider the coupon C that maximizes the company value V. By deriving equation [2.148] with respect to C, and by resolving the equation for C^* (optimal coupon), we have:

[2.153]

By taking the value of C* into account and replacing it within [2.147], [2.148] and [2.149]:

[2.154]

[2.155]

[2.156]

In the case where bankruptcy is determined exogenously, it is triggered thanks to protection clauses when the value of company assets is inferior to the value of the senior debt, notated as P. Let us consider that the senior debt corresponds to the market value of the debt when it is issued, notated as D₀. We thus consider here that P = D₀ = V_B. According to equation [2.125], the value of a protected debt at its issuance is:

[2.157]

When α = 0, that is, when the bankruptcy costs are zero:

• – the protected debt is less risky and pays the risk-free rate;
• – the fiscal advantage coming from protected debt is inferior to that of unprotected debt;
• – the value of assets related to bankruptcy for a protected debt is superior to the value of assets related to bankruptcy for unprotected debt.

Leland has shown that for α = 0, the optimal level of bankruptcy V_B^∗ is the same for protected debt and unprotected debt. Additionally, for all α = 0:

[2.158]

And if α = 0, the debt is without risk, so:

[2.159]

[2.160]

The conclusions of optimal financial structure analyzed by Leland (1994) are as follows:

• – the maximum value of a company when there are protection clauses is usually inferior to that of a company whose debt is not protected;
• – the value of a company when there are protection clauses is maximized if the tax on the company rises, the interest rates rise and bankruptcy costs fall;
• – the optimal debt for protected debt is always inferior to that of unprotected debt;
• – the interest rate paid for optimal debt is inferior for protected debt than for unprotected debt.

François and Morellec (2004)⁶⁷ analyze the impact of bankruptcy proceedings in American companies (via “Chapter 11 Bankruptcy”) on their financing policy. The authors observe that the possibilities for renegotiating debt contracts prompt shareholders to default more easily, and to augment credit spreads. Because of this, these proceedings enhance the power of creditors to make decisions on financial policies for companies in difficulty. Moreover, the time constraints and the costs associated with this process of renegotiation affect the extent of these effects. In this context, it would be expedient to examine the impact of decisions to refinance debt on the value of equity.

2.4.1. Risks of refinancing

He and Xiong (2012) underscore the fact that assuming short-term debts aggravates the risk of refinancing. Acharya et al. (2011)⁷⁴, who are also interested in the risk of refinancing, show that a high frequency of use of short-term debt can lead to a diminished ability of the company to take on debt. Krishnamurthy (2010)⁷⁵ observes that the massive use of short-term debt such as negotiable creditor or repo securities was a key factor in the bankruptcy of Bear Stearns and Lehman Brothers.

In other words, the authors wish to use credit spread to show the dependency between the liquidity premium – tied to the illiquidity of the secondary market – and the default premium, determined by the credit risk. Empirically, this link, which the authors call the refinancing risk, is particularly visible when we observe the subprime financial crisis of 2007–2008 that led numerous companies to serious financial difficulties, which in turn amplified their credit risk.

The He–Xiong model adopts the exogenous framework of Black and Cox (1976) with regard to the notion of bankruptcy and determines the credit risk of a company by carrying out a joint evaluation of debt and equity. In this context, when a bond loan reaches maturity, the company proceeds to a new issuance with the same nominal value and the same maturity as the previous one. The market value (preserved) is either inferior or superior to the face value and the loss or the gain of refinancing is absorbed by the equity. Consequently, the price of the stock is determined by this future expected refinancing gain or loss and the net cash flow of the shareholders is:

[2.161]

where:

• – NC_t: value of shareholders’ net cash flows after the issuance of a new bond loan (replacing the former);
• – δ: dividend distribution rate (constant);
• – V_t: company value;
• – π: tax rate on companies;
• – C: yearly coupon;
• – d(V_t, m): market value of the new bond loan with respect to the company value Vt and the maturity of the debt, m;
• – p: nominal value of the bond loan to repay yearly.

