DISCUSSION
PAPER SERIES
IN ECONOMICS
AND MANAGEMENT
Optimal Debt Service:
Straight vs. Convertible Debt
Christian Koziol
Discussion Paper No. 04-15
GERMAN ECONOMIC ASSOCIATION OF BUSINESS
ADMINISTRATION – GEABA
Optimal Debt Service:
Straight vs. Convertible Debt
Christian Koziol∗
May 2004
JEL Classification: G12, G32
∗
Dr. Christian Koziol, Chair of Finance, University of Mannheim, L5, 2, D-68131 Mannheim,
Germany, Phone: +49-621-181-1521, Fax: +49-621-181-1519, Email: c.koziol@uni-mannheim.de.
0
Optimal Debt Service:
Straight vs. Convertible Debt
May 2004
Abstract
In this paper, we analyze the optimal default strategy of a firm when debt
is convertible into equity. For this purpose, we consider a convertible consol
bond in a time-independent model in the presence of bankruptcy costs and
tax deductability. The optimal default and conversion strategy result from a
game between equity and debt holders. We show that an optimal default of
convertible debt occurs earlier than a default of otherwise identical straight
debt. Although the value of convertible debt is lower when the firm follows the
optimal debt strategy rather than the strategy for straight debt, the values of
the investment and option component of the convertible debt can be higher.
Furthermore, we find that the important difference between the default barrier
for convertible and identical but non-convertible debt rises with the conversion
ratio, the promised coupon, a lower tax rate, and a lower payoff rate.
JEL Classification: G12, G32
1
Introduction
The determination of the optimal default point is a complex decision for every firm.
Firms not only default when their firm value falls below an exogenous barrier such as
in the case of overindebtedness but they also have the possibility to stop payments to
the debt holders which results in a default due to the inability to pay. The possibility
for an insolvency without an overindebted balance sheet is a valuable option for the
equity holders because it can prevent from running the firm despite negative equity
values. As a consequence, the determination of the optimal time to default is an
important task for every firm, because it affects the wealth of the equity holders.
Black/Cox (1976) and Leland (1994) among many others deal with the problem
of strategic default by deriving an endogenous default strategy that maximizes the
1
equity value. The result is that interests should be paid as long as the equity value
exceeds the size of the required capital injections by the equity holders. Other approaches extend the standard model from Leland by relaxing the assumption that
a default necessarily results in a costly liquidation of the firm but allow for reorganization (see e.g. Anderson/Sundaresan (1996), Mella-Barral/Perraudin (1997),
Acharya/Huang/Subrahmanyam/Sundaram (2002), and Fan/Sundaresan (2000)).
The introduction of strategic debt renegotiation prevents the liquidation of the firm
and results in a higher value of the firm due to the elimination of bankruptcy costs.
All these models have in common that they all consider a simple capital structure
with equity and straight debt only. However, corporate debt is often equipped with
additional rights such as a conversion feature for the investors. The standard view
to convertible bonds is to decompose it into an investment component and an option
component (see e.g. Ingersoll (1977)). The value of the investment component is
that of a non-convertible but otherwise identical bond. The idea behind this decomposition is that both components can be separately priced, where the investment
value is typically obtained as the value of straight debt of an appropriate benchmark
firm. In the ideal case, the benchmark firm has identical assets and the same debt
obligation but without any conversion rights. According to this view, the big variety
on straight debt models can be applied to price the investment value by regarding
the debt value of the benchmark firm as a proxy for the investment value.
This practical approach to pricing convertible bonds, implicitly assumes that the
conversion right does not affect the default strategy. At first glance, this assumption
seems to be plausible because a default typically occurs when the firm is under
pressure and a conversion is not desirable, while a conversion is optimal if the danger
of a default is low. However, to derive the optimal default strategy for firms with
convertible debt, a game-theoretical analysis is required which fundamentally differs
from simply assuming that a conversion does not affect the default strategy. Strictly
speaking, the optimal default strategy of convertible debt is the result of a game
between the firm, which represents the equity holders, and the convertible debt
holders. In this dynamic game, the firm must continuously decide about a default,
where the debt holders have to decide upon conversion. It is not clear whether the
optimal default strategy resulting from this complex game differs from the strategy
for straight debt. However, the relation between these two strategies is important
when analyzing the default risk of convertible debt and especially when pricing the
investment component of a convertible bond by regarding the identical but otherwise
non-convertible bond value.
2
The goal of this paper is to determine optimal default strategies when debt is convertible and to analyze the factors that cause a difference between the optimal
strategy and the strategy for straight debt. To illustrate the importance of the optimal default strategy, we compare the optimal strategy with the strategy of straight
debt by regarding the price impact on the debt value, the investment value, and the
option value. For this purpose, we consider a time-independent model with a perpetual bond paying a continual coupon per unit time in the presence of bankruptcy
costs and tax deductability.
As a result, we obtain that a default of convertible debt always occurs before a
default would be triggered when the debt was not convertible. The default strategy
has a considerable effect on the values of convertible debt, the investment value and
the option component. We can show that this relation between the optimal default
strategy and the strategy of straight debt results in a higher value of convertible debt
when the firm follows the straight debt strategy rather than the optimal strategy.
On the contrary, the two components of a convertible bond, the investment value
and the option value, can be higher or lower when the straight debt rather than the
optimal strategy is chosen by the firm. Therefore, the practical approach to pricing
the investment value of a convertible bond by regarding an otherwise identical but
non-convertible debt is not justified. When examining the important difference
between the optimal default barrier and the barrier for straight debt, we find that
the difference rises with the conversion ratio, the promised coupon, a lower tax rate,
and a lower payoff rate.
The paper is organized as follows: In Section 2, we describe the model framework
and show how to determine the optimal default strategy and the related values of
convertible debt and equity. Properties of the optimal default strategy are derived in
Section 3. Section 4 provides a comparative static analysis of the difference between
the optimal default barrier and the default barrier for straight debt. Section 5
concludes. Technical developments are in the appendix.
3
2
Model Framework and Values of Equity and
Convertible Debt under the optimal Strategy
2.1
Model Framework
We consider a standard model of time-independent security values with bankruptcy
costs and tax deductability to analyze the optimal debt service of convertible debt.
Time-independent models are widely-accepted to determining optimal default strategies (see e.g. Black/Cox (1976), Leland (1994), Mella-Barral/Perraudin (1997), and
Fan/Sundaresan (2000)), because these models capture the relevant determinants
for a default and result in tractable formulae for asset prices. As an extension to
these approaches, we consider a convertible rather than a straight consol bond. A
convertible consol bond is a claim of the firm that continuously pays a coupon C,
per instant of time as long as the firm is solvent and no conversion takes place.
A conversion occurs upon the notice of the debt holders into a fraction γ of total
equity.1 Since a conversion is associated with a loss of the coupon claim, the firm
after conversion is not leveraged anymore and therefore the conversion value equals
γ · V , where V denotes the asset value of the firm. Before conversion when the firm
is still leveraged, the asset value V has the character of the value of an identical
but otherwise non-leveraged firm after tax. In the case of a default, which occurs if
the firm does not pay the coupon to the debt holders, the firm is liquidated. The
debt holder receive the whole proceeds from a liquidation, which equal the asset
value V minus bankruptcy costs α · V , while the stock holders are left with nothing.
The relative bankruptcy costs α range from zero to one. In the case of a default,
we do not allow for any renegotiations as presented by Fan/Sundaresan (2000) to
avoid bankruptcy costs, because these mechanisms would unnecessarily complicate
the analysis without providing us with further insights about the default strategy
of convertible debt. If desired, one could easily incorporate these possibilities of
renegotiation to prevent the occurrence of bankruptcy costs.
1
We assume that the whole debt issue must be converted at the same point in time in one block
or not at all. Depending on the fact whether perfect competition exists among the convertible
bond holders or a monopolist controls the whole conversions other strategies rather than the block
strategy can be optimal (see e.g. Constantinides (1984), Spatt/Sterbenz (1988), and Bühler/Koziol
