the design of safety covenant-protected debts
TRANSCRIPT
The Design of Safety Covenant-Protected Debts
Ridha M. Mahfoudhi∗
∗Department of Finance, Laval University. Pavillon Palasis-Prince, B.P.3885,Quebec City (QC) G1K 7P4, Canada; Phone (418) 656 6818; Email:[email protected]. I am grateful to my dissertation adviser KlausFischer for his constant guidance and encouragement. I am indebted to Pascal Françoisand Francis Longstaff for their precious comments that improved the quality of this pa-per. I would like also to thank Khemais Hammami, Érik Lueders, Issouf Soumaré andseminar participants at Laval University for their helpful comments and suggestions.The paper is a substantially modified version of an earlier draft titled “EndogenousBankruptcy and the Value of Protection”. Financial support from CIRPÉE is grate-fully acknowledged.
The Design of Safety Covenant-Protected Debts
Abstract
This paper examines corporate debt contracts embedding a safety covenant according
to which default occurs once the firm’s worth falls below a specified threshold. We
develop a contingent-claims model that allows claimholders to maximize their interests
subject to this contractual provision. The resulting debt design rationalizes liquidation
decisions, imposes an endogenous debt capacity constraint to the firm, and enables
debtholders to attenuate liquidation inefficiencies arising from bankruptcy procedures.
Analysis also reveals that the risk of premature liquidation combined with this design
leads equityholders to stress the growth potential of the value of firm’s assets when
choosing the optimal debt contract..
JEL Classifications: G12, G13, G33.
Keywords: Corporate debt; Debt covenants; Default risk; Bankruptcy; Contract de-
sign.
I. Introduction
Observed corporate debt contracts exhibit a wide variety of protective provisions. The most com-
mon is the positive net-worth agreement stipulating that the firm’s worth must always exceeds the
debt principal. The need of this contractual agreement can be justified in the situation where eq-
uityholders have the possibility to sell a fraction of firm’s assets to meet the promised debt service.
In such a case, indeed, the common debt covenant stating default based on the omission of interest
payments becomes ineffective. The use of the positive net-worth agreement can be also motivated
by the fact that it permits to debtholders to impose standards more severe than the promised
interest payments, which increases the debtholders’ protection against bad states of nature and
moral risk.
In this paper we propose an optimal design for safety-protected debts. Rather than imposing
the debt principal as an exogenous standard for the firm’s assets as the positive net-worth agree-
ment does, we allow debtholders to optimally set a safety threshold that consists of a minimum
worth requirement. We call the safety threshold covenant the debt provision according to which
default occurs once the value of firm’s assets falls below a contractual safety threshold initially
specified by debtholders. The debt design we introduce determines this optimal safety threshold
endogenously given the debt contract’s parameters. Our debt design is inspired from the Black and
Cox’ (1976) model analyzing safety covenants similar to the positive net-worth agreement, entitling
debtholders to force the firm into bankruptcy once the value of the firm falls to the discounted debt
principal amount. The model we propose is motivated by the observation that a safety threshold
exogenously set, as assumed by the last authors, leads to suboptimal firm liquidation decisions. To
resolve this problem, we adopt an optimal contract design approach by allowing both debtholders
and equityholders to maximize their own claims into two-path equilibrium model. At the first equi-
librium path, debtholders adopt an optimal exercise policy of the safety covenant right by resolving
a continuation/liquidation tradeoff. This, in contrast to the Black and Cox’ model, results in an
optimized safety threshold representing an endogenous barrier for default. In such a setup, we find
that the optimal option to liquidate hold by protected debtholders is analogue to an American put
written on the firm’s assets. At the second path, we solve for the optimal debt contract maximizing
the value of equity subject to the first path-optimized safety threshold. By doing so, our model
recognizes, as in practice, the preservation of equityholders’ interests in the design of debt contracts.
The role of debtholders is generally ignored in structural models of default. In the absence of
contractual agreements entitling debtholders the ability to precipitate and force firm’s liquidation,
it is natural to abstract from this question since the bankruptcy law often allows equityholders and
managers the possibility to entirely decide the action plans of distressed firms. When considering
protected debt contracts, however, debtholders are expected to affect the value of corporate claims.
One motivation of this paper is to assess the impact on corporate claims of allowing debtholders
the ability to precipitate early liquidation of the firm by virtue of safety covenants.
The results we find show that our protected debt’s design permits to debtholders to: (i) avoid
suboptimal premature liquidations; (ii) impose an endogenous debt capacity to the firm; and (iii)
attenuate the negative impact of liquidation inefficiencies arising from bankruptcy procedures.
The model also suggests that when the firm defaults, protected debtholders implementing our debt
design would accept to renegotiate the debt contract’s parameters (not the safety covenant itself) in
the same fashion than the commonly used strategies for debt renegotiation. Furthermore, analysis
reveals that the threat of premature liquidation constrains the equityholders’ choice of the optimal
debt contract, leading them to stress the growth potential of the firm’s assets.
Our analysis adopts a contract design approach by permitting claimholders to choose the optimal
parameters of the debt contract that maximize their own interests. This approach is similar to that
2
employed by Anderson and Sundaresan (1996). The reason is that both the former model and
ours define the optimal debt contract as the contract achieving equilibrium (Nash in the Anderson
and Sundaresan’ bargaining game model and subgame-perfect equilibrium in our two-path model)
between claimholders. However, two salient features make our model quite different from the
former. First, contrary to our model where default occurs by virtue of the safety covenant, the
Anderson and Sundaresan’ model examines a different debt contract by considering a cash flows-
based bankruptcy. Second, the purpose of the last authors is to analyze the problem of debt
renegotiation arising upon bankruptcy between unprotected debtholders and the firm’s owners,
while our aim is to investigate safety-protected debt contracts. Indeed, the main contribution of
this paper is to provide an optimal design model and formal analysis for these debt contracts that
are widely implemented in practice by institutional lenders and senior debtholders, but not enough
theoretically examined.
The remainder of the paper is organized as follows. Section II provides a brief synthesis of related
work. Section III presents the debt design model. Section IV and V consider the implications and
the performances of the debt design. Section VI examines the optimal debt contract to be chosen
by equityholders. Section VII concludes.
II. Related literature
Since the insightful work of Merton (1974), the theory of contingent claims helps to model and
price default risk. Given the fact that Merton’s model takes the enforcement conditions of debt
contracts as exogenously satisfied, the default risk premia it predicts are systematically below those
observed in corporate debt market. Since then, three main factors are recognized to be critical in
improving the theoretical predictions of structural models of default.
First, the possibility of premature bankruptcy of firms influences the value of outstanding cor-
3
porate debts, particularly those promising intermediate coupon payments. Longstaff and Schwartz
(1955) capture this situation by developing a model in which firms endowed with a multiple-debt
structure bankrupt once the corporate worth falls to a constant reorganization boundary. Briys
and de Varenne (1997) extend this model to a stochastic reorganization boundary. In a more recent
paper, Collin-Dufresne and Goldstein (2001) refine the model of Longstaff and Schwartz by allowing
the default boundary to fluctuate around the value of firm’s assets which permits to incorporate
the stationary property of financial leverages.
