3. Reputation Effects on Convertible Bond Call Policies
3.2 Equilibria of the Convertible Bond Calling Game
The equilibrium concept used in the model is that of a Bayesian equilibrium defined as below.
Definition: A Bayesian equilibrium is a set of conditional calling probabilities gt ,
G Bt , , and
1,2
, and asetofstockholders’beliefspd ,
C NC
d , , and
1,2
; where (i)gt maximizes the firm’s expected valve and (ii)pd satisfiesBaye’srule.The model constructed in the previous section has multiple Bayesian equilibria, which are consistent with any kind of beliefs. These equilibria contain pure strategies under both degenerate and non-degenerate probability distributions of the outsiders’beliefs. The features of these equilibria are discussed below.
Lemma 3.1 For each historical performance1{S,F}, there exists a pooling equilibrium where no firm will call the convertible bonds - that is,
t gt 0.
Proof
Suppose that, by observing a firm calling a bond, the stockholders believe the firm to be type-B. Therefore,pC 0. It follows that: After the algebraic calculation, the boundary [ ,1)
D E
D
is obtained so that (A3.3)-(A3.1)>0 and (A3.4)-(A3.2)>0.
Q.E.D.
This implies that if the proportion of the stocks converted from the bonds is large enough ( [ ,1)
D E
D
), then no matter what type the firm is, the expected value for a non-calling firm will be higher than that for a calling firm. Thus, the firm will not be better off to announce a call-back of the convertibles at time 1.
Lemma 3.1 is a common result in the reputation model. If we make those beliefs sufficiently unfavorable to one side of the players, then we can support a pooling equilibrium. In this case, calling a bond is a zero-probability event, and we can refer to such an event with the belief that the firm is of the lowest quality, thus making “no calling”an equilibrium. An equilibrium with a zero-probability event is defined as a degenerate equilibrium. The degenerate pooling equilibrium is due to
the assumption of the investors’degenerate beliefs regarding a firm’s type.
Proposition 3.2 proposes another degenerate equilibrium.
Lemma 3.2 For each historical performance1{S,F}, there exists a pooling equilibrium where both types of firms will call the convertible bonds -that is,gt 1t.
Proof
In Lemma 3.1 the assumptions gB 0 and gG 0 form a pooling equilibrium gt 0. Consider the other assumption that the stockholders believe the firm to be type-G when observing a firm calling. Therefore, let pC 1. It follows that: After the algebraic calculation, the boundary (0, ]
D E
D
is obtained so that
(A3.5)-(A3.7)>0 and (A3.6)-(A3.8)>0.
Q.E.D.
Here forms another degenerate equilibrium under investors’degenerate beliefs.
Lemma 3.2 implies that if the proportion of the stocks converted from the bonds is small enough ( (0, ]
D E
D
), then no matter what type the firm is, the expected
value of a calling firm is higher than that for a non-calling firm. The firm will hence choose to announce a call-back to the convertibles at time 1. Lemma 3.1 and Lemma 3.2 analyze the conditions for two degenerate equilibria to exist. In this paper we focus on two non-degenerate equilibria under investors’non-degenerate beliefs. Propositions 3.1 and 3.2 will show the features of these two equilibria.
Proposition 3.1 For a given level of the firm’s reputation, there exists a separating equilibrium where type-G firms will not call the convertible bonds, while type-B firms will call as long as the reputation effects dominate the dilution effects - that is, gG 0 and gB 1. Proof
Comparing the marginal benefit (MB ) from calling convertible bonds for a type-G firm with that for a type-B firm, (A9) is obtained.
} Assume that the reputation effects dominate the dilution effects so that we have
)
V . There exists a subgame equilibrium where type-G firms will not call, while type-B firms will.
Q.E.D.
In Proposition 3.1 a separating equilibrium exists as the investors believe that the reputation value which a firm will obtain from a future successful project will exceed the dilution loss due to the voluntary conversion at time 2. The benefit from calling the convertible bonds is negative for a good management quality firm, which prompts a type-G firm not to call the convertible bonds. On the other side, the benefit from calling the convertible bonds is positive for a bad management quality firm, which indicates that a type-B firm will make a conversion-forcing call at time 1. As a type-B firm recognizes its future failure of the project, it calls back the convertible bonds at time 1 to force conversion rather than to repay the debt, D, at time 2.
Proposition 3.2 There exists a separating equilibrium where those firms who have obtained good reputations will not call the convertible bonds, while the others who have suffered bad reputations will call as long as past reputation dominates - that is,gSt 0 and gtF 1t{G,B}. Proof
No matter what type the firm is, the value for a calling firm with a successful past is:
and the value for a calling firm with a failed past is:
N
Furthermore, the expected value for a non-calling firm with a successful past is:
N
and the expected value for a non-calling firm with a failed past is:
N
Comparing the marginal benefit (MB ) from calling convertible bonds for firmst with good reputations with that for firms with bad reputations, (A3.14) is obtained. Assuming that the past reputation dominates,4 the second and the third terms of the right-hand side will be negative, which makes (A3.14) also be negative. As (A3.14) is negative, VSt NVSt 0 and VFt NVFt 0. There exists a subgame equilibrium where the good reputation firms will not call, while the bad reputation firms will.
Q.E.D.
4 The outsiders believe that firms with a successful past are more likely to be type-G firms, no matter whether the projects are successful or a failure in the second period.
In Proposition 3.2 another separating equilibrium exists as the investors believe that the firm with a good past is more likely to be a good firm. Under such a belief, the reputation value which a firm obtained from its past success will exceed the dilution loss associated with the voluntary conversion at time 2. Thus, the benefit from calling the convertibles is negative for a good reputation firm, which induces a good reputation firm to not call the convertible bonds. On the other side, the benefit from calling is positive for a bad reputation firm, which makes a bad reputation firm call the convertible bonds at time 1. Again, the choice by a bad reputation firm is due to a fear of eroding the firm’s endowment wealth when it is necessary to repay the debt at time 2.