(2002)). Since the focus of our study lies on the optimal default decision, which is typically relevant
if a conversion is relatively unlikely, it is appropriate to work with this standard assumption.
Moreover, the convertible debt value and therefore also the default strategy are unaffected by the
conversion assumption if markets are frictionless and firms have no additional debt outstanding.
4
In the special case of very high relative bankruptcy costs such that (1 − α) < γ, a
conversion which results in γ · V is beneficial for the convertible debt holders relative
to a liquidation which amounts to (1 − α) · V . In this situation, the firm can force
a conversion promptly by stopping the coupon payment, because the debt holders
have an incentive to prevent a liquidation by a conversion. To avoid a discussion of
this trivial and untypical situation, we focus on cases with (1 − α) > γ, in which
debt holders can obtain a higher value by taking over the firm and liquidate it than
by receiving a fraction of the firm value through conversion.
The asset value V follows a geometric Brownian motion
dV /V = (µ − β) · dt + σ · dz,
where z is the value of a standard Wiener process, µ denotes the instantaneous
expected rate of return on the firm gross of all payout, and σ is the standard deviation
of the instantaneous return of the asset value. The firm’s payout ratio is β ≥ 0.
Since taxes are deductable, the equity holders obtain a tax benefit equal to τ · C
when paying the coupon, where τ indicates the tax rate. Hence, before a conversion
and a default, the instantaneous net payment to the equity holders amounts to
(β · V − C (1 − τ )) dt, which is the whole payout minus coupon payment plus tax
benefit. We note that β · V − C (1 − τ ) can have a negative sign which means that
the firm’s payout β · V is not sufficient for the after-tax coupon payment. Then,
the equity holders must satisfy the residual claim by an extra payment from their
private wealth or declare a default.2
In the special case without any net payout, β = 0, and non-convertible
debt, γ = 0, our model is consistent with that presented by Leland (1994).
Sirbu/Pikovsky/Shreve (2002) also consider a model to pricing convertible consol
bonds. The difference to our framework is that they consider a different payoff structure and an exogenous default barrier. To achieve our goal to analyze the effect of
a conversion right on the default barrier, a model like ours with an endogenous
default barrier is required rather than assuming an exogenous default barrier. A
further advantage of this approach is that we obtain analytical solutions for given
default and conversion strategies which are convenient for a further analysis.
2
In principle an equity issue would be also possible to satisfy the coupon payment. However,
the issuance of new stocks sometimes affects the conversion ratio of the convertible debt via anti
dilution clauses in a complex way. Since it is not necessary to present a discussion of different
sophisticated anti dilution clauses to see the main factors for the optimal debt service, we think of
the case of an extra payment of the equity holders, when the firm’s payoffs do not suffice for the
coupon payment, to deal with a constant conversion ratio γ.
5
2.2
Pricing of Equity and Convertible Debt
The values of equity S (V ) and convertible debt D (V ) arise from the following
two well-known differential equations as long as no prior conversion or default has
occurred yet
1 2
σ · SV,V + (r − β) · V · SV − r · S + β · V − C · (1 − τ ) = 0,
2
1 2
σ · DV,V + (r − β) · V · DV − r · D + C = 0.
2
r denotes the time-constant interest rate p.a.
for all maturities.
Other ap-
proaches such as those presented by e.g. Kim/Ramaswany/Sundaresan (1993) and
Longstaff/Schwartz (1995) introduce stochastic interest rates as a second source of
uncertainty. However, this extension does not provide further insights for our goal to
analyze the default strategy of convertible debt, but would substantially complicate
the further derivations.
We denote the two critical firm values at which the firm goes bankrupt and at which
a conversion occurs by VB and VC . Since under the optimal default strategy by the
firm and under the conversion strategy by the debt holders, both the equity value
S (V ) and the debt value D (V ) are continuous in the asset value V , we can write
the following four boundary conditions for the values of the securities
lim S (V ) = 0,
V →VB
lim D (V ) = (1 − α) · VB ,
V →VB
lim S (V ) = (1 − γ) · VC ,
V →VC
lim D (V ) = γ · VC .
V →VC
The boundary conditions in the case of default (V → VB ) reflect that debt holders
get the whole liquidation proceeds and equity holders give up any rights or obligations.3 After a conversion the firm is not levered anymore such that the firm
value equals the asset value V . Hence, the conversion value of the debt holders is
γ · V while the claim of the equity holders comprises of the remaining firm value
(1 − γ) · V .
3
This liquidation rule is consistent with those typically used in the literature (see e.g. Leland
(1994)). Under practical bankruptcy rules, however, there can be a positive liquidation value for
the equity holders in the case that the liquidation proceeds exceed the notional value of the consol
bond. This possibility can be incorporated in the boundary conditions in a straightforward way.
As this extension would result in several different cases without providing further insights about
default barriers of convertible debt, we abstain from this feasible extension in what follows.
6
These two differential equations together with the boundary conditions allow us to
write the values of equity S (V, VB , VC ) and debt D (V, VB , VC ) for arbitrary strategies
VB and VC . The unique solutions for S (V, VB , VC ) and D (V, VB , VC ), as long as no
default or conversion has occurred yet, are given by
µ
¶
C
C
S (V, VB , VC ) = V − (1 − τ ) + P B (V, VB , VC ) · −VB + (1 − τ )
r
r
µ
¶
C
+ P C (V, VB , VC ) · −γ · VC + (1 − τ ) ,
r
¶
µ
C
C
D (V, VB , VC ) = + P B (V, VB , VC ) · (1 − α) · VB −
r
r
µ
¶
C
+ P C (V, VB , VC ) · γ · VC −
,
r
where
P B (V, VB , VC ) =
µ
µ
V
VB
¶Y
V X − VCX
,
VBX − VCX
¶Y X
V
V − VBX
,
P C (V, VB , VC ) =
VC
VCX − VBX
q
4 (r − β)2 + 4 (r + β) σ 2 + σ 4
X=
,
σ2
1−X
r−β
Y =
.
−
2
σ2
One can easily verify these solutions for S (V, VB , VC ) and D (V, VB , VC ) by inserting
them into the differential equations and checking the boundary conditions.
Since the equity value S (V, VB , VC ) is negative for V below VB , we see that the firm
can never be alive for V < VB such that the optimal default strategy is to stop
paying coupons when the asset value intersects VB from above. On the contrary, if
no conversion has occurred despite of V > VC , the debt value D (V, VB , VC ) is lower
than the conversion value γ · V . Therefore, the optimal conversion strategy for the
debt holders is to convert when V hits VC from below.
2.3
Optimal Default and Conversion Strategies
The optimal barriers VB∗ and VC∗ for default and conversion result from a Nash
equilibrium of a game between the equity and debt holders. The equity holders
choose the default strategy VB in order to maximize the equity value. The debt
holders select a conversion strategy VC with regard to the debt value. A Nash
equilibrium is characterized by the property that, given the optimal strategy of the
7
debt holders, VC∗ , the equity holders have no incentive to deviate from their strategy
VB∗ and vice versa. Formally, we obtain the size of VB∗ and VC∗ by applying the two
smooth-pasting conditions for the equity and debt value
¯
∂S (V, VB∗ , VC∗ ) ¯¯
= 0,
¯
∂V
V =VB∗
¯
∂D (V, VB∗ , VC∗ ) ¯¯
= γ.
¯
∂V
∗
V =VC
The first condition means that the equity holders trigger a default when the change
of the equity value in the asset value V of an alive firm coincides with that of a
defaulted firm which is zero. In addition, the equity holders account for the optimal
behavior VC∗ of the convertible debt holders. According to the second condition, the
debt holders convert when the increase of their holdings with V is as high as the
increase with the conversion value γ ·V . We note that this solution is not necessarily
pareto optimal, because a cooperation of both equity and debt holders can result
in a higher firm value which might be advantageous for both of these two groups.
Then, one group, however, could be better off by deviating from the cooperating
strategy.
The required values VB∗ and VC∗ can be numerically obtained from these equations.
At this point, we emphasize that an equilibrium solution with
VC∗ > VB∗
always exists. The existence of a strategy and the validity of this relation follows
from properties presented in the next section.
If and only if the payout rate β equals zero, a conversion
cannot be optimal for the
¯
∂D (V,VB∗ ,VC∗ ) ¯
¯
debt holders. Formally, this is because
is below γ for every VC∗ but
∂V
¯
V =VC∗
it converges to γ for
VC∗
→ ∞. Thus, we must evaluate the limit V¯ C∗ → ∞ when
∂D (V,VB∗ ,VC∗ ) ¯
¯
β is zero. Conversely, when β is positive, the derivative
is higher
∂V
¯
∗
V =VC
than γ for a sufficiently high value of VC∗ such that the optimal conversion barrier
VC∗ must be finite. This outcome is consistent with optimal exercise strategies of
American call options on stock. When the stock pays a proportional dividend, a
premature exercise is optimal for sufficiently high stock values, where in the absence
of dividends a premature exercise is not advantageous. We note that even though
both the convertible debt and an American call on a non-dividend-paying stock
with infinite lifetime are not exercised, the call value and the conversion right have
a strictly positive value.
8
As a result, we can write the value of equity S (V ) = S (V, VB∗ , VC∗ ) and debt D (V ) =
D (V, VB∗ , VC∗ ) as follows:


0,



 V − C (1 − τ ) + P B (V, V ∗ , V ∗ ) · ¡−V ∗ + C (1 − τ )¢
r
¡B C ∗ C B r ¢
S (V ) =
∗
∗

+P C (V, VB , VC ) · −γ · VC + r (1 − τ ) ,




(1 − γ) · V,
D (V ) =











C
r
(1 − α) · V,
¡
¢
∗
+ P B (V, VB , VC∗ ) · (1 − α) · VB∗ − Cr
¢
¡
+P C (V, VB∗ , VC∗ ) · γ · VC∗ − Cr ,
V ≤ VB∗
VB∗ < V < VC∗
V ≥ VC∗
(1)
V ≤ VB∗
VB∗ < V < VC∗
γ · V,
(2)
V ≥ VC∗
In the case of β = 0, the conversion barrier is not finite and the evaluation of the
limit VC∗ → ∞ results in a simpler representation:


0,
V ≤ VB∗
³
´
´
³
Y
S (V ) =
¢
¡
)
 V − C(1−τ ) + V∗
· −VB∗ + C(1−τ
− γ · V Y V X − VB∗ , VB∗ < V
r
VB
r


(1 − α) · V,
V ≤ VB∗
³ ´Y ¡
D (V ) =
¢
¡
¢
 C + V∗
· (1 − α) · VB∗ − Cr + γ · V Y V X − VB∗ , VB∗ < V
r
V
B
Given that the firm is solvent and no conversion has occurred yet, VB∗ < V < VC∗ , we
can interpret P B (V, VB∗ , VC∗ ) as the current value of binary up-and-out put that pays
one unit at a future point in time when the asset value hits the default barrier V B∗
without being in the conversion region, V ≥ VC∗ , before. Otherwise, this claim pays
nothing. Accordingly, P C (V, VB∗ , VC∗ ) can be seen as the present value of a binary
down-and-out call that pays one unit only if V hits VC∗ and no default has occurred
yet. Using this view, we see that the convertible debt value consists of the value of
a default-free consol bond C/r plus a claim that swaps this consol bond into the
liquidation value at the default barrier plus a further claim that swaps the consol
bond into the conversion value at the conversion barrier. Accordingly, the equity
value S (V ) comprises of the asset value of the firm minus the value of after-tax
coupon payments of a default-free consol bond plus a claim that swaps the asset
value into the value of the default-free (after-tax coupon) consol bond at the default
barrier and a further claim that swaps the conversion value of the convertible debt
into the after-tax consol bond at the conversion barrier.
We note that the firm value v (V ) consisting of the sum of the values of equity S (V )
and debt D (V ) generally deviates from the asset value V . For the firm value v (V ),
9
we obtain:
v (V ) = S (V ) + D (V )
= V −P B (V, VB∗ , VC∗ ) · α · VB∗
|
{z
}
present value of bankruptcy costs
C
+ τ (1 − (P B (V, VB∗ , VC∗ ) + P C (V, VB∗ , VC∗ ))).
{z
}
|r
present value of tax shield
In line with other models considering tax deductability and bankruptcy costs, the
firm value v (V ) is the value of the firm’s assets minus the present value of bankruptcy
costs plus the present value of the tax shield.
Representations (1) and (2) indicate that for a given default and conversion strategy
(VB∗ , VC∗ ) the relative bankruptcy costs α are only at the costs of the debt value, where
the tax benefits only contribute to the equity value S (V ). Since the tax shield is
primarily in favor of the equity holders rather than the debt holders, it is not clear
whether the equity holders can benefit from the tax deductability in a such severe
way that the value of one stock is higher than the value of a number of convertible
bonds that allows to receive one stock in total. At first glance, we would expect
the value of one stock to be lower than the corresponding number of convertible
bonds, because the convertible debt holders have the possibility to convert anytime.
To answer this question, we compare the debt value, which is convertible into a
fraction of γ of the asset value, with a fraction of
γ
1−γ
of the equity value, which
also represents a fraction of the asset value equal to γ after a conversion. Thus, the
equity proportion of D (V ) corresponds to that of
γ
S
1−γ
(V ). The difference of these
two values amounts to:
D (V ) −









(1 − α) · V,
V ≤ VB∗
P B (V, VB∗ , VC∗ ) · VB∗ · (1 − α)
γ
γ
+ (P C (V, VB∗ , VC∗ ) · VC∗ − V ) 1−γ
S (V ) =

1−γ

C·(1−(P B (V,VB∗ ,VC∗ )+P C (V,VB∗ ,VC∗ ))) 1−γτ


+

r
1−γ



0,
VB∗ < V < VC∗
V ≥ VC∗
γ
In fact the difference D (V )− 1−γ
S (V ) can become negative, for example if α = 1 and
τ is close to one which means that the debt holders have to carry high bankruptcy
costs, while equity holders enormously benefit from the tax shield. Then, for V B∗ <
γ
is always negative and can have a
V < VC∗ , the term (P C (V, VB∗ , VC∗ ) · VC∗ − V ) 1−γ
bigger size than the positive term
C
r
(1 − (P B (V, VB∗ , VC∗ ) + P C (V, VB∗ , VC∗ ))) 1−γτ
.
1−γ
10
This finding is a fundamental difference to the standard framework without frictions
such as taxes and bankruptcy costs, in which convertible bonds are typically examined. The value of a convertible bond in a frictionless market must exceed the value
of stocks times the conversion ratio, as long as the firm has no additional claims
outstanding.4
2.4
Values of Investment and Option Component
To analyze and price convertible bonds, several approaches decompose the convertible debt value D (V ) into a straight bond and an option component.5 The bond
component, which is typically denoted by the investment value of the convertible
bond, is equal to the value of the convertible debt if the debt holders do not make
use of the conversion option, but the firm still follows its default strategy V B∗ .6 Thus,
the investment value IV (V ) of the convertible debt is given by
(
(1 − α) · V,
V ≤ VB∗
¡
¢
IV (V ) =
limVC →∞ Cr + P B (V, VB∗ , VC ) · (1 − α) · VB∗ − Cr
VB∗ < V


(1 − α) · V,
V ≤ VB∗
´
³
Y ¡
=
.
¢
C
∗
∗
 C + V∗
·
(1
−
α)
·
V
−
V
<
V
B
B
r
V
r
(3)
B
The idea behind this decomposition is to price the investment value by regarding an
appropriate benchmark firm with identical assets and coupon obligations but with
straight rather than convertible debt outstanding. Thus, the investment value is
obtained by standard procedures for straight debt due to this practical approach.
Then, the additional option component can be separately priced. We will analyze
how severely the investment value depends on the fact whether the firm follows the
optimal default strategy for convertible debt or the firm follows the optimal strategy
for straight debt which is the strategy of the benchmark firm. The first strategy will
be denoted by optimal strategy while the other strategy, that ignores the conversion
feature, is the straight debt strategy.
Accordingly, the residual amount D (V ) − IV (V ) is the option value OV (V ), which
4
See e.g. Bühler/Koziol (2002).
See e.g. Ingersoll (1977), Tsiveriotis/Fernandes (1998), Takahashi/Kobayashi/Nakagawa
(2001), and Hung/Wang (2002).
6
Of course, if the firm was aware of the fact that no conversion took place, it could increase the
equity value by adjusting the default level.
5
11
amounts to