Second, the financing of debt service payments may lead equityholders to tradeoff the tax
advantage of debt and immediate liquidation of the firm. Leland (1994) and Leland and Toft
(1996) examine this problem. They allow equityholders to maximize their claim by implementing
an optimal policy of early default trigger, i.e., strategic default. Furthermore, given the fact that
bankruptcy is costly for debtholders, debt renegotiation might be a plausible outcome upon default.
This point was first raised by the empirical study of Franks and Torous (1989) and analyzed by An-
derson and Sundaresan (1996) and Fan and Sundaresan (2000). The consequences of strategic debt
renegotiation in term of the effective debt service are also studied by Mella-Barral and Perraudin
(1997). In this same strand of literature, François and Morellec (2004) examine the implications of
the reorganization procedures under Chapter 11 of the U.S. Bankruptcy Code for the value of debt
claims and financial leverages.
Third, and most important for our purpose, taking into account safety covenants embedded in
debt contracts would add much to the analysis of corporate debt. Black and Cox (1976) show that,
contrary to what might be conjectured, early bankruptcy does not act necessarily against debthold-
ers’ interests. Debt contracts offering to debtholders the right to force immediate liquidation when
the firm is doing poorly minimize losses and, hence, guarantee a floor value for the debt claim.
In the same spirit, Kim, Ramaswamy and Sundaresan (1993) examine the enforcement of debt
4
covenants on debt service payments. They conclude that the ability of debtholders to force early
liquidation whenever the firm defaults on coupons would account for the magnitude of observed
default risk premium.
III. The model
A. Assumptions
Equityholders in our model are allowed to sell a fraction of the firm’s assets to meet debt service
payment as long as the safety threshold covenant was never binding. This ensures that the safety
threshold covenant provides to debtholders an effective protection when the common debt covenant
stating default based on the omission of interest payments does not.
When default occurs, equityholders, in practice, can proceed to the liquidation of the firm
under Chapter 7 of the Bankruptcy Code or, alternatively, undertake debt reorganization under
the Chapter 11 of the same code. In contrast to the immediate liquidation of defaulting firms filed
for Chapter 7, reorganization under Chapter 11 may result in a delayed firm liquidation. Franks
and Torous (1989) provide a detailed discussion of the scenarios offered under the U.S. Bankruptcy
Code. Papers such as Broadie, Chernov and Sundaresan (2004) and François and Morellec (2004)
examine the impact of bankruptcy procedures on corporate claims. This paper has nothing to say
about those issues. As in Black and Cox (1976), Leland (1994) and Longstaff and Schwartz (1995),
we assume that debtholders receive the ownership of assets in place once the firm enters financial
distress.
The value of the continuously traded assets of the issuer firm follows a geometric Brownian
motion described by the stochastic differential equation,
dVtVt
= µ(Vt, t)dt+ σdZt, (1)
5
where µ(Vt, t) and σ denote the instantaneous expected rate of return and the volatility of the value
of firm’ assets, respectively; Zt is a standard Brownian motion. Typically, protected debt contracts
restrict firms to not distribute dividends up to the maturity date. This is why the dynamics of the
firm value in equation (1) assume a zero payout rate. Obviously, including a positive dividend rate
is easily feasible but this would not add more intuition to our model. Further, given the purpose
of this paper, we shall assume a flat default-free term structure. That is, the instantaneous riskless
interest rate r is constant throughout our analysis.
We now are ready to setup the theoretical valuation framework. Consider a protected debt
contract paying a continuous coupon payout c and promising a principal amount B at a finite
maturity date T . It follows from the arbitrage pricing argument of Merton (1974) and Black and
Cox (1976) that the debt value, F , providing the assumptions above, satisfies the partial differential
equation
rVt∂Ft∂Vt
+1
2σ2V 2t
∂2Ft∂V 2t
+∂Ft∂t
+ c = rFt. (2)
Following the theory of dynamic asset pricing (Harrison and Pliska (1981)), there exists a risk-
neutral measure Q under which discounted arbitrage-free prices of traded securities are driven
by martingale processes. In particular, the risk-neutral process of the firm value defined on the
probability space (Ω,F , Q) is given by,
dVtVt
= rdt+ σdZQt , (3)
with ZQt represents the Q-process equivalent to the historical process Zt in equation (1).
B. The exercise policy of the safety covenant right
According to the debt contract we aim to analyze, default occurs once the value of the firm’s assets
falls below a contractual safety threshold. While specifying this minimum worth requirement,
6
debtholders are exposed to two competing effects influencing their incentives for forcing the firm
into bankruptcy earlier than debt maturity.
On one hand, the economic viability of the firm changes randomly over time given the fact that
the value of firm’s assets is stochastic. Therefore, at any time to maturity, triggering firm default
immediately and receiving the liquidation value of the firm’s assets might be a suboptimal decision.
This occurs if, while waiting, the value of the firm increases enough such that financial distress
could be avoided and the discounted value of subsequent debt payments exceeds the payoff received
from the immediate liquidation. This effect, we call the firm’s assets volatility effect, provides to
debtholders incentives for continuation. Intuitively, this effect must increase with the volatility of
the firm’s assets and the remaining time to maturity.
On other hand, when the current value of firm’s assets is relatively low, a high interest rate
reduces the present value of subsequent debt payments making debtholders reluctant to continue.
In such a situation, a safety first’ s type of incentives dominates the first kind of incentives for
continuation and, hence, triggering immediate default of the firm is more likely the optimal decision
for debtholders. However, since the instantaneous growth rate of the firm’s assets in an economy
endowed by the risk-neutral measure is equal to r, a high interest rate equally leads to high expected
forward values of firm’s assets, which lowers the debtholders’ incentives for triggering immediate
default. Accordingly, the interest rate generates two counter-vailing effects implying two opposite
schemes of incentives. The first, due to the role of r as the discount rate, encourage debtholders
to demand immediate liquidation whenever the current value of the firm is deeply low; while the
second, due to the role of r as the drift rate, provides to debtholders incentives for continuation
whenever the current value of the firm is moderately low.
Recognizing these effects, the exercise of the safety covenant right must be optimized such that
it resolves the basic tradeoff between continuing and triggering immediate default. We formulate
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this optimal exercise policy by using an optimal stopping-time approach from which the optimal
safety threshold is determined endogenously.
