1.1 – A parsimonious model for debt rescheduling and its consequences on the pricing of corporate securities

9In presence of bankruptcy costs, the liquidated value received by debt holders is strictly lower than the assets’ value. Denoting by ? ? [0,1] the realization rate, this value is equal to . To alleviate bankruptcy costs , debt holders can accept to reschedule their debt, e.g., by granting a delay. Longstaff (1990) demonstrates that there exists an optimal extension period ?. His procedure is based on creditors’ wealth which mainly depends on the firm’s assets value at time T₁ i.e. the default severity. By contrast, our own procedure highlights the key importance of both and the volatility at time T₁. Under the risk neutral measure Q, the net gain function is given by:

10

11Unnecessary references to the face value and the interest rate have been omitted. This function evaluates, at time T₁ the swap of the known payoff for the new debt that promises at T₁ + ? if and if otherwise. Straightforward computations then yield:

12

13where and N is the standard cumulative distribution function. The function H involves the firm’s asset value, the level of business risk, the default severity and liquidation costs that prevail at the time of default. It is a concave function of T₂ and may serve as an objective function. The optimal extension maturity is obtained by computing:

14

15The price of the rescheduled debt is then evaluated by [4]:

16

17The rescheduling also leads to the creation of a new claim received by equity holders at time T₁. Because our set up is similar to that of Black, Scholes and Merton, this claim resembles a call option except that its expiration is given by Equation 4. Its price at time T₁ is given by:

18

19Pricing formulae for corporate securities any time before T₁ can then be derived. If we denote by the risk neutral density of conditional on its t-value V_t, the t-time price of the equity is:

20

21and the debt price is:

22

23These two equations are important for our analysis. We can note that the first term of the right hand side of equation 7 is the Black, Scholes and Merton’s equity price whose theoretical expiration is time T₁. The second term is the (discounted) expected value of the claim received at time T₁ from rescheduling (see also equation (6)). In the same way, the first term of equation 7’ stands for repayment in full of the debt contract. The second term replaces the usual expected value of the liquidated asset obtained in case of default [5].

24Both equations highlight the way rescheduling affects security prices. If the firm is in trouble some time before debt maturity, the standard call option in equation 7 and the first term of equation 7’ can be worth little compared to the component implied by rescheduling. They also indicate that the firm’s asset volatility at time T₁ is an important variable for share holders’ wealth.

1.2 – A general model for business risk targeting driven by potential debt rescheduling

25The previous subsection assumes that only rescheduling happens at time T₁. This means that share holders are inactive during the period preceding debt maturity. This is not completely satisfactory since equity holders can anticipate problems with full repayment of their liability and the future debt rescheduling [6]. Because the firm’s asset volatility is a critical variable of their wealth, they can question the opportunity to adjust the business risk level so as to benefit from their position (remind that asset sales are proscribed by the covenant held by creditors). We assume a single adjustment of the business risk because of the financial distress. This volatility targeting may be viewed as being driven by future potential rescheduling. To emphasize the role of the firm’s asset volatility ?, the equity price and the debt price have been denoted by E_t(V_t, ?) and D_t(V_t, ?) respectively.

26We know that ? impacts prices directly and indirectly through, e.g., the extended maturities . When debtors decide to adjust the volatility at time t, they choose the level of business risk so as to maximize the equity price at that time. Properties of the equity price with respect to the volatility must be explored to find the potential optimal level of business risk ?^*_t. This value satisfies the first-order condition and requires that E_t(V_t, ?) < E_t(V_t, ?^*_t) for any other admissible volatility ? [7]. Whether and to which extent they are worth adjusting the firm’s assets volatility are questions we treat in different environments. Note that we also investigate the role of debt holders’ behaviour given that they possess no covenant to prevent the activism of equity holders on the unobservable business risk of the firm [8].