The first term represents the dividends paid by the company. The second term represents the payment of a coupon after taxes. The third and fourth terms represent the gain or loss due to the refinancing of the debt. Thus, when d(V_t, m) falls, the shareholders must cover the losses. Then, the value of equity reaches zero, the bondholders can recover their credit if the company assets are liquidated.

He and Xiong extend the scope of their research to include the structure of a bond market, similar to the one presented by Amihud and Mendelson (1986)⁷⁶, that is, bondholders are subjected to liquidity crises (with a Poisson occurrence ∝). During a liquidity crisis, the bondholder must sell their shares on the secondary market at a proportional cost k. In other words, the investor recovers a fraction 1–k of the market value of the bond. This transaction cost compounded by the intensity of the liquidity crisis determines the liquidity premium present in the credit spread of the company.

In this context, by deriving the evaluation of the bond, retaining V_B as the bankruptcy condition of the company, d(V_t, τ, V_B) as the value of a bond with a maturity of τ < m, c as the annual coupon, p as the face value of the bond and r as the risk-free rate, the partial derivative equation is as follows:

[2.162]

The left-hand side of the equation represents the necessary return the bond must generate. Within the right-hand side of the equation, the first term represents the payment of the coupon; the second term represents the loss caused by the occurrence of a liquidity crisis with the probability ∝ dt (when the event occurs, the bondholder is subjected to a transaction cost k(V_t, τ) when parting with the financial security at hand) and the last three terms capture the expected change in value due to a change in the maturity τ (third term) and a fluctuation in the value of the company V_t (fourth and fifth terms). We can write:

[2.163]

The transaction cost causes the future value rate to rise markedly, which is the required return. Additionally, He and Xiong distinguish two conditions at the limits to define the price of bonds. At V_B, bondholders share the value of company liquidation proportionately. Thus, each bond gives:

[2.164]

where a is a fraction of the company value resulting from the liquidation that the bondholders can recover.

When τ = 0, the maturity of the bond is reached and its holder gets the capital p if the company does not go bankrupt:

[2.165]

Consequently, by combining equation [2.162] with the conditions at the limits [2.164] and [2.165], the value of the bond is as follows:

[2.166]

Where:

[2.167]

[2.168]

[2.169]

[2.170]

[2.171]

[2.172]

[2.173]

[2.174]

[2.175]

Keeping the price of the bond from equation [2.166] in mind, the return y from the bond is obtained by resolving the following equation:

[2.176]

The right-hand side of the equation corresponds to the price of the bond, including the constant payment of a coupon over time and the nominal repayment at maturity, supposing that there is no default situation or other transaction in the meantime. Given that the price of the bond in equation [2.166] includes the transaction costs and the effects of bankruptcy costs, the credit spread (which corresponds to the difference between y and r) contains a liquidity premium and a default premium.

Furthermore, the value of equity E(V_t) goes through the following differential equation:

[2.177]

The left-hand side of the equation represents the necessary return on equity which must correspond to the sum of terms on the right side of the equation:

• – the first two terms of the equation on the right capture the variation expected from the value of equity due to the fluctuation in the company value V_t;
• – the third term corresponds to the cash flows generated by the firm in units of time;
• – the fourth term corresponds to the detachment of the coupon after tax;
• – the fifth and sixth terms concern the gain or loss of the refinancing after paying for the bonds close to maturity and creating a new bondholder loan.

He and Xiong affirm that, even in the absence of any constraint on the capacity to proceed with increases in capital, the deterioration of market liquidity of the debt can lead to a higher level of bankruptcy because of the increase in losses. The shareholders are thus ready to embrace the losses and pay back the creditors inasmuch as the value of the equity is positive. The value of an option to maintain company business thus justifies the absorption of costs tied to refinancing losses. In this case, the liquidity premium of company bonds, the probability of bankruptcy and the default premium increase.