To illustrate this optimal exercise policy, consider a simplified two-period example. Let t0
represent the current date and T the maturity date at which the debt principal, B, is assumed to
be paid in full, otherwise the firm bankrupts. A coupon c is to be paid out to debtholders at two
intermediate dates, t0 and t1. For simplicity, assume that early default can be triggered at t0 and
t1 only. Therefore, the dynamics of the debt value, F , given an endogenously determined safety
threshold L, are as follows:
[Insert Figure 1 here]
This simplified example shows that early default occurs at time s (s = t0, t1) if the firm value
falls below the specified safety threshold, Ls. Indeed, by fixing this safety threshold optimally,
debtholders proceed to a basic tradeoff between waiting to the next date and triggering immediate
default. At the maturity date, default occurs if the value of the firm’s assets does not exceed the
debt principal.
Now, the same default rule formulated in the context of our continuous-time model is described
by the following equation,
Ft = (FC (Vt, T − t) + c)1Vt≥Lt + Vt1Vt<Lt, (4)
where FC (Vt, T − t) denotes the date t-continuation value of debt; and 1ω is the indicator func-
tion of the event ω. At maturity, the debt value FT is equal to min (VT , B). We discuss below
the fundamental relationship between the maturity condition of default and the early default rule
(4). Since the liquidation of the firm implies the extinction of subsequent debt payments, the debt
value in equation (4) requires that the safety covenant was never violated prior to date t, otherwise
Ft = 0 for any t > 0 if there exists a date s such that, Vs < Ls.
8
Given equation (4), the optimal safety threshold must be endogenously determined, at any time
to maturity (T − t), from the value-matching condition,
FC (Lt, T − t) + c = Lt. (5)
More intuitively, the optimal safety threshold Lt constitutes the date t-critical value of firm’s assets
for which debtholders are indifferent between continuing and demanding immediate liquidation of
the firm. According to (2), (3) and (5), Lt is Ft’s measurable, and, thus, does not depend on Vt.
This property combined with equation (5) yields that the optimal safety threshold is function of
the time to maturity.
Since the continuation value of debt reduces to the principal amount at the maturity date, we
are able to derive the maturity-condition of default from the stopping rule (4) as follows:
FT = B1VT≥LT + VT1VT<LT . (6)
Thus, applying the value-matching condition at debt maturity, we obtain, LT = FC (LT , 0) = B,
which yields FT = min (VT , B). This means that at debt maturity, the default rule (4) respects the
terminal condition of default.
C. Debt and equity values
Assume for instance that the optimal safety threshold as if known. Let V0 denotes the value of
firm’s assets at time zero. The following proposition provides the exact debt value we get from the
partial differential equation (2) subject to the stopping rule (4).
Proposition 1 Let debtholders implement the contract (4)-(5). Therefore, the protected debt value
is given by
F0 =
Z T
0e−rtcN (d2 (V0, Lt, t)) dt+
¡e−rTB − p0 (V0, B, T )
¢, (7)
9
where d2 (a, b, t) = 1σ√t
¡log¡ab
¢+¡r − .5σ2¢ t¢ ;N (·) is the standard normal distribution function;
p0 (V0, k, t) is the value of the European put option on the firm’s assets with a strike price k and a
maturity date t > 0.
Proof. See Appendix.
This result means that the value of protected coupon debt is equivalent to the value of an
unprotected discount debt (Merton, 1974), the term between brackets in the right-hand side of (7),
plus a premium, the integral in the right-hand side of (7), valuing the expected present value of
coupons received as long as the safety threshold covenant was never violated. In fact, a similar
pricing result is obtained for American options, since it is known that the value of an American put
option is the sum of two components. The first component is the value of an equivalent European
put option pricing the maturity payoff if an early exercise is not allowed. The second one consists
of an integral, involving the endogenous boundary for early exercise, interpreted as a premium for
the American right of early exercise.
So far, the optimal safety threshold is implicitly defined through the value-matching condition
(5) as the critical value of firm’s assets at which debtholders are indifferent between triggering
immediate liquidation and continuing. Interestingly, an intuitive interpretation of the optimal
safety threshold follows from Proposition 1. Indeed, rewriting the valuation formula (7) as follows,
F0 =
½Z T
0e−rtcdt+ e−rTB
¾−½Z T
0e−rtcN (−d2 (V0, Lt, t)) dt+ p0 (V0, B, T )
¾, (8)
yields that the protected debt value can be expressed as the value of an equivalent riskless debt
(the first term between brackets in (8)) minus that of the option to liquidate (the second term
between brackets in (8)), i.e., the option to buy-back the firm’s assets upon default.1 Under the1 It is well known that the value of any risky debt can be expressed as the value of an equivalent riskless debt
minus some term interpreted as the value of the option to liquidate (or option to default) reflecting the price discount
due to the risk of default.
10
continuation/liquidation tradeoff, the early exercise of the safety covenant right is optimized, which
in turn maximizes the value of the option to liquidate. Given the convention c = rB, it then follows
that
Z T
0e−rtcN (−d2 (V0, Lt, t)) dt+ p0 (V0, B, T ) = P0 (V0, B, T ) ,
which, therefore, implies
F0 = G0 − P0 (V0, B, T ) , (9)
where G0 =¡cr
¡1− e−rT ¢+ e−rTB¢ is the value of an equivalent default-free debt and P0 (V0, k, t)
denotes the value of an American put option on the firm’s assets with a strike price k and a maturity
date t > 0 (For more details on the analytical valuation of American options, see Kim (1990) and
Carr, Jarrow and Myneni (1992)). In general, for any considered coupon payout, equation (8)
reduces to:
F0 = G0 −hP0
³V0,
c
r, T´+³p0 (V0, B, T )− p0
³V0,
c
r, T´´i
. (10)
Similarly to the European put-to-default derived from Merton’s (1974) model of unprotected debt,
the American put here (or more generally, the American put adjusted by the differential between
the two European puts in formula (10)) represents the optimal option to liquidate available to
protected debtholders. Equation (9) (and more generally, equation (10)), therefore, reveals that
the optimal safety threshold can be viewed as the endogenous boundary for early exercise that
maximize the value of the option to liquidate.
Now, it follows from the debt value (7) and the firm budget-balancing equation that the time
zero-value of equity, S0, verifies,
S0 +
Z T
0e−rtcN (d2 (V0, Lt, t)) dt = EQ
£e−rT max (VT −B, 0)
¤, (11)
11
with EQ is the Q-expectation conditional on V0. This means that the residual claim of equity-
holders, the European call option on the firm’s assets expressed in the right-hand side of equation
(11), is exactly priced by the equity value plus the present value of expected coupon payments. The
economic intuition is that debt service can be viewed as a sunk cost equityholders bear in order to
keep their residual claim “alive” up to the maturity date of debt.
To obtain the numerical values of debt and equity claims, we can price the option to liquidate
(i.e., the American put option) hold by protected debtholders using standard binomial and finite
elements methods, or alternatively, solve for the optimal safety threshold in an intermediate stage.
