Contingent convertible bonds with floating coupon payments:
fixing the equilibrium problem
Daniël Vullings *
* Views expressed are those of the author and do not necessarily reflect official positions
of De Nederlandsche Bank.
Working Paper No. 517
August 2016
De Nederlandsche Bank NV
P.O. Box 98
1000 AB AMSTERDAM
The Netherlands
Contingent convertible bonds with floating coupon payments:
fixing the equilibrium problem*
Daniël Vullingsa
a
Rijksuniversiteit Groningen. E-mail: [email protected].
August 2, 2016
Abstract
Contingent convertible bonds (CoCos) are increasingly popular financial instruments used by
banks to satisfy capital requirements. CoCos with market-based conversion triggers in particular
receive much attention in the literature. The pricing of CoCos with such a market trigger is
problematic as the market value of equity itself depends on the firm’s capital structure. This
results in a not-unique arbitrage-free price for the CoCos. We propose a new type of CoCos with
a market based trigger and floating coupons. The coupons increase near the trigger value to
compensate CoCo holders for the possibility of bankruptcy before conversion. This leads to a
unique no-arbitrage price before conversion. The properties of the innovative CoCo contract are
studied for different dynamic models of a bank’s assets, such as the (Black-Scholes) Merton
and stochastic volatility jump diffusion model. In particular, we illustrate how the CoCo
coupons vary as functions of jump intensities and volatilities.
Keywords: Contingent Convertible bonds, market trigger, floating coupons.
JEL classifications: G13, G21, G28.
*
This paper was written while the author was at De Nederlandsche bank and the university of Gronin- gen. The views expressed
in this paper are those of the author and do not necessarily reflect official positions of De Nederlandsche Bank. I am
grateful to Dirk Broeders and Diego Ronchetti for valuable feedback. I, furthermore, want to thank Michael van Baren,
Klaas Arjen Bootsma, Lammertjan Dam, and Sweder van Wijnbergen, as well as participants of the seminar at De
Nederlandsche Bank (2016) for helpful comments and suggestions.
1
Introduction
During the crisis, bankers and researchers have started to look for new ways to help banks
meet the stricter capital requirements as defined in Basel III. Contingent Convertible
bonds (CoCos) are becoming increasingly popular among banks to serve this purpose,
resulting in a sharp increase in the volume of CoCos being issued. Due to being a hybrid
between debt and equity, CoCos are considered to be a promising instrument to help
meet these requirements. In their current form, CoCos offer coupon payments similar
to regular bonds. However, when a certain trigger event takes place, the bonds are
either converted to equity or written down. The goal of this conversion is to quickly
and efficiently recapitalize banks in distress, which might at that time not be able to
obtain new capital by issuing more shares. Since the recent crisis, both practitioners and
academics have engaged in a heated debate on the proper design of CoCos and the risks
that arise from using these kinds of instruments.
Due to the Basel III regulation and banks’ issuance of more CoCos every year, it is
crucial to design CoCos that enhance the stability of the financial system. In this paper
we will consider a new, hypothetical, design of a CoCo contract with a market trigger.
Crucially, we will have floating coupon payments that incorporate market information
and are such that the market value of the CoCos should equal the face value. Upon conversion, the number of shares obtained is such that the total value of equity in possession
of the new shareholders is equal to the face value of the original CoCos. This design
resolves the multiple equilibria or no equilibrium problem, from now on referred to as
the multiple equilibria problem (see Sundaresan and Wang (2015)). This design could,
furthermore, reduce the potential increasing risk-taking incentives addressed in Berg and
Kaserer (2015) and Chan and van Wijnbergen (2016) and the problems with pricing CoCos due to the hybrid structure as in Wilkens and Bethke (2014). The lack of a unique
equilibrium price for the CoCos and equity could cause high volatility in these prices
and lead to discomfort among investors. As CoCos are designed to bring more stability
to the financial sector, this is an undesirable feature that should have a high priority to
be resolved. The new design proposed here is of a hypothetical nature and depends on
assumptions on the stochastic process underlying the asset value and the probability of
default. Simulation techniques are required to find the appropriate coupon values. The
trigger value is not considered, but we assume it is high in order to make sure that the
CoCo coupon payments, that increase when a bank faces more financial distress, cannot
cause the bank to run into trouble.
The floating coupon payments may give the impression that the proposed CoCos are
similar to equity, with coupon payments resembling dividend payments. The distinction
between our CoCo design and equity is sizeable though. The crucial difference between
the proposed CoCos and equity is that the market value of the CoCos remains constant,
and equal to the face value of the CoCos, until conversion has taken place. This implies
that CoCo holders can go to the market at any time and sell the CoCos for the face
value. Even when conversion takes place, CoCo holders will get the number of shares
that ensures that the value of the shares held by the former CoCo holders equals the
original face value of the CoCos and therefore the current market value of the CoCos as
well. The former CoCo holders can, therefore, get their original face value back upon
conversion by selling the newly obtained shares of stock. This structure of the CoCo
has several implications that make it crucially different from equity. First, the value of
the equity varies over time, whereas the value of this contract will be fixed. Second, the
1
equity holders are the first to suffer when the bank faces financial distress. When the
bank suffers from financial distress, the debt remains unaffected and the value of the
CoCos is designed to be constant. Hence, the share price will drop and equity holders
take a loss. CoCos are less risky, as the value of the CoCos remains constant until
conversion takes place. The third difference is that whereas dividends paid to equity
holders will typically decrease when the return on investments disappoints, the coupon
payments for the CoCo holders will increase to compensate the CoCo holders for the
increased probability of default. Hence, there are clear distinctions between equity and
CoCos with the proposed design. The perpetual nature of the CoCos, the juniority, and
the floating coupon payments make the CoCos different from bonds as well. The coupon
payments of the CoCos will be strictly higher than the coupons for senior debt due to
the additional risk incurred by investing in CoCos and this higher return will make the
CoCos interesting to investors.
The coupon payments to CoCo holders are determined by market prices of both equity
and debt. As this contains all the information from the market, investors have little reason
to believe that the value of the CoCos differs from the face value. The proposed design
makes the instrument interesting for investors, standing in between debt and equity.
When investors consider equity to be too risky, while they aim for higher returns than
bonds have to offer, the proposed CoCos are an alternative to equity and debt. The
multiple equilibria problem is resolved in our design, as there is no transfer of value
between equity holders and CoCo holders. The proposed design is mostly hypothetical.
As the design depends crucially on the probability of default, the volatility of the asset
value, and the equity value, a good measure for the probability of default and the volatility
of the asset value should be developed before the instrument can be introduced in the
financial sector. However, market-based proxies for the asset value are derived from the
credit default swap (CDS) market. Credit default swaps can be used to estimate the
probability of default and the volatility of the asset value. Although this will result in a
rough estimate, it is based on the perception of the market of the probability of default
and therefore the best indicator we can find. Future research should look into the use of
CDS rates to design instruments that are inspired by the one we propose here. The aim of
this paper is to show what the theoretic design of the proposed CoCo, with its desirable
characteristics, should look like and how this differs for different stochastic processes
that govern the asset value. This will give valuable insights in the value of CoCos,
its dependency on the default probability, and the equity value. It could, furthermore,
form the basis for the design of a new class of CoCos that do incorporate some of the
advantages, while relaxing the exact requirements for the CoCos as proposed in this
paper.
The main contribution of this paper is the proposal of a new hypothetical instrument
that resolves the multiple equilibria problem. The traditional desirable features of CoCos,
namely to recapitalize banks in distress and reduce potential bail out costs, are meanwhile
retained. We are the first to consider a CoCo with floating coupon payments that cause
the market value of the CoCo to be constant and equal to the face value of the CoCo.
Other problems addressed in the literature, such as the increased risk-taking incentives
(see Berg and Kaserer (2015) and Chan and van Wijnbergen (2015)) and the threat of a
death spiral (see Maes and Shoutens (2012)) are potentially reduced as well. The reason
for this is that, upon conversion, there is no longer a transfer of value between equity and
CoCo holders. As this transfer of value was the cause of these issues, the new design will
likely be much less sensitive to these problems.
2
Although the designed CoCo is of a hypothetical nature, valuable insights in the value
of CoCos and its sensitivity to different stochastic processes that govern the asset value
of a bank, will be obtained. This can inspire the development of a new class of CoCos
that are less sensitive to the problems that CoCos are currently vulnerable to. Due to
the advantages of the new design, the proposed CoCo could be the inspiration for the
development of a new class of CoCos and contribute to the stability of the financial
system. Another important contribution of this research is the insight gained in the
value of CoCos. The value of traditional market triggered CoCos is strongly linked to
the coupon payment. When a higher coupon payment is necessary to keep the value of
the CoCo constant, a CoCo with a constant payment will decrease in value. Hence, this
paper will demonstrate how CoCos behave under different circumstances and how stable
the value of CoCos is.
The structure of the paper is as follows: in Section 2 we will consider the literature on
CoCos; Section 3 describes the multiple equilibria problem; the desired coupon payments
are determined in Section 4; Section 5 considers the simulation of the stochastic process
that governs the asset value and gives a numerical illustration of the characteristics of
the proposed CoCo; the policy implications and conclusions are provided in Section 6.
2
Literature review
Since the financial crisis much research has focussed on CoCos. In this section we will
discuss the basic properties of CoCos and give a short overview of the most crucial debates
in the literature. See Flannery (2014) for a more complete summary of the literature.
An early mention of what we now call a CoCo is in Flannery (2002), where the author
introduces so-called “Reverse convertible debentures”. These instruments convert when
the equity to debt ratio falls below a pre-specified value to allow banks to recapitalize in
a timely manner and prevent financial distress. Due to the lower costs of debt compared
to equity, there is a strong incentive for large banks to operate under high leverage.
Debt is less costly as bonds are less risky for investors and the coupon payments are
tax-deductible. Furthermore, it is expected that governments will bail out systematically
important banks, which makes debt safer and therefore less costly. A natural solution
to this problem could be to force banks to hold more equity. However, bankers argue
that this will make banks uncompetitive as equity is expensive, even though this claim
is rejected by, e.g., Admati, Demarzo, Hellwig, and Pfleiderer (2010). It is clear that this
argument is, at least to some extent, considered to be valid, as CoCos have become an
accepted instrument to reinforce capital buffers as stated in Basel III.1 After the crisis
both regulators and banks became more interested in CoCos due to the new regulatory
standards banks needed to live up to. This has led to an enormous growth of academic
articles on this topic.
Despite the vast number of papers on this topic, there is a wide discrepancy between
what the literature argues to be well-designed CoCos and the CoCos that are actually issued by banks. Most papers stress the importance of a conversion to equity (CE) contract
based on a market value, see e.g. Sundaresan and Wang (2015) and Pennacchi, Vermaelen, and Wolff (2014). Calomiris and Herring (2013) state several conditions CoCos have
to adhere to in order to have the desired effect. Two conditions are the previously named
market trigger and a sufficiently dilutive conversion to equity. A market trigger is re1
Basel Committee on banking supervision (2010)
3
quired as accounting values traditionally lag behind the true financial condition of the
bank.2 Market prices appear to show signs of distress much more timely, increasing the
odds of timely conversion. Another advantage of market triggers is that conversion does
