On Equilibrium when Contingent Capital has a
Market Trigger: A Correction to Sundaresan and
Wang Journal of Finance (2015)
George Pennacchi
Alexei Tchistyiy
August 18, 2016
Abstract
This paper points out a logical error in Sundaresan and Wang (2015) (SW)
that invalidates their Theorems 1 and 2. It also presents a counter-example yielding closed-form solutions that shows unique stock price equilibria can exist for a
broader range of contingent capital (CC) contractual terms. Speci…cally, when a
CC’s conversion terms bene…t CC investors, a unique equilibrium can exist rather
than the multiple equilibria claimed by SW.
Department of Finance, University of Illinois, College of Business, 4041 BIF, 515 East Gregory Drive,
Champaign, Illinois 61820. Phone: (217) 244-0952. Email: [email protected].
y
Department of Finance, University of Illinois, College of Business, 461 Wohlers Hall, 1206 S. Sixth
Street, Champaign, Illinois 61820. Phone: (217) 333-3821. Email: [email protected].
1
Introduction
This paper notes an error in Sundaresan and Wang (2015) that invalidates their Theorems
1 and 2. Section II of Sundaresan and Wang (2015), hereafter SW, presents a continuoustime structural model of a bank that issues senior debt, contingent capital (CC), and
shareholders’equity. The conversion of CC from debt to equity is assumed to be triggered
by the market value of the bank’s shareholders’equity or stock price. Their analysis claims
that a unique equilibrium for the bank’s stock price exists only for a knife-edge case: the
market value of the CC must equal its par value at all times prior to, and including, the
time it converts to new equity. A requirement for this unique equilibrium is that the
CC conversion terms provide CC investors with a value of new equity exactly equal to
the CC’s par value. SW describe such conversion terms as requiring “no value transfer”
between CC investors and the bank’s initial shareholders.
We show that SW’s requirement is too severe. Rather, a unique stock price equilibrium can exist when the CC’s market value exceeds its par value, which occurs when
conversion terms give CC investors a value of new equity exceeding the CC’s par value.
The implication is that a unique equilibrium exists when conversion terms dilute (penalize) the bank’s original shareholders, which are situations that SW claim lead to multiple
stock price equilibria. Our results agree with the general analysis of Glasserman and
Nouri (2012) who also …nd that unique stock price equilibria exist for a broader set of
CC contacts than that claimed by SW. The main contribution of our paper is to present
a straightforward counter-example to SW’s claims that illustrates the error in their logic.
To show that more general CC contract terms permit unique stock price equilibria,
we present a special case of SW’s continuous-time model that leads to closed-form solutions for the equilibrium market values of the bank’s CC and shareholders’equity. Our
example permits us to analytically derive the conditions for existence and uniqueness of
an equilibrium stock price, which turn out to be broader than those claimed by SW. We
also show that our di¤erent result is due to a logical error in the proof of SW’s Theorems
1 and 2 which state necessary and su¢ cient conditions for a unique equilibrium. The
impact of their error is nontrivial. The abstract of SW states “The ‘no value transfer’
restriction precludes penalizing bank managers for taking excessive risk,” and their Section III criticizes other research based on this claim. In contrast, our results show that
proposals such as Calomiris and Herring (2013) and Pennacchi et al. (2014) that penalize
a bank’s initial shareholders with heavy dilution at conversion can lead to a unique stock
price equilibrium.
1
2
A Counterexample
This section develops a model that is consistent with the continuous-time framework of
Sundaresan and Wang (2015) yet, as will be shown in our Corollary 1, contradicts their
Theorems 1 and 2. Our notation and argumentation follow their Section II, which we
encourage the reader to compare. All of our proofs are given in the Appendix.
2.1
Model Assumptions
The value of a bank’s assets follows the geometric Brownian motion process
dAt = At dt + At dzt
(1)
where and are constants and zt is a Brownian motion process. The assets generate
cash ‡ows at the rate of a > 0, i.e., the total cash ‡ow during a short period dt is aAt dt.
Let r > 0 be the constant risk-free rate of interest. In risk-neutral probability measure,
we should have = r a.
We assume that the bank has issued a senior bond with a par value B and a maturity
date of T . The coupon rate of the senior bond, b, is equal to the risk-free rate r. Bank
regulators are assumed to close the bank whenever the value of its assets falls to B. Thus
the bank’s closure (bankruptcy) date satis…es
= infft
0 : At
Bg:
(2)
We also assume no bankruptcy costs. Hence, senior debt is default-free and priced at par
Bt = B:
(3)
Besides the senior bond, the bank capital structure consists of shareholders’ equity
and CC having a maturity date of T and a par value C. Prior to conversion, CC pays
…xed coupon interest continuously at the annual rate of c. The initial shareholders own
n shares of stock where St is the price per share (if it exists). CC is assumed to convert
automatically to m 0 additional shares when the equity value nSt falls to the level K
or lower for the …rst time. Let = [0; +1) be all points in time starting from the initial
date 0 and including the CC’s maturity date T during which the equity value is compared
to the trigger. The …rst time a stock price is found to be equal to or lower than the trigger
2
is
= minft 2
: nSt
Kg:
(4)
The quantity m is referred to as the conversion ratio and K as the conversion trigger. The
conversion trigger is hit if the bank’s per share stock price falls to K=n, which is referred
to as the trigger price.