The maturity of the debt plays a major role in the determination of refinancing risk. A short maturity reduces the bondholder’s risk while it increases the company’s risk of refinancing by obliging shareholders to quickly cover the losses from refinancing debt. The deterioration of market liquidity can have a significant effect on the credit risk depending on a company’s rating and the maturity of its debt. For example, if an unexpected crisis brings about an increase of 100 basis points in the liquidity premium, the default premium of a company B with a debt maturity of one year increases by 70 basis points, which contributes to an increase in the total credit spread of 41%. Additionally, the same type of liquidity crisis increasing the default premium causes:

• – for a company BB, with a debt maturity of six years, an increase in credit spread of 22.4%;
• – for a company A, with a debt maturity of one year, an increase in credit spread of 18.8%;
• – for a company A, with a debt maturity of six years, an increase in credit spread of 11.3%.

It would be useful to refine the study by trying to analyze the impact on the value of equity from the repayment of a debt with intermediate due dates.

2.4.2. Reimbursing loans at intermediate intervals and the impact on the value of equity

Geske (1977)⁷⁷ proposed a method for optional valuation of liabilities which includes n – 1 payments of individual coupons paid before the reimbursement of capital. Then, Geske (1979)⁷⁸ developed another model which refers to the one by Black and Scholes (1973). The latter considered that a share could be taken to be an option on the value of the company. From this perspective, a call on a share is an option on an option. Let us suppose V is the value of the company, S is the price of the share, D is the face value of the debt and K is the exercise price of the call on equity. Let us write t* for the maturity date of the call on equity and T for the maturity date of the debt. The following figure illustrates Greske’s principles, which lead us to consider the premium of an option.

On the intermediary date t*, the holder of the call exerts their option on the share if the call is in the money, that is, if S_t* > K. In the contrary case, if S_t* = K (or if S_t* < K), then the call on the action will not be completed. Like the value of the share, S depends on the value of company assets, V, such a situation occurs when the value of the company V is equal to (or less than) V*. Thus, V* is the value of V such that S_τ − K = 0. The holder of the call pays K at t = t* if, on this date, V > V* in order to maintain the possibility of paying D at t = T to get company assets. In this case:

[2.178]

where:

[2.179]

[2.180]

[2.181]

[2.182]

[2.183]

N(.) and Φ(.) are, respectively, the bivariate and univariate normal distribution functions.

By taking these principles as a starting point (all the while making a correction), Geske and Johnson (1984)⁷⁹ consider the case of a junior debt whose face value is M₂ with a maturity of T₁. In addition, Geske and Johnson suppose that T₂ > T₁ and that the senior debt is refinanced using equity. Thus, in order to maintain the percentage of control, on the date T₁, shareholders will all have to subscribe (at the level of their shares) to the new issuance. In this way, the present value of their shares after the refinancing of the senior debt will correspond to the expected discounted future company value, from which it is necessary to subtract M₂ (no possibility of bankruptcy is foreseen) as well as the future discounted payment on date T₁. This payment will only be made if no bankruptcy occurs in T₁, that is, if S_T1 − M₁ > 0 (where S_T1 is the value of equity at the moment T₁). Considering V* the critical value of the company in case of bankruptcy on the date T₁ and J_T1 the value of the junior debt on the date T₁, and since V_T1 = M₁ + J_T1, we get:

[2.184]

where:

[2.185]

[2.187]

[2.188]

[2.189]

The value of the senior debt B is the discounted value of M₁, given the full payment in T₁, to which we add the expected present value of the company value if the senior debt is not completely repaid. We then have:

[2.190]

where:

[2.191]

[2.192]

The correction by Geske and Johnson (1984) vis-à-vis the works by Geske (1977) comes down to the fact that the latter mistakenly gave the value of company assets each time V < V*. But, when M₁ < V < V*, the value of the senior debt should correspond to M₁ and the value of the junior debt should correspond to the remainder of the value. In this case, the value of equity is zero and the company goes bankrupt. Finally, the value of the junior debt J is the discounted value of M₂, given that no bankruptcy situation is expected, to which we must add the expected present company value if a bankruptcy situation occurs in T₂ and the payments arising in T₁ to repay the senior debt:

[2.193]

with: S + B + J = V.

Charitou and Trigeorgis (2004)⁸⁰ try to analyze the default situations of 420 American companies between 1986 and 2001 by constructing a model inspired by the theory of options by Black and Scholes (1973), Merton (1974) and Vacisek (1984)⁸¹ applied, in practice, by Moody’s. Their results indicate that volatility plays a significant role in the explanation of bankruptcy situations, up to five years before they occur.