not bring new information to financial markets, as everyone can observe the decreasing
stock price. This prevents investors from panicking due to a sudden realization that a
certain bank is in poor shape, a problem considered in Chan and van Wijnbergen (2015).
The conversion to equity is necessary to give equity holders an incentive to prevent conversion, as conversion can heavily dilute equity holders. Despite this, 60 percent of the
recently issued CoCos is of the principal write down (PWD) type, where instead of being
converted to equity, CoCos are written down with a pre-defined percentage. In Chan
and van Wijnbergen (2015) and Chan and van Wijnbergen (2016) it is argued that this
feature is highly undesirable, as it could increase the systemic risk and give equity holders
an incentive to increase the risk taken instead of decreasing it. Despite this, the opposite
effect can occur as well, as described in Martinova and Perotti (2015), where it is shown
that under certain assumptions the risk-taking incentive is higher with CE as compared
to PWD Cocos. This paper does make the restrictive assumption that the trigger is
exogenous from the risk-taking, which is unrealistic and makes PWD CoCos look more
favourable than justified. The third condition as mentioned in Calomiris and Herring
(2013) is that CoCos should form a significant part of the book value of equity of the
bank, in order to make sure that conversion is a real concern. Another condition often
named in the literature considers the investors in CoCos. When banks buy the CoCos of
other banks, conversion can cause a death spiral in which conversion of one bank, could
cause other banks to run into trouble themselves. Regulators should therefore discourage
banks to buy CoCos from other banks.
In Avdjiev, Bolton, Jiang, Kartasheva, and Bogdanova (2015) the first empirical evaluation of CoCos is given along with an overview of the different CoCos that have been
issued. The authors find that investors appear to be driven by the search for high yields
and do not take the risk of conversion into account. This is highly problematic, as investors might start to panic when CoCos do convert, leaving banks in a more troublesome
situation than they would have been in without CoCos. Academics are still working on
understanding the new risks that come along with issuing CoCos. The complexity and hybrid nature of CoCos has led to several pricing methods, which are compared in Wilkens
and Bethke (2014). A problem with developing a general pricing method is that there is
a large variety in the CoCos issued by different or even an individual bank. This feature
makes an empirical analysis difficult, as the lack of generality results in small samples to
work with.
The main problem we tackle in this paper involves an ongoing debate on the uniqueness of the equilibrium price of the shares for a bank that issues CoCos with a market
trigger. Sundaresan and Wang (2015), from now on SW, note that under realistic conditions there can occur multiple equilibria or no equilibrium due to a wealth transfer
between equity and CoCo holders. Depending on the belief of the agents whether conversion will take place, several equilibrium share prices are valid. This can lead to jumps in
the price or to agents manipulating the market price as described in Maes and Shoutens
(2012), where the authors call this phenomenon a death spiral. These death spirals could
be induced by CoCo holders that sell the stock of the bank short as a hedge for the CoCo.
If, upon conversion, wealth is transferred from the equity holders to the CoCo holders
due to heavy dilution, CoCo holders have an incentive to force conversion. Pennacchi
2
See Haldane (2011) for figures.
4
et al. (2014) propose to enhance CoCos with a call option given to the original shareholders. The original shareholders can, upon conversion, decide to exercise the option
to buy the newly issued shares from the former CoCo holders. The authors claim that
this results in a unique equilibrium. However, SW note that the proposed trigger makes
conversion exogenous as debt is assumed to be always priced at par. This assumption
implies that the value of debt is constant and unaffected by conversion. The asset value
is, therefore, unaffected by conversion and this makes the trigger insensitive to conversion as well, which leads to a unique equilibrium. The result follows from the unrealistic
assumption of debt being always priced at par. Hence, the problem of multiple equilibria
is not resolved and requires further research.
The problem of there being no unique equilibrium as addressed in SW has been
refuted by several other authors. Calomiris and Herring (2013) claim that there is no
problem when CoCos are properly designed. According to the authors, when conversion is
sufficiently dilutive, banks will always issue more equity, eliminating the belief of investors
that conversion is a realistic option. SW argue that this only holds for unlevered banks,
as levered banks do not always have the option to issue more equity. Hence, the lack
of a unique equilibrium in certain circumstances remains valid. Other critiques have
come from Glasserman and Nouri (2012), from now on GN, and Pennacchi and Tchistyi
(2015), who both argue that a unique equilibrium does exist. Both papers assume that
market prices of both CoCos and shares will adjust continuously. Investors will see
conversion coming and adjust the share price continuously and this results in a unique
equilibrium. SW, on the other hand, use a framework where the adjustment of market
prices is discrete and this causes investors to be caught by surprise by price developments,
as they cannot immediately adjust to the development, and this results in the multiple
equilibrium problem. The question is, therefore, whether prices adjust in continuous or
in discrete time. It is unclear which view of the world is correct. However, given that
investors cannot respond to news immediately and will in practice always need time to
adjust to the new situation, it is arguable that prices adjust in discrete time, although
it could be in small intervals. Hence, multiple equilibria could still well be a threat,
even though there are scenarios where this is not the case. In section 3.2 we discuss the
assumptions made by GN and show that due to the nature of CoCos theses assumptions
are likely not to hold. An instrument that mitigates the problem completely is, therefore,
desirable, as this definitively resolves the threat of multiple equilibria prices.
The multiple equilibria problem is one of the most crucial problems to resolve as it
makes CoCos with a market trigger potentially extremely volatile, a highly undesirable
feature for an instrument originally designed to bring stability to the financial sector. As
a general consensus in the literature is that market triggers should be used, new research
is needed to resolve the multiple equilibria problem for CoCos with market triggers.
3
Multiple equilibria problem
In this section we will first provide an example to illustrate why the multiple equilibria
problem occurs. Next, we will consider the critique of other authors that claim to have
resolved the problem and argue why the multiple equilibria problem is not fully resolved.
5
3.1
Illustrative example
In order to illustrate what causes the multiple equilibria problem, we will reiterate the
multiple equilibria problem as described in SW and repeat the example (see Sundaresan
and Wang (2015) p. 888-891) used to show why multiple equilibria occur. We consider
a firm with equity, senior debt, where deposits from savers are part of the senior debt,
and CoCos. In this framework At represents the asset value in period t, St is the stock
price in period t of a single equity share, n is the number of shares outstanding, K is the
trigger value of equity in the CoCo contract, B̄ the face value of the bonds of senior debt
holders, C̄ is the face value of CoCos, m shares are granted to CoCo holders when the
value of equity falls below K, and T is the date of maturity for the CoCos. In the next
section we will introduce a perpetuity CoCo, but for now we stick to the example in SW.
Following SW, we define a discrete-time model with one period, which implies that
T = 1. In this case, the initial asset value is A0 and the terminal asset value is the random
variable A1 . The PDF and CDF of the risk-neutral distribution of the asset value are
given by f (·) and F (·). The initial value of the regular bond, B0 , and the unconverted
CoCo bond, C0u , are given by
Z B̄
A1 f (A1 )dA1
(1)
B0 = B̄(1 − F (B̄)) +
0
C0u
Z
B̄+C̄+K
= C̄(1 − F (B̄ + C̄ + K)) +
(A1 − B̄)A1 dA1 .
(2)
B̄
R B̄
Note that F (B̄) is the probability of default and 0 A1 f (A1 )dA1 is what regular bond
holders get in case of default. CoCo holders get the face value of the CoCo on maturity,
i.e. when the asset value in period 1 exceeds the sum of the debt, the CoCos and the
trigger, which explains the value for the CoCo bond. In case of conversion at time t, the
value of the CoCos is clearly equal to mSt . The stock price, with and without conversion
respectively, equals
(A0 − B0 )
n+m
(A
0 − B0 − C0 )
.
S0u =
n
S0c =
(3)
(4)
Now consider the example in SW where B̄ = 90, C̄ = 10, K = 5, m = 2, and n = 1.
In this case we have that m = nKC̄ . The probability distribution for A1 is discrete with
P (A1 = 80) = 0.25, P (A1 = 100) = 0.5, and P (A1 = 120) = 0.25. The expected value
of A1 equals 100, hence A0 = 100, as investors are rational and we assume a risk neutral
world. Using (1), (2), and (4) we find that without conversion B0 = 87.50, C0u = 5.83
and S0u = 6.67 is an equilibrium as all equations are satisfied and S0u > K. However,
given that conversion does take place, we have B0 = 87.50, C0u = 8.33 and S0c = 4.17,
which is an equilibrium as well as again all equations are satisfied and S0c < K. The
reason for the multiple equilibria is that in this case, there is a wealth transfer from
equity holders to CoCo holders. When conversion takes place, too many shares are given
to CoCo holders, causing the expected returns for the equity holders and therefore the
share prices to decrease. SW show, furthermore, that there is no equilibrium if there is a
transfer of value from the CoCo holder to the equity holders.
There is no easy way around this problem as noted by SW. A unique equilibrium is
achieved when, upon conversion, there is no transfer of value between CoCo holders and
6
equity holders. In order to make this intuitively clear, consider a CoCo with coupon payments equal to zero that converts to a number of shares such that the value of the newly
obtained equity is equal to the face value of the CoCo upon conversion. Furthermore, we
assume the issuing bank has a positive risk of default before maturity. The value of this
CoCo, before maturity or conversion, is lower than the face value. Hence, CoCo holders
gain from conversion, as then they will get the face value of the CoCo, which is higher
than the current value of the CoCo. The equity holders, on the other hand, suffer from
conversion. This is because they expected to have to pay less to the CoCo holders, as
there was a possibility of not having to pay them at all in case of default. Suppose now
that the asset value is such that the equity value is just above the trigger, say at K + for any > 0. Conversion does not occur as the share price is above the trigger level.
As upon conversion the equity holders suffer in favour of the CoCo holders, the equity
value drops by, say, 2 upon conversion. Hence, if conversion would occur, the share price
drops below the trigger price and, given this lower share price, conversion is justified.
Given the underlying asset value, both scenarios are reasonable and there are, therefore,
two equilibrium prices. This can only be resolved when the number of shares granted to
CoCo holders upon conversion is such that the value of the shares given to CoCo holders
upon conversion equals the market value of the CoCos. This implies that the number of
shares given to the CoCo holders should be equal to the ratio of the market value of the
CoCos to the share price after conversion. Hence, in order to have a unique equilibrium,
we need
m=
Cτ
,
Sτ
where τ is the date of conversion, defined by τ = min{t ∈ Λ : nSt < K}, with Λ being
the set of all points in time where the equity value is compared to the trigger. The
problem with defining an m that always satisfies this equation is that the number of
shares granted to CoCo holders should not depend on the future market value of CoCos.
Such a contract would introduce a problem as the market value of a CoCo depends on
the contract, whereas the contract then depends on the market value of the CoCo. Such
a contract cannot be priced properly and is therefore undesirable. This example was
simple and assumes discrete time. SW show that in a more complete continuous-time
framework, although only discrete trading is allowed, the same problem arises.
3.2
The case of Glasserman and Nouri (2012)
In Glasserman and Nouri (2012) the authors show that under certain conditions, in a
framework where continuous trading is possible, there is a unique equilibrium where SW
found multiple equilibria. In this section we will show that the result in GN that there
does exist a unique equilibrium is only conditionally true. Furthermore, we will argue
why the conditions might be violated due to the nature of CoCos.
GN describe a framework in which there are three firms that are identical but for
their capital structure. The first firm has only equity and senior debt, the second firm
has equity and already converted CoCos, and the third firm has equity, and yet to be
converted CoCos. CoCos convert when the share price drops below L. The capital
structure of the three firms can now be denoted by
A1,t = n1 S1,t + B1,t
7
(5)
A2,t = n2 S2,t + C2,t
A3,t = n3 S3,t + ñ3 S3,t ,
(6)
(7)
where ni for i ∈ 1, 2, 3 is the number of shares for firm i, Si,t is the share price for firm