After contractual coupons on the senior bond and CC are paid, the cash ‡ow generated
from the assets of the bank will be paid to equity holders as dividends. Therefore, before
conversion, the total dividend paid to equity holders during a short period dt is (aAt rB
cC)dt. After conversion and before the senior bond’s maturity or the bank’s closure by
regulators, the total dividend paid to equity holders (including those new equity holders
after conversion) during an in…nitesimal period dt is (aAt rB)dt.
Our setup is a special case of SW’s more general model. SW allow the bank’s asset
value At to jump and also permit a time-varying cash ‡ow rate at , risk-free rate rt ,
volatility t , and bond and CC coupon rates bt and ct . All of these parameters are constant
in our model. SW also allow bankruptcy costs and a more general default barrier t . We
assume no bankruptcy costs and t = B.
A major bene…t of our simpli…ed setting is that we are able to derive closed-form
solutions for stock and CC prices that are relatively easy to interpret and allow us to
verify the existence and uniqueness of equilibrium analytically. Yet SW state on their
page 897 that their results hold for our case in which there are no bankruptcy costs and
where asset values follow a geometric Brownian motion without jumps:
It is important to point out that, even without bankruptcy costs or jumps,
multiplicity or absence of equilibrium may arise when the pricing restriction
in Theorems 1 and 2 is violated. Thus, even in a Modigliani-Miller world,
violation of the pricing restriction can lead to multiple equilibria or no equilibrium. Since Theorems 1 and 2 still hold if the asset value follows a geometric
Brownian motion, a CC that violates the pricing restriction may cause stock
prices to jump in a capital market even when the underlying asset prices have
no jumps.
Thus, by showing that our model yields unique equilibria that violate the pricing
restriction in SW’s Theorems 1 and 2, we provide a counterexample to their claims.1
1
Glasserman and Nouri (2012) show that there exist unique stock price equilibria for a broader range
of CC contractual terms than that claimed by SW, even when bank assets follow a jump-di¤usion process.
3
2.2
The Equilibrium Stock Price
At any time t before the CC converts (t < ), the per-share value of common equity is,
in rational expectations,
St
R minf ; ;T g
1n
aAs rB
= Et
(AT B C)e r(T t) 1minf ; g>T + t
n
R minf ;T g
1 n
+Et
aAs rB e
(AT B)e r(T t) 1 T < +
n+m
cC e
r(s t)
r(s t)
ds 1
ds
o
<minf ;T g
o
.
o
.
(5)
The value of the CC before conversion is
1 n r(T t)
e
C 1minf ;
n
m n
(AT B)e
+Et
n+m
Ct = Et
g>T
+
r(T t)
R minf
t
1
; ;T g
T<
e
+
r(s t)
R minf
Because of no bankruptcy costs, and the fact that
hR
T
At = Et t e r(s t) aAs ds + e
cCds
;T g
r(T t)
o
aAs
rB e
r(s t)
ds 1
<minf ;T g
(6)
i
AT ;
(7)
it follows from equations (5) and (6) that
(8)
nSt + Ct + B = At :
(
After the CC converts to m shares and before the senior bond matures or defaults
t < minf ; T g), the per-share value of common equity becomes
St = u t
1
At
n+m
B
(9)
where ut is de…ned as the “post-conversion” share price. Given the conversion trigger K
and conversion ratio m, a pair of value functions, (St ; Ct ), that satisfy equations (4), (5),
(6), and (9) is called an equilibrium. The equilibrium is unique if (St ; Ct ) has a unique
value for every realization of At .
We next show that since the equity value process is adapted to the Brownian motion
4
zt , and dividends are paid continuously, St in (5) must be continuous over time.
Lemma 1: If there is an equilibrium stock price, then St is continuous in t.
Lemma 1, whose formal proof is in the Appendix, says that there should be no jump in
the stock price, including no jump at the time of conversion. To understand the intuition,
note that the per-share value of common equity in (5) is the expected discounted value
of the dividends per share before and after conversion, conditional on the information
generated by the Brownian motion z. Since this information is continuous, the stock
price cannot jump.
Because St is continuous, its values just before conversion, S , and just after conversion, S + , must be the same and equal to the trigger price K=n. Now since the
post-conversion price, ut , in (9) is unique for every At , de…ne Auc as the asset value such
that this post-conversion stock price equals K=n. From (9) this asset value is
Auc = K
n+m
+B .
n
(10)
The fact that St equals ut at the time of conversion and from Lemma 1 that St must
be continuous has the following implication: when an equilibrium stock price exists, it
must lead to conversion only when At = Auc . Conversion at any di¤erent asset level
implies a predictable jump in the stock’s value that is inconsistent with equilibrium.