0, ¶

µ

´Y
³

¢
¡



P B (V, VB∗ , VC∗ ) − VV∗
· (1 − α) · VB∗ − Cr
B
OV (V ) =
¡
¢
∗


+P C (V, VB , VC∗ ) · γ · VC∗ − Cr ,


³ ´Y ¡

¢


γ · V − Cr − VV∗
· (1 − α) · VB∗ − Cr ,
V ≤ VB∗
VB∗ < V < VC∗ .
B
V ≥ VC∗
Figure 1: Values of Debt and Equity
The diagram shows the value of convertible debt D (V ) and equity S (V ) as a function of
the asset value V . Moreover, it plots the debt components, the investment value IV (V )
and the option value OV (V ). The parameter values are C = 5, γ = 0.3, α = 0.5, β = 0.04,
σ = 0.5, τ = 0.3, and r = 0.05. The critical asset values are VB∗ = 19.16 and VC∗ = 1184.97.
250
S(V)
D(V), S(V),
IV(V), OV(V)
D(V)
OV(V)
200
150
100
IV(V)
50
0
200
400
600
800
1000 1200
V
Figure 1 shows the values of equity S (V ), convertible debt D (V ), and the debt components IV (V ) and OV (V ) as a function of the asset value for a typical example.
We can see that the equity value is zero in the case of default, but for higher asset
values it strictly increases first with a recognizable convex shape and it is almost
a linear function afterwards. The convertible debt value linearly increases with V
first because it equals the liquidation value. At V = VB∗ , D (V ) has a kink and it
further increases with a concave shape. After a turning point, D (V ) is a convex
function which tends to the conversion value γ · V . The investment value of the convertible debt IV (V ) coincides with D (V ) in the case of default. Then, it increases
less sharply than D (V ) such that IV (V ) is a concave function that approaches
the value of a non-defaultable bond C/r. The option value of the convertible debt
OV (V ) is a strictly convex and increasing function of V when the firm is alive and
zero otherwise.
12
The functional behavior of these asset values is similar to the typical cases presented
e.g. in Brennan/Schwartz (1977). For low asset values the convertible debt is
primarily characterized by its investment value because it behaves like straight debt,
whereas for high asset values the convertible debt rather behaves like a proportion
of equity. Moreover, we can see that the curvature of the option value is more
pronounced than that of equity which is a typical property of option values.
Since the goal of this paper is to analyze the effect from the conversion feature on
the default strategy VB∗ , we need the strategy VB0 , if the debt was not convertible, as
the reference case (straight debt strategy). In the case of straight debt, the debt value
³ ´Y ¡
¢
C
0
0
under the strategy VB for V ≥ VB is r + VV0
· (1 − α) · VB0 − Cr which directly
B
follows from the representation of the investment value. Accordingly,
the equity
µ
³ ´Y ¶
C
value amounts from the difference between the firm value V + 1 − VV0
τ−
r
B
³ ´Y
³ ´Y ¡
¢
C
C
V
V
0
0
·
α
·
V
and
the
debt
value
·
(1
−
α)
·
V
−
. Evaluating the
+
0
0
B
B
V
r
V
r
B
B
smooth-pasting condition of the equity holders, we obtain the following representation for the optimal default strategy of non-convertible debt
VB0 =
2 (r − β) σ 2 − σ 4 + X
C
(1 − τ )
.
r
2 (r − β) σ 2 + σ 4 + X
0
Now, we regard the consequences if the firm follows the strategy VB for nonconvertible debt, but debt is convertible.
The corresponding optimal strategy
VC0
of the convertible
debt holders
follows from the smooth-pasting condition
¯
0
0
¯
∂D (V,VB ,VC )
¯
= γ. Having determined VB0 and VC0 once, the investment IV 0 (V )
∂V
¯
0
V =VC
and the option value OV 0 (V ) under this different strategy is