i at time t, B1,t is the market value of the senior debt at time t, C2,t is the value of the
CoCos at time t, and ñ3 is the number of shares given upon conversion to the CoCo
holders. The date of maturity for the senior debt and the CoCos is equal to T . We
assume, for notational ease, that the asset value of all three firms is equal, hence we have
that A1,t = A2,t = A3,t = At for all t. Even though this assumption is not made in GN,
this assumption does not matter for the results as all components of the asset value will
be scaled if the asset value differs. This implies that for the share price, which is our main
interest, differences in the firm sizes do not change the result. The asset value is governed
by the dynamics of a Merton jump diffusion model. It is assumed that the equity holders
prefer non-conversion. This implies that the share price of firm three is lower than the
share price for firm two, hence S2,t > S3,t . Furthermore, as there is a positive probability
that there will be a conversion, the share price of firm one is higher than the share price
of firm two, hence S1,t > S2,t .
GN prove that as long as S1,t , S2,t , and S3,t are adapted to the information set generated by the history of the dynamics of the underlying Merton jump diffusion model, Ft ,
there is a unique equilibrium. More specifically GN show that under some conditions that
limit the distance between S1,t and S3,t , S2,t and S3,t reach the trigger simultaneously.
This implies that investors will see conversion coming and adjust the equity and CoCo
prices to the fact that conversion will occur. We will show that, due to the nature of
CoCos, this assumption could well be violated. In the following we will write the share
price and the CoCo price as functions of the underlying asset value At . In order for the
assumption to be satisfied, we need that
m2 S2 (At ) =EtQ [[At − C(At )]1{S2 (At )>L ∀ t≤T }
+ [At − ñ3 S3 (At )]1{∃ a t<T such that S2 (At )≤L} ]
=EtQ [At − C(At )1{S2 (At )>L ∀ t≤T }
− ñ3 S3 (At )1{∃ a t<T such that S2 (At )≤L} ],
(8)
(9)
has a unique value. Note that the t index implies that the expectation is taken conditionally on the information set Ft . The condition can alternatively be stated as that we
need to be able to find a value for
P [S2 (At ) > L].
(10)
The assumption is satisfied if S2 (At ) depends on the characteristics of the asset value
alone. Write αt as the state variable that is given by
At
.
(11)
αt = ln
At−1
From the evolution of the asset value αt can be observed. Expression (10) has a value if
f Q (αT |αt )
(12)
can be determined. If there is no other state variable in the underlying process that
governs the asset value, this can be done and the assumptions in GN hold. Hence, in this
8
case there is a unique equilibrium under conditions where SW found multiple equilibria.
However, suppose now that there is a second state variable, say βt , that influences the
share price. In this case we cannot determine (10). Only when we know the specific value
of the state variable, so βt = σ̃t2 , we can write (10) as
P [S2 (At ) > L|βt ]
(13)
and determine this probability. Hence, the assumption that S2,t is adapted to Ft will
only hold when we can observe the specific value of the volatility. The question is whether
the absence of βt is a reasonable assumption to make.
Suppose a firm has issued a perpetual CoCo that does not give coupon payments and
converts to the face value upon conversion. Furthermore, assume that the firm has a
positive probability of default. The value of the CoCo is strictly less than the face value
as the CoCo holder either gets the face value or some lower value in shares. The latter
can occur when either default and conversion occur simultaneously or when the equity
value is, after a jump in the asset value, lower than the face value of the CoCo. This
implies that the CoCo holders want conversion to occur as soon as possible, while equity
holders prefer non-conversion. Suppose that the asset value is such that the share price
has the lowest value that does not result in conversion. Although the market price of the
CoCo will adjust to the high probability of conversion, it will be strictly lower than the
face value of the CoCo as there is still a positive probability of default. To illustrate this,
note that
Ct = EtQ [C̄1δ1 >δ2 + 01δ1 <δ2 ]
= C̄Pt (δ1 > δ2 ) < C̄,
(14)
(15)
δ1 = min(t : 0 < St < L)
(16)
δ2 = min(t : St ≤ 0),
(17)
where
t
t
with L being the trigger value, δ1 the time of conversion, and δ2 the time of default. As
we allow for jumps in the asset value, there is a positive probability that a sizeable jump
causes default before conversion. This implies that Pt (δ1 < δ2 ) > 0, from which (15)
follows. Furthermore, as we deal with a perpetual CoCo, either conversion, default, or
both will be triggered at some point. We conclude that the market value of the CoCo
is strictly lower than the face value in this scenario. Hence, upon conversion there is a
transfer of value and the share price will jump below the trigger. In this scenario both
conversion and non-conversion are reasonable and the information in Ft is not sufficient
to determine the share price.
Unless we make an assumption on what value the share price will take, it is not
adapted to Ft . In order to be adapted we need to be able to determine (13). If we would
know, given Ft alone, what value the share price takes, we can determine (13). The
share price is in this case indeed adapted and GN show that this will result in a unique
equilibrium. The unique equilibrium occurs as investors know in this scenario whether
conversion will occur and the prices of the CoCo and equity will adjust to match this
expectation. However, it is questionable whether Ft is enough to determine the share
price. Before conversion, investors can not be sure whether conversion will occur since at
any time before conversion both default and conversion can occur, as Ft does not give
9
information on future movements of the asset value. This results in a market value for
CoCos that is lower than the face value. Hence, there will be a transfer of value when
investors are not sure whether conversion will happen. As without conversion there is no
transfer of value and with conversion there is a transfer of value, the share price will be
lower in the second case. If the asset value is such that in the case without conversion
the share price is just above the trigger, then the share price will be below the trigger
when conversion does occur.3 If investors can anticipate which of the two it will be, the
prices of CoCos and equity will adjust and there is no transfer of value. However, it is
questionable whether investors would be able to anticipate the outcome as there is no
information in Ft that gives an indication of what situation will occur. Another state
variable, what we call βt , could well determine which of the two values the share price
will take. The share price is adapted to Ft and βt . When βt is unknown, the share
price cannot be determined for certain ranges of the asset value. Hence, the result in GN
is true conditionally on the assumption that we know the value of this additional state
variable.4 If the probability of a jump that causes immediate bankruptcy is very small,
a jump in the share price due to a transfer of value upon conversion could be impossible
as well. However, this is only the case when the jumps in the asset value are restricted.
Again, the results in GN are true conditional on an additional assumption.
The presence of an additional state variable, β, that violates the assumption of the
stock price being adapted to Ft is, due to the transfer of value upon conversion, reasonable. This is because Ft does not give information on whether conversion or nonconversion in the scenario described above will occur. Investors will not know whether
conversion occurs and the share price is therefore not adapted to Ft alone. The prices
of the CoCo and equity can therefore not anticipate conversion and the result in GN
does not hold in this scenario. It should be noted that regular CoCos do have coupon
payments though. These coupon payments could be such that there is no transfer of
value in the example described above. Alternatively, if the jumps in the asset value are
restricted such that the probability of default before conversion is very small, the stock
price could be adapted to Ft as well. In these cases the assumptions in GN are valid and
there will be a unique equilibrium. However, unless firms give special care to finding the
appropriate coupon payments, it is unlikely that actual coupon payments will be such
that there is no transfer of value. Furthermore, as coupon payments are often constant,
changes in the structure of the firm or in circumstances will result in coupons that violate the condition. Due to the risk that the assumptions in GN are violated, the multiple
equilibrium problem remains a concern for CoCos with a market trigger. In order to
ensure that CoCos are a safer and more stable instrument it is, despite the results in GN,
necessary to solve the multiple equilibrium problem by adjusting the structure of CoCos
to avoid the transfer of value.
3
Whether the share price will jump below the trigger depends on the size of the transfer of value. This
will depend on the stochastic process underlying the asset value. If the risk of default is high, even when
the share price is near the trigger value, the transfer of value is high and this could result in multiple
equilibria. If the probability is low though, the transfer of value might never be big enough to result in
multiple equilibria.
4
Note that the domain of the state variable could consist of only one value, which implies that we
always know the value. If this is the case, the results of GN hold and there is a unique equilibrium.
10
4
Floating coupon payments
In this section we will define the model we use to design the new CoCos. Before defining
the model, first some basic characteristics for the proposed CoCo need to be determined.
We consider CoCos without a date of maturity, with a market trigger, and a conversion to
equity (CE) scheme. A perpetual CoCo is most interesting, as in Basel III it is determined
that CoCos will only be considered as additional Tier 1 (AT1) capital if there is no date
of maturity. Since one of the main reasons CoCos have become so popular is the new
regulation as determined in Basel III, CoCos make most sense when they count as AT1
capital, as otherwise regular debt is a cheaper option to attract funds. The results will
not differ much due to this assumption and the analysis can easily be generalized to
include CoCos with a date of maturity. Additionally, Basel III requires banks to allow
regulators to trigger the conversion of CoCos. This is a discretionary feature which cannot
be modelled and will, therefore, be left out of the model. We consider CE contracts as
the problem we address only occurs with CE contracts.
We introduce the model in section 4.1 for a general CoCo structure. Then, we show
how the coupons should be designed in order to offset asset value movements.
4.1
Model for the firm’s asset value
The notation used in our model is similar to SW and we consider a risk neutral framework
as well. Again we consider a firm with equity, senior debt and CoCos. Define At to be
the asset value of the bank of interest at time t with some underlying dynamic process,
which will be discussed in section 5. For now it is enough to assume a model along the
lines of Merton (1976) as is done in SW:
dAt = µAt dt + σAt dzt + At (yt − 1)dqt ,
where zt is a standard Brownian motion, eyt ∼ N (µy , σy2 ), while qt is a poisson processes
with arrival rates λ. The mean return on the assets is given by µ and σ is the asset
volatility. Due to the poisson process the asset value will display jumps, causing the
multiple equilibria problem addressed in SW. The occurrence of jumps in the stock prices,
and therefore in the asset value, has strong empirical support, see e.g. Bakshi, Chao, and
Chen (1997) and Chernov, Ghysels, Gallant, and Tauchen (2003). As we deal with a risk
neutral probability measure, we have that µ = r − η − λE[y − 1]. Here, ηAs is the outflow
of asset value from which the dividends and coupon payments to equity holders and bond
holders are paid. Hence, we have ηAs = a(s, As )As , where a(s, As )As is the sum of the
coupon payments to senior bondholders, CoCo holders, and the dividend payments to
equity holders, given by b(s, As )B̄, c(s, As )C̄, and (a(s, As )As − b(s, As )B̄ − c(s, As )C̄)
respectively. c(s, As ) is implicit at this point. We will choose it in the next section in
order to satisfy the conditions to prevent a transfer of value between equity holders and
CoCo holders.
We assume that investors are rational and that the bank defaults when the value of
the equity holders and CoCo holders is wiped out, or put differently, when At < B̄. In
case a bank is hit by a sudden shock that causes immediate bankruptcy, CoCos function
as a buffer to reduce the social costs of bankruptcy. Several events are important for the
value of the firm and the debt, equity, and CoCos. First, we have the date of default,
given by δ, which can be defined as
δ = min{t ∈ Λ : At < B̄},
11
where Λ is defined as before. Defaults are costly and we define the costs of default to be
equal to ωAδ , Aδ being the value of assets at the date of bankruptcy, and ω ∈ [0, 1]. When
default occurs, we therefore have that the remaining value of the senior bond holders upon
default is given by (1 − ω)Aδ . Senior debt matures at infinity. As banks will in general
issue new bonds once current bonds expire, this simplification is not unreasonable. If
the level of debt changes, the coupon payments can simply be determined for the new
situation. Note that default could occur before conversion takes place due to the jumps
in the asset value.5
Next, the date of conversion is defined by
τ = min{t ∈ Λ : nSt ≤ K},