27It is important to point out that our model for business risk targeting makes several implicit assumptions. First, there is no agency problem between equity holders and the management of the firm so as to put emphasis on the relation between equity holders and creditors. Second, the firm’s management is able to render operational desired changes in asset volatility. Third, adjustment costs are not considered. Note that we discuss this issue in the appendix. In the following sections, we investigate whether equity holders can benefit from their position by finding an appropriate volatility ?^*_t before maturity. We proceed by enriching a simple setting which first considers a business risk adjustment at time T₁ – ? for small value of ?.

2.1 – The existence of the optimal business risk

37As explained in section 1, the existence of the optimal business risk level depends on properties of the equity price. Figure 1 plots equity prices described by equation 11 as a function of the business risk for different values of ?. Base case parameters are as follows. The face value, due at time T₁, is worth F₁ = 40. Without loss of generality, the firm’s asset value is set to and the level of the interest rate to r = 6%. The quasi leverage ratio is equal to . For illustration purposes, seven values for the realization rate ? are considered between 20% and 80% [9] and volatility ranges from 2.5% to 22.5%.

Figure 1

The price of the new equity when the maturity is optimally chosen by debt holders

The price of the new equity when the maturity is optimally chosen by debt holders

38As expected, rescheduling is a good arrangement for equity holders since they get a strictly positive value whatever the business risk is. We observe that the equity price is lower and lower as the realization rate increases. A reason for this is that the optimal extension maturity is a strictly increasing function of liquidation costs 1 – ?. The main point of this figure is that the equity price is not a straightforward function of volatility as usual. The graph displays a quasi concave structure with respect to the firm’s assets volatility. The humped shape is clearly in contradiction with usual conclusions of the Black-Scholes and Merton approach stating that equity is a strictly increasing function of the underlying firm’s assets volatility. Here, equity is affected differently by the level of business risk because volatility also influences the extended maturity .

39A business risk targeting procedure is then equivalent to the calculation:

40

41where one must calculate:

42

43Equity holders can identify the optimal business risk to the extent that a unique solution to the above optimization program exists. The quasi concave feature of with respect to ?, observed in Figure 1, suggests that the optimal business risk level may be found as the finite value such that:

44

45As a matter of fact, an analytical expression exists for this derivative, but the dependence of the extension maturity to volatility makes it really complex. It does not permit any closed form solution for either. This is the reason why we favour a numerical approach and conclude the existence of a finite optimal business risk level for all the sets of parameters we investigate.

2.2 – Effects of targeting business risk

46The target business risk level has a direct impact on the price of rescheduled debt. We now investigate debt price sensitivity to the business risk level and whether business risk targeting is made at the expense of debt holders. Adjustment of the initial business risk profile ?₁ towards the optimal volatility leads to either an upward or a downward movement. And the magnitude can be more or less significant [10].

47Figure 2 is very similar to Figure 1 except that we deal with the price of rescheduled debt. This graph shows that this is a decreasing function of the business risk (everything else being equal). Hence, debt holders will benefit or suffer from the modification of the business risk profile depending on the way the business risk is shifted towards the target level… If the initial level of business risk ?₁ is lower (resp. greater) than , equity holders will shift upward (resp. downward) the current level of risk and debt holders will be worse off (resp. better off).

Figure 2

The price of the renegotiated debt

The price of the renegotiated debt

48In view of Figure 2, we can conclude that creditor s can either face opportunity costs or benefit from opportunity gains. So the business risk targeting is not necessarily achieved at the expense of debt holders. In some cases, both parties benefit from it.

49In our set up, the reason for this is clear. Debt rescheduling is not a zero-sum game. The firm’s total value at time T₁ is not invariant to the firm’s asset volatility. By neglecting ?, the firm’s total value verifies with bankruptcy costs equal to depends on volatility both directly and indirectly via the optimal extension maturity. An optimal financial rearrangement may be worthwhile for all parties. The next paragraph places more emphasis on the price effects of shifting volatility from ?₁ to .

2.3 – Opportunity costs or opportunity gains?

