Contingent onvertibles. Solving or seeding the
next banking risis?
Christian Koziol∗
Johen LawrenzŸ
First draft: January 2010; This draft: February 2011
∗
Aknowledgment: We thank Manuel Ammann, Ken Behmann, Mark Grinblatt, Hendrik
Hakenes, Karl Ludwig Keiber, Jan Pieter Krahnen, Kolja Loebnitz, Kristian Miltersen, Alex
Mürmann, Stefan Pihler, Berend Roorda, Yves Shläpfer, Norman Shürho, and seminar
partiipants at the European Winter Finane Summit 2010, Symposium for International Corporate Finane and Governane in Twente 2010, University of Hannover, German Finane Assoiation Annual Meeting 2010 Hamburg, and the German Eonomi Assoiation of Business
Administration Frankfurt 2010. As always, all remaining errors are our own.
∗
Professor Dr. Christian Koziol, Chair of Risk Management and Derivatives, University of
Hohenheim, D-70593 Stuttgart, Germany, Email .kozioluni-hohenheim.de
Ÿ Dr. Johen Lawrenz, Department of Banking & Finane, Innsbruk University, A-6020
Innsbruk, Austria, Email johen.lawrenzuibk.a.at, Phone +43-512-507-7582.
Contingent onvertibles. Solving or seeding the
next banking risis?
First draft: January 2010; This draft: Otober 2010
Abstrat
A reent proposal to enhane banking stability reommends the use of
ontingent onvertibles (CoCos). Sine these hybrid seurities are mandatorily onverted into equity when banks are in need of a reapitalization,
they are redited for reduing banks' likelihood of nanial distress. In this
paper, we show within a ontinuous-time framework that this allegedly
beneial impat hinges ritially on the assumption of omplete ontrats.
We show that although CoCos always distort risk taking inentives, CoCos an reate wealth for all involved investors. Our main ontribution is
to demonstrate that a higher investors' wealth from CoCo naning an
ause a higher bank's probability of nanial distress so that the banking
system as a whole will be destabilized. Thus, individually rational deisions an have systemially undesirable outomes. Further results indiate
that CoCos should be used only in onjuntion with devies to mitigate
risk shifting inentives. This objetive an be aomplished by a striter
regulation and a higher onversion ratio.
JEL lassiation: G32, G21, G28
Keywords:
Contingent apital ertiates, Reverse onvertibles, Restruturing
mehanisms, Regulatory hybrid seurity
1
Introdution
In the aftermath of the nanial market risis of 2008/09, several ulprits have
been named that ontributed to the near melt-down of the nanial system.
While still subjet to ongoing debate, there seems to be onsensus that one major
problem was related to the high leverage ratios of banks, whih left them with
severe problems of reapitalization when market onditions worsened abruptly.
One proposal, that has reeived muh attention in reent times to help alleviating the problem of exessive leverage ratios is to indue banks to issue so-alled
ontingent onvertible (CoCo) bonds, also known as enhaned apital notes or
ontingent apital. The key feature of these hybrid seurities is that they pay
oupons like normal bonds but are automatially onverted into ordinary shares
one the equity ratio falls below a predetermined threshold. This proposal has
gained momentum in November 2009 when Lloyds Banking Group launhed a
apital raising whih inluded the issuane of ¿7.5 billion in ontingent onvert-
1
ibles.
Governments as well as regulators seem to put muh hope in these new
2
seurities.
Only reently, the Bank for International Settlement announed that
as part of their reform pakage, they will review the role of ontingent apital
3
within the regulatory apital framework,
and in Switzerland, an expert group
reommended that banks will have to hold 19% apital, where nine perent are
required in ontingent apital.
4
Advoates of ontingent onvertibles onsider these hybrid seurities as a transparent, eient and less ostly resolution mehanism for distressed banks, beause
1 The ase of Lloyds has gained attention for two reasons. First, it was the biggest issuane
of CoCo bonds sofar and seond, the British government has a 43% stake in Lloyds. See
Finanial Times Lloyds to oer sweeteners to bondholders, November 1, 2009, and
Finanial Times UK experiment raises prospets of new asset lass, November 5, 2009.
e.g. the
2 As doumented in speehes by Ben Bernanke, hairman of the US Federal Reserve, Paul
Tuker, deputy governor of the Bank of England and Lord Turner, hairman of the Finanial
Servies Authority. See e.g. the speeh by Ben Bernanke to the US House Finanial Servies
Committee on Otober 1, 2009, the
2009 and
The Wall Street Journal
Finanial Times The sweet x of CoCos?, November 12,
Poliy Makers Disuss Bank Capitalization, November
17, 2009.
3 See the press release at January 11, 2010.
4 See e.g.
Capital proposal targets UBS and Credit Suisse, Otober 4, 2010.
Finanial Times
1
they provide an inrease of the equity ratio at pre-ommitted terms when the
bank is in a diult situation due to a severe loss of their asset value. Therefore,
this feature redues the danger of a ostly default or, alternatively, supervisory
intervention. At rst glane, the idea seems impeable. During good times, the
bank takes advantage of the benets of debt naning, suh as e.g. exploiting
naning and/or tax advantages, while in bad times, when debt obligations impose the risk of nanial distress, these seurities automatially onvert to equity
and mitigate the default risk.
In some sense, ontingent onvertibles seem to be the ake, one likes to have
and eat it too.
However, a fundamental doubt that CoCo bonds do the trik,
omes from theoretial onsiderations about the optimality of debt naning. A
large body of literature argues that within an inomplete ontrats setting, debt
is an optimal naning arrangement beause it oers xed payments in good
states, while in bad states it stipulates transfer of ownership. The threat of losing ownership, or more generally ontrol rights, exerts a disiplining eet on the
deision-makers of the rm. Hart and Moore (1998) undersore the important
distintion between ash ow rights and ontrol rights for the theoretial explanation of debt ontrats. However, by onstrution, CoCo bonds postpone the
transfer of omplete ontrol rights.
Thus, ontingent onvertibles may distort
deision-makers' inentives.
This paper is, to the best of our knowledge the rst theoretial ontribution that
tries to shed light on potential drawbaks of ontingent onvertibles due to distorted risk inentives. Within a ontinuous-time strutural model, we onsider
a ommerial bank engaged in the deposit taking business whih satises additional naning needs by aessing apital markets.
In line with the pratial
evidene from Lloyds, we analyze the impat of exhanging straight bonds with
ontingent onvertible bonds. In our analysis, we distinguish between two ases:
A omplete ontrat and an inomplete ontrat setting. In the former, we assume the investment poliy of the bank to be given (or to be ontratible), while
in the latter ase bank managers have disretion over the hoie of the bank's
investment risk. From our analysis, we nd that CoCo bonds are unambiguously
beneial if the bank annot hange its business risk. However, we also nd that
results hange dramatially if we onsider inomplete ontrats. We show that
2
CoCo bonds always distort risk taking inentives and indues deision-makers to
at less prudent. As a main result, we demonstrate that although CoCo bonds
are optimal in the sense that they are fairly pried and rm value maximizing,
the distorted risk-taking inentives an atually inrease the bank's probability
of nanial distress as well as the expeted distress osts substantially. Contrary
to the initial intention, CoCo bonds an reate negative externalities for the
eonomy in the sense that individually rational deisions may have systemially
undesirable outomes.
While normal onvertible bonds have been analyzed extensively in the literature (see e.g. Brennan and Shwartz, 1977; Ingersoll, 1977; Brennan and Shwartz,
1980; Brennan and Kraus, 1987; Nyborg, 1995), there are only few reent aademi ontributions dealing speially with
ontingent
onvertibility provisions.
Flannery (2005, 2009) argues that ontingent onvertibles are an eetive mehanism to exert market disipline, beause by issuing CoCo bonds shareholders
internalize, i.e. bear the full ost of their risk taking deisions rather than rely
on the (ostless) regulatory bail-out option. In a related analysis, Landier and
Ueda (2009) disuss several options for a bank restruturing, among whih they
mention onvertible debt. In line with the reasoning by Flannery (2009), they
onlude that onvertibles an derease the probability of default (see Landier
and Ueda (2009), p. 25). Aharya et al. (2009) simply state in their reent ontribution that assesses the strengths and weaknesses of the US nanial reform
legislation: Contingent apital is learly a good idea .5
More reently, Glasserman and Nouri (2010), Pennahi (2010), and Sundaresan
and Wang (2010) put forward theoretial models of CoCo bonds.
Glasserman
and Nouri (2010) and Pennahi (2010) both analyze the priing of CoCo bonds
when onversion an take plae ontinuously (Glasserman and Nouri, 2010) or
when the bank's asset value follows a jump-diusion proess and interest rates
are stohasti (Pennahi, 2010).
Sundaresan and Wang (2010) fous on the
design of the onversion trigger, and argue that for a wide lass of trigger mehanism there does not exist a unique equilibrium for equity and bond pries, thus
opening up the possibility of prie manipulation.
5 Aharya et al. (2009), p. 43.
3
The work by Flannery (2005, 2009) has reeived onsiderable attention and CoCo
bonds have been mentioned favorably in poliy reommendations by e.g. Stein
(2004), Kashyap et al. (2008), Kaplan (2009), and Due (2009). While ontingent onvertibles are welomed by pointing out their ability to overome problems
of high leverage, Hart and Zingales (2009) reognize that the proposal by Flan-
6
nery (2005) eliminates some of the disiplinary eets of debt.
However, sine
their fous is on the implementation of a new apital regulation for systemially
important banks, they do not explore that issue in detail. As a further poliy
reommendation, the so-alled Squam Lake Working Group on Finanial Regulation advoates the use of ontingent apital as a transparent, eient and less
7
ostly resolution mehanism for distressed banks.
One of the few onerns that have been raised against CoCo bonds so far points
towards a potentially destabilizing eet whih might our when large institutional investors, who are not allowed to hold shares, are fored to sell their
onverted bond position.
This eet might exaerbate the share prie deline
8
and spur doubts about the bank's stability.
With respet to a potential moral
hazard problem, the aademi literature so far seems to onsider risk-shifting
problems of CoCo bonds to be marginal at most.
Flannery (2005, 2009) on-
ludes from verbal reasoning rather than from a formal model-based approah
that risk-taking inentives an be ontrolled through the internalization of riskshifting osts.
Pennahi (2010) admits the presene of risk-shifting inentives
but argues that they are less pronouned as with subordinated debt. Glasserman
and Nouri (2010) analyze how higher risk impats the yield spread, but do not
explore the stability onerns of CoCo bonds.
Thus, to the best of our knowledge, our ontribution is the rst that fouses on
potentially adverse impliations of CoCo bonds from distorted risk-taking inentives. In the light of the reent banking risis, it need not be stressed that the
analysis of banks' risk taking behavior is of ruial importane. Our results are
6 See Hart and Zingales (2009), p. 5.
7 The Squam Lake Working Group resembles fteen distinguished aademis as e.g. Darrell
Due, Douglas Diamond, John Cohrane, Robert Shiller and Raghuram Rajan. See Squam
Lake Working Group (2010).