where conversion will take place when the equity value of the firm, nSt , drops below the
trigger level K, and Λ is defined as before. Similar to SW, if conversion never takes place,
so if nSt > K for all t ∈ Λ, we define τ = ∞.
We, furthermore, define a third date that is crucial in order to make sure there is no
transfer of value between equity holders and debt holders. Let
γ = min{t ∈ Λ : B̄ < At < B̄ + C̄},
where γ denotes the time at which a shock occurs that is such that all equity value is
wiped out but there is some value of the CoCos left. In this case we assume that the CoCo
holders will take over the bank and obtain all the shares. This situation only occurs when
a jump in the asset value takes place that causes the asset value to fall in the interval
B̄ < At < B̄ + C̄. The assumption is not uncommon in the literature and is made in, for
example, Pennacchi (2010) as well.
Next, we can define the value of the senior debt, the equity and the CoCos. Let the
risk free interest rate be given by rt , resulting in the discount factor
R(t, s) = e−
Rs
t
ru du
.
(18)
We consider the period from now until the maturity, T , of the senior debt, where the
senior debt includes deposits. The current value of the senior debt is then given by
Z δ
Bt = (1 − ω)Aδ R(t, δ) +
b(s, As )B̄R(t, s)ds,
t
where b(s, As ) is the coupon payment to bondholders.
In order to resolve the multiple equilibria problem in SW, we define the coupon
payments, c(s, As )C̄ to CoCo holders in such a way that the market value of the CoCos is
equal to C̄ at any point in time where conversion can be triggered. Due to the definition
of the conversion quantity, this will ensure that upon conversion, the CoCo holders will
get SC̄τ Sτ = C̄ worth of shares. Hence, no transfer of value between equity and CoCo
holders takes place, resolving the multiple equilibria problem. In order to obtain the
required coupon payments we first define the value of a share
1
1
St = Et
It + Et
{Jτ R(t, τ )1τ <min{δ,γ} } ,
(19)
n
n+m
5
We assume default to be exogenous. In our framework there are no costs of bankruptcy or conversion
for equity holders due to limited liability and there being no transfer of value between equity and CoCo
holders upon conversion (as will be shown in the rest of this section and the appendix). We, furthermore,
abstract from the possibility of endogenous bankruptcy as discussed in Diamond and Rajan (2011).
12
where It and Jt are given by
Z min{τ,δ,γ}
It =
(a(s, As )As − b(s, As )B̄ − c(s, As )C̄)R(t, s)ds,
t
Z min{δ,γ}
(a(s, As )As − b(s, As )B̄)R(t, s)ds.
Jt =
t
The first element on the right hand side of (19) represents the expected dividend payments
before conversion, given by It . The second element on the right hand side represents the
dividend payments after conversion, given by Jt , when the equity holders are diluted.
The total value of the CoCos is given by
"Z
#
min{τ,δ,γ}
m
{Jτ R(t, τ )1τ <min{δ,γ} }
Ct = Et
c(s, As )C̄R(t, s)ds + Et
n+m
t
+ Et [(Aγ − B̄)R(t, γ)1γ<min{τ,δ} ],
(20)
where 1a>b is one if a > b and zero otherwise. The first element in (20) represents
the expected coupon payments to CoCo holders before conversion. The second element
consists of the expected dividend payments to the equity holders that owned CoCos which
were converted to equity. The third element gives the expected value CoCo holders receive
when the equity holders are wiped out completely and the CoCo holders become the new
equity holders.
Note that the main difference with SW is that we assume no date of maturity for the
CoCos and the bonds. The value of both the equity and the CoCos is merely the sum of
the expected future payments in a risk neutral world discounted by the risk free interest
rate. We have that (a(s, As )As − b(s, As )B̄ − c(s, As )C̄) are the dividends paid to the
equity holders, where a(s, As ) is the outflow of asset value.
4.2
Floating coupons offsetting movements in the total asset
value
In the previous section it was clear that part of the value of the CoCos comes from the
coupon payments. If the coupon payments are defined in such a way that the market value
of the CoCo is equal to the face value when conversion can occur, then the conversion
ratio that ensures that there is no transfer of value can easily be determined.
Note that CoCos are a form of junior debt when default takes place before conversion
and that when the equity value is wiped out, in the absence of default, that the CoCo
holders will become the sole equity holders. This allows us to rewrite (20):
"Z
#
min{τ,δ,γ}
c(s, As )C̄R(t, s)ds + Et [(Aγ − B̄)R(t, γ)1γ<min{τ,δ} ]
Ct = E t
t
+ Et [C̄R(t, τ )1τ <min{δ,γ} ].
(21)
There are two possible scenarios that influence the value of the CoCos: either the complete
equity value is wiped out before time τ and the CoCo holders will only get max(Aγ − B̄, 0),
or the CoCo holders will retain the value C̄ either in the form of CoCos or, in case of
conversion, in shares. In a scenario with an interest rate of zero, the coupon payments
only serve the purpose of covering the risk that either default occurs or the equity holders
13
are wiped out before conversion. Note that the CoCo holders face a much lower risk than
equity holders, as the value of the CoCos will decrease only after conversion. Before
conversion the value of the CoCo is constant, making CoCos debt like, while it becomes
equity after conversion. When a large bank is in financial distress, first the equity holders
will suffer and the CoCo holders will only be held accountable once the financial situation
becomes so bad that the share price breaches the trigger. In the process the bank will be
recapitalized and, given a trigger that is set high enough, be relieved from the financial
pressure.
Before we determine the desired coupon payment, we first note that as conversion can
only take place at certain discrete time intervals, the periods included in Λ, the coupons
can be determined at these moments as well. This is the case because the market value
of the CoCos only needs to be equal to the face value of the CoCos when conversion
can take place. This provides us with a natural transition from the continuous coupon
payments in the model to discrete coupon payments that are applicable in practice.
The expected value received upon conversion, default, or when the equity holders are
wiped out, divided by the value of the CoCos is given by
Et [(Aγ − B̄)R(t, γ)1γ<min{τ,δ} ]
.
(22)
Γt = 1 − Et [R(t, τ )1τ <min{δ,γ} ] −
C̄
The coupon payments paid to CoCo holders before conversion can be derived by solving
the following equation
"Z
#
min{τ,δ,γ}
c(s, As )R(t, s)ds ,
Γt = Et
t
where we use (21) and the fact that Ct = C̄. Γt can be interpreted as the expected
difference after conversion or default of the CoCo at time t between the face value of the
CoCo and what the CoCo holders will receive. From now on we will refer to Γt as the
conversion loss. If the probability of conversion instead of default is high at time t, the
conversion loss is small as CoCo holders expect a value close to the face value of the CoCo.
If, on the other hand, the probability of instant default is high at time t, the conversion
loss will be high. In the extreme case when the probability of conversion is one and the
discount rate is zero, the conversion loss will equal zero and the coupons can be set equal
to the risk free rate. On the other extreme, when the probability of the CoCo holders
being wiped out completely is one, the conversion loss is one and the coupon payment in
period t should equal the face value of the CoCo. The coupons should be constructed in
such a way that they cover the difference between the expected value after conversion or
default and the face value of the CoCo. If this can be achieved, the value of the CoCo
will equal the face vale of the CoCo independent of the asset value.
We define the coupon payment such that, given the current value of underlying assets,
we have that for s > t that c̃(s, As ) = c(s, As )R(t, s). Then we have that the following
should be satisfied for a given c(t, At ),
Et [(Aγ − B̄)R(t, γ)1γ<min{τ,δ} ]
1 − Et [R(t, τ )1τ <min{δ,γ} ] −
C̄
"Z
#
min{τ,δ,γ}
= Et
c̃(s, As )ds
(23)
t
14
Appendix A.1 shows the derivation of the desired coupon payment. We find:
c̃(s, As ) = Γs − Es [1min{τ,δ,γ}−s−1>0 Γs+1 ].
(24)
By setting the coupon rate as in (24), we have that Ct = C̄ for all t < min{τ, δ, γ}. The
intuition behind the coupon payment is as follows: the CoCo holder knows at time t
that in the future coupon payments are such that the expected loss in the next period,
Et [1min{τ,δ,γ}−t−1>0 Γs+1 ], is compensated. However, the conversion loss is higher as there
is a chance that the next period will not be reached and a loss is incurred during this
period. The investor needs to be compensated for this additional expected loss in the
form of the coupon payment.
It is clear that the coupon payments yield much information about the value of the
CoCo without the coupons, or a traditional CoCo with fixed coupons, as well. The
coupons are constructed in such a way that the value of the CoCo without a coupon plus
the coupon is equal to the face value. Hence, if the coupon payment is high, the value of
the CoCo without the coupon, or with a constant coupon, will be much lower than for
the case where our floating coupon payment is low. In section 5 the coupon payments
of CoCos for different scenarios will be determined and the relation between the value
of a standard CoCo and the coupon payment will be used to get insight in the value of
traditional CoCos as well.
An objection to the proposed design could be that in times of financial distress, when
the asset value drops, the increasing coupon payments could increase the probability of
default. However, by setting the trigger for conversion well before a bank is in realistic
danger of default, conversion will take place before the bank is in any sort of danger
and the coupon payments will therefore not form a serious problem regarding the risk of
default. When the trigger for conversion is set sufficiently high, default only occurs before
conversion when a major shock hits the bank and a jump in the asset value causes the
default. In this case the coupon payments will clearly be stopped as the CoCo holders
will either become equity holders, and dividend payments can be stopped at any time,
or the bank defaults.
It is important to determine a suitable trigger level that is sufficiently high. Regulators
provide certain conditions, partially depending on the market price of equity, when they
consider the financial situation of a bank to be too dire to pay dividends. When paying
dividends poses a risk for the solvency of a bank, coupon payments for CoCos could
as well. The trigger could therefore be placed such that conversion takes place before
dividends and the coupon payments of CoCos form a danger for the solvency of the bank.
As CoCos are an instrument mainly used to satisfy the demand of regulators, a joint
effort to determine certain characteristics of the CoCo makes sense.
5
Numerical experiment
Determining the theoretical value of the coupon payment is only the first step, as the
numerical value should be obtained as well. In order to determine the numerical value
of the coupon payment, expression (24) needs to be estimated. The first challenge is to
find a measure for At , as the underlying asset value is unknown and unobservable. A
common solution in the literature is to define the underlying asset value to be equal to the
sum of the market value of equity and all debt, see e.g. Feng and Volkmer (2012). The
dependency of the coupon payments on the unobservable asset value is the main drawback
15
of the proposed design. As the asset value cannot reliably be determined, the coupon
payments will be inaccurate as well. Despite this, as investors cannot determine the asset
value either, the above proposed measure is the best guess we have and although results
derived with this measure may be inaccurate, investors do not have a better measure and
therefore have no reason to adjust the price of the CoCo. In this section we will provide
several strategies to estimate and simulate the dynamics of the asset value for different
models, assuming that we can observe the asset value.
In Section 5.1 we describe the data generating process. In Section 5.2 the simulated
coupon payments assuming a Merton jump diffusion model for the underlying asset value
are given. Section 5.3 gives the coupon payments when a stochastic volatility jump
diffusion process underlying the asset value is assumed.
5.1
Simulation design
In order to simulate the dynamics of the asset value, we will assume that the asset value
follows a general stochastic process. We consider a similar continuous-time dynamic
model as in SW, who used the model introduced in Merton (1974) and extended by
Black and Cox (1976) and Heston (1993) to include stochastic volatility. This results
in the Scott and Bates model developed in Bates (1996) and Scott (2002). We include
jumps in the volatility as well and arrive at the following model, first developed by Duffie,
Pan, and Singleton (2000):
p
(25)
dAt = µAt dt + Vt At dzt + At (yt − 1)dqt
p
dVt = κ(θ − Vt )dt + σv Vt dWt + xt dpt ,
where zt and Wt are standard Brownian motions, eyt ∼ N (µy , σy2 ), xt is exponentially
distributed with mean µx , while qt and pt are poisson processes with arrival rates λq and
λp , and κ and θ represent the speed of mean reversion and the long-run mean variance
respectively.√Note that Vt is the variance of the asset value at time t, while the volatility
is given by Vt . The stochastic process is general and is supported by empirical evidence
in e.g. Bakshi et al. (1997) and Chernov et al. (2003).
We will give a numerical illustration of what the coupon payments will look like given
different stochastic processes for the asset value. This will give insight in the sensitivity
of the coupon payments to different parameter settings and the range between which the