(1 − α) · V,
V ≤ VB0
0
³
´
Y ¡
IV (V ) =
,
¢
C
0
0
 C + V0
·
(1
−
α)
·
V
−
V
<
V
B
B
r
VB
r


(1 − α) ¶
· V,

µ

³ ´Y

¡
¢



· (1 − α) · VB0 − Cr
P B (V, VB0 , VC0 ) − VV0
B
OV 0 (V ) =
¢
¡
0


+P C (V, VB , VC0 ) · γ · VC0 − Cr ,


³ ´Y ¡

¢


γ · V − Cr − VV0
· (1 − α) · VB0 − Cr ,
B
V ≤ VB0
VB0 < V < VC0
V ≥ VC0
As before, if β = 0 holds, we have to evaluate the limit VC → ∞, since it is always
better to wait with a conversion.
13
3
Analysis of optimal Debt Service
In this section, we compare the optimal default strategy of the firm VB∗ with that
of an identical firm with straight debt VB0 , i.e. the conversion feature is not taken
into account. Similarly, we consider the conversion strategies VC∗ and VC0 . Then,
we regard the consequences for the debt value, the investment value and the option
value when the firm follows the optimal strategy rather than the optimal strategy
for non-convertible debt.
As a first property of the default and conversion strategy, it is important to control
whether equilibrium solutions for VB∗ and VC∗ as well as VB0 and VC0 , respectively, can
be always obtained.
Property 1 (Existence of Equilibrium Strategies) An optimal default and
conversion strategy, VB∗ and VC∗ , always exist. Also under the optimal straight debt
strategy VB0 , there is an optimal conversion strategy VC0 .
The proof is shown in Appendix A.1.
For the relation between the optimal and the straight debt strategy, we obtain the
following properties:
Property 2 (Default Barrier) The optimal default barrier VB∗ if debt is convertible is higher than VB0 for non-convertible debt. Moreover, the default barrier VB∗ is
below
C
r
· (1 − τ ).
The intuition for the first part of this assertion, which is proven in Appendix A.2, is
that the equity holders suffer from the conversion right of the debt holders because
the conversion right means an additional claim on the firm. Therefore, the incentive
of the equity holders to keep the firm alive by paying the coupon is lower when debt
is convertible. Therefore, the critical default barrier VB∗ of convertible debt is higher
than VB0 under the strategy for straight debt.
Property 3 (Conversion Barrier) The conversion barrier VC∗ when the firm optimally defaults is below the optimal conversion barrier VC0 when the firm follows the
default strategy VB0 for non-convertible debt. Moreover, the conversion barrier VC∗ is
above
C
.
r·γ
The proof of this property is in Appendix A.3. The idea behind this assertion is
that since the firm follows a suboptimal default strategy VB0 rather than VB∗ , the
14
convertible debt holders can longer benefit from it by choosing a conversion barrier
VC0 higher than VC∗ .
As a consequence of the second part of this property, we can see that VB∗ must be
below VC∗ . This is because the upper bound for the default barrier is below the lower
bound for the conversion barrier.
We can study the importance of the default strategy by regarding the consequences
of it on the debt value, the investment value, and the option value.
Property 4 (Relation between Debt Values) If the firm follows the strategy
for straight debt VB0 rather than convertible debt VB∗ , then the debt value D (V, VB0 , VC0 )
is higher than D (V, VB∗ , VC∗ ) for every asset value V .
We prove this property in Appendix A.4. At first glance, it is comprehensible
that the debt value is higher when the firm does not follow the optimal strategy.
However, the primary reason for this property is that the non-optimal strategy of
the firm results in a higher firm value v (V ) due to lower bankruptcy costs.
Figure 2: Difference between Debt Values
The left diagram shows the difference between the convertible debt value under the straight
debt and the optimal strategy D (V, VB0 , VC0 ) − D (V, VB∗ , VC∗ ) as a function of the asset value
V . The right diagram plots the difference related to the debt value D (V, V B0 , VC0 ). The
parameter values are C = 5, γ = 0.3, α = 0.5, β = 0.04, σ = 0.5, τ = 0.3, and r = 0.05.
The critical asset values are VB∗ = 19.16 and VC∗ = 1184.97 for the optimal strategy and
VB0 = 17.05 and VC0 = 1202.18 for the straight debt strategy.
0.1
2
absolute
difference D
relative
difference D
0.05
1
0
0
200
400
600 800 1000 1200
V
0
200 400 600 800 1000 1200
V
Figure 2 shows the difference between the debt value D (V, VB0 , VC0 ) under the straight
debt strategy and the debt value D (V, VB∗ , VC∗ ) under the optimal strategy. If the
asset value is low such that in both cases a default occurs, the debt values under both
15
strategies coincide. Then, the difference increases with the asset value V if only a
default under the optimal but not under the straight debt strategy is optimal. When
no default occurs under both strategies the difference D (V, VB0 , VC0 ) − D (V, VB∗ , VC∗ )
declines with V because the default strategy is less relevant the higher the asset
value V . For high asset values under which a conversion is optimal, both debt
values coincide again and therefore D (V, VB0 , VC0 ) − D (V, VB∗ , VC∗ ) is zero.
The difference as a percentage of the debt value D (V, VB0 , VC0 ) has a similar structure.
We can see that the difference can lie above ten percent and it has a considerable
magnitude over a broad range of asset values. Especially for low asset values, when
a default is relatively probable, the difference is important, but it is less striking for
asset values close to a conversion.
Although the value of debt is higher when the firm follows the straight debt
strategy VB0 , the investment value IV (V, VB∗ , VC∗ ) as well as the option value
OV (V, VB∗ , VC∗ ) of debt under the optimal strategy can be higher than IV (V, VB0 , VC0 )
and OV (V, VB0 , VC0 ), respectively.
Property 5 (Relation between Investment Values) The
investment
value
under the straight debt strategy IV (V, VB0 , VC0 ) is higher than under the optimal
strategy IV (V, VB∗ , VC∗ ) for all V if IV (VB∗ , VB0 , VC0 ) > (1 − α) · VB∗ . Otherwise,
IV (V, VB∗ , VC∗ ) is higher than IV (V, VB0 , VC0 ) for V > VB∗ , but it is lower for firm
values V slightly above VB0 .
This proof is shown in Appendix A.5. Intuitively, we would expect that the investment value IV (V, VB0 , VC0 ) under the suboptimal strategy exceeds the value
IV (V, VB∗ , VC∗ ) under the optimal strategy as it is true for the convertible debt value.
This relation nevertheless holds for the investment value for all V , as long as the
liquidation value under the optimal strategy is not too high. However, if the proceeds from a liquidation at the barrier under the optimal strategy VB∗ · (1 − α) are
very high and lie even above the investment value IV (VB∗ , VB0 , VC0 ) for this firm value
when the firm follows the suboptimal strategy, we see that a liquidation is better for
the debt holders than to keep the firm alive. This is the reason why the investment
value under the optimal strategy can also exceed the value under the strategy for
straight debt.
Figure 3 shows the difference between the investment values under the straight
debt strategy and the optimal strategy. In this case, the important determinant
IV (VB∗ , VB0 , VC0 ) − (1 − α) · VB∗ is positive. For low asset values, IV (V, VB0 , VC0 ) −
IV (V, VB∗ , VC∗ ) behaves like the difference of the debt values presented in Figure 2.
16
Figure 3: Difference between Investment Values
The left diagram shows the difference between the investment value under the straight debt
and the optimal strategy IV (V, VB0 , VC0 ) − IV (V, VB∗ , VC∗ ) as a function of the asset value V .
The right diagram plots the difference related to the investment value IV (V, V B0 , VC0 ). The
parameter values are C = 5, γ = 0.3, α = 0.5, β = 0.04, σ = 0.5, τ = 0.3, and r = 0.05.
The critical asset values are VB∗ = 19.16 and VC∗ = 1184.97 for the optimal strategy and
VB0 = 17.05 and VC0 = 1202.18 for the straight debt strategy.
0.1
2
relative
difference IV
absolute
difference IV
1
0
0.05
0
200 400 600 800 1000 1200
V
0
200 400 600 800 1000 1200
V
For higher asset values under which no default occurs, the difference decreases with
V and tends to zero, as both investment values tend to the value C/r of the nondefaultable consol bond. A comparison of the differences of the debt values and the
investment values, however, reveals that the difference of the investment value is
much more affected by the default strategy than the debt value itself. Even for high
asset values under which a conversion is optimal, the difference is considerable. The
reason why the investment value is much more affected by the default strategy than
the convertible debt value for high asset values is that the convertible debt rather
exhibits an equity character for high values of V which is almost independent of the
default strategy.
Figure 4 displays the difference IV (V, VB0 , VC0 ) − IV (V, VB∗ , VC∗ ) in the case in which
the important determinant IV (VB∗ , VB0 , VC0 ) − (1 − α) · VB∗ has the opposite sign than
in Figure 3. To have a liquidation value (1 − α) · VB∗ higher than the investment
value IV (VB∗ , VB0 , VC0 ) at VB∗ , we need low relative bankruptcy costs α and a high
value of VB∗ relative to VB0 . For this purpose, we set α = 0 and consider an especially
high equity proportion of the debt value γ = 0.75. In addition, we lower β to 0.01
which also results in a higher difference between VB∗ and VB0 .
Figure 4 shows that in the case of default under the straight debt strategy but not
under the optimal strategy, the difference IV (V, VB0 , VC0 ) − IV (V, VB∗ , VC∗ ) is first
17