Interestingly, various numerical approaches, including Huang, Subrahmanyam and Yu (1996), Ju
(1998) and Bunch and Johnson (2000), offer accurate analytical estimation of American options
values by approximating the early exercise boundary.2
D. The term structure of default risk premium
We now examine variations of the default risk premium over the time to maturity dimension. Figure
2 illustrates credit spreads associated with newly-issued protected debts with a range of maturities
growing up to 20 years. Panel A plots the term structure of the credit spread for three levels
(50, 100 and 150%) of the quasi-leverage ratio (B/V0). For high quasi-leverage ratio, predicted
credit spreads drop rapidly from 638 to 165 basis points (bps) as soon as the time to maturity
rises from 5 years to 15 years, but slowly decrease thereafter to close to 115 bps at 20 years.
With a moderate quasi-leverage ratio, our model generates a humped term structure. For shorter
maturities, predicted credit spreads reach the peak of 600 bps at about one year and decrease
2A numerical implementation of the model using an exponential approximation for the optimal safety covenant
standard is available upon request. The valuation method is inspired from Ju (1998) and consists of deriving the
value of debt analytically after approximating the endogenous default boundary by a multipiece exponential function.
Results show that the method yields highly accurate estimations.
12
beyond. For longer maturities, spreads remain considerably high so that they fall to only 67 bps
at 20 years from maturity. With the lowest leveraged firms, predicted spreads are relatively low,
but still significantly non negligible. They increase slowly but constantly before hitting 30 bps at
a maturity of 20 years. The term structures shown in Panel A are consistent with the empirical
patterns documented by Sarig and Warga (1989).
[Insert Figure 2 here]
Panel B plots the term structure of credit spread for different coupon payouts: 5, 7.5, and 10$.
For the tow lowest coupons, predicted term structures are clearly humped. However, as the coupon
payout drops from 7.5 to 5$, credit spreads decrease more rapidly beyond a critical point situated
at about two years. For example, at 20 years from maturity, spreads scarcely reach 38 bps when
the paid out coupon is 5$, and are at about 87 bps for the greater coupon payout of 7.5$. However,
when the coupon is equal to 10$, spreads increase monotonously over the time to maturity, but
with an upward-slope that drops once a switch point of 2 years is reached. In summary, as long
as the time to maturity is behind this critical point, credit spreads increase monotonously. The
intuition is that both the firm volatility and the interest rate effects, that provide incentives for
continuation, tend to disappear near debt maturity. This makes the discounted value of debt
service payments relatively high, increasing in consequence the likelihood of default. Once the time
to maturity exceeds this switch point of two years, these two effects become dominant and the
previous pattern is inverted, except with the highest coupon where spreads are still increasing up
to 20 years. In that case, in fact, the likelihood of default is maintained very high so that the term
structure is monotonously increasing. This is consistent with Helwege and Turner (1999) reporting
that speculative-grade bonds mostly exhibit an upward-sloping yield curve.
13
IV. The debt design’s implications
A. The optimal safety threshold
To solve for the optimal safety threshold, we implement a binomial tree that recursively executes
the stopping rule (4).3 At each stopping-time, the approximated value for this default threshold
is determined as the highest value of firm’s assets for witch debtholders trigger immediate default
of the firm. Figure 3 plots this optimal safety threshold for two selected values of the interest rate
and the volatility of firm’s assets.
[Insert Figure 3 here]
The endogenous default barrier representing the optimal safety threshold constitutes a frontier
between a downside region for optimal early default and an upside region over which continuation is
optimal. As illustrated by Figure 3, the optimal safety threshold shifts down when the volatility of
the firm’s assets rises from 10 to 40 percent. This result should not be interpreted as if debtholders
prefer riskier firms. Rather, the economic intuition behind relates to the positive relationship
between the early exercise boundary of American options and the volatility of underlying asset.
Indeed, the higher the volatility of firm’s assets, the culminating is the probability of observing
large future rates of growth of the firm value, and, hence, the dominant becomes the opportunity
cost of triggering immediate default. Resultantly, increasing the volatility of the firm’s assets lowers
the reliability of the decision of triggering immediate default, thus, boosting debtholders’ incentives
for continuing. This is a byproduct of the optimal exercise policy of the covenant right that, by
minimizing the opportunity cost of triggering immediate default, makes high growth potential-
3A discrete-time coupon payout, cd, equivalent to the continuous-time payout, c, is used when performing compu-
tations. The discrete-time coupon is cd = cr
¡1− e−r∆t¢, where ∆t denotes the length of the time increment in the
binomial tree.
14
firms (i.e., riskier firms) benefiting from more considerable ‘out-of-money’ effect than low risk-
firms typically do. Similarly, Kahl (2002) shows in his theoretical model of financial distress that
debtholders (the bank financers in his model) will be reluctant to trigger immediate liquidation of
the firm whenever the payoffs promised by investments in place are volatile.
When r rises from 4 to 10 percent, the optimal safety threshold reacts by shifting down. The
point is that a high interest rate leads to a high expected forward value of firm’s assets, and, hence,
to a low risk of observing the firm value dropping subsequently. As Figure 3 shows, this kind
of interest rate effect, due to the role of interest rate as the risk-neutral drift, is quite dominant.
The opposite effect, however, due to the role of interest rate as the discount rate, momentarily
occurs near maturity (approximately, less than 3 months from maturity) and only for the highest
considered value of the firm’s assets volatility. In that case, we observe that the optimal safety
threshold associated with the highest interest rate of 10% is above that associated with the lowest
interest rate of 4%. This is because at small intervals from maturity, the expected forward value of
the firm’s assets becomes less sensitive to the interest rate so that increasing r in such a situation
essentially diminishes the discounted value of subsequent debt payments, which in turn lowers the
debtholders’ incentives for continuation.
Similarly to the endogenous boundary for early exercise of American options, the default barrier
here is convex with an infinitely large slope near maturity. This is due to the fact that the volatility
effect decreases non-linearly and more rapidly over time than the interest rate effect, which is, in
contrast, linearly absorbed over time.
[Insert Figure 4 here]
Furthermore, Figure 4 shows that increasing the coupon payout makes the optimal safety thresh-
old shifting up. The intuition behind this result is simple. Since the value of firm’s assets changes
15
randomly over time, high promised debt service payments implies a high opportunity cost of trig-
gering immediate default. Consequently, increasing coupon payout leads debtholders to increase
the payoff they require upon default trigger. This is in line with what we would observe in prac-
tice. Indeed, high coupon payouts are mostly associated with high debt principal amounts, which,
everything else being equal, imply high safety thresholds.
It is worthwhile to note that the dynamics illustrated above of the optimal safety threshold
match perfectly debt renegotiation strategies employed by claimholders during financial distress
(e.g., see Franks and Torous (1989) and Fan and Sundaresan (2000)). This is due to the fact that
debtholders in our model set the default trigger policy endogenously in function of both the firm
characteristics and debt contract parameters. More specifically, whenever we allow for substantial
fall of coupon, principal amount renegotiation, or debt maturity stretching, the default barrier in
our model representing the optimal safety threshold is pushed down, therefore, creating additional
room for continuation. Further, it can be seen that shifting up the likelihood of observing future
large changes in the value of the firm’s assets by increasing volatility boosts the out-of-money effect
of continuation which permits to the claimholders to mutually avoid immediate bankruptcy. In
such a situation, indeed, claimholders’ incentives for risk shifting are compatible and the typical
asset substitution problem is offset. This happens in our model because we implicitly assume that
shifting the growth potential (riskiness) of the firm will be costless for equityholders. Otherwise,
in fact, the Myers’ (1977) under-investment problem emerges.