coupons will vary for different underlying stochastic processes of the asset value. The
analysis will focus on a simple Merton process with jumps, after which we investigate
whether there are major differences with the stochastic volatility with jumps in both the
asset value and the volatility as given in (25). The simulations are performed in Mathematica due to its convenience for simulating the stochastic volatility. We will not estimate
the value of the parameters from empirical data. Instead we use a range of parameter
values to illustrate the consequence for the floating coupon payments. Note that the scale
of the coupons is not of significant interest, as we are mainly interested in the sensitivity
of the coupon payments to different parameter settings. We will, therefore, assume that
the risk free interest rate is zero, as using a non-zero value does not add value to the
illustration. We, furthermore, assume that the cost of bankruptcy is zero, hence ω = 0.
There is a predetermined and constant rate for the outflow of asset value from which the
dividends to equity holders and the coupon payments to both senior bondholders and
CoCo holders will be paid. The outflow of the assets is assumed not to depend on the
16
coupon payments. This is assumed as otherwise an endogeneity problem would occur.
The assumption makes sense as given the process we assume for the underlying asset
value, the expected outflow of assets is equal to the asset value. This implies that if
CoCo holders require high coupon payments, this is financed by equity holders, whom
will receive lower dividend payments. When CoCo holders require low coupon payments,
the equity holders benefit by obtaining higher dividend payments. This is intuitive as
equity holders want high risk and therefore benefit in good times while they are the first
to suffer in bad times.
We do not compute values for the dividend and coupon payments to equity and debt
holders. Over the bank’s lifespan the total expected outflow of assets, which is paid to
the equity, CoCo, and debt holders, is in expectation equal to the value of the firm. This
implies that the firm is fairly priced. How the payments from the firm are divided is
beyond the scope of this research, as we are only interested in what the payments to the
CoCo holders look like.
The obtained coupon payments to CoCo holders give an impression of what the proposed CoCos will look like. This will give a notion of the characteristics of CoCos that
are perpetual and convert to a number of shares equal to the face value. Note that the
coupons are a measure for the change in value of the CoCos due to differences in the
asset value as well. The difference between the coupons for different asset values will give
the sensitivity of the CoCo to changes in the asset value, which gives an idea about how
the prices of CoCos can be expected to change when banks face financial distress. The
level of the conversion loss for different levels of the asset value will be given as well, as
this is a measure for the loss that CoCo holders are expected to suffer upon conversion
or default compared to the face value of the CoCo.
5.2
Merton process with jumps
The Merton process with jumps is a common way to model stock prices or the asset value
of a firm and forms the basis for more extensive models. The model is given by
dAt = µAt dt + σAt dzt + At (yt − 1)dqt ,
(26)
where we recall that zt is a standard Brownian motion, eyt ∼ N (µy , σy2 ), and qt is a
σ2
poisson process with arrival rate λ. We set µy = − 2y in order to make sure that the
mean of the jump is zero.
1
,
The asset value will be simulated on a weekly basis, so t grows in steps of size 52
whereas the coupon payments will be given on a yearly basis. The size of the steps is
arbitrary though as the parameters could be scaled to any time window. We set the
default level at an asset value of 210, and the drift is µ = −0.003.6 The face value of the
CoCos is set at 5. Note that the difference between the asset value and the default level
is simply the value of equity plus the value of the CoCos. Hence, a trigger based on asset
values can easily be translated to a trigger based on the value of equity. We assume here
that for a given moment the total debt is priced fairly, implying that the lenders expect
the discounted future payments to equal the face value of the debt. In reality the value
of the debt will vary with the asset value. Although these variations can be included in
the model, we abstain from this as the deviations are bank specific and including them
6
The drift is negative as there should be a positive outflow of assets. To satisfy the assumptions for
risk neutrality and given that the risk free interest rate is zero, this results in a negative drift.
17
0.20
Probability
Probability
0.15
0.15
0.10
0.05
240
245
250
255
260
265
0.10
0.05
270
Asset Value
252
254
256
258
260
Asset Value
Figure 1: Base case: σ 2 = 0.01, λ =
5.2, σy = 0.07, Au = 255, and 100, 000
iterations.
Figure 2: σ 2 = 0.0005, λ = 2.6, σy =
0.05, Au = 255, and 100, 000 iterations.
does, therefore, not yield much added value to obtain a general picture of what the CoCo
coupon payments will look like. The changed conditions will, furthermore, be taken into
account when new debt contracts are made. The new debt holders will demand a different
interest rate due to the new risk profile of the bank. The new debt contract will have
a market value equal to the face value. When debt matures and new debt is issued,
the old debt that had interest payments based on past asset values is replaced by new
debt that has interest payments based on the current asset value. Newly issued debt will
have a market value similar to the face value and this will keep the difference between
the market value of debt and the face value small, implying that the assumption has a
limited impact. As the trigger level is high, the probability of default is relatively low,
which will result in a small difference between the market and the face value of debt as
well.
The simulation is done with an algorithm similar to Zhou (2001). First, the conversion
loss will be simulated with 50, 000 iterations for the asset values between the conversion
level plus 0.1 and 264.1 with steps of one. Hence, if the conversion level is 230, the
conversion loss will first be computed for an asset value of 230.1, next for a value of
231.1, then for a value of 232.1 and so on. In order to estimate the second term on the
right hand side of (24), we simulate a one period change in the asset value 10, 000 times
and use the estimates for the conversion loss to compute the expectation. A downside of
this approach is that we treat the conversion loss as discrete. This will introduce some
bias, although the influence is likely to be limited as the conversion loss is a smooth
function of the asset value, as can be seen from, for example, figure 4. Although the
expectations can be computed in a rather straightforward manner, some problems can
occur. The coupons can, for example, become slightly negative due to the approximation.
When this occurs, the coupons will be set to zero, which is in our case the risk free rate.
Furthermore, in order to reduce computation time, we will truncate the distribution of the
asset value from above at a value of 264.1. All mass from the probability density function
that is above this value will be put at 264.1. Figures 1 and 2 give histograms of the asset
value after 1 step for different settings of the parameters. Figure 1 is computed using the
most volatile process considered in this section along with the highest jump frequency
and variance. As even for this scenario few values exceed the 264.1, the truncation will
not have a huge impact on the result. For regular cases, such as depicted in figure 2, the
asset value will typically not reach the 264.1 and the truncation is therefore harmless.
In order to get an understanding of the behaviour of the coupon payments for a wide
18
Coupon payments Conversion Loss
Trigger value
Jump frequency
Jump volatility
Asset volatility
=
+
+
Ambiguous
=
+
+
Ambiguous
Table 1: The columns give the effect of an increase of the parameter in the row on the
coupon payments and the conversion loss. A plus implies that when a parameter in the
first column and a given row increases, so does the variables in the column. A minus
implies the opposite and ambiguous implies that it depends on the parameter value.
variety of banks, it is necessary to consider several scenarios. We will vary the most
interesting variables, namely σ 2 , σy2 and therefore µy , the trigger level, and the frequency
of jumps in the asset value. All variables will be varied between three different values
and then a simulation is performed for all possible scenarios, resulting in eighty-one
simulations to be performed. The outer values chosen for the parameters are extreme
cases. This is done to illustrate the behaviour of coupon payments to CoCo holders for a
wide variety of parameter values. This will result in high CoCo payments for cases where
there is a high probability that the complete value for the CoCo holders is wiped out.
In other cases, the coupon payments are expected to be virtually constant at zero for
different asset values as there is almost no risk of a loss of value due to jumps occurring
rarely. The variance, σ 2 , will be varied between 0.01 and 0.000025, the frequency of the
jumps is defined on a weekly basis and varied between 5.2 and 1.3, the variance of the
shocks is varied between 0.07 and 0.03, and the trigger level is varied between 220 and
240. Again we emphasize that the scale of the coupon payments has little meaning as
the parameters are chosen rather arbitrarily. The real interest lies in how the coupon
payments differ when one parameter is adjusted while the others remain constant. This
will give valuable insights in how the value of CoCos, for which the coupons are a measure,
is influenced by different parameter settings.
In figures 3 and 4 the scenario where all parameters take the moderate value is given
in order to create a benchmark which can be used to compare scenarios. For this scenario
we have σ 2 = 0.0005, λ = 2.6, σy = 0.05, and the trigger level is 230. Although the
coupon payments and the value of the conversion loss, denoted by Gamma in the figures,
have not completely converged to the true value yet, the relations between the asset
value and the coupon payments and the conversion loss are clear. The coupon payments
decrease, as expected, in the asset value. Whereas for values near the trigger, the coupon
payments are high, for high asset values the coupon payments are low and appear to be
stable. Close to the trigger value, CoCo holders are compensated for the high probability
that conversion or default occurs. The values for the conversion loss first increase for
asset values near the trigger, until a maximum is reached, after which the conversion
loss starts to decrease again and finally stabilizes for high asset values. Recall that the
conversion loss can be interpreted as the loss CoCo holders expect compared to the face
value of the CoCo in the absence of coupon payments. Near the trigger it is likely that
conversion will occur instead of default as a small downwards movement is enough to
trigger conversion. As upon conversion there is no loss of value for the CoCo holders, this
explains the low conversion loss for asset values near the trigger. For slightly higher asset
values, the distance to the trigger is high enough for conversion to not happen easily.
Jumps in the asset value, on the other hand, are more likely to hurt the CoCo holders at
19
these asset values than for cases with a higher asset value, which explains the shape of
the graph. Table 1 gives an overview of the effect that changes in the parameters have
on the coupon payments and the conversion loss.
Figures 5 to 8 give the coupon payments and the conversion loss when the trigger
level is adjusted to 220 and 240 respectively. Figure 5 displays higher coupon payments
with a steeper decrease near the trigger compared to the previous scenario. For similar
asset values, the coupon payments are similar though. The coupon payments are only
higher for values between 230 and 233 in case of a trigger at 230. In the case with the
higher trigger, these asset values imply that there is a relatively high probability of either
conversion or default before the next coupon payment, implying that there are fewer
coupon payments expected. To compensate for this, the coupon payments should be
higher. Figure 6 gives the values of the conversion loss for different asset values. The
shape of the graph is similar to the graph for the case with the higher trigger level. Only
the level shows major differences. This makes sense as with a lower trigger to conversion,
default instead of conversion will occur more often, implying that the value for CoCo
holders will be wiped out more often when trigger levels are lower, resulting in a higher
conversion loss. Figure 7 gives the coupon payments for the case when the trigger is set at
240. There is no clear pattern visible in the graph. The explanation for this is that when
the trigger level is this high, there is little thread of default before conversion. Figure 8
shows that the expected loss upon conversion or default is lower than one percent, and
conversion will happen in the far majority of cases. As there is a low risk of taking losses
for CoCo holders, the coupon payments should be low as well, which they indeed are.
Figures 9 to 12 give the coupon payments and levels of the conversion loss for cases
with both a higher and a lower frequency of jumps. The graphs yield few surprises, as
the shape of the figures is similar. The main differences are the differences in levels,
as a higher jump frequency result in a higher expected loss upon conversion or default.
When the jump frequency increases it appears that the expected loss upon conversion
or default near the trigger is much more similar to other values than for cases when the