Finanial Times Stability onerns over CoCo bonds, November 5, 2009, and Finanial
Times Report warns on CoCo bonds, November 10, 2009.
8 See
4
an important warning signal that CoCo bonds may reate negative externalities,
in the sense that the (destabilizing) risk-shifting problem indued by CoCo bonds
may overompensate the (stabilizing) eet of providing a pre-ommitted reapitalization to banks.
The artile is organized as follows.
The next setion sets up the general
model framework and determines the optimal naning behavior with straight
bonds and CoCo bonds. In setion 3, we analyze the impat of CoCo naning
if omplete ontrats an be written, i.e. when the bank has no disretion over
the risk tehnology. Setion 4 analyzes the ase where ontrats are inomplete,
i.e. the bank an hange the investment risk. Setion 5 onludes.
2
General Model Framework
2.1
Initial Bank
In line with Bhattaharya et al. (2002), Deamps et al. (2004) and Rohet (2004),
we onsider a bank with assets in plae that ontinuously generate instantaneous
ash ows before taxes equal to
xt , whih are assumed to be driven by a Geometri
9
Brownian Motion
dxt = µxt dt + σxt dzt ,
where
µ and σ
are onstant drift and diusion parameters, and
(1)
zt
denotes a stan-
dard Wiener proess. We will apply the arbitrage-free valuation priniple where
(for priing purposes) the proess in (1) is onsidered under the risk-neutral measure
Q.
In the following setions, we will analyze both, the ase where the proess
of the ash ows governed by
µ
and
σ
is exogenously given, and the ase where
bank managers have disretion over the ash ow proess and an swith to a
9 While being a standard assumption in the literature, the assumption of Geometri Brownian
Motion implies that EBIT is always non-negative. However, this is no major restrition sine
it does not hange the general behavior of a bank as EBT (i.e. EBIT after interest payments)
an still beome negative, whih then requires outside naning. Furthermore, in pratie,
the EBIT of banks is virtually never observed to be negative.
5
dierent risk parameter
σ.
An inrease in the bank's ash ow risk an be inter-
preted either as a relaxation of monitoring ativities in the sense of less areful
risk management ativities (see e.g. Deamps et al., 2004), or as a shift in the
investment poliy towards more risky seurities as in the lassi asset substitution
problem (see e.g. Jensen and Mekling, 1976; Green, 1984; Leland, 1998).
As a seond salient harateristi, we adopt the frequently made assumption in
the nanial intermediation literature that banks' assets are sold at a signiant
disount when the bank is losed.
The disount may reet the illiquidity of
banks' assets due to speialized human apital (as e.g. argued in Diamond and
Rajan, 2000, 2001) or adverse seletion osts due to opaity, i.e. information
asymmetry.
10
In this sense, at the time of losure
to have a liquidation value of
11
rate.
xT
Λ = λ r−µ
,
where
We onsider a orporate tax rate equal to
12
the private level.
r
τ
T,
banks' assets are assumed
denotes the risk-free interest
and ignore additional taxes on
Sine the liquidation value annot exeed the after-tax risk-
less value of a perpetual ash-ow stream,
This formulation of
λ
λ
has to lie between zero and
omprises as a speial ase the frequently employed all
equity value after taxes and bankrupty osts whih is well-known as
λ = (1 − α) · (1 − τ )
(1 − τ ).
for proportionate bankrupty osts
xt
λ r−µ
with
α.13
On the liabilities side, the main harateristi of ommerial banks is their ability
to take deposits. We assume the bank to have a given volume of deposits that
require an aggregate ontinuous, instantaneous interest payment of
d
d.
We take
to be exogenous. The reasoning behind the fat that in our model, the bank
annot arbitrarily hoose the deposit volume is as follows. On the one hand, the
available amount of deposits is limited so that a given maximum amount an
hardly be exeeded without oering unreasonable high deposit rates.
On the
other hand, deposits are a possibility to reate a lose relationship to ustomers
10 See e.g. Stein (1998) for an adverse seletion model, and Morgan (2002) for evidene on the
opaity of banks' assets.
11 As it is standard, r is assumed to be onstant and µ < r to ensure nite solutions.
12 This is only for reasons of parsimony and does not hange subsequent results as long as there
is a tax advantage to debt naning after orporate and personal taxes. Furthermore, we
assume that losses ause immediate (negative) tax payments, whih ontrasts with a more
realisti asymmetri tax-regime, but whih is imposed for tratability reasons.
13 See e.g. Leland (1994), Goldstein et al. (2001), and Morelle (2004).
6
and an therefore be understood as an option to future valuable deals (e.g. after
investing into deposits, a ustomer might want to nane the aquisition of a
ostly good by a loan from the bank). Thus, the bank must keep the deposits on
a given level to avoid an ineient redution of the future business. As a result,
our model aounts for the fat that banks have a positive deposit volume with
an exogenous nature.
As long as the bank is solvent, depositors reeive their ontratual interest payment
d.
The residual ash ow
bank's equity holders. Sine
xt
(xt − d) (1 − τ )
is paid out as dividends to the
an be onsidered as the earnings before interest
and taxes (EBIT), the dierene
xt − d
denotes EBT, and thus
(xt − d) (1 − τ )
denotes the after tax dividend payment. We assume that total revenues are entirely paid out to investors.
debt struture.
Moreover, we do not allow for adjustments of the
With this stati view, we aount for the fat that espeially
during times of a banking risis, banks only have very limited possibilities to
adjust the apital struture. Given that the apital struture ould be arbitrarily
adjusted at no osts, a default ould be prevented in our model (see e.g. Koziol
and Lawrenz, 2009).
A typial approah within this model lass from the orporate nane literature (see e.g. Fisher et al., 1989; Leland, 1994; Goldstein et al., 2001) is to assume
perfet markets and unregulated rms. Therefore, equity holders are willing to
injet apital as long as they onsider it to be a worthwhile investment, and default of a rm an only our for reasons of indebtedness. The default event is
a result of the optimizing behavior of owners and therefore endogenous. While
this may be a reasonable approah for modeling the default deision of standard
non-nanial rms, it may be less appropriate for banks for basially two reasons.
First, banks are regulated and not at least due to mandatory minimum apital
requirements, the default deision is subjet to an exogenous onstraint.
This
is onsistent with e.g. Deamps et al. (2004). Seond, as witnessed during the
reent banking risis, banks in nanial distress fae various diulties or osts
in raising apital. Thus, severe nanial fritions in times of stress also impose
exogenous onstraints on the naning of banks, and an result in default for
reasons of illiquidity. The importane of the inability to raise apital in times of
7
stress need not to be emphasized in light of the reent turmoil on international
nanial markets and is the very reason that underlies the proposal for ontingent
onvertibles.
For these reasons, we onsider nanial distress as being triggered by an
ogenous onstraint
ex-
that may be interpreted in both ways. Either as regulatory
intervention, or as the inability to raise further apital. In tehnial terms, nanial distress is modeled as a stopping time, whih is dened as the rst time, the
xt
ash ow proess
hits the boundary
ξ
from above, i.e.
Tξ = inf{t; xt ≤ ξ}.
It is reasonable to expet that the nanial onstraint, i.e. the boundary
ξ
will
depend on the extent of the bank's debt liabilities. In terms of a regulatory intervention threshold, it is onsistent with a minimum apital requirement. Therefore,
we onsider the exogenous boundary to be related to the bank's debt liabilities
in a linear way:
ξ(π) = φ π,
where we use
π
(2)
to generally denote the bank's instantaneous payments to all
debt holders, and
φ>0
as a onstant that determines the severity of nanial
onstraints.
Note that in order to be binding, the exogenous threshold
as high as the endogenous default threshold
ξ ∗ (π)
ξ(π)
must be at least
that would be optimal for the
shareholders without any exogenous restritions.
Under the assumptions of the model, it is standard to show that the valuation
of equity holders' laim, denoted as
Q
St = E
Z
Tξ
−r(s−t)
e
t
= (1 − τ )
Note that
β
xt
ξ
St ,
is given by
14
(1 − τ )(xs − π) ds
xt
π
−
r−µ r
−
ξ
π
−
r−µ r
xt
ξ
β !
,
(3)
is to be interpreted as a probability-weighted disount fator, or
as the state prie of one unit of aount onditional on the event that
14 See e.g. Fisher et al. (1989), Mella-Barral (1999) and Deamps et al. (2004).
15 To ease notation, we will be sloppy sometimes, and write ξ without argument.
8
xt
hits
ξ .15
β
is the (negative) solution to the quadrati equation
given by
β = −(µ − σ 2 /2 +
q
σ2
β(β
2
2 r σ 2 + (µ − σ 2 /2)2 )/σ 2 .
− 1) + µβ − r = 0,
For ease of notation, we use the following onvention. Write V(y, π) = (1
β
y
τ ) r−µ
− πr , and D(y, y ′) = yy′ , then (3) is more ompatly expressed as
St = V(xt , π) − V(ξ, π)D(xt, ξ).
−
(4)
π
If the bank has only deposits, the total interest payment to debt holders,
equals
d.
We assume deposits to be insured, where the bank has to pay a fair
deposit insurane premium. Sine the work of Merton (1977), it is well known
that the value of the deposit insurane protetion orresponds to the value of a
orresponding put option, whih we denote by
I .16
For a given deposit volume
d,
the insurane premium is found to be
It = max
λ
d
ξ − , 0 · D(xt , ξ).
r−µ
r
(5)
Obviously, insurane makes deposits riskless. If depositors disount future interest payments at the risk-free rate
equals
D = d/r .
the aggregate deposit value, denoted as
Therefore, sine the bank pays
deposits for the bank is
2.2
r,
It
D,
for protetion, the value of
D − It .
Bank With Bond Finaning
While deposits are given, the bank an aess apital markets to satisfy additional naning needs. In the following, we will ompare two types of bonds, a
straight bond and bonds exhibiting the mandatory onvertibility feature.
For
tratability, we assume a straight bond to be a xed-oupon onsol bond that
promises a ontinuous instantaneous oupon payment
tal interest payment
π
of the bank is therefore:
debt liabilities will raise the default threshold
b
to bondholders. The to-
π = d + b.
ξ,
In general, the higher
sine due to the higher interest
payment obligation, the bank runs into nanial diulties already at higher ash
ow levels. In ase of default, most banking systems stipulate seniority among
debt laims for depositors with bondholders being subordinated. Aording to
16 An alternative onsideration of 'heap' deposits would not ruially aet our analysis beause the important relationship between the equity value and risk still remains.
9
this priority struture, the available reovery value should be split in a way that
bondholders obtain part of the liquidation value only if depositors have been
ompensated in full.