B. The endogenous debt capacity of the firm
Since the endogenous default barrier representing the optimal safety threshold delimits an optimal
region for continuation, the optimal safety threshold at the issue date of debt can be considered as
the critical level of assets in place below which protected debtholders do not agree to finance the
16
firm. Fixing the contractual values of coupon and debt maturity, such a mechanism gives us the
maximum debt principal for which financing is feasible. This upper bound for the principal amount
constitutes, therefore, the endogenous debt capacity of the firm.
Since the optimal safety threshold is a decreasing function of time-to-maturity, the firm is able
to reduce its debt capacity constraint by selecting longer maturities. This maturity effect is often
observed in corporate debt markets and is common for various debt instruments. The severity of
the debt capacity constraint also depends on the coupon payout, the volatility of the firm’s assets
and the interest rate level. Table 1 reports the endogenous debt capacity of the firm for different
levels of the risk-free interest rate (Panel A) and the volatility of the firm’s assets (Panel B).
[Insert Table 1 here]
We observe that the endogenous debt capacity of the firm rises as the interest rate and/or the
volatility of firm’s assets increases. Additionally, keeping everything else equal, the endogenous debt
capacity is decreasing in the coupon payout. These properties directly follow from the dynamics
for the optimal safety threshold illustrated by Figures 3 and 4. The positive relationship between
the debt capacity and the risk of the issuer firm requires, however, a particular attention. One
may interpret this as if larger firms having mature activities and, thus, endowed with low volatile
assets are less able to contract protected debt financing than small firms likely having high growth
proceeds and, thus, more volatile assets. However, given the definition of endogenous debt capacity
we introduced, increasing the value of firm’s assets at debt issue date considerably reduces this
financing constraint. Therefore, both the size and the growth potential of the firm attenuate this
endogenous debt capacity constraint, which mitigates the latter statement.
Even though coupons are exogenously fixed in the experiments above, an alternative approach
could consist of expressing coupon as a fraction of the debt principal amount. By doing so, we shall
17
be able to generate, simultaneously, the endogenous debt capacity of the firm and the associated
capacity attained for debt service payments.
V. The debt design’s evaluation
A. Experimental evidence
The main feature of our debt design consists of optimizing the exercise of the safety covenant
right hold by protected debtholders. As it has shown, this can be achieved by determining the
optimal safety threshold representing the minimum worth requirement. We now propose to assess
the efficiency of this debt design. The test consists of evaluating the ex-post performances of the
optimally designed safety threshold covenant. The protected debt contract studied by Black and
Cox (1976) imposing the discounted debt principal as the worth requirement, is used as a benchmark
contract.
[Insert Table 2 here]
The test methodology consists of generating in a first stage a simulated sample of 100, 000 firms,
form which only firms that have not defaulted according to the safety threshold covenant (our debt
design) are retained. The second stage consists of implementing the worth requirement provision
of the benchmark contract over the first stage’s subsample of survived firms. This yields us the
exclusive subsample of the firms that would have survived according to our debt design, but failed
to liquidation based on the benchmark contract. Providing this subsample, we compute for each
firm the opportunity cost of the early liquidation due to the benchmark contract. This opportunity
cost consists of the (normalized) gap between the present value, computed at the bankruptcy time,
of the nominal payments that would be received by debtholders under our debt design (ex-post
conditional on no default) and the liquidation value (the stopping firm value) debtholders get under
18
the benchmark contract. Namely, let θ = inf©t > 0 : Vt < e
−r(T−t)Bªdenotes the (empirical)
bankruptcy time under the benchmark contract. Therefore, the opportunity cost is defined as
follows: ³R Tθ e
−rtcdt+ e−r(T−θ)B´− Vθ
Vθ.
The results reported in Table 2 show that imposing the worth requirement provision of the bench-
mark contract will lead to suboptimal liquidation decisions. When compared to our safety threshold
covenant, this provision leads to an opportunity cost exceeding 12%. The main feature accounting
for the efficiency of our debt design is that the safety threshold involved takes into account the
random shocks affecting the economic viability of the issuer firm.
The same experience conducted with the benchmark contract embedding the positive net-worth
provision (according to this provision, default occurs once the firm value falls below the nominal
debt principal, i.e., θ = inf t > 0 : Vt < B), yields different results. Statistics show that the
average opportunity cost is not significantly different from zero. This is due to the fact that the
positive net-worth requirement ensures a sufficiently high floor value for the debt claim, i.e., the
debt principal amount. Resultantly, debtholders do not bear a significant opportunity cost by
foregoing the subsequent nominal debt payments when triggering premature liquidation of the
firm. However, under this benchmark contract, experience shows that the percentage of the second
stage’s subsample of defaulting firms in the first stage’s subsample of surviving firms ranges from
76% to 91%. This means that the zero opportunity cost of the positive net-worth provision comes
at the cost of an excessively high failure rate of firms. In the light of this result, one can conclude
that, when compared to debt contracts embedding a positive net-worth provision, our debt design
achieves a Pareto dominant welfare improvement.
19
B. Liquidation inefficiencies
It is assumed so far that when default occurs, equityholders place the firm under Chapter 7 of the
Bankruptcy Code, thus, permitting to debtholders to receive, in full, the liquidation value of the
firm’s assets. In real-world situations, however, the absolute priority rule is often violated. Franks
and Torous (1989) and Weiss (1990) report that important deviations have occurred in distressed
firms’ data. Further, as explained in Bebchuk (2002), similar violations also occur under Chapter
11. In addition to this moral risk problem, it is also documented that liquidation procedures are
costly (see Warner (1977) and Betker (1997)).
In the light of this evidence, it is important to examine the behavior of the debt design when
liquidation inefficiencies are confronted. In doing so, we shall assume that the payoff received by
debtholders upon bankruptcy is inferior to the total liquidation value of the firm’s assets. This
payoff, we denote by g (Vt, T − t), is considered as a function of the value of the firm’s assets and
the remaining time to maturity at default. In a rational anticipations setup, absolute priority
violations as well as liquidation fees are expected by debtholders and, thus, the default rule (4)
changes to,
Ft =
[FC (Vt, T − t) + c]1Vt≥Lt + g (Vt, T − t)1Vt<Lt, 0 ≤ t < T,
B1VT≥B + g (VT , 0)1VT<B, t = T.