Charitou and Trigeorgis thus consider equity to be perpetual call over the value of company assets. Nonetheless, contrary to the Merton model, which concentrates exclusively on payment defaults on the debt principal at its maturity, Charitou and Trigeorgis assume the non-payment by default of any planned installment, whether creditor interest or repayment of borrowed capital. In order to account for the probability of an intermediary default, the adjusted model lowers the conditions for default at maturity. The point is to resolve what is missing from the Vassalou–Xing (2004) model⁸² which, if Vacisek’s default limit is adopted, does not explicitly consider the probability of intermediary default often leading to negative expected growth rates (which seems incompatible with the theory of asset evaluation).

On the contrary, the Charitou–Trigeorgis model more closely resembles the one by Leland (2004)⁸³. The latter analyzes different implications for limits on critical default and the relative performance of two default models: the first is being exogenous (corresponding to the Merton model) and the second turns out to be endogenous and refers to the works by Leland and Toft (1996)⁸⁴. According to the latter, shareholders must decide if it is better to repay the debt or default on the payment. The Charitou–Trigeorgis model is analogous to the endogenous approach with the difference that they account for, at the start, the possibility of defaulting considering the cash flow hedge. It appears crucial for both researchers to consider the capacity of companies to face their obligations with cash and cash equivalents or flows coming from operating activities. Indeed, Leland does not include variables of liquidity. Additionally, he does not do an empirical study but rather creates simulations of data using Moody’s notations which he compares with expected default probabilities (from simulations). Finally, the Leland model relies in large part on the financial structure of companies and the use of leverage, while Charitou and Trigeorgis include this intermediary default probability. In this context, Charitou and Trigeorgis distinguish two scenarios:

• – the probability of a voluntary default from shareholders (on interest and repayment of capital on the debt I at a moment T’):

[2.194]

with:

[2.195]

where:

• – μ: global return rate expected on the value of company assets (which replaces the risk-free rate);
• – D: total distribution rate by the company to all its interested parties (including dividends and coupons) expressed as a percentage of the economic value of the company V;
• – V*: critical value of the company (lower than the value of the option);
• – σ: volatility of company assets;
• – τ’: intermediary maturity remaining on D;
• – T’: intermediary maturity of D;
• – E: economic value of equity similar to a European call.

The higher the interest and repayment costs of debt I, the higher the probability of voluntary default at time T’ will be. Additionally, the formula shows that shareholders can choose to default just before maturity if the value of the option is insufficient, that is, it is not enough to pay interest and the nominal amount. Liquidation is then very much voluntary. On the contrary, the default situation can be triggered by creditors if the company (profitable even from the point of view of the shareholders, i.e. (V*,τ′) > I) does not have enough cash flow or liquid assets to repay the debt costs (interest and nominal). The default situation is then involuntary:

• – the probability of involuntary default (on interest and repayment of the debt capital I because of a lack of liquidity at moment T’)

[2.196]

with:

[2.197]

[2.198]

where:

• – c: proportion of cash from operating activities;
• – CFC: cash flow hedge;
• – Cash: cash and cash equivalents.

If the company generates a constant proportion of cash from operating activities, an involuntary default situation would be triggered at T’(< T) if c.V_T + cash < I.

Charitou and Trigeorgis thus transpose the optional valuation model by Geske (1977) when the flows tied to repayment of a loan end up being paid at an intermediary date τ’ when the maturity of the debt comes later. If at τ’, V is smaller than V* such that E(V*,τ₁) – I = 0, the shareholders will voluntarily default on the payment of the loan maturity. In this case, based on the notations by Geske, K is replaced by I and C is replaced by E (corresponding to the value of equity). The shareholders do indeed have the option to pay I on the intermediary date τ’ to keep the possibility of paying M at t = T in order to get company assets. In this case:

[2.199]

and the default risk at the intermediary date T is

where a₁, a₂, b₁, b₂ and ρ are the values defined by Geske (1979).