In pratie, absolute priority annot always be enfored,
and the atual split-up is often the result of a bargaining proess. It would be
straightforward to expliitly model the sharing rule as a (Nash) bargaining solution following e.g. Fan and Sundaresan (2000) and Mella-Barral and Perraudin
(1997). However, for our purposes, the preise sharing rule is not ruially important for the following reason. As long as equity holders' laim in default is
not aeted by the sharing rule,
17
the split-up among debt holders is irrelevant
to equity holders and therefore does not aet any deisions made by the bank
owners.
We take this fat as justiation to pursue a redued-form modeling
approah, and to assume that a ertain fration
θ
of the liquidation value
to bondholders. Similar to (3), the value of the straight bond at time
as
Bt ,
t,
λξ
goes
denoted
an be expressed as
b
b
Bt = + θλξb −
D(xt , ξb ).
r
r
Note that the knowledge of
total debt value
B0 + D − I0
θ
is not required for our analysis, beause only the
net of insurane osts will be required, whih is inde-
pendent of the distribution of the bank between the depositors and bondholders
in the ase of a default. The equity value an still be expressed as in (4), with
π = d + b,
b
and a presumably higher
ξb
whih we distinguish with the supersript
to indiate the straight bond ase.
While we take the deposit volume as exogenous, we assume the bank to optimally adapt their remaining naning needs by issuing a orresponding amount
of bonds (proxied by the aggregate oupon payment). Therefore,
variable, and the optimal bond oupon
b∗
b
is a hoie
is the result of the maximization of the
value of all laims net of the value of the deposit insurane premium
b∗ = arg max{Stb + Bt + D − It }
b
As usual, the maximization of the bank value
Stb + Bt + D − It
the maximization of the equity holders' wealth
Stb + Bt − It .
is onsistent with
This is a result of
17 This ondition is most obviously satised if priority between debt and equity holders is
enfored, and equity holders' laim in default is zero.
10
the notion that equity holders keep the stoks, pay the deposit insurane
ash injetions and reeive a speial dividend equal to the bond value
oupon
Bt .
with
(Sine
d/r
and is therefore independent of the
ξb = φ · (d + b),
the rst derivative of the bank value
the deposit value is insured,
D
It
equals
b.)
With the general onstraint
Vtb = Stb + Bt + D − It
with respet to
b
is
τ
ζ
∂ Vtb
= − D(xt , ξb ),
∂b
r
r
where
ζ =
(1−β)((r−µ)τ +φr((1−τ )−λ)
. From the properties
r−µ
λ ≤ (1 − τ ) < 1,
for
it follows that
D(xt , ξb) → 0,
that
(6)
D(xt , ξb)
ζ >τ >0
ξb
and
D(xt , ξb ) → 1.
The fat
(i.e. in the debt level) onrms the
existene of an optimal debt level. Solving the rst order ondition yields
xt
b =
φ
∗
0 ≤
whih implies a positive derivative
while the derivative is negative for
inreases monotonially in
β < 0, φ > 0
−1/β
τ
− d.
ζ
b∗
as
(7)
The rst term on the right-hand side of (7) determines the debt apaity of the
bank whih depends upon its growth potential, risk, urrent ash ow level, the
liquidation osts
λ
and the strength of nanial onstraints
veried from (7), high liquidation osts (small
onstraint (high
λ)
φ) urtail the bank's debt apaity.
φ.
As it an be
as well as a strong nanial
Note that in the optimum, a
bank issues less bonds if it has already a high volume of deposits. We assume that
the exogenously given deposit volume is suh that the bank has not exhausted
its debt apaity, i.e.
2.3
b∗ > 0.
Bank With CoCo Finaning
As an alternative to raise apital with a straight bond, the bank an issue ontingent onvertible (CoCo) bonds. Before onversion, CoCo bonds are equivalent
to ordinary straight bonds in that they pay a xed oupon rate. We denote the
ontinuous instantaneous oupon payment of CoCo bonds as
c.
The main dif-
ferene to straight bonds is the onversion feature. The ontrat of CoCo bonds
stipulates that onversion will take plae one the (ore) apital ratio of the bank
falls below a ertain threshold.
Upon onversion the former bondholders will
11
hold an equity laim, i.e. they reeive newly issued shares, whih entitles them
to a fration of the bank's prot. In our model setup this is neatly reeted by
introduing the parameter
γ,
whih denotes the fration of the bank's after-tax
prots that goes to former CoCo bondholders. If the number of shares (after onversion) is normalized to 1,
γ
is also neatly interpreted as the number of shares
going to former CoCo bondholders. The fat that former equity holders reeive
the remaining fration
1 − γ,
whih is less than before onversion, is onsistent
with the dilution eet of a seasoned new issue. Obviously, former bondholders
also reeive proportionate voting rights.
To formalize the onversion feature, we dene the ash ow barrier
χ
as the
threshold at whih the bond is onverted into equity. Sine after onversion, the
debt only onsists of deposits, the bank has a smaller interest payment obligation.
The redued debt obligations lower the default threshold
supersript
c
to indiate the ase of the CoCo bond.
threshold denition in (2), we have
of
χ
χ = φ · (d + c)
ξc ,
where we add the
By applying the general
and
ξc = φ · d.
This hoie
ensures that a onversion takes plae at a time when the bank would fae
nanial distress if it had not issued bonds with a mandatory onvertibility fea-
χ
ture. Furthermore, the hoie of
shields for given interest payments
CoCo bonds at time t, denoted by
also ensures that the highest volume of tax
d+c
18
an be generated.
The valuation for
Ct , and the value of initial (old) equity holders
is given by
c
(1 − D(xt , χ)) + γ D(xt , χ) (V(χ, d) − V(ξc , d) D(χ, ξc))
r
= V(xt , d + c) − V(χ, d + c) D(xt , χ) +
Ct =
Stc
(1 − γ) D(xt , χ) (V(χ, d) − V(ξc , d) D(χ, ξc))
As
xt → ∞, D(xt , χ) → 0,
and
Ct
approahes
c/r ,
(9)
i.e. for very high ash ow
levels, the value of the CoCo bond approahes its risk-free value.
χ, D(xt , χ) = 1,
(8)
At
xt =
and CoCo bondholders are holding an equity laim worth
γ (V(χ, d) − V(ξ, d) D(χ, ξc)),
i.e. a fration
γ
of total equity.
Eventually, we are interested in evaluating if a CoCo bond is an attrative
alternative for the bank.
At rst glane, it seems meaningful to ompare the
18 We explore the impliations of dierent hoies for the onversion threshold in setion 4.3.
12
bank's equity value when it has a straight bond outstanding to the ase when it
raises the
same amount of debt with a CoCo bond.19
However, suh an approah
ignores the fat that a CoCo bond issue an hange the overall debt apaity
of the bank. Therefore, we need to solve a related maximization problem as in
the previous setion, i.e. the optimal CoCo bond oupon
where
Vtc = Stc + Ct + D − It
c∗
has to solve
maxc Vtc ,
is the bank value (inluding the deposit insur-
ane obligation) under CoCo naning. Plugging in from (8) and (9), the rst
derivative of
Vtc
with respet to
c
is
τ
ζc
∂ Vtc
= − D(xt , χ),
∂c
r
r
where
ζc =
τ (π−c β)
. Again,
π
be veried that for
as
c
∗
D(xt , χ)
ζc > τ > 0
D(xt , χ) → 0,
(10)
so that as in the previous setion, it an
the derivative is positive, while it hanges sign
tends to one. Thus, a maximum in the oupon
c
exists. Although
annot be determined analytially, we will show in the next setion, that the
optimal oupon of a CoCo bond will be higher than the optimal oupon of a
straight bond.
3
CoCos as Solution Against Banking Crises Complete Contrats
With the framework set up in the previous setion, we an now address the issue
whether a CoCo bond is an attrative alternative for banks and has the potential
to mitigate future banking rises.
We analyze this issue along two dimensions.
First, we onsider whether the use of CoCo bonds is worthwhile for bank owners.
Seond, we analyze whether the use of CoCo bonds is beneial to the eonomy
in the sense that it dereases the risk that a bank runs into nanial distress
measured by both the default probability and the osts of distress.
19 For the analysis of rating-trigger step-up bonds, Bhanot and Mello (2006) take suh an
approah. For a disussion, see also
?.
13
3.1
Inentive Compatibility
As a rst result, we show that for a given deposit volume, the bank has a higher
debt apaity if it issues CoCo bonds relative to straight bonds. In other words,
the optimal oupon of a straight bond issue is lower than the optimal oupon of
a CoCo bond. To derive this result, we establish the following laim.
For a given ash ow level x0 , deposit volume d, all admissible parameter values (i.e. 0 < µ < r, β < 0, 0 < φ, 0 < τ < 1, and λ < (1 − τ )), and
any given oupon level k suh that ξ = φ · π < xt , it holds
Lemma 1
∂ Vtc ∂ Vtb <
.
∂ b b=k
∂ c c=k
The proof is in appendix A.1.
From lemma 1, two important onlusions an
∂ Vtc
be drawn. Sine the slope
of the bank value with CoCo bond naning is
∂c
∂ Vtb
positive for a given oupon k for whih the orresponding slope
= 0 of the
∂b
bank value under straight bond naning is zero, it immediately follows that the
optimal oupon
c∗
of the CoCo bond must be larger than
b∗
of the straight bond.
Seond, the bank's rm value under the optimal oupon hoie is higher for CoCo
bonds than for straight bonds, i.e.
V b (b∗ ) < V c (c∗ ).
This inequality is a result of the fat that for
c = 0, Vtb
must equal
Vtc
sine
χ = ξ.
Thus, both rm value funtions originate in the same point. By integrating, we
obtain
hold
V b (k) < V c (k) from lemma 1 for any given oupon k
b
∗
c
∗
V (b ) < V (c )
due to
∗
∗
c >b
and therefore it must
.
Eonomially, the higher bank value is a result of the fat that bonds with a
higher oupon an be issued that reate additional tax shields. For the ase of a
CoCo bond, this advantage of debt is not ountered by the typial disadvantage
of higher expeted distress osts, beause due to the onvertibility feature, a
onversion takes plae before the bank will run into nanial distress.
As a
result, additional tax shields are reated by CoCos without inreasing the default
probability. It is worth noting, that the same qualitative result will also hold, if an
14
alternative trade-o model for debt naning is onsidered. Thus, the assumption
of a tax shield is not ruial, and ould be substituted by some other advantage
of debt naning.
3.2
Severity of Finanial Distress
From the systemi perspetive that a regulator will take on, two aspets of nanial distress are interesting. On the one hand, the severity of nanial distress
an be evaluated as the expeted (present value of ) distress osts. On the other
hand, it is also important to quantify the likelihood that a distress event will
our within a given time period. Formally, the bank enters nanial distress in
xt
our setup when the ash ow
hits the distress threshold, whih is
ase of straight bond naning and
ξc
ξb = φ · (d + b).