(12)
In this setup, the optimal safety threshold is the solution for the value-matching condition,
FC (Lt, T − t) + c = g (Lt, T − t) . (13)
By letting the safety threshold endogenously determined from (13), this means that debtholders
are still able to optimize the exercise policy of the covenant right. The following corollary provides
the valuation result we get in this generalized framework.
20
Corollary 1 Let debtholders implement the contract (12)-(13). Therefore, for any continuous and
differentiable payoff function g (Vt, T − t) : <+ × [0, T ] −→ (0, Vt], the protected debt value is given
by
F0 =
Z T
0e−rtcN (d2 (V0, Lt, t)) dt
+
Z T
0EQ·µrg − rVt ∂g
∂Vt− 12σ2V 2t
∂2g
∂V 2t− ∂g
∂t
¶1Vt<Lte
−rt¸dt
+ e−rT©BN (d2 (V0, B, T )) + EQ
¡g (VT , 0)1VT<B
¢ª.
(14)
Proof. See Appendix.
Given the value-matching condition (13) and the debt pricing formula (14), it is easy to see
that the optimal safety threshold will be readjusted to take into account expected liquidation
costs. To illustrate this point, Figure 5 plots this safety covenant for the simple payoff function
g (Vt, T − t) = (1− w)Vt. The writedown parameter w takes the values of 0%, 2.5% and 5%.
[Insert Figure 5 here]
Therefore, both theoretical and numerical results reveal that debtholders are qualified to min-
imize the negative impact of anticipated liquidation inefficiencies when implementing the debt
design described by equations (12) and (13). The idea is that by, endogenously, shifting up the
optimal safety threshold when moving to an imperfect liquidation environment, as shown in Figure
5, debtholders are placed to control for the liquidation costs to be lost. This is because the value-
matching condition, defining the optimal safety threshold, internalizes deviations from the total
liquidation value of firm’s assets occurring upon bankruptcy. Interestingly, when considering the
intuitive case where the payoff function g(·) is equal to a constant fraction of the firm’s assets (i.e.,
the one considered in Figure 5), we can easily show that the expression for the debt value resulting
from the pricing formula (14) is analogue to that entailed by formula (7) of Proposition 1, recog-
nizing that the optimal safety threshold is set from the value-matching condition (13). This means
21
that, equipped with the debt design (12)-(13), debtholders are able to internalize liquidation costs
as if they are completely offset, while still being exposed to a higher default risk comparatively
to the benchmark case of no inefficiencies (credit spread must be higher for nonzero liquidation
costs since the resulting upward shift of the optimal safety threshold will inevitably increase the
likelihood of default).
The capacity of debtholders of reducing their exposure to inefficiencies arising from bankruptcy
procedures is an important implication of our debt design model. It confirms the well-known
idea from the economic theory of contracts that an optimally designed contract must attenuate
inefficiencies the contracting parties (or the principal) would not face in the first-best outcome’s
world.
VI. The optimal debt contract
Our aim now consists of solving for the protected debt contract that maximizes the value of equi-
tyholders’ claim subject to the safety threshold covenant optimally designed by debtholders as our
model shows. The resulting optimal debt contract in fact achieves a subgame-perfect equilibrium
between claimholders. In studying the dynamics of the optimal debt contract, we assume that abso-
lute priority violations represent a fraction α ∈ (0, 1) of the total liquidation cost, [Vt − g (Vt, T − t)].
This leads to the following value for equity claim:
S0 = EQ£e−rT max (VT −B, 0)
¤− Z T
0e−rtcN (d2 (V0, Lt, t)) dt
+αEQ£e−rT (VT − g (VT , 0))1VT<B
¤(15)
+α
Z T
0EQ·µrg − rVt ∂g
∂Vt− 12σ2V 2t
∂2g
∂V 2t− ∂g
∂t
¶1Vt<Lte
−rt¸dt.
As one can see, equity value here exceeds that obtained with zero liquidation costs (see equation
(11)) by the expected amount of deviations from absolute priority (the two terms multiplied by the
22
parameter α in the right-hand side of (15)). The optimal debt contract formally consists of the set
of variables I ≡ B, c, T that,
maximizesI=B,c,T
S0
³I, V0, r,σ,α, g(·), (Lt)0≤t≤T
´,
subject to: (i) the optimal safety threshold Lt (I, r,σ,α, g(·)) for any t ∈ [0, T ]; and (ii) the endoge-
nous debt capacity constraint, V0 ≥ L0 (I, r,σ,α, g(·)).
In general, multiple solutions to this optimal contract problem are easy to derive providing
more refined structures for debt financing and investment constraints. For simplicity, we restrict
the debt principal and maturity to take given values so that the optimal debt contract is indicated
in full by the contractual coupon payout. The optimal debt contract is determined after entitling
equityholders to benefit of a tax advantage on the interest payments as long as the firm is solvent.
Since debtholders’ claim is completely unaffected by this consideration, the stopping rule (12)-(13)
as well as its impact on the equityholders’ choice of optimal debt contract will be preserved. By
doing so, in fact, we solely avoid the trivial case where equityholders do not have a rationale for
issuing coupon-paying debt.
[Insert Table 3 here]
The numerical results we obtain are summarized in Table 3. This table reports the optimal debt
contract described by the contractual coupon rate and the associated value of equity attained for
different combinations of the debt principal and the volatility of firm’s assets. The payoff function
we consider in our simulations is g = (1− w)Vt. Two experiments are conducted for two reasonable
values of 0.10 and 0.20 assigned to the writedown parameter w.
The first point to note is that increasing the riskiness of firm’s assets affects the optimal debt
contract differently depending on the financial leverage which is indicated in our experiment by
the debt principal-firm value ratio. We observe that the optimal coupon rate is a decreasing
23
function of the assets’ volatility for low-leverage firms while, in contrast, becomes increasing for
high-leverage firms. The rationale behind this result is due to the dynamics of the optimal safety
threshold governing the likelihood of firm default. Since equityholders of low leveraged firms are
waiting for a high residual claim upon debt reimbursement, their incentives for preserving the firm
from premature liquidation are high. Hence, as the growth potential of the asset value indicated
by volatility rises, as these incentives become more dominant than the immediate tax advantage,
therefore, leading equityholders to consolidate the continuation effect (‘out-of-money’ effect) of
assets’ volatility by lowering the contractual coupon rate. When moving to the case of high-leverage
firms, however, the residual claim of equityholders becomes less important so that equityholders
would prefer to not forego the tax advantage of coupon payments. In such a situation, as the
growth potential of the firm’s asset value increases, as the default threshold shifts down generating
additional region of continuation, and as equityholders are willing to exploit tax benefits of coupons.
This is because the continuation effect of incremental volatility will compensate for the opposite
effect caused by increasing contractual coupon. As a consequence of the flexibility an enhanced
growth potential of asset value (i.e., volatility increments) provides to equityholders in fixing the
optimal debt contract in function of the financial leverage, we observe that equity value reached
in equilibrium increases monotonously with the volatility of firm’s assets for both low and high-
leverage firms. In conformity with this idea, Table 3 clearly shows that whether the volatility of
firm’s assets is hold fixed, the maximized value of equity we reach is decreasing in the firm leverage.