jump frequency decreases, see figures 10 and 12. In the latter case, the conversion loss
is low for values close to the trigger, after which it steeply increases to its maximum,
while ending in a small decrease. For higher jump frequencies, the difference between the
conversion loss for asset values near the boundary and the maximum is smaller, although
the difference between the maximum and the conversion loss for high asset values is
bigger. The intuition behind this result is that when the jump frequency is higher, the
probability of conversion or default due to a jump is high even for low asset values.
Hence, for different asset values the probability of the bank defaulting before conversion
is similar. When the jump frequency is low though, the probability of default before
conversion is low for asset values near the trigger, as the regular volatility is likely to
trigger conversion first.
In figures 13 to 16 the coupon payments and levels of the conversion loss for different
levels of the asset value are given, where the level of σy is varied, along with µy to
preserve the risk neutrality condition. Figures 13 and 14 show that for higher levels of
σy little changes. Only the level of the coupon payments, mainly near the trigger, and
the conversion loss are higher due to the higher probability of instant default. The shape
of the graphs is similar to the base case though. For a lower σy we observe a similar
scenario as to when the trigger level was increased. For this scenario the probability of
instant default before conversion is low and the coupon payments are therefore low as
well. Figure 16 looks similar to white noise, implying that the asset value has almost no
20
Coupon payment
0.35
0.30
0.25
0.20
0.15
0.10
0.05
235
240
245
250
255
Asset value
Figure 3: σ 2 = 0.0005, λ = 2.6, σy = 0.05, and conversion = 230.
Gamma
0.045
0.040
0.035
235
240
245
250
255
260
265
Asset value
Figure 4: σ 2 = 0.0005, λ = 2.6, σy = 0.05, and conversion = 230.
21
Gamma
Coupon payment
0.28
1.5
0.26
0.24
1.0
0.22
0.20
0.5
0.18
225
230
235
240
245
250
255
Asset value
220
Figure 5: σ 2 = 0.0005, λ = 2.6, σy =
0.05, and conversion = 220.
230
240
250
Asset value
260
Figure 6: σ 2 = 0.01, λ = 2.6, σy = 0.03,
and conversion = 220.
Gamma
Coupon payment
0.05
0.04
0.0055
0.03
0.0050
0.02
0.0045
0.01
242
244
246
248
250
252
Asset value
254
245
Figure 7: σ 2 = 0.0005, λ = 2.6, σy =
0.05, and conversion = 240.
250
255
260
265
Asset value
Figure 8: σ 2 = 0.0005, λ = 2.6, σy =
0.05, and conversion = 240.
Gamma
Coupon payment
0.6
0.060
0.5
0.058
0.4
0.056
0.3
0.054
0.2
0.052
0.050
0.1
0.048
235
240
245
250
255
Asset value
235
Figure 9: σ 2 = 0.0005, λ = 5.2, σy =
0.05, and conversion = 230.
240
245
250
255
260
265
Asset value
Figure 10: σ 2 = 0.0005, λ = 5.2, σy =
0.05, and conversion = 230.
Gamma
Coupon payment
0.25
0.035
0.20
0.15
0.030
0.10
0.025
0.05
235
240
245
250
255
0.020
Asset value
235
Figure 11: σ 2 = 0.0005, λ = 1.3, σy =
0.05, and conversion = 230.
240
245
250
255
260
265
Asset value
Figure 12: σ 2 = 0.0005, λ = 1.3, σy =
0.05, and conversion = 230.
22
effect on the conversion loss for this scenario.
In figures 17 to 20 the variance of the asset value, σ 2 , is varied. Figure 17 and 19
yield the at first sight surprising result that an increase in volatility results in lower
coupon payments when the asset value is near the trigger. For asset values away from
the trigger, this effect is reversed and we do find that coupon payments increase with
the volatility. In order to understand the relation between the coupon payments and the
volatility, we have to consider expressions (22) and (24). When the volatility is high and
the asset value is near the trigger, conversion will happen quickly, which implies that
there is little opportunity for big jumps to cause immediate default. Hence, the second
term on the right hand side of (22) will be high. The third term is expected to be lower
as an increased volatility can add to jumps and increase the probability of default before
conversion. This implies that the conversion loss can both be higher and lower when
the volatility increases. The second term on the right hand side of expression (24) can
both be higher and lower as well. Whereas, the indicator function in the expectation will
decrease in the volatility, the conversion loss could increase. Due to the ambiguous effect
of the volatility on the conversion loss, the effect of the volatility on the relation between
the asset value and the coupon payments is ambiguous as well. The relation between
the asset value and the conversion loss can be seen in figures 18 and 20. The conversion
loss appears to decrease in the volatility and the relation between the asset value and the
conversion loss is sensitive to the volatility. Figures 21 and 22 give the relation between
the coupon payments, the asset value and the volatility as well as the relation between
the conversion loss, the asset value, and the volatility. Figure 22 gives a different picture
regarding the effect of the volatility on the conversion loss. It can be observed that for
high levels of the volatility the conversion loss increases again. The relation between the
conversion loss and the asset value changes for different levels of the volatility. For the
lowest values of the volatility, the conversion loss increases at first in the asset value, then
it decreases, and finally it stabilizes for high levels of the asset value. The conversion
loss increases in the asset value for moderate values of the volatility. For the highest
plotted value of the volatility, the conversion loss decreases in the asset value. Whereas
the relation between the asset value and the conversion loss was relatively constant for
varying values of the other variables, this is not the case for the volatility. The coupon
payments near the trigger are high for low levels of the volatility, low for a moderate
volatility, and high again when the volatility is high. For asset values away from the
trigger, the coupon payments increase in the volatility.
Given that the value of the CoCos is so sensitive to the underlying asset value, CoCos,
as they are structured at the moment, are difficult to price, especially for asset values
near the trigger. This implies that for CoCos without floating coupon payments, the price
development will be unpredictable. Steep decreases in the price of CoCos of a specific
bank as a result of financial setbacks and price deviations between similar CoCos issued
by different banks are to be expected. This is undesirable as investors will have more
difficulty valuing the risks that these instruments bring and this could cause markets to
become more volatile. Floating coupon payments that compensate investors for these
additional risks will be an attractive alternative to investors.
Despite the sensitivity of the coupons to the different parameter settings, the coupons
do appear quite constant for asset values away from the trigger. When a CoCo with
a structure similar to what we propose would be implemented in practice, the coupon
payments would mostly be stable. The proposed CoCo design will, therefore, in general
not differ much from regular CoCos with constant payments. For high asset values, the
23
Gamma
Coupon payment
0.14
0.8
0.13
0.6
0.12
0.4
0.11
0.10
0.2
0.09
235
240
245
250
255
Asset value
230
Figure 13: σ 2 = 0.0005, λ = 2.6, σy =
0.07, and conversion = 230.
235
240
245
250
255
260
265
Asset value
Figure 14: σ 2 = 0.01, λ = 2.6, σy =
0.07, and conversion = 230.
Gamma
Coupon payment
0.04
0.0028
0.03
0.0026
0.0024
0.02
0.0022
0.01
0.0020
235
240
245
250
255
Asset value
235
Figure 15: σ 2 = 0.0005, λ = 2.6, σy =
0.03, and conversion = 230.
240
245
250
255
260
265
Asset value
Figure 16: σ 2 = 0.0005, λ = 2.6, σy =
0.03, and conversion = 230.
Gamma
Coupon payment
0.018
0.20
0.15
0.016
0.10
0.014
0.05
0.012
235
240
245
250
255
Asset value
235
Figure 17: σ 2 = 0.01, λ = 5.2, σy =
0.05, and conversion = 230.
240
245
250
255
260
265
Asset value
Figure 18: σ 2 = 0.01, λ = 5.2, σy =
0.05, and conversion = 230.
Gamma
Coupon payment
0.075
1.0
0.070
0.8
0.065
0.6
0.060
0.4
0.055
0.2
235
240
245
250
255
Asset value
235
Figure 19: σ 2 = 0.000025, λ = 5.2, σy =
0.05, and conversion = 230.
240
245
250
255
260
265
Asset value
Figure 20: σ 2 = 0.000025, λ = 5.2, σy =
0.05, and conversion = 230.
24
Figure 21: Plot with coupon payments for varying levels of the asset value and the
volatility.
Figure 22: Plot with the conversion loss for varying levels of the asset value and the
volatility.
25
coupon payments will even be lower, as there is little risk for which the CoCo holders
have to be compensated. There will only be sizeable adjustments in the coupon payments
when the asset value drops to values near the trigger, which, in the absence of jumps,
will happen gradually. Big adjustments are only necessary when jumps occur, which have
serious implications for the value of CoCos, as they are currently designed, as well. Hence,
major jumps in the asset value are not more of a thread for CoCos with the proposed
design than for regular CoCos. The CoCo holders will, furthermore, in our design be
more comfortable when markets are volatile due to the stable value of the CoCos.
5.3
Merton process with jumps and Stochastic volatility with
jumps
In this section we compute the coupon payments for a more general stochastic process. In
reality the volatility is not constant, as was assumed in the previous section. It is instead
stochastic and displays jumps as noted by for example Chernov et al. (2003). Hence, we
have
p
(27)
dAt = µAt dt + Vt At dzt + At (yt − 1)dqt
p
dVt = κ(θ − Vt )dt + σv Vt dWt + xt dpt ,
using the same notation as before and recalling that xt is exponentially distributed with
mean µx , while pt is a poisson process with arrival rates λ. We simulate the asset value
using the Euler discretization scheme described in Broady and Kaya (2006) and assume
that jumps in the volatility and the asset
√ simultaneously as well. We, fur√ value occur
thermore, set κ = 1, θ = V0 , and σv = V0 , where V0 is the volatility at t = 0. The
mean of the jumps in the variance will be varied to investigate whether different jump
sizes have unexpected consequences. The main purpose of this section is to investigate
whether allowing for stochastic volatility and jumps in the volatility will have major consequences for the found coupon payments. The parameters chosen are therefore again
somewhat arbitrary and the found coupons should not be considered to be realistic values
for the coupon payments that the CoCo holders will obtain if the proposed CoCos would
be issued.
Figures 23 and 24 show the histograms for the asset values after one step. Due to
the model with stochastic volatility with jumps in both the asset value and the volatility,
from now on SVJD model, the tails of the distribution are fatter, which is considered
in the literature to be one of the main goals of introducing stochastic volatility with
jumps in the volatility (see Gallant, Hsieh, and Tauchen (1997) and Eraker, Johannes,
and Polson (2003)). From the histograms it becomes clear that the SVJD model has a
clear impact on the distribution of the asset value when compared to the Merton jump
diffusion model. In order to compute the coupon payments and the conversion loss for
different asset values, we will use the same approach as in section 5.2 where we replace
the previous model with the SVJD model. For the scenario with the highest variance
and the highest mean jumps in the variance, as in the scenario of figure 23, though with
regular size jumps in the asset value, the conversion loss will be determined up to a value
of 274.1, as these values can in this model be achieved more frequently after one period.
We will perform experiments with a starting variance of V0 = 0.0005 and V0 = 0.01. Our
main interest is to see whether including stochastic volatility with jumps will cause the
coupon payments and the conversion loss to be similarly related to the asset value as in
the case of constant volatility.
26
0.15
Probability
Probability
0.15
0.10
0.05
240
250
260
270
0.10
0.05
250
Asset Value
252
254
256
258
260
Asset Value
Figure 23: Base case: V0 = 0.01, λ =
5.2, σy = 0.07, Au = 255, µx = 0.05,
and 100000 iterations.
Figure 24: V0 = 0.0005, λ = 2.6, σy =
0.05, Au = 255, µx = 0.005, and 100000
iterations.
Figures 25 to 28 give the relation between the coupon payment and the conversion
loss and the asset value under the SVJD model. The plots in figure 25 and 26 have the
same parameter settings as the benchmark scenario except for the inclusion of stochastic
volatility and jumps in the variance with a mean of µx = 0.005 and µx = 0.05 respectively.
The general shape is similar for both the coupon payments and the conversion loss,
although there is a difference in levels. The coupon payment near the trigger is higher
for the SVJD model, although the difference is small. The explanation is that due to the
thicker tails of the distribution of the SVJD model, the indicator function in the second
term on the right hand side of expression (24) is more often zero and the coupons are
therefore higher. For asset values away from the trigger, the coupon payments are not
higher for the SVJD model. To explain this, we note that in figure 26 the level of the
conversion loss is lower in the SVJD model for all asset values. This causes a decrease in
both terms on the right hand side of expression (24). The impact of including stochastic
volatility with jumps on the coupon payments is therefore ambiguous. When the mean
of the jumps in the variance, µx , is increased, the coupon payments near the trigger are
higher still, whereas the coupon payments away from the trigger appear to be similar to
the benchmark case or even lower, as can be seen from figure 27. The relation between
the conversion loss and the asset value, displayed in figure 28, has clearly changed and
is in shape similar to the relation between the conversion loss and the asset value for