T
probability that the ash ow proess
x0 .
xt
hits the boundary
20
z̄ − µ̂ T
√
σ T
z̄ = log(ξ/x0 ), µ̂ = µ − σ 2 /2,
umulative distribution funtion.
immediately obvious that
Pξ,T
the default threshold
ξc = φ · d
is equivalent within our framework to the
From standard results,
Pξ,T = Prob{Tξ < T } = N
where
π = d,
The probability that the bank enters nanial
distress within a given time horizon
urrent level of
in the
for CoCo bond naning. Sine interest
payments after onversion are redued to
is lower than
ξb
from above given a
this an be alulated as
+ exp
and
ξ
N(·)
Note that
z̄
2 µ̂ z̄
σ2
N
z̄ + µ̂ T
√
σ T
,
(11)
denotes the standard normal
inreases in
ξ,
from whih it is
is monotonially inreasing in the threshold
Therefore, it follows for any time horizon
T
ξ.
that the default probability with
CoCo bond naning is smaller than the orresponding probability under straight
bond naning
Pξc ,T < Pξb ,T .
Apparently, the inequality holds for any arbitrary drift rate
µ
so that we do not
need to distinguish between the physial and risk-neutral drift rate.
As a seond measure of the severity of nanial distress, we an ompare the
expeted distress osts between straight bond and CoCo bond naning.
20 See e.g. Björk (1998), Ch. 13.
15
We
nd the same unambiguous positive eet of CoCo bonds and report details in
Appendix A.2. We summarize the ndings in this setion in
(Contingent onvertibles in a omplete ontrat setting) Under
the possibility to write omplete ontrats, CoCo bonds are inentive-ompatible
in the sense that an optimally designed bond inreases the bank's rm value and
mitigates the severity of nanial distress doumented by a lower probability of
nanial distress as well as lower present value of distress osts.
Proposition 1
Figure 1: Bank value and distress probabilities.
b
The left panel plots the rm value for a bank with straight bonds, Vt (solid line) and with
c
CoCo bonds, Vt (dashed line) as funtion of the oupon rate c and b. The right panel plots
the probability of nanial distress Pξ,T (on log-sale) for straight bonds (solid line) and
T . Note that Pξ,T is plotted under the
physial measure, and thus represents the atual probability of default. It is alulated by
P
exploiting the relationship µ = µ − ψσ , and assuming a market prie of risk equal to ψ = 0.5.
CoCo bonds (dashed line) for inreasing time horizons
Parameter values are:
xt = 1, d = 1.5, λ = 0.55, r = 0.08, µ = 0.05, τ = 0.35, σl = 0.15.
V
Pξ,T
0.1
28
26
0.01
24
0.001
PSfrag replaements22
PSfrag replaements 10-4
20
0.2
0.4
0.6
0.8
c,b ,
1.0
2
4
6
8
10
T
10-6
As an illustration of the ndings in proposition 1, gure 1 reports numerial
values for the rm value (left panel) of a bank having issued straight bonds (solid
line) and the value of a bank using CoCo bonds (dashed line). The graph reveals
that with CoCo bonds the debt apaity of the bank is substantially higher whih
in turn leads to an inreased rm value at the optimal oupon rate. The right
panel plots the probability for nanial distress (on log-sale) for a bank with
an optimal amount of straight bonds (solid line) and for a bank with an optimal
CoCo bond (dashed line).
Again, the numerial example onrms the general
nding that for any time horizon
T,
the probability of nanial distress is lower
for banks having issued CoCo bonds.
16
4
CoCos as Origin of Banking Crises Inomplete
Contrats
The previous setion has demonstrated that CoCo bonds have substantial beneial eets on banks as well as on the banking system.
Not only do CoCo
bonds raise the bank's debt apaity whih inreases rm value, it also dereases
the probability of running into ostly nanial distress beause of the automati
impliit reapitalization mehanism. A folk wisdom in nanial eonomis is the
saying that there is no suh thing as a free lunh. So, one may wonder whether
there are really only advantages to CoCo bonds, or if there is a hidden drawbak.
On an abstrat level, a large body of theoretial literature has stressed the role of
debt ontrats as being a disiplining devie on the deision making of managers
and owners. Jensen (1986)'s free ash-ow hypothesis or the role of short-term
debt as ommitment devie stressed by Calomiris and Kahn (1991) and Diamond
and Rajan (2000, 2001) are prominent examples.
From a ontrat-theoretial
perspetive, in partiular Innes (1990) and Hart and Moore (1998) are important
ontributions whih establish the optimality of a debt ontrat. They show that
under the ondition of inomplete ontrats, a naning arrangement is optimal
where the nanier reeives a xed payment in good states of the world, while
being handed over ontrol rights in bad states of the world. This naning arrangement resembles exatly a debt ontrat. The ruial insight is the fat that
managers (or the entrepreneur) is disiplined by the threat of being expropriated
in bad states. Note that CoCo bonds undermine preisely this disiplining feature
of debt ontrats. In bad states of the world, former owners do not loose their
ontrol rights immediately. On the ontrary, former bondholders are obliged to
onvert their debt laim into an equity laim.
Arguably, absent the threat of
losing ontrol rights, owners fae dierent inentives.
In this setion, we analyze this doubt formally by assuming ontrats to be inomplete in the sense that manager-owners an hange the investment poliy, or
more preisely that the hoie of investment poliy is observable, but not ontratible.
The issuane of CoCo bonds instead of straight bonds has arguably
an important impliation for the risk taking behavior of bank manager-owners.
Intuitively, sine the bank's equity holders enjoy the full benets from a ash ow
17
inrease but are proteted against a default in a less favorable state due to the
mandatory onversion, they might prefer a more risky strategy ompared to the
ase with straight bond naning. For this reason, we introdue the problem of
risk-shifting to the bank's naning deision and evaluate both the impat on
equity holder's wealth and the severity of running into nanial distress.
4.1
Risk Taking Inentives
In this setion, we assume that bank managers have an unique option to inrease
the riskiness of the bank's operating prot. In partiular, we an think about the
risk-shifting option as the possibility of managers to hoose dierent tehnologies
suh as the bank's possibility to relax their monitoring eort or to put less resoures into an eetive risk management system (shirking) as e.g. in Deamps
et al. (2004). While these ations avoid osts and ould therefore raise (expeted)
prots, unmonitored reditors have moral hazard inentives whih inreases the
riskiness of returns.
Alternatively, with respet to their (proprietary) trading
ativities, the risk-shifting option is to be interpreted as the investment in more
risky seurities whih inreases the overall asset value risk in the sense of the lassi asset substitution problem as in Jensen and Mekling (1976), Green (1984) or
Leland (1998).
In formal terms, we assume that the urrent ash ow risk is
managers have an irreversible option to inrease risk to
purposes relevant risk-neutral) drift
µ
σh ,
σl
and that bank
while the (for priing
21
remains unaeted.
To ompute the de-
fault probability under the physial (true) probability measure, knowledge of the
physial drift
µP
physial drift via
is also required. Sine the risk-neutral drift
P
µ = µ − ψ σ,
the physial growth rate
for a given positive market prie of risk
ψ > 0.
µ
P
µ
is related to the
rises with the risk
σ
The assumption that an inrease
in risk is ompensated by a orresponding inrease in the (physial) drift rate is
22
preisely onsistent with the notion of shirking.
We abstrat from manager-owner onits whih would only add another layer of
ageny onits and assume that deisions are made in the interest of the owners.
21 This modeling approah is standard within the asset substitution literature as e.g. in Leland
(1998), Erisson (2000), and
?.
22 See e.g. Deamps et al. (2004).
18
In this setion, we analyze the inentives of equity holders to shirk, i.e. to hoose
the high risk investment program. We rst show that the nanial onstraints
are ruial to understanding risk taking inentives.
One the rm has issued the bond, equity holders have an inentive to hoose the
high risk tehnology if this inreases the value of their laim. In formal terms, the
risk preferene an be evaluated by determining the sign of the rst derivative of
the equity laim with respet to the risk parameter
derivative of the equity value
Stb
Taking the rst partial
for straight bond naning w.r.t.
σ
yields,
∂St
∂
= −V(ξ, π)
D(xt , ξ, σ)
∂σ
∂σ
∂β(σ)
xt
·
.
= −V(ξ, π) D(xt, ξ, σ) · log
ξ
∂σ
xt
> 0,
immediately obvious that D(xt , ξ, σ) > 0 and log
ξ
While it is
∂β(σ)
straightforward to show that
∂σ
depends on the rst term
V(ξ, π),
> 0.
V(ξ, π)
(12)
it is also
Therefore, the sign of the derivative
whih was dened as
V(ξ, π) = (1 − τ )
Note that
σ.
ξ
π
−
r−µ r
.
(13)
is neatly interpreted as the present value from a riskless after-
tax perpetual net inome stream given a urrent earnings level of
interest obligation
π , V(ξ, π)
for suiently high levels of
is negative as long as
ξ , V(ξ, π)
ξ
ξ.
For a given
is suiently small, while
turns out to be positive.
Eonomially, with striter nanial onstraints (i.e. with a higher boundary
ξ ),
equity holders loose a more valuable ash ow stream relative to the given debt
liabilities if default ours, whih makes them relutant to the risk of running into
nanial distress. Thus, the earlier a bank is fored into nanial reorganization,
the more severe is the negative eet for the equity value. As a result, striter
nanial onstraints are an inentive for the bank to at more prudent and to
avoid exessive risks taking. We highlight the nding as
In the ase of straight bond naning, equity holders' risk preferenes
depend on the exogenous onstraints ξ . If onstraints are suiently weak in the
sense that the threshold ξ is low, equity holders have inentives to inrease risk,
i.e. ∂St /∂σ > 0. If onstraints are strong in the sense that ξ is suiently high,
equity holders have inentives to avoid risks, i.e. ∂St /∂σ < 0.
Lemma 2
19
From (13) we an immediately derive the ritial threshold where risk preferenes
swith, i.e. where the derivative is zero,
threshold
ξˆ diretly
∂St
∂σ
= 0,
for any
xt > ξ .
This ritial
follows from (2) and satises
r−µ
ˆ
ξ(π)
=π
.
r
For any
ξ
above
ξˆ,
(14)
owners dislike higher risks, while for any
holders have inentives to inrease risk. At
ξˆ,
ξ
below
ξˆ,
equity holders are indierent with
respet to the risk strategy. In other words, the equity laim is onvex in
ξ < ξˆ,
while being onave for
ξˆ < ξ
equity
and (pieewise) linear for
xt
for
ξ = ξˆ.
Note that if the bank does not fae any exogenous onstraints, the optimal endogenous threshold an be shown to be
ξ ∗ (π) = π
r−µ β 23
.
Sine the last fration
r β−1
is always smaller than one, the endogenous threshold is always below
ξˆ whih
shows that absent any nanial onstraints, equity holders always have the usual
all option-like inentive to inrease risk.