Intuitively, conversion when At > Auc implies a stock price immediately after conversion
of St = ut > K=n which exceeds the trigger price and is inconsistent with conversion
occurring. Similarly, conversion when At < Auc implies a stock price immediately after
conversion of St = ut < K=n, which if St must be continuous implies that conversion
should have occurred earlier. For either of these cases, the possibility of a jump in the
stock price at the time of conversion is inconsistent with an absence of arbitrage.
Proposition 1: If there is an equilibrium stock price, then conversion happens when
At falls to Auc for the …rst time; that is,
= inf ft 2 [0; T ] : At
Auc g .
(11)
Proposition 1 implies that if there is an equilibrium, then the pre-conversion stock
and CC prices equal equations (5) and (6), respectively, where is the …rst time that the
asset value At equals Auc in (10). Assuming A = Auc allows us to derive the closed-form
solutions for “candidate”equilibrium stock and CC prices given in the next proposition:
5
Proposition 2: If there is an equilibrium stock price, then the pre-conversion date t
values of the stock and CC equal
1
(At B C(At ; Auc ; q));
n
cC
cC
mK
C(At ; Auc ; q) =
+ e rq C
[1 F (q; At ; Auc )] +
r
r
n
(12)
S(At ; Auc ; q) =
where q
T
t is the CC’s time until maturity,
F (q; At ; Auc ) =
[x1t (q)] +
2a
At
Auc
G(q; At ; Auc ) =
y1t (q) =
x1t (q) =
ht =
(14)
[x2t (q)];
a+z
and
cC
G(q; At ; Auc );
r
(13)
a z
At
At
[y1t (q)] +
[y2t (q)];
Auc
Auc
ht z 2 q
ht + z 2 q
;
y2t (q) =
;
p
p
q
q
ht a 2 q
ht + a 2 q
;
x2t (q) =
;
p
p
q
q
q
1 2
1 2
At
2
2
ln
; z=
; a=
2
2
Auc
(15)
(16)
(17)
2
+ 2r
2
;
(18)
( ) is the cumulative standard normal distribution.
Consistent with the assumption that conversion occurs when At equals Auc , it can be
veri…ed from (12) and (13) that S(Auc ; Auc ; q) = Kn and C(Auc ; Auc ; q) = mK
. However,
n
in order for (12) and (13) to truly be equilibrium prices S(At ; Auc ; q) must also exceed
the conversion trigger price, Kn , whenever the asset value At exceeds Auc . Otherwise,
conversion would happen before At falls to Auc , contradicting Proposition 1. Deriving the
conditions under which S(At ; Auc ; q) in (12) remains above Kn whenever At > Auc is the
basis for proving the following theorem on the existence and uniqueness of equilibria:
Theorem 1: When a CC has a …nite maturity and
(i) if mK
maxfC; crC g, then there exists a unique equilibrium in which the CC’s
n
conversion occurs when the bank’s asset level drops to Auc for the …rst time and where the
equilibrium stock prices per share before and after conversion are given by (12) and (9),
respectively;
6
(ii) if
mK
n
< C, then there is no equilibrium stock price;
mK
(iii) if C
< crC , then there may be no equilibrium stock price if the CC’s
n
maturity is su¢ ciently long.
Theorem 1 states that the absence of a stock price equilibrium is possible only when
conversion terms bene…t shareholders by having a conversion value that is less than
< C) or less than the CC’s unconverted perpetuity value
the principal payment ( mK
n
mK
cC
( n < r ). In the former case, there is never an equilibrium stock price process because
conversion shortly before maturity creates a large one-time value transfer to shareholders
that instantaneously moves the stock price above the conversion trigger. In the latter
case, conversion has a weaker e¤ect on equity, since the value transfer is spread over time
and, as a result, both equilibrium and non-equilibrium outcomes are possible depending
> C and
on the parameters. In contrast, a conversion that bene…ts CC investors ( mK
n
mK
cC
> r ) negatively a¤ects the stock’s value, consistent with the equilibrium requirement
n
that the stock price remains at or below the trigger level immediately after conversion.
> maxfC; crC g, then the unique equilibrium CC price is
Corollary 1: If mK
n
C (At ; Auc ; q) < mK=n for all t strictly prior to conversion.
This corollary contradicts SW’s Theorems 1 and 2 which state that a necessary and
su¢ cient condition for a unique equilibrium is that Ct = mK=n at all times prior to
conversion. The simple intuition is that when the CC’s conversion value (mK=n) exceeds
its par value (C) and an upper bound for its coupon value (cC=r), its price strictly prior
to possible conversion must be strictly less than its conversion value.
2.3
Graphic Illustration
Theorem 1 states that unique stock and CC price equilibria exist whenever mK
maxfC; crC g.
n
We illustrate this case for various values of m, the number of new shares of stock granted
to CC investors at conversion. The following parameter values are assumed.