the Merton jump diffusion model with a variance of 0.01 (figure 18). Interestingly, the
coupons appear to behave more like the coupons of a Merton model with a very low
volatility, whereas the conversion loss behaves like the the conversion loss of a Merton
jump diffusion model with a high volatility. This illustrates the sensitivity of the coupon
payments and the conversion loss to different stochastic processes.
Figures 29 to 32 give the graphs for the coupon payments and the conversion loss
when V0 = 0.01. When the volatility is higher and µx is low, the coupon payments for
the SVJD model are very similar to the Merton jump diffusion model, as can be seen from
figure 29. From figure 30 we can observe that the conversion loss is very similar for both
models as well. When µx is higher, see figures 31 and 32, the coupon payments and the
conversion loss are higher, as was the case before. The level of the coupon payments for
an SVJD model with these parameters is similar to the level of the coupon payments for
a Merton model with V0 = 0.05 (see figure 21). The main consequence of the stochastic
volatility is, similar to the case with the lower V0 , that the coupon payments should be
similar to the coupon payments in a Merton jump diffusion model with a higher volatility
27
Gamma
Coupon payment
0.036
0.4
0.034
0.032
0.3
0.030
0.2
0.028
0.026
0.1
0.024
235
240
245
250
255
Asset value
235
Figure 25: σ 2 = 0.0005, λ = 2.6, σy =
0.05, µx = 0.005, and conversion = 230.
240
245
250
255
260
265
Asset value
Figure 26: σ 2 = 0.0005, λ = 2.6, σy =
0.03, µx = 0.005, and conversion = 230.
Gamma
Coupon payment
0.042
1.0
0.040
0.8
0.038
0.6
0.036
0.4
0.034
0.2
235
240
245
250
255
Asset value
235
Figure 27: σ 2 = 0.0005, λ = 2.6, σy =
0.05, µx = 0.05, and conversion = 230.
240
245
250
255
260
265
Asset value
Figure 28: σ 2 = 0.0005, λ = 2.6, σy =
0.05, µx = 0.05, and conversion = 230.
Gamma
Coupon payment
0.019
0.25
0.018
0.20
0.017
0.15
0.016
0.015
0.10
0.014
0.05
0.013
235
240
245
250
255
Asset value
235
240
245
250
255
260
265
Asset value
Figure 29: σ 2 = 0.01, λ = 2.6, σy =
0.05, µx = 0.005, and conversion = 230.
Figure 30: σ 2 = 0.01, λ = 2.6, σy =
0.05, µx = 0.005, and conversion = 230.
Coupon payment
Gamma
0.036
0.7
0.6
0.034
0.5
0.032
0.4
0.3
0.030
0.2
0.1
0.028
235
240
245
250
255
Asset value
235
Figure 31: σ 2 = 0.01, λ = 2.6, σy =
0.05, µx = 0.05, and conversion = 230.
240
245
250
255
260
265
Asset value
Figure 32: σ 2 = 0.01, λ = 2.6, σy =
0.05, µx = 0.05, and conversion = 230.
28
than the starting volatility of the SVJD model in order to achieve a unique equilibrium.
Despite the differences that we observe between the Merton jump diffusion model and
the SVJD model, the conclusions are similar. The coupon payments are stable for asset
values away from the trigger and they only increase close to the trigger. The level of
the coupon payments is, furthermore, similar for different stochastic processes when the
asset value is high. Near the trigger the coupons do differ and in order to implement
the proposed CoCo, care should be taken to determine the coupons when the asset value
approaches the trigger for conversion. This emphasizes the need for an accurate proxy for
the asset value. Future research could, furthermore, look into the estimation of Merton
jump diffusion models and SVJD models for real banks and compare to what extent these
result in different coupon payments.
6
Policy implications and conclusions
The main contribution of this paper is the description of a theoretical CoCo that has
many desirable properties. The results, furthermore, show the sensitivity of the CoCo’s
floating coupons to different parameter settings for the underlying asset value. Next to
the theoretical interest, the proposed CoCo could serve as a model for a new generation
of CoCos that, in a simplified form, work similarly. Even though the proposed CoCo
contract solves the multiple equilibria problem of SW, this is of little use when the
market is not interested in buying instruments with a similar structure, or when banks
are not interested in issuing them. Whereas, currently, CoCo holders receive fixed coupon
payments, and the market price of the CoCos fluctuates, for our CoCo design, the value
of the CoCos is constant as long as conversion has not occurred due to floating coupon
payments. This implies that CoCo holders can go to the market before conversion or
default and sell the CoCos to other investors at face value. As the coupon payments are
adjusted at specific points in time, the market value of the CoCos will only be exactly
equal to the face value at these points in time where conversion can be triggered and
the coupon payments are paid out. When the interval between the payments is small,
there will be little fluctuation in the value of CoCos in the interval between potential
conversion dates. The coupon payments will be adjusted to the default risk and ensure
that when the coupon payments are paid out, the market value of the CoCos is equal
to the face value. Investors will anticipate that the future coupon payments will adjust
to the new situation and therefore value the CoCo, even in between coupon payments,
close to the face value. When a current CoCo holder finds the risk no longer desirable,
other investors, with a different risk preference, will be willing to purchase the CoCos
at face value, as the coupon payments compensate for the additional risk. Furthermore,
as the market price of the CoCos is equal or close to the face value of the CoCo at any
time before conversion, CoCos are less risky than equity, although more risky than senior
debt. Investors that consider regular equity to be too risky, while they are seeking higher
returns than senior debt has to offer, will be interested in the proposed CoCos. The
strong increase in the magnitude of coupon payments near the trigger illustrates how
regular CoCos will lose their value when conversion is approaching. This could make the
instrument, as it is structured now, undesirable for investors. This problem is resolved
in our design and could therefore potentially attract more investors.
From the perspective of the bank, CoCos are a cheap substitute for capital, as CoCo
holders face lower risk and therefore require less compensation. Issuing the proposed
29
CoCos instead of new shares does increase the risk for the current share holders, as instead of sharing financial distress with other share holders, they will have to carry the
burden themselves. This does give banks more flexibility in increasing the risk profile of
equity holders, by issuing the proposed CoCos, without making the bank as a whole more
risky. Another advantage for banks to issue CoCos is that in many European countries,
the coupon payments on CoCos are tax-deductible, which clearly makes the instruments
interesting for banks as well. A crucial advantage compared to regular CoCos with a
market trigger is that the multiple equilibria problem is resolved, while the recapitalization features that standard CoCos have, are retained. The problem of the risk shifting
incentives as in Chan and van Wijnbergen (2016) are reduced as upon conversion there
is no transfer of value between CoCo holders and equity holders. Another advantage is
that the price, which is extremely difficult to determine for regular CoCos, is constant
in our framework.7 The clear price of the CoCo will prevent speculation and uncertainty
that currently plague CoCos, which makes the product interesting for regulators as well.
Furthermore, the trigger is based on the market value of equity and conversion does therefore not yield any new information that could cause unrest in the market. The trigger
incorporates the prediction of the market and this has proven to be a much more reliable
prediction than predictions resulting from either regulators or book values. New CoCos
could be inspired by the proposed design, where some of the conditions can be relaxed
to make the instruments less complex. Such an intermediate form could on itself be a
useful improvement over the current design.
Market manipulations are less of a risk in the proposed design as there is no transfer
of value upon conversion and investors in CoCos from a given bank have no longer an
incentive to force conversion. This solves a sizeable problem of traditional CoCos with
a market trigger. Still banks might dislike conversion based on hitting the trigger once.
By only considering conversion on set time intervals, a sudden drop in the market value
of equity that is not based on the true financial condition of the bank and is followed
by a steep increase, will likely not cause conversion though. Alternatively, Calomiris
and Herring (2013) suggest to use a 90-day moving average to smooth fluctuations in
the price. Future research should look into the exact conversion conditions though, in
order to come up with clearer guidelines that suit the proposed instrument. Despite this,
the proposed design has many desirable features and is, even if the multiple equilibrium
problem is not a severe threat, an interesting instrument to study as it could potentially
add to the stability of the financial sector.
In this paper we focussed on the design of a new hypothetical CoCo that resolves
some of the key problems that CoCos with a market trigger face. We investigated the
characteristics of a perpetual CoCo that converts to equity based on a market trigger.
The CoCo has floating coupon payments that are set in such a way that the market value
of the CoCos is equal to the face value of the CoCo at any time when conversion can
occur. This feature makes determining the conversion ratio of the CoCos to equity easy
and resolves the multiple equilibria problem addressed in SW. Numerical experiments
were conducted to investigate the behaviour of the coupon payments. The results show
that the coupons are highly nonlinear when the asset value is close to the trigger and that
the volatility of the underlying stochastic process has a major and non trivial influence
on the required coupon payments. This has several implications for the proposed design
of CoCos compared to the way they are currently structured. The value of CoCos that
7
The different pricing methods in Wilkens and Bethke (2014) show the lack of consensus on how to
price CoCos.
30
have a market trigger and constant coupon payments can be expected to decrease steeply
when the share price falls and approach the trigger level. The size of this drop in prices
is difficult to determine due to the dependence on the volatility of the asset value and
this could cause discomfort among investors. As the asset value can only be measured
roughly, proper pricing models for CoCos are difficult to design. The proposed CoCo
suffers from the same problem and due to the sensitivity to the characteristics of the
underlying process for the asset value, the instrument is for now mostly hypothetical.
A design inspired by the proposed hypothetical CoCo could be implemented though, as
investors are likely interested in an instrument that better protects them against the
possible losses that are associated with CoCos. This design will make CoCos more of
a hybrid between equity and debt, compared to existing CoCos, that is less risky than
equity but more risky than debt. As market-triggered CoCos are widely acknowledged
to be more desirable than book value-triggered CoCos, there is a high priority to resolve
problems faced by market-triggered CoCos. Hence, new designs that resolve the current
issues of market-triggered CoCos are highly desirable. Future research should focus on
developing more accurate methods to use CDS rates to derive the probability of default
of banks. The development of a technique to reliably estimate the probability of default
of banks will make the proposed design feasible and resolve the main limitation of this
paper.
The complexity of the proposed CoCo, and CoCos in general, in combination with the
sensitivity of the value of CoCos to the underlying process that governs the asset value,
is problematic. Investors might not be able to understand the nature of CoCos and this
could lead to unrest on financial markets. This uncertainty and lack of understanding
cannot be resolved, due to the ambiguous dependence between the unobservable asset
value and the value of CoCos. Given that the number of outstanding CoCos is increasing
every year and the green light regulators have given banks to issue CoCos in Basel III,
research on improving the design of CoCos is necessary. The proposed design gives
investors in CoCos more certainty and the CoCos a clearer place between debt and
equity in terms of risk, which is a major improvement. The multiple equilibrium problem
is resolved, which will result in more stable prices of CoCos. The increased risk-taking
incentive is reduced as well, due to the lack of a transfer of value between equity holders
and CoCo holders upon conversion. Future research should look into strategies to measure
the asset value more accurately in order to get a better impression of the value of CoCos.
The CoCo design proposed here could, moreover, be adjusted to make it more feasible for
practical applications. An optimal trigger condition should, additionally, be determined.
A
A.1
Appendix
Coupon payments
In this appendix we show that expression (24) gives the desired coupon value. In order
to find the desired coupon, first note that coupon payments are only paid out at discrete
intervals. Let these intervals be the same intervals as those that are used to evaluate
whether the thresholds for τ , γ, or δ are
breached. Then,
we define that for a given
R i+1
R i+1
interval the function g(s, As ) is such that i g(s, As )ds = i c̃(s, As )ds for all i, where
31
g(·) is continuous in s. Next, we recall the following:
Et [(Aγ − B̄)R(t, γ)1γ<min{τ,δ} ]
.
Γt = 1 − Et [R(t, τ )1τ <min{δ,γ} ] −
C̄
(28)
Expression (23) can then be rewritten as
"Z
#
min{τ,δ,γ}
Γt = Et
c̃(s, As )ds
t