The threshold
ξˆ
displays two intuitively reasonable harateristis.
threshold depends on the growth potential of the bank. For higher
First, the
µ, ξˆ is
lower,
thus, a bank with good growth prospets is able to raise apital for ash ow
levels where a omparable bank with a lower growth rate already faes nanial
diulties. Seond, it inreases with the interest payment
π,
whih is intuitive
sine with higher debt levels the bank is supposed to run into nanial diulties
earlier.
To gauge the impat of a CoCo bond on the risk taking behavior of banks, we
analyze how risk preferenes hange when the bank replaes the optimal straight
bond by an optimal CoCo bond. Optimal bond volumes refer to those levels,
and
c∗ ,
b∗
respetively, that maximize the bank value, whih need not neessarily
oinide with the rst-best solution that ould be obtained in a omplete ontrat
ase. From the previous setion, we know that given the onstraint
ξˆ,
the bank
owners have no inentives to inrease risk, i.e. to shirk when having issued a
straight bond. In what follows, we onsider the ritial level
ˆ
ξ(π) = ξ(π)
at whih
the bank for straight bond naning is indierent about its risk preferenes as the
nanial onstraint
ξ(π)
for the bank in order to have a meaningful omparison
for CoCo bond naning.
23 See e.g. Dixit and Pindyk (1994), Mella-Barral (1999), and Deamps et al. (2004).
20
Again, we need to alulate the rst derivative of the equity value
respet to
σ.
Stc
with
The basi idea of CoCo bonds is to provide banks with a preom-
mitted reapitalization at a time where they fae diulties in raising external
naning. In line with this reasoning, we assume the onversion to our at the
time when the external onstraint would be binding if there were no onvertibility option, i.e. at the threshold
ˆ + c).
χ = ξ(d
The onversion avoids immediate
nanial distress beause it lowers debt obligations. However, if operating prots
ontinue to deline, the bank eventually may fae nanial diulties when ash
ows deteriorate further to the lower threshold
From the denition of
χ
and by dierentiating
Stc
and
ξc ,
the terms
with respet to
ˆ .
ξc = ξ(d)
V(χ, d + c) and V(ξc , d) in (9) are zero,
σ,
we are left with
∂Stc
∂
= (1 − γ)V(χ, d)
D(xt , χ)
∂σ
∂σ
xt
∂β(σ)
·
.
= (1 − γ)V(χ, d)D(xt , χ) · log
(15)
χ
∂σ
xt
> 0 and
Again, the sign of the derivative depends on V(χ, d), sine log
χ
∂β(σ)
χ
d
>
0
. By the denition of V and χ, this equals V(χ, d) = (1−τ )
−
=
∂σ
r−µ
r
(1 − τ ) rc > 0 whih is always positive for γ < 1. Therefore, we derive the
important result that, as long as initial equity holders retain a positive fration
of ash ow rights (γ
< 1),
∂Stc
is always positive, whih means that equity holders
∂σ
always have an inentive to engage in risk-shifting ativities. We highlight this
result as
(Risk taking in inomplete ontrat setting) Suppose a bank has
ˆ + c) and ξc =
issued straight bonds and faes nanial onstraints (χ = ξ(d
ˆ ), whih leaves manager-owners indierent with respet to the risk strategy,
ξ(d)
b
t
= 0. If that bank replaes the straight bond by an optimally designed CoCo
i.e. ∂S
∂σ
bond where initial equity holders retain a positive fration of ash-ow rights,
c
t
i.e. γ < 1, manager-owners always have an inentive to inrease risk, i.e. ∂S
> 0.
∂σ
Proposition 2
The proposition shows that for the ase where exogenous nanial onstraints
are suh that
ˆ
ξ(π)
= φ̄ · π
with
φ̄ = (r − µ)/r ,
a CoCo bond for straight bond
swap leads to risk-shifting inentives. Although we onsider the ase, where the
21
bank initially has no inentives to hange its business risk as the most relevant
24
situation,
this partiular hoie does not limit the generality of the eonomi
insight in our main result in proposition 2. The next subsetion shows that the
results an be generalized to the ase of arbitrarily strit (or weak) nanial
onstraints.
Consider the general exogenous threshold
ξ = φ·π
for an arbitrary
φ > 0.
We
rst show that if the CoCo bond for straight bond swap is strutured in suh
a way that the oupon payment is kept onstant, i.e
b = c,
then a risk-shift
inreases the bank's equity value more strongly under CoCo bond naning than
under straight bond naning as indiated by the following lemma:
For any arbitrary thresholds χ = ξb and ξc with ξc < χ and for CoCo
bonds that replae straight bonds with the same oupon, i.e. b = c, a CoCo bond
inreases risk taking inentives of manager-owners, in the sense that
Lemma 3
∂Stb
∂Stc
<
.
∂σ
∂σ
The proof of lemma 3 is in the appendix. From lemma 3 together with lemma 1,
we an derive the following more general proposition about risk taking inentives
for an arbitrary onstraint and optimal debt level hoies.
(Generalization of risk taking inentives) If nanial onstraints
are weak, then the bank will always prefer the higher risk with both optimal straight
bonds as well as optimal CoCo bonds. For strong nanial onstraints, the rm
will always prefer the low risk for optimal straight bond naning, while it might
still prefer the higher risk under optimal CoCo bond naning.
In tehnial terms, for arbitrary onstraints ξ = φ · π and optimal hoie of the
b
∂Stc t ˆ, then ∂Stb ∗ < 0 and
debt level, if ξ < ξˆ, then 0 < ∂S
<
.
If
ξ
>
ξ
∗
∗
∂σ b=b
∂σ c=c
∂σ b=b
Proposition 3
∂Stc ∂σ c=c∗
≷0
The proof is in the appendix A.4.
Proposition 3 generalizes the ndings from
proposition 2 and lemma 3 to the ase of arbitrary nanial onstraints and optimal bond hoies. Importantly, the result shows that it will never be the ase
24 The argument is that the initial bank, that has disretion over its risk poliy, has already
adapted its desired risk level and is in 'equilibrium' with respet to the hoie over
22
σ.
that the bank has preferenes for the high risk under straight bond naning, but
preferenes for the low risk under CoCo bond naning.
The intuitive reason for why straight bonds impose lower risk-shifting inentives
on the equity holders than CoCo bonds is due to the wealth eet for the equity holders in partiularly poor states. In the ase of CoCo bonds outstanding,
there are poor states in whih onversion of the CoCo bonds takes plae and
the (old) equity holders still keep their valuable equity laims.
In the ase of
straight bond naning, however, the same state leads to default, in whih ase
the equity holders are left with nothing aording to absolute priority rules. As
a result, banks with CoCo bonds outstanding have a lower inentive to prevent
poor states so they are willing to aept a higher ash ow risk than the equity
25
holders for straight bond naning would be willing to aept.
This more general nding shows that the distortion of risk shifting inentives
whih are indued by a CoCo bond for straight bond swap is robust with respet
to the assumption on the exogenous nanial onstraints. This leads to an important impliation for the ase of a bank that benets from an impliit bail-out
option by the government. If a bank is onsidered to be too big to fail, and the
bank manager-owners antiipate the bail-out ommitment, they will already fae
inentives to engage in risk shifting ativities. Our result in proposition 3 show
that in suh a ase, the risk taking inentives for the bank with a CoCo bond an
even be inreased.
Taken together, proposition 2 and 3 onrm the intuition that in inomplete ontrat settings, debt serves as a disiplining devie, where the disiplining eet
stems from the threat of losing omplete ontrol rights in a bankrupty proess.
If, as in the ase of CoCo bonds, equity holders only lose ash-ow rights but
not omplete ontrol rights, the disiplining impat is mitigated, and managerowners fae distorted risk inentives. Hene, our analysis ontradits results from
25 The result is a neat analogy to the ase of regular onvertible bonds. As rst stated in Green
(1984), regular onvertible bonds an mitigate the lassi asset substitution problem (see e.g.
Jensen and Mekling, 1976), beause onversion takes plae at the disretion of bondholders
in good states. Contingent onvertible bonds are the preise polar ase, where onversion
takes plae mandatorily in poor states, and therefore it is intuitively reasonable to expet
them to aggravate the asset substitution problem.
23
Flannery (2005) who nds that shareholders onfront undistorted risk-bearing in-
26
entives.
Proposition 3 implies that we an distinguish three ases.
First, the ase
where banks with straight bonds already have risk-shifting inentives (e.g. due
to an impliit bail-out ommitment) so that swithing to a CoCo bond naning
reinfores risk inentives. Seond, the ase where banks with straight bonds have
an aversion against inreasing risks, and for whih the swith to a CoCo bond
does not result in a preferene for higher risks.
Third, the ase where banks
with straight bonds have a preferene for the low risk, but where the swith to
CoCo bonds reverses risk preferenes, i.e. where banks will engage in more risky
ativities one having issued CoCo bonds.
4.2
Impat of Distorted Risk Taking Inentives
In this setion, we analyze the impat of risk-shifting inentives on the net-wealth
of the bank, inentive ompatibility onditions and the severity of nanial distress. In partiular, we fous on the last ase identied at the end of the previous
subsetion, where CoCo bonds hange the risk taking behavior.
Sine we onsider an inomplete ontrats setting, the hoie of the risk-strategy
is observable but not ontratible.
Thus, in suh a full-information setup, in-
vestors an rationally antiipate the risk hoies of bank owners and will demand
a orresponding ompensation. This means that the bond will be pried by taking into aount the expeted hoie of the risk parameter
bond is pried by assuming
σl ,
σ.
Thus, a straight
while a CoCo bond will be pried by taking into
aount the risk-shifting inentives, i.e. by assuming
σh .
In general, it is a well
known result within this model lass that a higher risk redues the optimal rm
27
value,
beause within a trade-o model, the bank benets from advantages of
debt naning as long as it remains solvent but inurs losses when it suers from
a default. A higher business risk inreases the likelihood of a default, whih inreases the present value of the losses in the ase of a default without providing
26 See Flannery (2005), p. 12.
27 This nding has already been disussed in the basi model by Leland (1994).
24
any additional advantages.
In formal terms, it is straightforward to show that the rm value
negatively on
σ,
β
x
x
log
< 0,
ξb
ξb
(16)
where the laim follows from the repeatedly used fat that
σ.
depends
i.e.
∂Vtb
τ
= −π
∂β
r
on
Vtb
β
depends positively
A little more algebra shows that this nding arries over to the ase of
CoCo bonds. Taking the derivative yields,
∂Vtc
dτ (µ − r) + ξc r(λ − (1 − τ ))
=
∂β
r(r − µ)
β
cτ x
x
< 0.
log
r χ
χ
As
λ
is bounded above by
(1 − τ ),
and
µ < r,
β
x
x
log
−
ξc
ξc
(17)
the derivative is always negative.