Figure 4: Difference between Investment Values
The left diagram shows the difference between the investment value under the straight debt
and the optimal strategy IV (V, VB0 , VC0 ) − IV (V, VB∗ , VC∗ ) as a function of the asset value V .
The right diagram plots the difference related to the investment value IV (V, V B0 , VC0 ). The
parameter values are C = 5, γ = 0.75, α = 0, β = 0.01, σ = 0.5, τ = 0.3, and r = 0.05.
The critical asset values are VB∗ = 48.90 and VC∗ = 1756.61 for the optimal strategy and
VB0 = 19.20 and VC0 = 1900.70 for the straight debt strategy.
2
0.05
absolute 0
difference IV
-2
relative
0
difference IV
-0.05
-4
-0.1
0
500
1000
V
1500
2000
0
500
1000
V
1500
positive but tends to a negative local minimum afterwards. The reason is that
the liquidation value (1 − α) · V is such high that it even exceeds the investment
value IV (V, VB0 , VC0 ). Then, if no default occurs, the difference IV (V, VB0 , VC0 ) −
IV (V, VB∗ , VC∗ ) remains negative and tends to zero with the asset value. Therefore,
the fact that the firm serves the debt too long can result in a lower investment value
of the convertible debt. The difference in this case is considerable and lies between
two and twelve percent.
Now, we reconsider the practical approach to pricing the investment value of a
convertible bond by valuing the debt like straight debt of an otherwise identical
firm. According to this approach, the used investment value is IV (V, VB0 , VC0 ) rather
than the correct one IV (V, VB∗ , VC∗ ). However, the optimal default strategy of a firm
is fundamentally affected by the existence of a conversion feature. As seen in the
considered examples the magnitude of the difference between the real investment
value IV (V, VB∗ , VC∗ ) and the investment value IV (V, VB0 , VC0 ) from the practical
approach is considerable and can be even much higher than the difference between
the debt values itself.
Property 6 (Relation between Option Values) The option value for low asset
values under the straight debt strategy OV (V, VB0 , VC0 ) is higher than under the optimal strategy OV (V, VB∗ , VC∗ ). For high asset values, OV (V, VB∗ , VC∗ ) is higher than
18
2000
OV (V, VB0 , VC0 ) if IV (VB∗ , VB0 , VC0 ) ≥ (1 − α) · VB∗ holds and vice versa.
The proof of the property is in Appendix A.6. When a default only under the optimal
strategy rather than the straight debt strategy occurs, the option value under the
straight debt strategy is higher because it is zero under the optimal strategy. For
high asset values, the convertible debt value is hardly affected by the choice for one
of the two default strategies. However, the value of the investment component still
depends on the default strategy even for high asset values. Since the option value
is the residual claim between the debt value and the investment value, the option
value is higher (lower) under the optimal strategy when the investment value under
this strategy is lower (higher) than under the straight debt strategy.
Figure 5: Difference between Option Values
The left diagram shows the difference between the investment value under the straight debt
and the optimal strategy OV (V, VB0 , VC0 ) − OV (V, VB∗ , VC∗ ) as a function of the asset value
V . The right diagram plots the difference related to the investment value OV (V, V B0 , VC0 ).
The parameter values are C = 5, γ = 0.3, α = 0.5, β = 0.04, σ = 0.5, τ = 0.3, and
r = 0.05. The critical asset values are VB∗ = 19.16 and VC∗ = 1184.97 for the optimal
strategy and VB0 = 17.05 and VC0 = 1202.18 for the straight debt strategy.
0.1
0.2
absolute
0
difference OV
-0.2
relative 0.05
difference OV
0
-0.4
-0.6
-0.05
0
200 400 600 800 1000 1200
V
0
200 400 600 800 1000 1200
V
Figure 5 shows the difference between the option values OV (V, VB0 , VC0 ) −
OV (V, VB∗ , VC∗ ) in the case that corresponds to Figure 3. As long as no default occurs, the option value is strictly positive. Therefore, the difference strictly increases
with V when a default only under the straight debt but not under the optimal
strategy occurs. Then, OV (V, VB0 , VC0 ) − OV (V, VB∗ , VC∗ ) declines and attains a local
minimum. Afterwards, it approaches zero. Since the convertible debt values under
both strategies are nearly equal for high asset values but the investment value under the straight debt strategy is higher, the option value under this strategy must
be lower. This effect explains why the difference OV (V, VB0 , VC0 ) − OV (V, VB∗ , VC∗ )
19
changes the sign when V is varied. Moreover, we can see that the size of the differences in percentage terms is much higher than those of the investment values
when the asset values are low. Conversely, for high asset values, the option values
deviate by less than 0.1 percent such that the default strategy is less important for
the option value in this asset value region.
Figure 6: Difference between Option Values
The left diagram shows the difference between the option value under the straight debt and
the optimal strategy OV (V, VB0 , VC0 ) − OV (V, VB∗ , VC∗ ) as a function of the asset value V .
The right diagram plots the difference related to the option value OV (V, V B0 , VC0 ). The
parameter values are C = 5, γ = 0.75, α = 0, β = 0.01, σ = 0.5, τ = 0.3, and r = 0.05.
The critical asset values are VB∗ = 48.90 and VC∗ = 1756.61 for the optimal strategy and
VB0 = 19.20 and VC0 = 1900.70 for the straight debt strategy.
20
0.2
relative
difference OV
absolute
difference OV
10
0.1
0
0
500
1000
V
1500
2000
0
500
1000
V
1500
In the case of Figure 6, which corresponds to Figure 4, the difference of option
values is always non-negative. It first increases with V until VB∗ and declines to
zero afterwards. Over a relatively broad range of asset values the difference has a
considerable size above three percent, but it is again neglectable close to conversion.
When regarding the differences between the investment and option values in Figures
3, 4, 5, and 6, we find that either the investment value under the straight debt strategy dominates the investment value under the optimal strategy and the difference
of option values changes its sign or the difference of option values is always positive
and the difference of investment values has a change of the sign.
As a consequence of the impact of the default strategy on the value of the option
component, we see that it is important to account for the real strategy not only when
computing the investment value but also the option value. This is a further point
which is not in line with the practical approach to decomposing the convertible debt
value, because it suggests that the option value can be separately priced independent
20
2000
of the default strategy.
4
Comparative Static Analysis of the optimal Default Barrier
The potential for high differences between the convertible debt values, between the
investment values, and between the option values under the optimal and straight
debt strategy primarily depends on the difference between the optimal VB∗ and the
straight debt strategy VB0 . In general, the differences of the asset values are more
pronounced, the higher the difference of the strategies VB∗ − VB0 . Therefore, we
provide a detailed comparison of the optimal default strategy VB∗ with the straight
debt strategy VB0 in this section. For this purpose, we do a comparative static
analysis of the difference VB∗ − VB0 . The parameters, which are varied, are the equity
proportion γ, the size of the coupon C, the tax rate τ , the payout rate β, the
liquidation costs α, the interest rate r, and the volatility σ of the return of the asset
value V . The graphical results for a typical example are presented in Figure 7.
Equity Proportion γ
In this figure, we can see that the optimal default barrier VB∗ increases with γ.
Clearly, the straight debt strategy VB0 is not affected by γ as the straight debt value
does not depend on the conversion feature. Thus, the difference, VB∗ − VB0 , increases
with γ. The reason for this behavior is that the equity holders let the debt holders
to participate more in the firm value the higher γ. Since this higher participation
is at their own costs, the incentive for the equity holders to save the firm from a
default declines with γ which is indicated by a VB∗ increasing with γ.
Coupon C
A higher coupon results in increasing default barriers VB∗ and VB0 because it is more
costly for the equity holders to save the firm. Therefore, the incentive to default rises
with C. Moreover, the important difference VB∗ − VB0 between these two strategies
also increases with C. In general, the difference is more pronounced, the more
valuable the conversion right for V slightly above VB∗ is. Since the value of this right
increases with V such as the default boundaries VB∗ and VB0 , we can see why the
difference VB∗ also increases with C more sharply than VB0 .
Tax Rate τ
The higher the tax rate the lower the default barriers VB∗ and VB0 and also the lower
21
Figure 7: Variation of Default Barriers
These diagrams show the default barrier VB∗ under the optimal strategy and VB0 under the
straight debt strategy as a function of the equity proportion γ, the coupon C, the tax rate τ ,
the relative liquidation costs α, the payout rate β, the interest rate r, and the volatility σ of
the return of the asset value. The standard parameter values are C = 5, γ = 0.75, α = 0,
β = 0.01, σ = 0.5, τ = 0.3, and r = 0.05.
VB*,
VB' 60
VB*, 28
VB'
40
24
VB*
VB*
VB'
20
20
VB'
0
0.2
0.4
0.6
γ
0
0.8
5
0
10
C
15
20
20
25
VB*,
VB' 20
10
VB'
10
5
5
0
0.2
0.4