Overall, these results suggest that the riskiness of assets and the inherent potential of high firm’s
worth should influence considerably the equityholders’ choice of debt contract. Despite the fact
our debt design framework and the nature of debt contract we analyze are quite different from
those studied in Anderson and Sundaresan (1996), the findings above are substantially similar to
those obtained by the last authors. Since our model and the former both allows debt covenants
24
to be determinant for the occurrence of early default, this suggests that the threat of premature
liquidation leads equityholders to stress the growth potential of the firm’s assets when choosing the
debt optimal contract.
As these results permit us to see, the negative impact of highly leveraged issues of safety-
protected debt on claimholders’ interests can be attenuated whether the safety threshold is optimally
specified by debtholders. In such a situation, equityholders are able through an optimal choice of
the debt contract parameters to affect ex ante the debtholders’ incentives for continuation, and,
thus, to reduce likelihood of forced liquidation. This results in an enhanced aggregated value for
claimholders if compared to an exogenous default threshold, particularly for issuer firms with high
growth potential where the opportunity cost of early liquidation is relatively expensive.
Now, comparing changes of the optimal debt contract that result from shifting the writedown
parameter from 0.10 to 0.20 permits us to assess the impact of liquidation inefficiencies. Keeping the
volatility of firm’s assets and leverage constants, we observe that the optimal contractual coupon is
decreasing in w. The reason is that increasing liquidation costs will push up the safety threshold,
thus, implying higher likelihood of early default. To counter-balance this effect, equityholders can
initially lower the contractual coupon rate, providing the equilibrium relationship between coupon
and the default threshold. In doing so, they reduce the risk of forced liquidation at the cost of
less benefits from the tax advantage of coupon payments. For most of cases examined in Table
3, this tradeoff does not affect equity value attained in equilibrium. Rather, results show that
equityholders are mostly able to reach higher values for equity as the parameter w rises. This is
because the tax advantage of coupons to be lost whenever increasing liquidation costs might be
compensated by the associated increasing deviations from absolute priority equityholders benefit.
Although not reported here, experiments show that our findings remain qualitatively the same
after varying debt maturity. As it is the case in general for contingent claim models, the elementary
25
effects discussed above still influence the optimal debt contract in the same manner but with
intensities depending on the case parameters considered. An interesting point to raise, however,
is that two particular outcomes may occur whenever we consider short term debt. First, the
endogenous debt capacity constraint becomes rapidly binding for highly leveraged debt issues.
Second, for relatively low and moderate leverages, the optimal contractual coupon may shrink to
zero leading to the issue of a discount debt.
VII. Conclusions
We have examined in this paper a contingent-claims model that focuses on debt contracts with
an embedded safety threshold covenant. According to this debt covenant, default occurs once
the firm’s worth falls below a specified contractual threshold. We build our analysis in a two-
path equilibrium framework by allowing claimholders to maximize their own interests subject to
this contractual provision. In the first stage, debtholders adopt an optimal exercise policy of the
covenant right by resolving a continuation/liquidation tradeoff. We solve in the second stage for
the optimal debt contract maximizing the value of equity subject to the constraints issued from
the first-path’s optimized safety threshold.
The model offers an optimal design for safety-protected debts that allows debtholders to avoid
suboptimal premature liquidations by recognizing the effect of the volatility of the firm’s assets on
the viability of triggering immediate default. We find that the optimal option to liquidate hold
by protected debtholders is analogue to an American put written on the firm’s assets. It is also
shown that, by implementing the optimal design we propose, debtholders are able to minimize their
exposure to the liquidation inefficiencies emerging from bankruptcy procedures. Finally, analysis
shows that the threat of forced premature liquidation leads equityholders to stress the growth
potential of assets when choosing the debt contract’s parameters.
26
Overall, the model shows that the impact of safety covenants can be determinant for the value
of corporate claims. It also offers a promising framework to analyze corporate borrowing from the
perspective of institutional lending such as bank loans and project financings, where the ability of
financers to force the firm into bankruptcy generally exceeds that of bond holders.
27
Appendix: Mathematical proofs
Proof of Proposition 1: Consider the date t-discounted debt price Xt = e−rtFt. Using the extension
of the Itô’s formula yields,
XT −X0 =Z T
0
∂Xt∂Vt
dVt +
Z T
0
µ1
2σ2V 2t
∂2Xt∂V 2t
+∂Xt∂t
¶dt. (16)
Substituting Xt by its expression into (16) we obtain:
e−rTFT − F0 =Z T
0e−rt
∂Ft∂Vt
dVt +
Z T
0e−rt
µ1
2σ2V 2t
∂2Ft∂V 2t
+∂Ft∂t− rFt
¶dt. (17)
Replacing in the right-hand side of (17) the debt value, Ft, by its expression provided by the default
rule (4) we have that,
e−rTFT − F0 =
Z T
0e−rt
µ∂Ft∂Vt
1Vt≥Lt + 1Vt<Lt
¶hrVtdt+ σVtdZ
Qt
i+
Z T
0e−rt
µ1
2σ2V 2t
∂2Ft∂V 2t
+∂Ft∂t− rFt
¶1Vt≥Ltdt (18)
−Z T
0e−rtrVt1Vt<Ltdt.
Then, regrouping terms conditional on Vt ≥ Lt in the same integral and separating the resulting
Wiener integral leads to,
e−rTFT − F0 =
Z T
0e−rt
µrVt
∂Ft∂Vt
+1
2σ2V 2t
∂2Ft∂V 2t
+∂Ft∂t− rFt
¶1Vt≥Ltdt
+
Z T
0e−rtσVt
µ∂Ft∂Vt
1Vt≥Lt + 1Vt<Lt
¶dZQt . (19)
Given the partial differential equation (2), the first integral in the right-hand side of (19) reduces
toZ T
0−e−rtc1Vt≥Ltdt.
Now, replacing FT by its expression and taking expectations of both sides of equation (19)
under the risk-neutral measure yields,
EQ£e−rT min (VT , B)
¤− F0 = −Z T
0EQ£e−rtc1Vt≥Lt
¤dt. (20)
28
Hence, Proposition 1 follows by solving for F0 above. ¥
Proof of Corollary 1: We adopt the same proof strategy for Proposition 1. After substituting
Ft1Vt<Lt by g (Vt, T − t) in (17) we obtain,
e−rTFT − F0 =
Z T
0e−rt
µ∂Ft∂Vt
1Vt≥Lt +∂g (Vt, T − t)
∂Vt1Vt<Lt
¶hrVtdt+ σVtdZ
Qt
i+
Z T
0e−rt
µ1
2σ2V 2t
∂2Ft∂V 2t
+∂Ft∂t− rFt
¶1Vt≥Ltdt (21)
+
Z T
0e−rt
µ1
2σ2V 2t
∂2g (Vt, T − t)∂V 2t
+∂g (Vt, T − t)
∂t− rg (Vt, T − t)
¶1Vt<Ltdt..