In setion 3.1, we have established from lemma 1 that the optimal rm value
with CoCo bonds is always higher than with straight bonds, i.e.
V b (b∗ ) < V c (c∗ ),
if the risk-strategy is ontratible. As the bank shifts towards a more risky strategy, the bank rm value for a CoCo bond naning delines. If the risk-shifting
possibility is not severe, i.e.
σh
only slightly exeeds the initial risk
σl ,
the bank
value under CoCo naning will still be higher than under straight bond naning.
Conversely, if the risk-shifting possibility is severe, i.e.
σh
is large, the optimal
bank value will be lower than in the ase of a straight bond issue. The left panel
in gure 2 shows in line with (17) that the bank's rm value under CoCo naning,
Vtc
(dashed line), delines monotonially as its risk-shifting option is more
severe, i.e. as
σh
inreases, while the bank value under straight bond naning,
Vtb
(horizontal line), is obviously independent of
for
σh
for whih
long as
σh
Vtc
equals or exeeds
lies between the initial
merial base ase
σ̂ = 0.196),
Vtb
as
σl = 0.15
σh .
We dene a ritial value
σ̂ = sup{σh |Vtc (σh ) ≥ Vtb (σl )}.
and the ritial value
(in the nu-
the bank's rm value is higher under CoCo bond
naning. The right panel plots yield spreads (dened as
bonds and straight bonds.
σ̂
As
c∗ /Ct − r )
for CoCo
While the spread for a straight bond (depited as
horizontal dashed line) is only 50 basispoints, it is substantially higher for CoCo
bonds. Even if bank managers do not engage in risk shifting, the yield spread
25
Figure 2: Bank value and yield spreads
c
The left panel plots the dierene between the bank rm value with CoCo bonds, Vt and the
b
b
c
b
b
bank rm value with straight bonds, Vt , as a perentage of Vt i.e. ∆V% = (Vt − Vt )/Vt . The
horizontal line indiates the ritial level
σ̂
for whih the dierene is zero. The right panel
plots yield spreads in basispoints for straight bonds (horizontal solid line) versus spreads of
CoCo bonds (dashed line). Parameter values are:
xt = 1, d = 1.5, λ = 0.55, r = 0.08, µ = 0.05,
τ = 0.35, σl = 0.15, γ = 0.4.
ys
∆V%
700
σ̂
0.04
500
0.02
PSfrag replaements-0.02
600
400
0.16
0.18
σh300
0.20
0.22
0.24
PSfrag replaements200
100
-0.04
0.16
-0.06
0.18
0.20
0.22
0.24
σh
amounts to 325 basispoints. The spread rises with the risk shifting opportunity,
and bond investors demand a spread of 524 basispoints if the bank has inentives
to raise the risk to
σh = 0.20.
The substantially higher spreads are roughly in
line with evidene from the industry where CoCo bonds are expeted to trade up
28
to an additional 300 basispoints interest margin.
The higher spreads are due
to two fators. On the one hand, a bank with CoCo bonds has a higher leverage,
and on the other hand, CoCo bondholders demand an additional premium for
the fat that onversion mandatorily takes plae at a time when equity values are
low. Thus, the higher spreads partly reet this insurane premium.
Two points about the results are noteworthy: First, despite the fat that CoCo
bond naning is assoiated with a higher risk
σh , the overall bank value an still
be higher relative to the ase of straight bond naning. The reason for the value
inrease lies in the relaxation of nanial onstraints (i.e. a lower default barrier)
whih inreases the bank's debt apaity and enables the bank to exploit debt
advantages to a larger extent. In our setup, the bank is able to generate a higher
instantaneous tax shield. Note that the value inrease due to higher tax shields
is at the expense of the tax authority, whih in turn reeives less taxes.
Seond, although CoCo bonds inrease the bank rm value via a higher debt a28 See
Finanial Times UK experiment raises prospets of new asset lass, November 5, 2009.
26
paity, bond investors demand a substantially higher spread whih inreases with
the riskiness of the asset value.
Spreads are high beause by onstrution the
onversion of CoCo bonds takes plae at a time when onversion is not favorable
from the perspetive of investors, i.e. CoCo bond investors obtain a fration of
the equity laim preisely in states where equity values are low. Our ndings are
an important warning signal against arguments as e.g. mentioned in Flannery
(2005, 2009) that the internalization of risk-taking osts will deter banks from
taking too muh risk. Our results show that although banks bear the osts of
risk-taking via substantially higher spreads, they an still have an inentive to
engage in risk-shifting ativities.
The nding that a CoCo bond issue for
σh < σ̂
results in a higher overall
bank rm value implies that equity holders obtain a higher wealth by issuing
CoCo bonds.
Our analysis demonstrates the important insight that although
reditors antiipate a potential risk-shift so that they pay a fair prie for the
investment, owners are still better o. As a result, from an investors' perspetive,
CoCos are a Pareto improvement beause no investor suers and some obtain a
stritly higher wealth.
However, it is important to stress that although from the perspetive of the bank
CoCo bonds are Pareto-optimal, it is a priori not lear whether the use of CoCo
bonds is also desirable from a systemi point of view. To evaluate the systemi
eet of CoCo bonds, we onsider the probability of nanial distress for the
banking system in the following paragraph.
As in setion 3.2, we ompute the bank's probability of running into nanial
distress.
To ompare a bank with straight bonds to a bank with CoCo bonds,
not only the threshold
parameter
σ.
ξ
diers but, ontrary to the previous setion also the risk
Furthermore, from the denition of the risk-shift, we assume that
the risk-neutral drift remains onstant. Thus, as a result of the inreasing risk,
the physial drift rate inreases as well.
4.1 that the physial drift rate
µP
Reall from the disussion in setion
is related to
σ
by
µP = µ − ψ σ .
the atual inrease depends on the market prie of risk
fators, we extend the notation and write
27
Pξ,σ,T .
ψ.
Therefore,
To aount for these
Further, to ease interpretation,
we report the dierene in distress probabilities,
time horizon
T
between a bank with CoCo bonds and risk
Sine
for
∆PT
σh ,
σh = σl ,
σh
over a
and a bank with
σl .
straight bonds and risk
Note that for
∆PT = Pξc ,σh ,T − Pξb ,σl ,T
we know from proposition 1 that
inreases monotonially in
σh
∆PT
must be negative.
to a positive value, there is a value
for whih probabilities of nanial distress oinide. Therefore, we dene
another ritial value
non-negative, i.e.
σ̄
as the smallest value of
σh
for whih the dierene is
σ̄ = inf{σh |Pξc ,σh ,T ≥ Pξb ,σl ,T }.
Numerial results for the base ase senario are reported in the left panel of gure
3.
Figure 3: Probability of nanial distress.
The left panel plots the dierene in distress probabilities of a bank with CoCo bonds and
∆PT = Pξc ,σh ,T − Pξb ,σl ,T , for T
The right panel plots the ritial values
straight bonds, i.e.
of
ψ = 0.5.
Parameter values are:
= 10 years and a market prie of risk
σ̂ and σ̄ for dierent time horizons T .
xt = 1, d = 1.5, λ = 0.55, r = 0.08, µ = 0.05, τ = 0.35, σl = 0.15.
σh
∆PT
0.22
0.21
σ̂
σ̄
0.08
σ̂
0.20
0.06
0.19
0.04
0.18
PSfrag replaements
0.17
PSfrag replaements0.02
0.16
0.18
0.20
0.22
0.24
σ̄
σh
10
20
30
40
50
T
The graph onrms the general insight from setion 3.2, that CoCo bonds are
beneial in terms of probability of distress
as long as
over the hoie of the risk tehnology, i.e. for
the bank has no disretion
σh = σl .
However, if ontrats
are inomplete, and the bank has disretion to inrease the investment risk
σh ,
distress probabilities inrease with the severity of the risk-shift, and outweigh
the initially beneial eets. The ritial value
σ̄
above whih CoCo bonds are
detrimental in terms of probability of nanial distress turns out to be
(for a time horizon of
line. For
σh
T = 10
larger than
σ̄ ,
σ̄ = 0.183
years), whih is indiated as the vertial dashed
a bank with CoCo bond naning will atually have a
higher probability of nanial distress. The graphi also shows the ritial value
28
σ̂
up to whih CoCo bonds are rm value-inreasing as the vertial solid line.
From omparing the ritial values
σ̄
and
σ̂ ,
we obtain the important existene
result, that ases are possible for whih we nd
σ̂ .
In other words, there exist risk levels
σh
σ̄
to be stritly smaller than
whih the bank an hoose, for
whih CoCo bonds are an ex ante optimal strategy for the bank in terms of value
maximization, but for whih the probability of nanial distress is higher relative
to the ase of straight bond naning. The right panel in gure 3 plots the ritial
values
σ̄
and
σ̂
for dierent time horizons
T
and shows that the existene of suh
a onstellation is robust over longer time horizons. Furthermore in appendix A.2,
we show that we obtain the same qualitative result if we onsider the expeted
present value of distress osts as a seond measure for the severity of default.
The numerial analysis thus demonstrates the potentially adverse impat of CoCo
bond naning, whih we summarize as
(Wealth of investors versus banking stability) Suppose nanial
onstraints are suh that a bank with straight bonds hooses the low-risk tehnology
σl (the bank monitors), while a bank with CoCo bonds has inentives to hoose
the high-risk strategy σh (the bank shirks). Suppose further that bond investors
rationally antiipate distorted risk taking inentives and obtain a orresponding
ompensation. Then, it an be possible that a bank with CoCo bond naning
obtains a higher bank value, while the probability of nanial distress is higher
than for optimal straight bond naning.
Proposition 4
The intuition for proposition 4 an be explained in terms of a trade-o. On
the one hand, CoCo bonds are beneial in the sense that they relax the nanial
onstraints (i.e. lower the default barrier
ξc ),
whih allows the bank to exploit
debt naning advantages to a larger extent. Therefore, the bank value inreases
relative to the ase of straight bond naning. On the other hand, from propositions 2 and 3, we know that CoCo bond naning indues the inentive to
inrease the ash ow risk. Although the default barrier is lower in the ase of
CoCo bonds, the probability, that the more volatile ash ow proess hits the
default barrier, an atually inrease. Intuitively, the higher ash ow volatility
overompensates the relaxation of nanial onstraints.
Proposition 4 highlights a potentially dangerous adverse impliation of CoCo
29
bond naning and ontrasts with the main results in e.g. Flannery (2005, 2009)
and Landier and Ueda (2009). The reasoning in these papers relies on a stati
analysis and basially resembles our results for omplete ontrats.
Flannery
(2005, 2009) and Landier and Ueda (2009) stress the potential benets of CoCo
bonds in terms of the relaxation of nanial onstraints, i.e. the redution in the
probability of nanial distress.