50This business risk targeting is worthwhile for equity holders, but it may have some positive and negative effect on the debt price. In order to study the magnitude of these latter opportunity gains or costs, we proceed as follows. We assume that equity holders shift the volatility of the firm’s assets from ?₁ to and we assess the impacts by means of a couple of measures. The first measure is the difference of prices defined by: ; the second one is the relative price difference, computed by , where S is the price of the considered claim. In both measures, the price prevailing before business risk targeting is the benchmark price. So, when we apply these measures to the equity price, a positive value can be directly interpreted as the magnitude of the incentives for equity holders to change the "actual" level of business risk.

51Figure 3 gathers together two different graphs which focus on a specific measure. The upper graph plots price differences (in, say, euros) whereas the lower graph considers relative price differences by percentage. The abscissa displays different magnitudes for risk shifting.

Figure 3

Impact of the business risk adjustment and potential opportunity costs

Impact of the business risk adjustment and potential opportunity costs

52The shift is defined with the help of a variable u ranging from (expressed by percentage in the graphs). We set to center the abscissa on the target risk level. A negative value for u, say –10%, means that the value is actually 10%-lower than the optimal level and that equity holders will shift the volatility upward. A positive value for u, say + 10%, means that the value is actually 10%-greater than the optimal level and that equity holders will shift downward the business risk of the firm.

53Figure 3 offers different insights. We can first check that when the initial firm’s asset volatility ?₁ is equal to the optimal (i.e. u = 0), equity holders have no incentive to make any changes in the firm’s business and that, in all the other cases, equity holders benefit from changing the business risk level. When the initial volatility is lower , debtors are better off increasing the business risk level. This is especially perceptible on the lower graph that indicates an increase of more than 40% of the initial equity price. This benefit decreases as the modification tends to be moderate, i.e. as the initial ?₁ is initially closer to . Almost symmetrically, debt holders suffer from such risk shifting. The negative values observed on both graphs are measures of the opportunity costs faced by creditors. When the initial volatility is 66% lower than the optimal one, absolute costs are about 4 times higher than the equity holders’ gain. In relative values, however, risk shifting appears less detrimental for debt holders than it is beneficial for equity holders. When the initial volatility is greater than the optimal level of business risk , the situation is rather different since both parties benefit from business risk targeting. Compared to equity holders, debt holders are even better off in absolute values! In relative values, however, the gain is comparable. To conclude, we retain that business risk targeting can induce opportunity costs but it can also align interests of both parties. A natural question arising now is the economic significance of opportunity costs.

2.4 – Do opportunity costs really matter?

54To answer whether opportunity costs matter, we compare them to costs resulting from the asset substitution problem ex post reorganization. From standard analysis of corporate finance, unexpected post-default asset substitution leads to some agency costs for lenders. This context involves some degree of information asymmetry. It is well known that such risk shifting is beneficial for equity holders at the expense of debt holders. Of course, agency costs can be avoided indirectly by an appropriate covenant. Bhanot and Mello (2006), e.g., have recently suggested a trigger clause to monitor the rating of the debt. They serve however our purpose by providing benchmarks.

55In view of previous sections, is the level of business risk used by debt holders to compute extension maturity, i.e. to design the rescheduled debt. This level may or may not be equal to the pre-default level of a firm’s asset volatility ?₁. When , debt holders face some opportunity costs, that can be measured by relative price differences. After debt rescheduling, equity holders can (once again) increase the volatility of the firm’s assets (say, to ?₂). We assume that they act immediately after reorganization. The price of the new claims may be rewritten and to highlight the different volatilities. Agency costs implied by the upward shift of the business risk can then be assessed by a relative difference of debt prices. We compute .

56Table 1 provides and compares opportunity costs versus agency costs. Opportunity costs are implied by business risk targeting while agency costs are caused by the classical ex post asset substitution problem. Both consider comparable shifts in business risk. Shifts are computed by as percentage of the optimal business risk. Other parameters are , F₁ = 40 and r = 6%.