Parameter
Senior Debt Principal, B
Senior Debt Coupon Rate b
CC Principal, C
CC Coupon Rate, c
CC Maturity
Value
96
3.0%
5
3.0%
5 years
Parameter
Conversion Trigger, K
Initial Equity Holder Shares, n
Risk-neutral Cash‡ow Growth,
Volatility of Asset Returns,
Risk-free Interest Rate, r
7
Value
8
1
0.0%
4.0%
3.0%
Note that since C = cC=r = 5, K = 8, and n = 1, the number of shares that provide
CC investors with a value of equity exactly equal to the CC’s par value is m = nC=K =
5
= 0:625. SW claim that this is the only case for which a unique equilibrium exists. In
8
contrast, our Theorem 1 states that unique equilibria exist for all m nC=K = 0:625,
and there is no equilibrium stock price for m < nC=K = 0:625. Panel 1 of Figure 1
illustrates stock price equilibria based on equations (12) and (9) for values of m = 0:625,
m = 1, and m = 1:5, which are equivalent to the values of equity received by CC investors
of mK=n = C, mK=n = 1:6C, and mK=n = 2:4C, respectively. The …gure graphs the
stock prices for each bank asset value and shows that as conversion terms are more (less)
favorable to CC investors (initial shareholders), there is a rise in the bank asset conversion
level where the stock price …rst equals K = 8.2 Intuitively, for a given value of bank assets,
less favorable shareholder conversion terms reduce the stock’s pre-conversion value, leading
to earlier conversion.
Panel 2 of Figure 1 graphs the corresponding CC prices in equation (13) for the same
values of m. It shows that only when m = 0:625 does Ct = mK=n = C. Consistent with
Corollary 1, when m = 1 or m = 1:5, then Ct < mK=n strictly prior to conversion and
maturity.
3
Source of the Error
The con‡ict between our example and SW’s claimed pricing restriction can be traced to
an error in the proof of their Theorems 1 and 2 given on page 896:
THEOREM 1 (SW): For any given trigger Kt and conversion ratio mt , a
necessary condition for the existence of a unique equilibrium (St ; Ct ) is nCt =
mt Kt for every t 2 .
This necessary condition is also su¢ cient in the following sense:
THEOREM 2 (SW): For any given trigger Kt , there exists a conversion
ratio mt and a unique equilibrium ( St ,Ct ) satisfying nCt = mt Kt for every
t2 .
SW’s Theorem 3 on page 902 follows from their …rst two theorems:
2
For m = 0:625, 1, and 1:5, the asset value at which conversion occurs is Auc = 109, 112, and 116,
respectively.
8
THEOREM 3 (SW): Suppose a bank’s asset value follows geometric Brownian motion, dAt = (rt
t ) At dt + At dzt , where
t is the rate of cash ‡ow
from the asset, rt is the instantaneous risk-free rate, and zt is a Wiener
process. Given any conversion trigger Kt that is a continuous function of
time, the CC with coupon rate ct = rt , continuous veri…cation = [0; +1],
and conversion ratio mt = nC=Kt has a unique equilibrium value, which equals
the par value.
To be clear on the implications of these theorems, note that they require Ct = mt Kt =n
to hold for every t 2 , which means this pricing restriction must hold not only at the
time of conversion but at all times prior to conversion or maturity. But since the CC’s
market value must converge to its par value if it matures unconverted, Ct = C must hold.
This leads SW to conclude that a unique stock price equilibrium requires both of these
conditions to be met. That can only occur when Ct = mt Kt =n = C, or mt = nC=Kt so
that the CC must sell for its par value at all times prior to conversion or maturity. This
is why they state on page 902 just before Theorem 3:
The pricing restriction presents a challenge for the implementation of the
CC design:....However, if the unconverted CC is always priced at the par value,
the problem will be solved by setting mt = nC=Kt ....To make a CC priced at
par all the time before conversion, we need to focus on a structure that makes
the market value of the CC immune to changes in interest rates and default
risk. For example, if the CC had no default risk until conversion, by selecting
the coupon rate at each instance to be the instantaneously risk-free rate, we
can ensure that the CC will trade at par....In this case, CC will work well,
because we can determine the conversion ratio ex ante as mt = nCt =Kt =
nC=Kt . Since C and Kt are known ahead of time, CC can be designed to be
default-free during its life before conversion, even though the bank may have a
positive probability of default on its debt claims subsequent to the expiration
of CC. This idea is formalized in Theorem 3.
As shown our Theorem 1 and Corollary 1, our counter-example yields a unique equilibrium whenever mK
maxfC; crC g and, in particular, mK
> maxfC; crC g implies Ct < mK
n
n
n
strictly prior to conversion.3
3
If, as in SW’s Theorem 3, it is assumed that c = r, then it is easy to verify from (13) that when
mK=n > C then C < Ct < mK=n strictly prior to conversion or maturity, so that the CC price exceeds
its par value.