Although in partiular Flannery (2005, 2009)
also reognizes the potential risk-taking inentives, he onludes that this is not
a drawbak of CoCo bonds, sine CoCo bond holders will antiipate distorted
risk preferenes and demand a higher premium. Sine the bank has to internalize
risk-taking osts, this will ontrol the risk-shift. In ontrast, we show that this
reasoning does not need to be true in general. Our results suggest that although
the bank internalizes risk-taking osts, it an still be optimal for manager-owners
to inrease risk. More importantly, the risk-shift an be suh that it overompensates the redution in the default threshold, whih atually inreases the probability of nanial distress relative to straight bond naning.
Thus, ontrary
to the previous work on CoCo bonds, our results demonstrate that CoCo bonds
an reate negative externalities for the eonomy, and that individually rational
deisions may have systemially undesirable outomes.
Although the nding in proposition 4 is to be understood as an existene
result, whih issues an important warning signal against the euphori use of
CoCo bonds, we argue that it is also found to hold for a substantial range of
parameter values by providing robustness results in Appendix A.5. Thereby, we
fous on two ruial parameters whih might have a signiant impat on results:
The market prie of risk and the fration of total liabilities replaed by CoCo
bonds.
4.3
Inentive-ompatible onversion trigger
In this setion, we disuss a potential solution to regulate the moral hazard
problem inherent in CoCo bonds. From the results established in lemma 2, we
know that the severity of exogenous onstraints ontrols the risk-taking inentives
of manager-owners. Striter onstraints (in the sense of higher thresholds) indue
30
manager-owners to at more prudently, sine they fae the risk of loosing a more
valuable laim.
In this subsetion, we aim at designing the CoCo bond that
does not provide risk-shifting inentives.
The idea is that if onversion takes
plae at a higher threshold (i.e. at higher apital ratios), then the bank will loose
valuable tax shields earlier whih makes it relutant to inrease the risk of loosing
these debt advantages. Reall from proposition 2 that if the onversion threshold
is dened as
χ = φ̄ · (d + c),
the sense that
∂Stc
∂σ
> 0.
onversion threshold
onversion ratio
γ
the bank always faes risk-taking inentives in
Hene, we now reverse the question and determine the
χ suh that risk preferenes are unhanged.
By doing so, the
is taken as a onstant. The notion behind this is that the (old)
shareholders are not willing to aept a severe hange of the majority of ontrol
rights when a onversion is triggered. For this reason, typial onversion ratios
of lassial onvertible bonds are below 15 perent. Simplifying the ondition
∂Stc
=0
∂σ
for the inentive-ompatible onversion threshold
χIC
results in the following
representation:
IC
χ
c
= φ̄ · d +
.
γ
(18)
The inentive-ompatible onversion threshold depends on the onversion ratio
γ.
Sine
γ < 1,
it is immediately obvious that
Intuitively, the lower the onversion barrier
χIC
γ
χIC
is higher than
χ = φ̄ · (d + c).
is, the higher the onversion barrier
needs to be. This is a result of the fat that the more shares
1−γ
the (old)
equity holders still keep after onversion, the less they are hurt by a onversion.
Thus, a onversion needs to take plae more early in order to represent a reliable
threat for the equity holders against a risk inrease.
As a onsequene of the
higher onversion threshold, the CoCo bond oupon obligation c disappears earlier
whih eteris paribus results in lower tax shields and therefore in a lower overall
bank value.
Hene, this raises the question whether CoCo bond naning is
still worthwhile relative to straight bond naning from the perspetive of the
optimal bank value. Figure 4 reports the the dierene between the rm value
of a bank with inentive-ompatible CoCo bonds to straight bonds as a fration
of the straight bond ase, i.e.
∆V% = (V c − V b )/V b
31
for dierent levels of the
Figure 4: Bank value hange for inentive-ompatible CoCo bonds
The gure shows the dierene between the rm value of a bank with inentive-ompatible
c
b
b
CoCo bonds to straight bonds as a fration of the straight bond ase, i.e. ∆V% = (V − V )/V .
Parameter values are: xt = 1, d = 1.5, λ = 0.55, ψ = 0.5, r = 0.08, µ = 0.05, τ = 0.35,
σl = 0.15.
∆V%
0.05
0.10
0.15
0.20
0.25
γ
-0.005
PSfrag replaements
-0.010
onversion ratio
γ.
The graph reveals that as long as the onversion ratio is
below approximately 0.2, the inentive-ompatible CoCo bonds derease bank
value relative to the straight bond ase.
Only if CoCo bond holders obtain a
suiently large part of the equity apital (i.e. if
γ
exeeds approximately 0.2),
the inentive-ompatible CoCo bonds are still value-maximizing. Note however
that from a orporate governane perspetive former owners will be very relutant
to issue CoCo bonds, whih upon onversion give former bondholder too muh
voting rights.
0.2.
Thus, for pratial appliations
γ
is expeted to be well below
Hene, we nd that even though CoCo bonds an be strutured in suh
a way that they mitigate their assoiated moral hazard problem by setting the
onversion threshold suiently high, our analysis also indiates that this design
will most likely result in lower bank values relative to straight bonds, beause
former owners will be relutant to transfer a signiant part of voting rights.
4.4
Further Conerns and Poliy Impliations
The existing literature onsiders ontingent apital to be a desirable instrument
for future banking regulation. One of the few onerns that have been raised so far,
mentions that the onversion of CoCos may trigger an undesirable negative prie
spiral when large institutional investors, who are not allowed to hold equity, sell
32
29
their shares.
The argument resembles onerns raised in Hillion and Vermaelen
(2004), who have analyzed the priing of oating-pried onvertibles. It has to
be stressed that the potentially adverse impliations of CoCo bonds that we
have shown in this ontribution relies on the assumption that investors have full
information, fae no portfolio restritions and at fully rational. It is not unlikely
that in a situation of asymmetri information, the observation of mandatory
onversion will be interpreted as a negative signal by the market, whih in turn
an put additional pressure on the stok prie of the bank.
Sine we abstrat
from these additional market fritions, our results an even be onsidered as a
onservative evaluation of CoCo bond naning.
The poliy reommendation by the Squam Lake Working Group (2010) reognizes
the important disiplining eet of debt, and reommends that onversion should
be made ontingent not only on the individual rms apital ratio, but also on a
systemi event. However, from our analysis we annot onrm that the inlusion
of a systemati trigger entirely solves the dilemma.
To see this, onsider the
following reasoning. Making the onversion ontingent not only on the individual
nanial situation of the issuing bank, but also on a systemi trigger implies that if
the bank runs into nanial troubles, the onversion is not sure to our. However,
unless the systemi trigger is not entirely negatively orrelated, there will be
a positive probability that onversion ours.
Thus, although the additional
systemi trigger diminishes the expeted relaxation of nanial onstraints, the
qualitative eets desribed in the paper are still present (even though they might
be less pronouned).
Sine our analysis reveals the shirking inentive impliitly ontained in CoCo
naning as a dangerous drawbak, we strongly reommend the use of CoCo
bonds together with additional devies that mitigate the risk-shifting inentive
of the bank. As we have shown in setion 4.3, a higher onversion threshold is one
possible mehanism to restrit risk-shifting, although it may turn out to derease
overall bank value.
A seond alternative refers to the simultaneous issuane of bonds with other wellknown bond harateristis suh as the (normal) onversion feature or rating-
Finanial Times Stability onerns over CoCo bonds, November 5, 2009, and Finanial
Times Report warns on CoCo bonds, November 10, 2009.
29 See
33
trigger step-up feature, that usually disperse risk inentives and prevent a risk
shift. However, if the inlusion of additional seurities that have the potential
to reverse the risk-taking inentives makes the use of CoCo bonds still inentiveompatible from the perspetive of the issuing bank is an open issue and worth
further researh.
5
Conlusion
Contingent onvertibles have reeived muh attention in reent times as a naning arrangement that provides an automati reapitalization mehanism for banks
in times when raising new equity may be diult. The existing literature redits CoCos for reduing the probability that the bank runs into nanial distress,
thereby making the banking systems more robust. Regulators seem to put muh
hope in these seurities, as e.g. the Basel Committee on Banking Supervision has
announed that it will review the role of ontingent onvertibles within a reform
of the regulatory framework. However, to the best of our knowledge, there is not
yet a omprehensive theoretial analysis of the impat of CoCo naning. Our
ontribution tries to ll this gap and analyzes CoCo bonds within a dynami
ontinuous-time framework.
Our results indiate that the beneial impat of
CoCo bonds ruially hinges on the assumption if bank managers have substantial disretion over the bank's business risk. If the bank annot hange the risk
tehnology, or in other words, if omplete ontrats an be written, CoCos are
unambiguously beneial. However, results hange dramatially when we allow
for inomplete ontrats. We show that CoCo bonds always distort risk taking
inentives. Therefore, equity holders have inentives to take exessive risks. We
demonstrate that, although investors antiipate distorted inentives and demand
a orresponding higher premium, CoCos an still be value maximizing for equity
holders and therefore an optimal naning hoie. More importantly, our main
result shows that although CoCos an be Pareto-optimal from the perspetive of
the individual rm, they have the potential to substantially inrease the bank's
probability of nanial distress as well as expeted proportional distress osts.
Thus, CoCos may be an example where individually rational deisions an have
systemially undesirable outomes.
34
Hene, the major hallenge will be to nd a mehanism that prevents the riskshifting inentive imposed by CoCo bonds. In this ase CoCos will be bank value
and stability inreasing instruments.
Otherwise, they might bear the risk for
seeding the next banking risis.
35
A
Appendix
A.1
Proof of Lemma 1
To establish the laim in lemma 1, it is enough to ompare equations (6) and (10) for
a given ommon oupon k:
∂ Vtb ∂ b b=k
∂ Vtc ∂ c c=k
=
=
with:
τ
ζ
− D(xt , ξb ),
r
r
τ
ζc
− D(xt , χ)
r
r
(1 − β)((r − µ)τ + φr((1 − τ ) − λ)
ζ=
,
r−µ
ζc =
Note that ξb = φ · (d + k) and χ = φ · (d + k). Therefore, subtrating
yields
τ (π − c β)
π
∂ Vtb
∂b
from
∂ Vtc
∂c
∂ Vtc ∂ Vtb πφr(β − 1)(λ − (1 − τ )) − d(r − µ)τ β
D(xt , ξb ),
−
=
c=k
b=k
∂c
∂b
πr(r − µ)
where the numerator determines the sign. Sine λ is bounded between 0 and (1 − τ ),
c
b
and β < 0, it is veried that the numerator is positive and therefore ∂∂Vct − ∂∂Vbt > 0
c
b
whih onrms the laim, that ∂∂Vct c=k > ∂∂Vbt b=k .
A.2
Expeted distress osts
Besides the probability for the ourrene of nanial distress, a seond measure for
the severity nanial distress, in whih a systemi regulator will be interested, is the expeted present value of distress osts. First note that the value D(xt , ξ) of one monetary
unit paid out in ase of default is a monotonially inreasing funtion in ξ .