Table 1

Ex ante risk targeting versus ex post asset substitution: a comparison of relative costs

Opportunity Costs (ex ante reorganization) Agency Costs (ex post reorganization) Risk Shifting 10% 20% 30% 40% 50% 10% 20% 30% 40% 50% ? = 60% –1.499 –3.137 –4.919 –6.845 –8.913 –1.366 –2.617 –3.777 –4.861 –5.883 ? = 65% –1.302 –2.727 –4.281 –5.966 –7.782 –1.186 –2.275 –3.286 –4.234 –5.128 ? = 70% –1.108 –2.324 –3.653 –5.098 –6.662 –1.010 –1.940 –2.805 –3.617 –4.385 ? = 80% –0.732 –1.538 –2.423 –3.392 –4.448 –0.668 –1.286 –1.864 –2.409 –2.926 ? = 90% –0.366 –0.770 –1.215 –1.704 –2.243 –0.335 –0.647 –0.939 –1.217 –1.482 This table provides and compares costs associated to a shift in the business risk ex ante versus ex post the reorganization. Opportunity costs come from ex ante business risk targeting while agency costs are implied by the classic ex post asset substitution problem. Shifts are expressed in percentage of the optimal business risk. They are respectively computed by (? *1 ? ?1)/? *1 and (?2 ? ?*1 )/? *1. Other parameters are VT1 = 34. F1 = 40 and r = 6%.

Ex ante risk targeting versus ex post asset substitution: a comparison of relative costs

57Table 1 shows magnitudes of respective costs. As expected, we find negative values for every relative price difference we compute. We can observe in both cases, that the larger the risk shifting is, the more significant costs are. The same is true for liquidation costs. The main point is that opportunity costs appear larger than agency costs. In view of this, we can conclude that opportunity costs are economically significant.

58The above simple setting can be extended in different ways. We proceed in successive steps. The first and second steps question whether it is profitable for equity holders to adjust business risk long before the maturity of the debt contract. The next step addresses the reciprocal influence of stakeholders on the other’s decision process. The final one addresses whether rescheduling can be granted several times. Adjustment costs are discussed in an appendix.

3.1 – Time to maturity effects on the equity price

60This sub-section investigates potential "time to maturity" effects on the price of equity. By virtue of equations 7, the price of equity at time ?_AT < T₁ is described by:

61

62where is the risk neutral probability density function of conditional on its value at time ?_AT.

63Equation 15 highlights two distinct components in the wealth of equity holders. The first term corresponds to the standard call option they hold (even) in absence of rescheduling. The second one is the (discounted) weighted sum of claims they receive at time T₁ if the debt is rescheduled . It is well known from option theory that the first component (the standard call option) is a strictly increasing function of the level of business risk ?. By contrast, section 2 suggests that, individually, each claim of the weighted sum is a quasi-concave function of ? in the sense that it first increases, admits a maximum and then strictly decreases. Overall, the global property of equity is not easy to appreciate, so we experiment simulations of equation 15.

64We can note that the debt price is similarly given by:

65

66Simulations of this equation can illustrate consequences of a business risk targeting on the creditors’ wealth.

67Graphs of Figure 4 plot the equity price (on the left) and the debt price (on the right) as functions of the underlying business risk level for different times to maturity. Here debt holders extend the maturity of their debt at time T₁ only if necessary. Structural parameters are those of Figure 1. The time to maturity T₁ – ?_AT ranges from a quarter (i.e. 0.25) to 2 years. In both graphs, the continuous line depicts the case T₁ = ?_AT i.e. that of Figure 1.

Figure 4

The price of liabilities at adjustment time

The price of liabilities at adjustment time

68Figure 4 demonstrates that prices of corporate securities are more complex functions of the firm’s volatility, in the sense that their convexity clearly depends on the remaining time period T₁ – ?_AT. Observed shapes are direct consequences of the two components highlighted in equations 15 and 15’. We mainly comment hereafter those of the equity price; the right graph highlighting situations where debt holders can suffer from shifting.