min{τ,δ,γ}−t−1 Z i+t+1
X
= Et 
c̃(s, As )ds
(29)
i+t
i=0


min{τ,δ,γ}−t−1 Z i+t+1
X
= Et 
g(s, As )ds
(30)
i+t
i=0


min{τ,δ,γ}−t−1
= Et 
X
g(λi , Ai+t )(i + t + 1 − i − t)
(31)
i=0


min{τ,δ,γ}−t−1
= Et 
X
g(λi , Ai+t ) ,
(32)
i=0
where (31) follows from the mean value theorem, and we write subscript t in the expectation toh emphasize that we condition
on the current information. We have to show that
i
Pmin{τ,δ,γ}−1
g(λi , Ai ) for all t ∈ Λ. For an arbitrary t ∈ Λ we have
Γt = Et
i=0


min{τ,δ,γ}−t−1
X
Γt = Et g(λt , At ) + 1min{τ,δ,γ}−t−1>0
g(λi , Ai+t ) ,
i=1
which implies that


min{τ,δ,γ}−t−1
X
Et [g(λt , At )] =Γt − Et 1min{τ,δ,γ}−t−1>0
g(λi , Ai+t )
i=1
"
=Γt − Et 1min{τ,δ,γ}−t−1>0 g(λt+1 , Ai+t+1 )
min{τ,δ,γ}−t−1
X
+ 1min{τ,δ,γ}−t−1>1
(33)
!#
g(λi , Ai+t )
i=2
"
"
=Γt − Et 1min{τ,δ,γ}−t−1>0 Γt+1 − Et+1 1min{τ,δ,γ}−t−1>1
min{τ,δ,γ}−t−1
X
g(λi , Ai+t ) + 1min{τ,δ,γ}−t−1>1
i=2


min{τ,δ,γ}−t−1
#
X
g(λi , Ai+t )
i=2


=Et 1min{τ,δ,γ}−t−1>0 Et+1 1min{τ,δ,γ}−t−1>1
min{τ,δ,γ}−t−1
X
i=2
32

g(λi , Ai+t )

min{τ,δ,γ}−t−1
X
−1min{τ,δ,γ}−t−1>1
g(λi , Ai+t ) + Γt − Et [1min{τ,δ,γ}−t−1>0 Γt+1 ].
i=2
(34)
We can get rid of the final two terms in (34) by observing that




min{τ,δ,γ}−t−1
X
Et 1min{τ,δ,γ}−t−1>0 Et+1 1min{τ,δ,γ}−t−1>1
g(λi , Ai+t )
(35)
i=2

min{τ,δ,γ}−t−1
−1min{τ,δ,γ}−t−1>1
X
g(λi , Ai+t )
i=2



min{τ,δ,γ}−t−1
X
=Et Et+1 1min{τ,δ,γ}−t−1>0 1min{τ,δ,γ}−t−1>1
g(λi , Ai+t )
i=2

min{τ,δ,γ}−t−1
X
−1min{τ,δ,γ}−1>0 1min{τ,δ,γ}−t−1>1
g(λi , Ai+t )
i=2


min{τ,δ,γ}−1
=Et 1min{τ,δ,γ}−t−1>0 1min{τ,δ,γ}−t−1>1
X
g(λi , Ai+t )
i=2


min{τ,δ,γ}−1
− Et 1min{τ,δ,γ}−t−1>0 1min{τ,δ,γ}−t−1>1
X
g(λi , Ai+t )
(36)
i=2
=0,
where we used the tower property to arrive at (36). Hence, we find that if we set
Et [g(λt , At )] = g(λt , At ) = Γt − Et [1min{τ,δ,γ}−t−1>0 Γt+1 ],
we have that
"Z
#
min{τ,δ,γ}−t
Γt = Et
c̃(s, As )ds .
t
Finally, we set
c̃(t, At ) = g(λt , At )
for all t and we conclude that the coupon payments for period s should equal
c̃(s, As ) = Γs − Et [1min{τ,δ,γ}−t−1>0 Γs+1 ].
From theorem 3 in SW we conclude that for this coupon payment and m =
a unique equilibrium.
33
Cτ
Sτ
we have
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