A.2.1
Complete ontrats
Within the omplete ontrat setting, the risk poliy is ontrollable, and thus the same
for a bank with straight bonds or CoCo bonds. Therefore, it is immediately obvious
that
D(xt , ξc ) < D(xt , ξb ).
ξ
Distress osts, in our model setup, are given by (1 − λ) r−µ
so that its present value
ξ
amounts to D(xt , ξ) · (1 − λ) r−µ
. Sine a higher ξ results in both higher distress
36
ξ
osts (1 − λ) r−µ
given a distress and a higher state prie D(xt , ξ) for a distress, it
is immediately obvious that the following inequality holds:
(1 − λ)
ξc
ξb
D(xt , ξc ) < (1 − λ)
D(xt , ξb ).
(r − µ)
(r − µ)
The severity of nanial distress in terms of its expeted present value of distress osts
are stritly lower for CoCo bonds than for straight bonds.
A.2.2
Inomplete ontrats
If risk-shifting is possible, we need to keep trak of the risk parameter σ, and we extend
the notation to D(xt , ξ, σ).
Denoting the expeted present value of default osts as DC , it equals DC b = (1 −
ξb
ξc
λ) r−µ
D(xt , ξb , σl ) for a bank with straight bonds, and DC c = (1 − λ) r−µ
D(xt , ξc , σh )
orrespondingly for a bank with CoCo bonds. Note that in ontrast to the ase of omputing the atual probability of distress (for whih the physial drift rate is neessary)
we use the risk-neutral drift µ to ompute DC , sine we need the priing measure.
To gauge the impat, gure 5 reports the dierene in expeted distress osts as perentage of initial rm value, i.e. ∆DC = DC c /Vtc − DC b /Vtb .
Figure 5: Expeted distress osts.
The gure plots the dierene in expeted distress osts as perentage of initial bank value
c
c
b
b
between straight bonds and CoCo bonds, i.e. ∆DC = DC /Vt − DC /Vt . Parameter values
are:
xt = 1, d = 1.5, λ = 0.55, r = 0.08, µ = 0.05, τ = 0.35, σl = 0.15. γ = 0.4.
∆DC%
0.06
0.04
0.02
PSfrag replaements
0.16
0.18
0.20
0.22
0.24
σh
-0.02
A.3
Proof of Lemma 3
To establish the laim in lemma 3, we analyze the dierene between the partial derivatives for a general exogenous threshold ξ . Let ∆∂ denote the rst derivative of the
37
dierene in equity values ∆S = Stc − Stb , i.e. ∆∂ =
∂∆S
∂σ .
Reall from (4) and (9) that
Stb = V(xt , d + b) − V(ξb , d + b)D(xt , ξb )
Stc = V(xt , d + c) − V(χ, d + c) D(xt , χ) +
(1 − γ) D(xt , χ) (V(χ, d) − V(ξc , d) D(χ, ξc ))
Sine we onsider the ase where the oupon of CoCo bonds is equal to the initial
straight bond oupon, b = c, we also have χ = ξb (i.e. onversion takes plae when the
bank with straight bonds would fae nanial distress), and ∆S simplies to
∆S = (1 − γ) D(xt , χ) (V(χ, d) − V(ξc , d) D(χ, ξc ))
(A-1)
= (1 − γ) (V(χ, d) D(xt , χ) − V(ξc , d) D(xt , ξc ))
and therefore we get
∂
(1 − γ) (V(χ, d) D(xt , χ) − V(ξc , d) D(xt , ξc ))
∂σ
x ∂β(σ)
t
= (1 − γ) V(χ, d)D(xt , χ) · log
·
−
χ
∂σ
x ∂β(σ) t
.
V(ξc , d)D(xt , ξc ) · log
·
ξc
∂σ
∆∂ =
Now, for arbitrary χ > ξc , it an be veried that V(χ, d) > V(ξc , d), log xχt > log xξct ,
and D(xt , χ) > D(xt , ξc ), whih implies that ∆∂ > 0. This establishes the laim that
A.4
∂Stb ∂Stc <
.
∂σ b=b
∂σ c=b
Proof of Proposition 3
Reall from (12), the derivative w.r.t. σ of the equity value with straight bonds, is given
as
∂St
∂σ
∂
= −V(ξ, π)
D(xt , ξ, σ),
∂σ
D(xt , ξ, σ) = D(xt , ξ, σ) · log xξt · ∂β(σ)
∂σ > 0. Thus, the sign of
∂
where ∂σ
mined by V(ξ, π). For the general onstraint ξ(π) = φ π, V(ξ, π) is
V(ξ, π) = (1 − τ ) ·
φ r − (r − µ)
· π.
r(r − µ)
∂St
∂σ
is deter(A-2)
For φ > (r − µ)/r, V(ξ, π) is positive and inreasing in π , while for φ < (r − µ)/r,
V(ξ, π) is negative and dereasing in π .
Furthermore, from lemma 3, we already know that
∂
(1 − γ) (V(χ, d) D(xt , χ) − V(ξc , d) D(xt , ξc )) > 0
∂σ
38
(A-3)
Now, as in Proposition 3 distinguish the ase of weak and strong nanial onstraints,
i.e. the ases ξ < ξ̂ and ξ > ξ̂ respetively.
Consider rst weak onstraints, i.e. ξ < ξˆ, whih is equivalent to φ < (r − µ)/r. From
(A-2), we know that V(ξ, π) is negative and dereases in π . Therefore, we nd that
b
t
the derivative of equity value with straight bonds ∂S
∂σ is positive. From the impliation
of lemma 1, we know that the optimal oupon of CoCo bonds is larger, i.e. c∗ > b∗ ,
whih implies that V(ξ, (d + c∗ )) < V(ξ, (d + b∗ )) < 0. From (A-3), the derivative of
the additional term in the equity value funtion of CoCo bonds is always positive, from
whih we immediately derive the result, that
0<
∂Stb ∂Stc <
.
∂σ b=b∗
∂σ c=c∗
Now, onsider the ase of strong onstraints, i.e. ξ > ξ̂ , whih is equivalent to φ >
(r − µ)/r . From (A-2), we know that V(ξ, π) > 0 and inreases in π . Therefore, we nd
b
t
that the derivative of equity value with straight bonds ∂S
∂σ is negative. Again, from the
impliation of lemma 1, c∗ > b∗ , we obtain V(ξ, (d + c∗ )) > V(ξ, (d + b∗ )) > 0. However,
sine from (A-3), the derivative of the additional term in the equity value funtion of
c
t
CoCo bonds is always positive, it is not determined whih sign ∂S
∂σ has, while it is
b
t
unambiguous that ∂S
∂σ is negative. This shows that
∂Stb < 0 and
∂σ b=b∗
as asserted in the proposition.
A.5
∂Stc ≷0
∂σ c=c∗
Robustness Results
In this appendix, we argue that although the nding in proposition 4 is to be understood
as an existene result, it an be shown to hold for a substantial range of parameter values.
We fous on two ruial parameters whih might have a signiant impat on results:
The market prie of risk and the fration of total liabilities replaed by CoCo bonds.
First, as the market prie of risk inreases, the physial drift rate must inrease as well.
Due to a higher µP , the physial probability for a default delines. This is, however,
true for both CoCo and straight bond naning so that it is not diretly lear how this
aets the relationship between the default risk under the two types of bond naning.
Seond, as a higher fration of total liabilities is replaed by CoCo bonds, the threshold
for nanial distress dereases. Both eets will impat on results.
The left panel in gure 6 plots graphs for the ritial values σ̂ and σ̄ when the market
39
prie of risk ψ varies from -0.5 to 2. This range is very wide and obviously overs all
reasonable values for the market prie of risk.30 The graph shows that results are largely
unaeted by the hoie of the risk premium. Sine the risk premium results in a higher
drift under both forms of bond naning, it is not surprising that the net eet on the
ritial risk level σ̄ is marginal. The interval [σ̄, σ̂] is larger for small risk premia. Thus,
our base ase senario seems to be a onservative hoie. Hene, we an onlude that
our results are not signiantly driven by assumptions about the market prie of risk.
The right panel in gure 6 plots the ritial the ritial values σ̂ and σ̄ as a funtion
Figure 6: Robustness results onerning market prie of risk and debt struture
The gure shows the ritial values
σ̂
as a funtion of the market prie of risk ψ (left
B
B+D (right panel). Parameter values are:
0.08, µ = 0.05, τ = 0.35, σl = 0.15, γ = 0.2.
and
σ̄
panel) and the fration of bonds to total liabilities
xt = 1, d = 1.5, λ = 0.55, ψ = 0.5, r =
σh
σh
0.21
0.24
σ̂
0.20
0.22
0.19
0.20
PSfrag replaements0.18
0.18
PSfrag replaements
σ̄
0.16
0.17
0.0
0.14
0.2
0.4
0.6
0.8
1.0
ψ
σ̂
σ̄
0.1
0.2
0.3
0.4
B
B+D
B
of the fration of optimally-hosen bonds to the total liabilities B+D
. To ompute
the liability ratio we onsider exogenous deposit obligations with d between zero and
the maximum level dmax , whih exhausts the debt apaity of the bank and ompute
the orresponding optimal straight bond oupon b∗ ≥ 0.31 Then, these interest rate
obligations for deposits and the straight bond are translated into the values of D and
B
B as well as the resulting liability ratio B+D
.
30 Dimson et al. (2006) report that over the period 1900-2005, the ten-year average equity
premium never exeeded 20% (and was even slightly negative for some deades). Sine the
average volatility of equity markets is around 20%, an upper bound for the market prie of
risk is roughly 1. Dimson et al. (2006) further report that the average equity premium is
around 5-8% in the entire period, thus a value for the market prie of risk around 0.25 - 0.4
seems reasonable.
31 Tehnially speaking d
max is determined suh that the debt apaity of the bank is exhausted,
i.e. that
b∗ = 0
in (7).
40
B
The fration B+D
an be interpreted as the extent to whih a bank has exhausted
B
its debt apaity by deposits. For a value of B+D
lose to zero (one), the bank has
a small (high) amount of bonds outstanding that an be replaed by a CoCo bond.
As mentioned in the introdution, the largest issue of CoCo bonds has been initiated
by Lloyds in November 2009 with a volume of ¿7.5 billion. As stated in its annual
report from end of 2008, Lloyds had ¿245 billion in ustomer deposits and traded debt
seurities in issue. Thus, the CoCo bond issue represented roughly 3%.
Figure 6 shows that the ritial result σ̄ < σ̂ is obtained for liability ratios up to
approximately 22%. Thus, our nding in proposition 4 holds for a substantial range of
the parameters. In partiular, as evidened by the example of Lloyds, where the CoCo
bond issue represents 3% of the sum of deposits and traded debt seurities, the largest
issue sofar is very well within the range for whih our results hold.
41
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