69The upper graph mainly shows a couple of things. First of all, for moderate volatilities, prices are not much different from one another and approximate those computed in section 2. Second, when the remaining time to maturity T₁ – ?_AT is not zero, equity prices are locally concave with respect to volatility. Hence, the volatility revealed by the hump is not a global optimal business risk level (anymore) but only a local solution. The reason for this is unambiguous. The standard call is a strictly increasing function of volatility and, for T₁ – ?_AT ? ? > 0, it cannot be neglected in equation 15. So there always exists a high volatility that implies a higher equity price than the one observed at the hump. For many ?_AT – T₁, however, this local solution appears plausible because alternative levels of business risk are (operationally) questionable. As times to maturity get short, the business risk level revealed by the hump becomes effectively valuable. For longer times to maturity, the standard call induces a rather pronounced positive slope (with respect to volatility) for large volatilities.

70The equity price conforms here with standard implications of the Black, Scholes and Merton framework: equity holders are better off increasing the business risk level as much as possible [12]. Overall, simulations suggest that there is, ceteris paribus, no significant "time to maturity" effect on the volatility to target before rescheduling but that the timing of business risk targeting is a key decision to make.

3.2 – On the timing of business risk targeting

71Whether there exists an optimal timing for business risk targeting is now the natural question. To answer this precisely, it is necessary to model and price the American option held by equity holders and solve the associated optimal exercise problem. Clearly, this is challenging. A simpler approach consists in finding ad hoc criteria to answer slightly different questions such as "how long before maturity could it be profitable for equity holders to adjust their business ?" or "is current time a suitable time to adjust business ?". Such a pragmatic approach essentially neglects the value of delaying the volatility adjustment. We now suggest a couple of methods in this vein.

72The first method we consider is a two-step approach. It consists in a) questioning if the current time is appropriate for a volatility targeting and then b) finding the appropriate volatility. Given a contemporaneous volatility ?₁, equity holders can conclude that it is time to adjust business if their current wealth (defined by the equation 7) rests mainly on potential rescheduling rather than full debt repayment. The adjustment time then verifies:

73

74This criterion involves the contemporaneous firm’s asset volatility ?₁ so that the adjustment time is a function of it . It applies before any business risk adjustment and defends the idea that equity holders decide to make the adjustment as soon as they understand that they are better off with rescheduling (even without being in an optimal context of business risk). By denoting the total wealth of equity holders, the previous condition is equivalent to:

75

76Once given, equity holders must find an optimal volatility .

77The second method we propose exploits findings of Figure 4 and characterizes and simultaneously. To this end, we consider a specific firm whose operational activities imply a certain maximum level for business risk ?_H as in Leland (1998) and Ericsson (2000). By virtue of equation 7 and Figure 4, equity holders have chosen ?_H as proper business risk level long before maturity. As the time-to-maturity decreases, equity holders question the opportunity of a business risk targeting. They can think that a shift from ?_H to the optimal business risk level is worthwhile at time if the following equation is verified:

78

79Equation 17 states that equity holders understand at time that they should not maximize the firm’s risk as suggested by the standard approach.

80The way rescheduling is taken into account in equation 17 may appear less explicit than in equation 16. However, both approaches lead to a regime switch in volatility before maturity and both are driven by the perspective of rescheduling. We illustrate in Table 2 the different timing implied by the above two criteria. This table supports the intuition that the more severe the financial distress is, the earlier the adjustment is. It is interesting to note that adjustment times appear rather similar and that no criterion is systematically earlier than the other.

Table 2

A timing for business risk adjustments

First approach Second approach Firm Value ?H = 20% ?H = 30% ?H = 20% ?H = 30% V = 34 1.11 0.56 1.04 0.73 V = 30 1.98 1.05 1.95 1.37 V = 24 3.64 1.98 3.73 2.59 This table compares the timing for business risk adjustment implied by the criteria defined by equations 16 and 17. Other parameters are F1 =40, ? = 60% and r = 6%.