τ
0.6
0.8
1
0
VB*
20
0.4
α
0.6
0.8
10
VB'
15
VB*
5
VB'
0
0
0.04
β
0.08
0.12
40
20
VB*
VB'
0
0.2
0.4
σ
0.6
0.8
0
0.2
0.4
0.6
r
60
0
0.2
20
VB*,
VB' 15
VB*,
25
VB'
VB*,
VB'
VB'
VB*
15
0
VB*
VB*,
VB' 15
1
22
0.8
1
the difference VB∗ − VB0 . The reason for these relations is that the tax rate has an
opposite effect on the default barriers as the coupon. This is because we can think
of a higher tax rate τ as a higher deductability of the interest rate payments as the
net payment by the equity holders is not the full coupon payment but the after tax
payment C · (1 − τ ). Thus, the net payment declines with τ . Since the difference
VB∗ − VB0 and the barriers, VB∗ and VB0 , decline with a lower C they must also decline
with a higher τ .
Costs for Liquidation α
Since the relative liquidation costs α do not affect the wealth of the equity holders,
the straight debt strategy VB0 is invariant of α. However, the convertible debt holders
follow a conversion strategy that depends on α such that also the optimal default
barrier VB∗ is affected by the liquidation costs. However, this effect is rather marginal
that a dependency cannot be recognized in Figure 7.
Payout Rate β
The default barriers VB∗ and VB0 decrease with the payout rate β. This effect stems
from a higher dividend payment to the equity holders when β increases. The value
of equity typically increases with the size of the dividend payment at the costs of
the debt holders and therefore we can understand why the incentive of the equity
holders to prevent a default by making coupon payments increases with β. In other
words, the higher β, the higher the fraction of the asset value that is exclusively for
the equity holders, while otherwise the asset value is shared with the debt holders.
Moreover, the difference VB∗ − VB0 also decreases with β. This is due to the fact that
the conversion right becomes less interesting for the debt holders when β is high.
The reason for this effect is that the growth of the firm’s asset value is mitigated
by higher payoffs β such that the debt holders suffer from a lower future conversion
value γ · V . As a consequence, the value of the conversion right declines with β such
that VB∗ approaches VB0 with β.
Interest Rate r
When the interest rate r rises, the default barriers VB∗ and VB0 tend to zero. This is
due to the fact that the present value of promised coupon payments C/r declines
and tends to zero if r increases. Thus, it is cheaper for the equity holders to fulfil
their debt obligation when r is high such that their survival incentive goes up. The
difference VB∗ and VB0 , however, first increases with r and then approaches zero as
Figure 7 shows. This is a result of two counter-effects. The first effect is that the
conversion probability (under the risk-neutral measure) increases with r such that
23
the conversion right rises with r. On the other hand, both barriers VB∗ and VB0 tend
to zero with r such that the difference VB∗ − VB0 must decline for high r.
Volatility σ
In principle, the variation of the default barriers, VB∗ and VB0 , and their difference
VB∗ − VB0 with the volatility of the return of the asset value V is comparable with
a variation of r as seen before. Since the equity value typically benefits from a
higher volatility at the costs of the debt holders, the default barriers decline with σ
and both approach zero when σ tends to infinity. This effect also explains why the
difference VB∗ − VB0 tends to zero for sufficiently large volatilities σ. As the present
value of the conversion right, which the equity holders are short, increases with the
volatility, we see why the difference VB∗ − VB0 increases first before it goes to zero.
5
Conclusion
This paper rests on the observation that a broad literature discusses optimal default strategies of straight bonds, but on real markets debt is often equipped with
additional rights such as a conversion feature. Therefore, the goal of this paper is
to derive and to analyze the optimal default strategy of convertible debt. For this
purpose, we use a time-independent model framework, similar to that presented by
Leland (1994), with bankruptcy costs and taxes.
This model reveals how the default strategy of convertible debt is affected by the
conversion feature. We can see why an optimal default of convertible debt occurs
earlier than a default of non-convertible debt. Moreover, the value of convertible
debt is higher when the firm follows the optimal straight debt strategy, but the
values of the investment and option component can be lower. As a result, the
optimal default strategy is a relevant pricing factor for convertible debt. Hence,
the practical approach to pricing the investment value of a convertible bond, by
regarding the value of straight debt of an otherwise identical firm, can result in
severe inaccuracies of the investment value. Accordingly, the debt and option value
are also impacted by this problematical approach especially when a default is likely.
The differences of the asset values primarily arise from the difference between the
optimal default strategy and the straight debt strategy. As a comparative static
analysis shows, the difference between the two default barriers VB∗ − VB0 rises with
the conversion ratio γ, the promised coupon C, a lower tax rate τ , and a lower
payoff rate β. These are testable implications which can be empirically evaluated in
24
another paper.
In addition, we can extract further insights of this paper when using the model
framework to analyze the default strategy of firms whose debt is equipped with
other non-straight features. Analogously, we can show that the default strategy of
a firm with debt that is attached with warrants is in principle similar to that of
convertible debt. Conversely, if debt is callable by the firm, we can argue that a
default occurs later due to the call feature. The intuition is that the equity holders
have an additional call right when they save the firm by providing the coupon
payments which provides them with a higher incentive to fulfil the coupon obligation
relative to the case of non-callable debt.
A
A.1
Proofs of the Properties
Proof of Property 1
For every arbitrary strategy of the equity holders VB ≤
C
,
r
an optimal strategy of
VC∗
the convertible
debt holders
exists which satisfies the smooth-pasting condition
¯
∗
¯
∂D (V,VB ,VC )
¯
= γ. If β = 0, the optimal strategy is infinite as discussed above.
∂V
¯
V =VC∗
¯
∂D (V,VB ,VC∗ ) ¯
¯
Otherwise, a finite solution exists. This is because the derivative
∂V
¯
V =VC∗
is negative for
VC∗
close to VB and it is above γ for
VC∗
→ ∞. As a consequence of the
VC∗
intermediate
above VB exists for the condition
¯ value theorem, a finite solution
∂D (V,VB ,VC∗ ) ¯
¯
= γ. Since VB0 is below Cr , we can see that under the straight debt
∂V
¯
∗
V =VC
strategy VB = VB0 an equilibrium strategy VC0 always exists.
We will use the finding that an optimal conversion barrier VC∗ (VB ) exists, given that
C
,
r
to prove the existence of the optimal
strategy.
¯
∗
∂S (V,VB ,VC (VB )) ¯
¯
has the
The important term of the smooth-pasting condition
∂V
¯
V =V¯B
∂S (V,VB ,VC∗ (VB )) ¯
¯
opposite sign at VB = VB0 than for VB = Cr (1 − τ ). Since
is a
∂V
¯
V
=V
B
¯
∂S (V,VB ,VC∗ (VB )) ¯
¯
continuous function in VB , the smooth-pasting condition
=0
∂V
¯
the firm follows the strategy VB ≤
V =VB
must be satisfied for a VB between VB0 and
VC∗ (VB ) is the Nash equilibrium strategy.
25
C
r
(1 − τ ). Then, this point VB and
A.2
Proof of Property 2
We formally show VB∗ ≥ VB0 by arguing that a firm with non-convertible debt would
not choose a default barrier VB0 above the optimal barrier VB∗ for convertible debt.
This is because the equity value of the firm with straight debt exceeds the equity
value of a firm with convertible debt. This relation even holds when the firm with
straight debt follows the optimal strategy of the firm with convertible debt. Due to
the higher equity value, the incentive for a default is lower and VB0 is below VB∗ .
To see the fact that the equity value of a firm with non-convertible debt is higher
than that of a firm with convertible debt, we compare the equity value S (V, V B∗ , VC∗ )
of the firm with convertible debt with the equity value of a firm with non-convertible
debt that also follows the default strategy VB∗ . Since the representations of these two
equity values have different structures, we use a lower bound for the equity value of
the firm with non-convertible debt. The equity value of a firm with straight debt
under the optimal straight debt strategy is higher than the equity value if the firm
stops following an optimal default strategy when V hits VC∗ before VB∗ . In the case
that V hits VC∗ before VB∗ , the default option is given up and the firm always pays
the coupon even for negative equity values. The value of this lower bound for the
equity value given V > VB∗ is
µ
¶
C
C
∗
∗
∗
V − (1 − τ ) + P B (V, VB , VC ) · −VB + (1 − τ )
r
r
which is the sum of the asset value minus after-tax coupon payments plus the loss
of the value of the assets (long) and after-tax coupon payments (short) when V
hits VB∗ before VC∗ . The advantage of this lower bound for the equity value in the
straight debt case is that it can be easily compared with S (V, VB∗ , VC∗ ). The difference
between the lower bound and S (V, VB∗ , VC∗ ) results in
µ
¶
C
∗
∗
∗
−P C (V, VB , VC ) · −γ · VC + (1 − τ ) > 0.
r
The positive sign is a consequence of −γ · VC∗ +
3. Therefore, we can conclude that
VB0
C
r
< 0 which follows from Property