9
SW’s Theorems 1 and 2 are overly-restrictive due to an error in the proofs of these
theorems. In their proofs, SW …rst de…ne Ut as the total value of equity of a bank that
has identical assets and senior debt but has issued no CC. SW show on pages 914-15 in
their Lemma A1 that this hypothetical bank’s total equity value must equal the sum of
the CC-issuing bank’s total equity value plus CC value prior to conversion: Ut = nSt + Ct .
In our notation, Ut = (n + m) ut where recall that we de…ned ut as the post-conversion
bank’s per share stock price. Now the error in SW’s logic comes in the section of their
proof where they give a necessary condition for a unique equilibrium. We repeat this part
of their proof given on page 915 including the necessary condition that they label (A10):4
inf ft 2
: Ut
Kt + Ct g = inf ft 2
: Ut
Kt (n + mt )=ng
(A10)
The above equation holds for all possible paths of Ut if and only if Kt +Ct =
Kt (n + mt ) =n for all t 2 , which implies mt = nCt =Kt for all t 2 .
Therefore, to have a unique equilibrium, the conversion ratio must satisfy mt =
nCt =Kt for t 2 .
SW’s condition (A10) is correct and is equivalent to our Proposition 1 which states
that if there is a unique equilibrium, conversion must happen at exactly the time that the
post-conversion stock price equals the trigger price. To see this, substitute nSt + Ct for
Ut in the left-hand side of (A10) and rearrange both inequalities to obtain
inf ft 2
: St
Kt =ng = inf t 2
:
Ut
= ut
n+m
Kt =n
(19)
which states that conversion occurs when the post-conversion stock price, ut , …rst equals
the trigger. This is equivalent to our Proposition 1’s condition = inf ft 2 [0; T ] : At Auc g.
SW’s logical error comes in their next statement where they mistakenly infer that
(A10) holds for all possible paths of Ut “if and only if Kt + Ct = Kt (n + mt ) =n for all
t 2 .” There is nothing to support the inference that Kt + Ct = Kt (n + mt ) =n for all
times prior to conversion. Rather, Kt +Ct = Kt (n + mt ) =n needs to hold only at the time
of conversion, not strictly before conversion. From this incorrect inference they conclude
that a necessary and su¢ cient condition for an equilibrium is Ct = mt Kt =n, which when
mt and Kt are constants implies that Ct = mK=n is constant. Since Ct must equal C at
maturity, they claim Ct = C = mK=n can be the only equilibrium.
4
Independent of us, a revised version of Glasserman and Nouri (2012) also notes in their footnote 15
that the logic following equation (A10) in Sundaresan and Wang (2015) is ‡awed.
10
Indeed, our model shows that (A10) can hold yet Kt + Ct = Kt (n + mt ) =n only at
conversion, not for all t 2 . Figure 2 illustrates this using the same model parameter
values used earlier in Figure 1. It graphs the second case given in Figure 1 where mK=n =
1:6C > C or m = 1 so that conversion terms are strictly favorable (unfavorable) to CC
investors (initial shareholders) and conversion occurs when Auc = 112. Since K = 8 and
m = n = 1, then K(n + m)=n = 16. As a function of bank assets, Figure 2 graphs the
total post conversion equity value, Ut , (blue dotted line) and the sum of the conversion
trigger and CC value, K + Ct (red solid line). Intuitively, as bank assets decline, K + Ct
rises since conversion that bene…ts CC investors becomes more likely.
Consistent with (A10) and our Proposition 1, the …gure shows that Ut …rst equals
K+Ct at the same (conversion) time that Ut …rst equals K(n+m)=n. Yet the …gure clearly
shows that SW’s inference that a unique equilibrium requires K + Ct = K (n + m) =n or
Ct = mK=n at all times prior to conversion is incorrect. It demonstrates that (A10) can
still hold with K + Ct < K (n + m) =n strictly prior to conversion, which explains why
there is a broader range of conversion terms that penalize initial shareholders and permit
a unique stock price equilibrium.
4
Conclusion
Our model of a bank that issues CC with a stock price trigger makes assumptions consistent with SW’s continuous-time framework. The model yields closed-form solutions for
CC and stock price values that allow us to analytically derive conditions for the existence
of a unique equilibrium. Unique equilibria exist whenever CC conversion terms are favorable (unfavorable) to CC investors (initial shareholders), resulting in the CC’s value
exceeding par. Our result is a counter-example to SW’s claim that a unique equilibrium
requires perfectly-neutral “no value transfer” conversion terms and a CC market value
always equal to par. We explain the con‡icting results by identifying an error in the
proofs of SW’s Theorem 1 and 2, and the model is used to illustrate their incorrect logic.
Our correction has implications for CC proposals that seek to reduce a bank’s riskshifting and debt overhang incentives by heavily diluting the bank’s initial shareholders
at conversion. CC can be designed to penalize banks for taking excessive risk without
necessarily generating a multiplicity or absence of stock price equilibria.