A timing for business risk adjustments

5.1 – Equity holders’ influence on rescheduling

91If equity holders can influence the financial reorganization at time T₁, they will affect the objective function to maximize. Their relative bargaining power denoted by ? hereafter verifies 0 ? ? < 1. 1 – ? represents the debt holders "remaining" bargaining power. ? is of course less than one, otherwise equity holders can design rescheduling alone; in simulations, it does not exceed 0.5. The objective function to consider for rescheduling is given by:

92

93where

94 and are respective net gain functions [13]. The corresponding optimal extension maturity is then computed by:

95

96subject to the constraints that both H_debt and H_eq remain positive. Since H_eq is, by definition, positive for all parameter values, H_debt should be positive. Otherwise this means that debtors could swindle creditors. As long as the extension maturity described by equation 19 exists, share holders obtain new rescheduled claims denoted by and . These claims are similar to and except that their (common) expiration date is now . Of course, when ? = 0, i.e. when creditors have all the bargaining power, the above rescheduling design drops to the standard one, and the extension maturity vanishes to .

5.2 – Creditors’ influence on business risk targeting

97If debt holders can influence the choice of a business risk profile, they will modify the objective function to consider. Their intervention power, denoted by ?, verifies 0 ? ? < 1. 1 – ? is associated to the equity holders’ position. ? is less than one, otherwise creditors would control the firm [14]. If the design of rescheduled claims remains at the creditors’ discretion, then the new objective function to consider for business risk targeting is:

98

99E_tV_t, ?₁) and D_t(V_t, ?₁) are described in equations 7 and 7’ respectively and we know that both involve .

100The corresponding optimal business risk is then computed by:

101

102Of course this equation nests our previous analysis. For ? = 0 and t = T₁ – ? with ? rather small; we find:

103

5.3 – Reciprocal influences in the general context

104In the most general context, equity holders influence rescheduling and creditors influence the business risk targeting. The complete picture is then obtained when both influences are considered in the same framework. The general objective function for business risk targeting is defined by:

105

106where E_t(V_t, ?₁, ?) and D_t(V_t, ?₁,?) are respectively given by :

107

108and

109

110

111where

112

113as described in equation 18. Business risk targeting is then solved when stake holders can find a business risk level ?^*_t such that:

114

115Equations 23 and 24 form a system that must be solved simultaneously. In what follows, we extend the setting of section 2 to undertake the analysis of this system.

5.4 – Reciprocal influences in the simple setting

116This part questions whether the optimal business risk level defined by equation 24 exists and aims to appreciate quantitatively the impact of reciprocal influences. We restrict our attention to a setting that permits direct comparison with previous investigation. As in section 2, we consider t = T₁ – ? with ? a rather small value. Equation 23a then becomes:

117

118 and are similar to and except that their common expiration date satisfies:

119

120where , see equation 18. Figure 6 explores whether admits a maximum value with respect to ? and effects of the reciprocal influence on business risk targeting.

Figure 6

Reciprocal influences and the G-function

Reciprocal influences and the G-function

121Figure 6 gathers together six different graphs. The left graphs plot of equation 25 as a function of the business risk and the influence of equity holders on financial reorganization (? ranges from 0 to 0.25). Three levels of debt holders’ influence on the business risk targeting are considered: ? = 0, 0.25 and 0.5. The right graphs plot as a function of the business risk and the influence of creditors on business reorganization (? ranges from 0 to 0.25). Three levels of equity holders’ influence on rescheduling are considered (? = 0, 0.25 and 0.5). In upper graphs, either ? or ? is null meaning that the influence is one-sided. In the lowest left graph, ? is set to 0.5 meaning an identical bargaining power during the business reorganization. In the lowest right graph, ? is set to 0.5 meaning an identical bargaining power during the financial reorganization. Middle graphs consider intermediate situations with either ? or ? equal to 0.25.

122All these graphs reveal unambiguously that G admits a maximum with respect to the volatility whatever the influences are. Hence, reciprocal influences have no effect on the pertinence of a business risk targeting before rescheduling. There exists an optimal volatility to target in the sense given by the objective function. Very interestingly, the function is increasing with both parameters ? and ? meaning that negotiations with significant ? and ? are valuable. Stake holders have therefore incentives to design both reorganizations with some degree of consensus. We can finally observe that the debt holders’ influence on the business reorganization lowers the business risk level to target. This is in lines with Figure 2 where debt prices increase as the firm’s assets volatility decreases.