cannot lie above VB∗ because a lower default
barrier VB0 ≤ VB∗ would result in higher equity values for V ∈ (VB∗ , VB0 ). Since the
choice of the optimal strategy is independent on V in this time-independent model,
the equity values for all V are higher for VB0 ≤ VB∗ rather than VB0 > VB∗ .
The second assertion that VB∗ ≤
C
r
(1 − τ ) is valid is intuitively understandable if the
debt value was not convertible. Then the simple strategy always to pay the coupon,
VB0 = 0, results in a firm value equal to the sum of the asset value V and the present
26
value of the tax shield
C
τ,
r
but no bankruptcy costs. Since the equity value is the
firm value minus the value of the non-defaultable debt, it equals V + Cr τ − Cr and is
therefore always positive for V > (1 − τ ) Cr .
If debt is convertible, the default strategy VB =
C
r
(1 − τ ) simplifies the equity value
to
µ
C
S V, (1 − τ ) , VC
r
¶
C
= V − (1 − τ ) + 0
r
µ
¶ µ
¶
C
C
+ P C V, (1 − τ ) , VC · −γ · VC + (1 − τ ) .
r
r
¡ C
¢
Clearly, for this default barrier the equity value S V, r (1 − τ ) , VC is a continuous
C
(1 − τ ). The fact that a reasonable conversion
r
C
C
barrier VC must lie above r·γ > r (1 − τ ) = VB will become clear in Appendix
¡
¢
¡
¢
A.3. If the asset value increases V ≥ Cr (1 − τ ) , the terms P C V, Cr (1 − τ ) , VC
¡
¢
and V − P C V, Cr (1 − τ ) , VC · VC increase with V . This assertion is obvious as
¢
¡
the value of the binary call P C V, Cr (1 − τ ) , VC is increasing in V . To see the
¡
¢
assertion for the second term, we can characterize P C V, Cr (1 − τ ) , VC · VC as a
claim that is equal to the asset value if V hits VC before Cr (1 − τ ) but zero otherwise.
function in V which is zero at V =
An increase of one marginal unit of V results in an increase of the wealth of the
¢
¡
holder of P C V, Cr (1 − τ ) , VC · VC by one marginal unit given that the binary call
¡
¢
P C V, Cr (1 − τ ) , VC will end in the money, but it is unaffected in the case that
¡
¢
P C V, Cr (1 − τ ) , VC expires worthless. Therefore, the current value of the claim
¡
¢
P C V, Cr (1 − τ ) , VC · VC cannot benefit by more than one marginal unit when V
increases by one marginal unit. Since for V ≥ Cr (1 − τ ) the relation
¡
¢
µ
µ
¶
¶
¶ µ
∂S V, Cr (1 − τ ) , VC
C
∂
C
C
P C V, (1 − τ ) , VC ·
≥
(1 − τ ) − (1 − τ )
∂V
∂V
r
r
r
µ
µ
¶
¶
C
∂
V − P C V, (1 − τ ) , VC · VC > 0
+
∂V
r
¢
¡ C
holds due to γ ≤ 1, we have that S V, r (1 − τ ) , VC is zero at V = Cr (1 − τ ) and
higher afterwards. Hence, the optimal default barrier VB∗ cannot lie above Cr (1 − τ ),
¡
¢
because for V ∈ Cr (1 − τ ) , VB∗ the equity value would be zero, even though it could
be positive for VB =
A.3
C
r
(1 − τ ).
Proof of Property 3
To show VC∗ ≤ VC0 , we argue that a conversion for V < VC∗ is not optimal under the strategy VB0 for non-convertible debt. Suppose the firm follows the strategy VB0 (≤ VB∗ ) and VC0 = VC∗ . Then, the firm value v (V, VB∗ , VC∗ ) is lower than
27
v (V, VB0 , VC∗ ) because the present value of bankruptcy costs is lower and the present
value of the tax shield is higher. As the equity holders follow a suboptimal strategy
VB0 , the equity value S (V, VB0 , VC∗ ) must be lower than S (V, VB∗ , VC∗ ). The debt value
is the difference between the firm value v (V ) and the equity value S (V ) such that
we can see that D (V, VB0 , VC∗ ) is higher than D (V, VB∗ , VC∗ ). Since a conversion for
V < VC∗ would result in a debt value D (V, VB0 , V ) = γ ·V smaller than D (V, VB∗ , VC∗ ),
VC0 cannot be lower than VC∗ .
C
can be seen from the addend P C (V, VB∗ , VC∗ ) ·
The fact that VC∗ is higher than r·γ
¡
¢
γ · VC∗ − Cr in the representation of D (V, VB∗ , VC∗ ). If γ · VC∗ − Cr < 0, the debt value
was lower than without a conversion right. Therefore, converting not before V hits
VC∗ ≥
A.4
C
r·γ
is obviously a better strategy for the convertible debt holders.
Proof of Property 4
The relation D (V, VB0 , VC0 ) ≥ D (V, VB∗ , VC∗ ) is a consequence of the fact that
D (V, VB0 , VC0 ) ≥ D (V, VB0 , VC∗ ) = v (V, VB0 , VC∗ ) − S (V, VB0 , VC∗ ) ≥ v (V, VB∗ , VC∗ ) −
S (V, VB∗ , VC∗ ) = D (V, VB∗ , VC∗ ) holds. The rationale for the second inequality is,
as argued above, that v (V, VB0 , VC∗ ) ≥ v (V, VB∗ , VC∗ ) is valid due to a lower present
value of bankruptcy costs and that S (V, VB0 , VC∗ ) ≤ S (V, VB∗ , VC∗ ) holds because of
the suboptimal strategy VB0 for the equity holders. Hence, even with the suboptimal
strategy VC∗ of the debt holders as a reaction to the default strategy VB0 of the equity
holders is the debt value D (V, VB0 , VC∗ ) higher than D (V, VB∗ , VC∗ ). Since the strategy
VC0 is even better for the debt holders than VC∗ , D (V, VB0 , VC0 ) must be higher than
D (V, VB∗ , VC∗ ).
A.5
Proof of Property 5
We find the validity of the property that IV (V, VB∗ , VC∗ ) ≥ IV (V, VB0 , VC0 ) for all
V > VB∗ given that IV (VB∗ , VB0 , VC0 ) ≤ (1 − α) · VB∗ if we write the investment value
IV (V, VB0 , VC0 ) as
IV
(V, VB0 , VC0 )
C
= +
r
µ
V
VB∗
¶Y µ
IV
(VB∗ , VB0 , VC0 )
which is the value of the non-defaultable bond
IV (VB∗ , VB0 , VC0 ) −
C
r
C
r
C
−
r
¶
,
minus a claim that pays
if the asset value V hits VB∗ for the first time. Comparing
this representation with (3), we can see that IV (V, VB∗ , VC∗ ) > IV (V, VB0 , VC0 ) if
IV (VB∗ , VB0 , VC0 ) < (1 − α) · VB∗ and vice versa. This means if the liquidation value
28
(1 − α)·VB∗ is relatively high a default can result in a value higher than the investment
value under the straight debt strategy. However, if the liquidation value (1 − α) · V B∗
at V = VB∗ is lower than the investment value IV (VB∗ , VB0 , VC0 ), the investment value
IV (V, VB0 , VC0 ) exceeds IV (V, VB∗ , VC∗ ) for all V . This property is a result of the fact
that the slope of IV (V, VB0 , VC0 ) in V at VB0 is at least (1 − α) and IV (V, VB0 , VC0 ) is
a strictly concave function in V .7 Thus, IV (V, VB0 , VC0 ) and (1 − α) · V have exactly
one intercept point for V > VB0 . As for IV (VB∗ , VB0 , VC0 ) > (1 − α) · VB∗ , the intersect
point V lies above VB∗ , IV (V, VB0 , VC0 ) is also higher for all V with VB0 ≤ V ≤ VB∗
which explains the assertion.
A.6
Proof of Property 6
For asset values V < VB0 ≤ VB∗ , the option value under both strategies is worthless
due to a default. For a default only under the optimal strategy, VB0 < V < VB∗ , the
option value OV (V, VB0 , VC0 ) is strictly positive but OV (V, VB∗ , VC∗ ) is zero. For very
high asset values for which a conversion is optimal under both strategies, the option
value is the conversion value γ·V minus the investment value. Hence, OV (V, VB0 , VC0 )
is higher than OV (V, VB∗ , VC∗ ) if IV (V, VB0 , VC0 ) is lower than IV (V, VB∗ , VC∗ ) and vice
versa which is the case when IV (VB∗ , VB0 , VC0 ) ≤ (1 − α) · VB∗ as seen in the property
for the investment value.
References
[1] Acharya, V. V., Huang, J., Subrahmanyam, M. G., Sundaram, R. K.,
“Does Strategic Debt Service Matter?,” Working Paper.
[2] Anderson, R. W., Sundaresan S. 1996, “Design and Valuation of Debt
Contracts,” Review of Financial Studies, 9, 37–68.
[3] Black, F., Cox, J. C., 1976, “Valuing Corporate Securities: Some Effects of
Bond Indenture Provisions,” Journal of Finance, 31, 351–367.
[4] Brennan, M. J., Schwartz, E. S., 1977, “Convertible Bonds: Valuation and
optimal Strategies for Call and Conversion,” Journal of Finance, 32, 1699–1715.
7
The fact that the slope of IV (V, VB0 , VC0 ) is above (1 − α) follows from the equivalent straight
debt model. At V = VB0 , the slope of the equity value is zero due to the smooth-pasting condition
where the slope of the straight debt value equals that of IV (V, VB0 , VC0 ). Since the slope of the debt
value must correspond to that of the firm value which is at least (1 − α), we see why the slope of
IV (V, VB0 , VC0 ) is also at least (1 − α).
29
[5] Bühler, W., Koziol, C., 2002, “Valuation of Convertible Bonds with Sequential Conversion,” Schmalenbach Business Review, 54, 302–334.
[6] Constantinides, G. M., 1984, “Warrant Exercise and Bond Conversion in
Competitive Markets,” Journal of Financial Economics, 13, 371–397.
[7] Fan, H., Sundaresan, S. M., 2000, “Debt valuation, Strategic debt service
and optimal dividend policy,” Review of Financial Studies, 13, 1057–1099.
[8] Hung, M. W., Wang, J. Y., 2002, “Pricing Convertible Bonds Subject to
Default Risk,”, Journal of Derivatives, 10, 39–52.
[9] Ingersoll, J. E., 1977, “A Contingent-Claims Valuation of Convertible Securities,” Journal of Financial Economics, 4, 289–322.
[10] Kim, J., Ramaswamy, K., Sundaresan, S., 1993, “Does default Risk in
Coupons affect the Valuation of Corporate Bonds? A contingent Claims Model,”
Financial Management, 22, 117–131.
[11] Leland, H., 1994, “Corporate Debt Value, Bond Covenants, and optimal Capital Structure,” Journal of Finance, 49, 1213–1252.
[12] Longstaff, F., Schwartz, E., 1995, “A simple Approach to valuing risky fixed
and floating rate Debt,” Journal of Finance, 50, 789–819.
[13]
[14] Mella-Barral, P., Perraudin W. 1997, “Strategic Debt Service,” Journal of
Finance, 52, 531–556.
[15] Sirbu, M., Pikovsky, I., Shreve, S. E. 2002, “Perpetual Convertible
Bonds,” Working Paper.
[16] Spatt, C. S., Sterbenz, F. P., 1988, “Warrant Exercise, Dividends, and
Reinvestment Policy,” Journal of Finance, 43, 493–506.
[17] Takahashi, A., Kobayashi, T., Nakagawa, N., 2001, “Pricing Convertible
Bonds with Default Risk,” Journal of Fixed Income, 11, 20–29.
[18] Tsiveriotis, K., Fernandes, C., 1998, “Valuing Convertible Bonds with
Credit Risk,” Journal of Fixed Income, 8, 95–102.
30