11
5
Appendix
5.1
Proof of Lemma 1
Let the probability space ( ; F; fFt ; t 2 [0; T ]g ; P ) be such that the Brownian motion
zt generates the information ‡ow fFt ; t 2 [0; T ]g and all processes are adapted to this
information ‡ow. Consider the following quantity, Y , which equals the discounted per
share payo¤ to shareholders at date T plus dividends paid from date 0 to date T :
Y
1 n rT
e (AT
=
n
1 n
+
e
n+m
B
rT
C) 1minf
(AT
B) 1
; g>T
+
+e
T<
R minf
0
r
; ;T g
R minf
rs
e
;T g
e
aAs
r(s
)
rB
aAs
cC ds
o
rB ds 1
(A.1)
o
<minf ;T g
Since Y is a realized stream of cash‡ows discounted as of date 0, it does not depend on
time t but it is a random variable for all times t < T . De…ne Xt Et [Y ] where Et denotes
the risk-neutral expectation conditional on Ft . By the law of iterated expectations, Xt is
a martingale adapted to the …ltration generated by Brownian motion zt . Hence, Xt is a
continuous process since all martingales adapted to a Brownian …ltration are continuous.
Comparing Xt to St in (5) yields
Xt =
Rt
0
e
rs
1
aAs
n
rB
cC
1fs
g
+
1
aAs
n+m
rB
1fs>
g
ds + e
rt
St ;
(A.2)
which can be rewritten as
St = ert Xt
Rt
0
e
rs
1
aAs
n
rB
cC
1fs
g
+
1
aAs
n+m
rB
1fs>
g
ds :
(A.3)
Since Xt and the time integral in (A.3) are continuous in t, St must be continuous in t.
5.2
Proof of Proposition 1
Since the post-conversion price ut in (9) is strictly monotone in At , conversion at any
asset level other than Auc would lead to a jump in the stock price, which cannot be an
equilibrium according to Lemma 1.
To …nish the proof we need to show that conversion must happen the …rst time that
12
At = Auc . Our argument is that if it did not, it would lead to a contradiction. Suppose
that conversion did not happen the …rst time At = Auc . Due to the Brownian motion
process generating At , immediately after this …rst passage time it must be that At < Auc
for some …nite time while the equilibrium stock price St > K=n since conversion did
not occur by assumption.5 Moreover, St must remain strictly above K=n as long as At
is strictly below Auc since conversion can only occur when At = Auc . However, when
At < Auc there is positive probability that At continues to decline and reach B, the
promised payment on the bank’s senior debt. But by assumption this is the time that the
bank is closed by regulators.6 At this date shareholders lose all claims on the bank’s assets,
necessitating that St = 0. But this positive probability event contradicts the requirement
that St remains strictly above K=n as long as At is strictly below Auc . Consequently,
conversion must happen the …rst time that At = Auc .
5.3
Proof of Proposition 2
Our model has similarities to the default-risky, …nite maturity debt model of Leland
and Toft (1996). Let f (s; At ; Auc ) be the risk-neutral probability density of the …rst
passage time of At to Auc at date t + s, and let F (s; At ; Auc ) be corresponding cumulative
distribution function. Then the only possible candidate equilibrium CC value at date t is
Z q
C(At ; Auc ; q) =
e rs cC[1 F (s; At ; Auc )]ds + e rq C[1 F (q; At ; Auc )]
0
Z q
mK
f (s; At ; Auc )ds:
(A.4)
+
e rs
n
0
The …rst term is the discounted risk-neutral expected value of the coupon ‡ow, which is
paid at s periods in the future with probability (1 F (s; At ; Auc )). The second term is the
risk-neutral expected discounted value of repayment of principal, and the third term is the
risk-neutral expected discounted value of the shares given to CC investors at conversion
if conversion occurs. Integrating the …rst term by parts yields
C(At ; Auc ; q) =
cC
+e
r
rq
C
cC
[1
r
F (q; At ; Auc )] +
5
mK
n
cC
G(q; At ; Auc );
r
(A.5)
That there must be some time that At < Auc following inf t 2 f[0; 1) : At Auc g is due to Brownian
motion having in…nite variation and Auc not being a re‡ecting or absorbing barrier.
6
While this assumption is speci…c to our model, any sensible model of a levered …rm leads to bankruptcy when At hits some critical lower bound.
13
where
G(q; At ; Auc )
Z
q
e
rs
(A.6)
f (s; At ; Auc )ds:
0
Harrison (1990) and Rubinstein and Reiner (1991) show that F and G equal (14)-(18).
Finally, the pre-conversion per-share stock value must be equal to the asset value
minus the value of the senior debt and CC:
S(At ; Auc ; q) =
5.4
1
(At
n
B
(A.7)
C(At ; Auc ; q)):
Proof of Theorem 1
(i) The CC’s conversion value exceeds its principal and its coupon value in
perpetuity:
mK
cC
maxfC;
g;
n
r
that is, conversion always bene…ts CC investors.