Hence, replacing FT by its expression,¡B1VT≥B + g (VT , 0)1VT<B
¢, developing and taking ex-
pectations under the risk-neutral measure yields formula (14). ¥
29
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32
Panel AV0 = 100$, T = 5 years, σ = 20 percent
r(%) c = 2.5$ c = 5$ c = 7.5$ c = 10$4 192 144 113 1016 213 162 128 1038 237 181 145 11710 263 202 163 13412 291 225 183 152
Panel BV0 = 100$, T = 5 years, r = 6%
σ c = 2.5$ c = 5$ c = 7.5$ c = 10$0.15 172 136 111 910.20 213 162 128 1030.25 272 201 154 1220.30 354 301 190 1480.35 471 319 239 184
Table 1: The Endogenous Debt Capacity of the Firm.
This table reports the endogenous debt capacity of the firm computed from a binomial tree of300 time steps for different values of the interest rate and the volatility of firm’s assets. This debtcapacity is defined as the maximum debt principal permitted for which financing is feasible, i.e.,the critical value of B for which the initial firm value V0 exactly equalizes the time zero-level of theoptimal safety threshold.
33
Time step 1/6 1/12 1/24 1/52First stage’s subsample (#) 71, 930 70, 284 69, 256 68, 366(% in the whole sample) (71.93%) (70.28%) (69.25%) (68.36%)
Second stage’s subsample (#) 23, 255 23, 602 23, 779 24, 026(% in the first stage’s subsample) (32.33%) (33.58%) (34.33%) (35.14%)
Average opportunity cost (%) 14.50% 13.58% 12.89% 12.30%
Table 2: The Debt Design’s Efficiency
This table reports the results we get for the experimental test of efficiency of our debt design.After generating a simulated sample of 100, 000 firms, the first stage’s subsample is restricted toonly contain firms that survived under our safety threshold covenant. The second stage’s subsampleexclusively include firms from the first stage’s subsample that are failed to liquidation under thebenchmark contract studied by Black and Cox (1976). The average opportunity cost associatedwith the benchmark contract indicates the (relative) efficiency of our debt design. The variablesused in this experience are those of the basic case’s firm: V0 = B = 100$, r = 6%, σ = 20 percentand T = 5 years.
34
w = 10 percent w = 20 percentB σ c (%) Equity ($) B σ c (%) Equity ($)60 0.20 7.96 57.4345 60 0.20 5.46 56.854060 0.30 5.40 59.0141 60 0.30 4.02 58.968660 0.40 4.92 62.1340 60 0.40 3.30 62.320980 0.20 3.33 43.7155 80 0.20 1.70 43.675280 0.30 2.85 48.0430 80 0.30 1.49 48.403580 0.40 2.71 53.0975 80 0.40 1.18 53.6439100 0.20 1.40 32.7644 100 0.20 0.53 33.4867100 0.30 1.44 39.3042 100 0.30 0.71 40.0872100 0.40 1.47 45.8591 100 0.40 0.81 46.8208120 0.20 0.58 24.3466 120 0.20 0.17 25.7679120 0.30 0.76 32.2864 120 0.30 0.28 33.8109120 0.40 0.89 39.9057 120 0.40 0.39 41.1901
Table 3: The Optimal Debt Contract
This table reports the optimal contractual coupon rate and the equity value attained for differentlevels of the debt principal and the volatility of firm’s assets. The liquidation payoff functiong (Vt, T − t) we consider has the explicit form of (1− w)Vt. The parameter α takes the value of50% and the effective tax rate is assumed equal to 10%. The rest of variables are r = 6%, V0 = 100$and T = 5 years.
35
1( )t
00 tt VF =
1 0
10 0
( )r t ttt t e FF c − − = +E
0 0t tV L<
0 0t tV L≥11 tt VF =
1
1 1
( )r T tTt t e FF c − − = +E
1tt tV L<
1 1t tV L≥TT VF =
T BF =
TV B<
TV B≥
0( )t ( )T
Figure 1: The Dynamics of the Debt Value in a Simplified Two-Period Example. In thisillustrative example, the time line is divided into three dates. The debt pays a discrete cou- pon cat t0 and t1, and a principal amount B at the maturity date T . The variable Ls deno- tes the
date s-safety covenant standard to be determined endogenously. .
36
The Term S tructure of Credit S preadsPan e l A
0
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0 5 10 15 20
Tim e to m aturity (years )
Cre
dit s
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The Term S tructure of Credit S preadsPane l B
0
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0 5 10 15 20
Time to maturity (years )
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Figure 2: The Term Structure of Credit Spreads. The lines plot the term structures ofcredit spread in an economy characterized by an interest rate r = 6% and a volatility offirm’s assets σ = 20 percent. Panel A consider the term structure for firms with assets
initially valued at V0 = 100$, paying out a coupon c =5$, and having a quasi-leverage ratio(B/V0) of 50% (short dashed line), 100% (long dashed line), and 150% (solid line).Panel B consider the term structure for firms with assets initially valued at V0 = 120$and issuing a protected debt with a facial value B =100$ and a coupon payout c equal
to 5$ (short dashed line), 7.5$ (long dashed line), and 10$ (solid line). .
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Figure 3: The Endogenous Default Barrier. The lines plot the endogenous default barrierrepresenting the optimal safety covenant standard for different values of the interest rate and thevolatility of firm’s assets: r = 4% and σ = 10 percent (short dashed line), r = 10% and σ = 10percent (medium dashed line), r = 4% and σ = 40 percent (long dashed line), and r = 10% andσ = 40 percent (solid line). It is assumed that the principal amount B =100$ and the coupon
pay-out c =5$. .
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The Endogenous Defulat Barrier for Different Coupon Payouts
70 $
80 $
90 $
100 $
012345
Time to maturity (years )
Figure 4: The Endogenous Default Barrier for Different Coup ons. T he lines plot the endogenousdefault boundary representing the optimal safety covenant standard associated with a couponpayout c equal to 10$ (short dashed line), 7.5$ (long dashed line), and 5$ (solid line), respectively.It is assumed that the interest rate r = 6%, the volatility of firm’s assets σ = 20 percent and theprincipal amount B = 100$.
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The Endogenous Default Barrier for Different Values of w
70 $
80 $
90 $
100 $
012345
Time to maturity (years )
Figure 5: The Endogenous De fault Barrier for Different Values of the Writedown Parameter w .The lines plot the endogenous default boundary for different values of the writedown parameter wequal to 5% (short dashed line), 2.5% (long dashed line), and zero percent (solid line), respectively.It is assumed that the interest rate r = 6%, the volatility of firm’s assets σ = 20 percent, thecoupon payout c = 5$ and the principal amount B = 100$.
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