When At > Auc = K n+m
+ B, substituting (13) into (12) yields
n
1
(At B C(At ; Auc ; q))
n
1
>
Auc B C(At ; Auc ; q)
n
K
1 mK
=
+
C(At ; Auc ; q)
n
n n
K
1 mK cC
=
+
(
)(1 G(q; At ; Auc ))
n
n
n
r
S(At ; Auc ; q) =
(A.8)
e
rq
Note that the cumulative distribution function F (q; At ; Auc )
G(q; At ; Auc )
F (q; At ; Auc );
(C
cC
)(1
r
F (q; At ; Auc )) :
1. In addition,
(A.9)
since G(q; At ; Auc ) is given by (A.6) with r > 0.
Because of mK
maxfC; crC g and (A.9), the term in square brackets in the last line
n
of (A.8) is non-negative. The implication is that S(At ; Auc ; q) > Kn for any q
0 and
any At > Auc ; that is, the stock price remains above the conversion trigger as long as the
asset level remains above Auc . Thus, S(At ; Auc ; q) is the unique equilibrium price prior to
conversion.
14
(ii) The CC’s conversion value is less than its principal:
mK
< C;
n
that is, CC investors receive less than the principal value at conversion.
Note that an equilibrium stock price must be equal to Kn when At = Auc and must
be greater than Kn for all At > Auc . This requires that the stock price is increasing in At
near Auc . However, we now show that this is not the case when mK
< C.
n
Taking the derivative of S(At ; Auc ; q) in (12) with respect to At yields
@S(At ; Auc ; q)
@At
=
At =Auc
=
@C(At ; Auc ; q)
@At
1
1
n
1
1+e
n
mK
n
@G(q)
@At
=
At =Auc
1
Auc
=
At =Auc
1
Auc
2a (a
(a
C
cC
r
If At = Auc , then ht = 0, [x2t (q)] = 1
p
1
[y1t (q)] = [z q]. As a result, we have
@F (q)
@At
rq
At =Auc
cC
r
@F (q)
@At
@G(q)
@At
At =Auc
q) + 2
z) + 2z (z
(a
p
[a
p
q)
p
q
p
(A.10)
:
[x1t (q)] =
p
At =Auc
q) + 2
q], and
(A.11)
;
(z
[y2t (q)] =
p
q)
p
q
;
(A.12)
where ( ) denotes the standard normal density function. Importantly, note that as q ! 0,
@F (q)
@At
At =Auc
!
1 and
15
@G(q)
@At
At =Auc
!
1:
Now equation (A.10) can be rewritten as follows:
n
@S(At ; Auc ; q)
@At
At =Auc
1
e
cC
r
As q ! 0, the second term converges to
converge to zero
while (1
if
mK
n
e
rq
)
(q)
since @F
@At
rq <<
and
At =Auc
p
q and
rq
At =Auc
@F (q)
@At
cC
r
C
mK
n
@F (q)
@At
1 when
@G(q)
@At
@F (q)
@At
@F (q)
@At
mK
n
= 1+ C
mK
n
At =Auc
(A.13)
At =Auc
@G(q)
@At
:
At =Auc
< C. The last two terms in (A.13)
are of the order of magnitude of
At =Auc
At =Auc
@G(q)
@At
At =Auc
(z 2 a2 ) q
p p
2 2
q
p1 ,
q
! 0. Thus,
< C, then
lim
q!0
@S(A; Auc ; q)
@At
=
At =Auc
1.
(A.14)
Therefore when mK
< C, S is declining in the bank’s assets at a time su¢ ciently close
n
to maturity when the asset level is near Auc . This means that the candidate stock price
falls below the trigger before the asset level drops to Auc . According to Proposition 1, it
cannot be an equilibrium stock price. Consequently, an equilibrium stock price does not
exist for this case.
(iii) The CC’s conversion value exceeds its principal but is less than its
coupon value in perpetuity:
C
mK
n
cC
:
r
When CC has a long maturity and its coupon rate exceeds the risk-free rate, there
may be no equilibrium stock price even though the CC’s conversion value exceeds its
principal. This follows because as q ! 1, the model is equivalent to the perpetual
maturity CC model analyzed in Pennacchi and Tchistyi (2015). Their Theorem 1 shows
that for some, albeit unrealistic, parameter values for which condition (ii) in their Lemma
2 is not satis…ed, no stock price equilibrium exists.
16
5.5
Proof of Corollary 1
Corollary 1 considers the situation where mK
strictly exceeds maxfC; crC g and conversion
n
has not yet occurred so that At > Auc .7 Equation (13) can be rearranged as:
C(At ; Auc ; q) =
=
<
=
<
=
<
cC
+e
r
mK
+
n
mK
+
n
mK
n
mK
n
mK
n
mK
n
mK cC
G(q; At ; Auc )
n
r
mK cC
cC
e rq [1 F (q; At ; Auc )]
[1 G(q; At ; Auc )]
C
r
n
r
mK cC
mK cC
e rq [1 F (q; At ; Auc )]
[1 G(q; At ; Auc )]
n
r
n
r
mK cC
[1 G(q; At ; Auc )] e rq [1 F (q; At ; Auc )]
n
r
mK cC
[1 F (q; At ; Auc )] e rq [1 F (q; At ; Auc )]
n
r
mK cC
1 e rq [1 F (q; At ; Auc )]
n
r
rq
C
cC
[1
r
F (q; At ; Auc )] +
The …rst inequality follows from the fact that mK
> C. The second inequality follows
n
from the fact that G(q; At ; Auc ) < F (q; At ; Auc ) when At > Auc and q > 0. The last
inequality follows from the fact that F (q; At ; Auc ) < 1 and mK
> crC .8
n
7
We also assume q > 0 so that maturity has not yet occured. If q = 0, then trivially C (At ; Auc ; 0) =
C < mK=n by assumption.
8
Inequality G(q; At ; Auc ) < F (q; At ; Auc ) < 1 is explained in the proofs of Proposition 2 and Theorem
1.
17
References
Calomiris, C. and Herring, R.: 2013, How to design a contingent convertible debt requirement that helps solve our too-big-to-fail problem, Journal of Applied Corporate Finance
25, 39–62.
Glasserman, P. and Nouri, B.: 2012, Market-triggered changes in capital structure: Equilibrium price dynamics. Columbia University working paper.
Harrison, J. M.: 1990, Brownian Motion and Stochastic Flow Systems, …rst edn, Robert
E. Krieger, Malabar, Florida.
Leland, H. E. and Toft, K. B.: 1996, Optimal capital structure, endogenous bankruptcy,
and the term structure of credit spreads, Journal of Finance 51, 987–1019.
Pennacchi, G. and Tchistyi, A.: 2015, A reexamination of contingent convertibles with
stock price triggers. University of Illinois working paper.
Pennacchi, G., Vermaelen, T. and Wol¤, C.: 2014, Contingent capital: The case of COERCs, Journal of Financial and Quantitative Analysis 49, 541–574.
Rubinstein, M. and Reiner, E.: 1991, Breaking down the barriers, Risk Magazine 4, 28–35.
Sundaresan, S. and Wang, Z.: 2015, On the design of contingent capital with a market
trigger, Journal of Finance 70, 881–920.
18
Figure 1: Stock and CC Price Equilibria for Different Conversion Terms
This graph shows unique equilibrium stock and CC prices as a function of the bank’s asset value, At, when a CC’s
maturity is T=5 years, its coupon rate is c=r=3%, its principal is C = 5, and its conversion trigger stock price is K/n
=8. The value of senior debt equals 96, the risk-neutral cash flow growth rate is 0, and the volatility of asset returns
is 4%. The solid lines are the equilibrium stock and CC prices prior to conversion (equations (13) and (12)). The
dotted lines in the first panel are the stock prices post conversion (ut in equation (9)) and the dotted lines in the
second panel are the CC’s conversion values.
Theorem 1 shows that unique equilibria prices exist for mK / n  C , and the figure graphs the three cases
mK / n  C ( where m=0.626), mK / n  1.6C (where m = 1), and mK / n  2.4C (where m = 1.5). The first
case is the only one that SW claim would result in a unique equilibrium. The other cases represent counter-examples
to their claim. For these three cases, conversion occurs at the asset values Auc = 109, 112, and 116, respectively.
19
Figure 2: Equilibrium Values of Ut and K + Ct
Graphed as a function of bank assets, At, is the total post conversion equity value, Ut, (blue dotted line) and sum of the conversion trigger and CC value, K + Ct
(red solid line). These values assume that a CoCo’s time until maturity equals 5 years, its coupon rate equals the risk-free rate, c=r=3%, its principal is C = 5,
and its conversion trigger stock price is K/n =8. The value of senior debt equals 96, the risk-neutral cash flow growth rate is 0, and the volatility of asset returns is
4%. The conversion terms are assumed to be mK / n  1.6C  C (where m = 1). This corresponds to the second case in Figure 1 where conversion occurs at
the asset level Auc = 112. Consequently with m=n=1, K(n+m)/n = 16.
SW’s equation (A10) and our Proposition 1 show that a requirement for a unique equilibrium is that Ut must first equal K + Ct at the same time that Ut first equals
K(n+m)/n. The graph shows this is true since conversion occurs the first time that the bank’s assets fall to 112, which is also the first time K + Ct equals
K(n+m)/n. However, SW incorrectly infer this requirement implies that K + Ct = K(n+m)/n or Ct = mK/n for all times prior to conversion. The graph also clearly
shows this need not be the case, which explains why there are more conversion terms that lead to unique equilibrium that what SW claim.
20