Contingent convertible bonds with ﬂoating coupon payments: ﬁxing the equilibrium problem

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Contingent convertible bonds with

floating coupon payments: fixing the

equilibrium problem

Dani¨

el Vullings, 2221551

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Master’s thesis Research Master: Economics, Econometrics, and Finance Supervisor: Dr. D. Ronchetti

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Contingent convertible bonds with floating coupon

payments: fixing the equilibrium problem

Dani¨

el Vullings 2221551

July 18, 2016

Abstract

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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 con-version, 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 Co-Cos 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.

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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.

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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 is-sued 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, Vermae-len, 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

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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 con-version, 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 eval-uation 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 in-vestors 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 hy-brid 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 unique-ness 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 con-ditions 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 conver-sion 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

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et al. (2014) propose to enhance CoCos with a call option given to the original share-holders. 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 conver-sion 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

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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, ¯B the face value of the bonds of senior debt

holders, ¯C 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, Cu 0, are given by B0 = ¯B(1 − F ( ¯B)) + Z B¯ 0 A1f (A1)dA1 (1) C₀u = ¯C(1 − F ( ¯B + ¯C + K)) + Z B+ ¯¯ C+K ¯ B (A1− ¯B)A1dA1. (2)

Note that F ( ¯B) is the probability of default and R₀B¯A1f (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

S₀c= (A0− B0)

n + m (3)

S₀u = (A0− B0− C0)

n . (4)

Now consider the example in SW where ¯B = 90, ¯C = 10, K = 5, m = 2, and n = 1.

In this case we have that m = n ¯_KC. 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 Su

0 = 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.

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equity holders. In order to make this intuitively clear, consider a CoCo with coupon pay-ments 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

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A2,t = n2S2,t+ C2,t (6)

A3,t = n3S3,t+ ˜n3S3,t, (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 ˜n3 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

gener-ated 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

m2S2(At) =EtQ[[At− C(At)]1{S2(At)>L ∀ t≤T }

+ [At− ˜n3S3(At)]1{∃ a t<T such that S2(At)≤L}] (8)

=E_tQ[At− C(At)1{S2(At)>L ∀ t≤T }

− ˜n3S3(At)1{∃ a t<T such that S2(At)≤L}], (9)

has a unique value. Note that the t index implies that the expectation is taken

condi-tionally on the information setFt. 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

αt = ln At At−1 . (11)

From the evolution of the asset value αt can be observed. Expression (10) has a value if

fQ(αT|αt) (12)

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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 = ˜σt2, 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[ ¯C1δ1>δ2 + 01δ1<δ2] (14) = ¯CPt(δ1 > δ2) < ¯C, (15) where δ1 = min t (t : 0 < St< L) (16) δ2 = min t (t : St≤ 0), (17)

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 inFt 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 toFt. 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

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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,

rea-sonable. This is because Ft does not give information on whether conversion or

non-conversion 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 toFt 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 vio-late the condition. Due to the risk that the assumptions in GN are viovio-lated, 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}

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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 = µAtdt + σAtdzt+ At(yt− 1)dqt,

where zt is a standard Brownian motion, eyt ∼ N (µy, σ2y), 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) ¯B, c(s, As) ¯C, and (a(s, As)As − b(s, As) ¯B − c(s, As) ¯C)

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 < ¯B. 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

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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 : ¯B < At < ¯B + ¯C},

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

¯

B < At< ¯B + ¯C. 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−Rtsrudu. (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

Bt = (1 − ω)AδR(t, δ) +

Z δ

t

b(s, As) ¯BR(t, s)ds,

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) ¯C to CoCo holders in such a way that the market value of the CoCos is

equal to ¯C 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 C¯

SτSτ = ¯C 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

St = Et 1 nIt + Et 1 n + m{JτR(t, τ )1τ <min{δ,γ}} , (19)

5_{We assume default to be exogenous. In our framework there are no costs of bankruptcy or conversion}

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where It and Jt are given by It = Z min{τ,δ,γ} t (a(s, As)As− b(s, As) ¯B − c(s, As) ¯C)R(t, s)ds, Jt = Z min{δ,γ} t (a(s, As)As− b(s, As) ¯B)R(t, s)ds.

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

Ct= Et " Z min{τ,δ,γ} t c(s, As) ¯CR(t, s)ds # + Et m n + m{JτR(t, τ )1τ <min{δ,γ}} + Et[(Aγ− ¯B)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) ¯B − c(s, As) ¯C) 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):

Ct = Et " Z min{τ,δ,γ} t c(s, As) ¯CR(t, s)ds # + Et[(Aγ− ¯B)R(t, γ)1γ<min{τ,δ}] + Et[ ¯CR(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− ¯B, 0),

or the CoCo holders will retain the value ¯C either in the form of CoCos or, in case of

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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

Γt= 1 − Et[R(t, τ )1τ <min{δ,γ}] − Et[(Aγ− ¯B)R(t, γ)1γ<min{τ,δ}] ¯ C . (22)

The coupon payments paid to CoCo holders before conversion can be derived by solving the following equation

Γt= Et " Z min{τ,δ,γ} t c(s, As)R(t, s)ds # ,

where we use (21) and the fact that Ct = ¯C. 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 ˜c(s, As) = c(s, As)R(t, s). Then we have that the following

should be satisfied for a given c(t, At),

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Appendix A.1 shows the derivation of the desired coupon payment. We find: ˜

c(s, As) = Γs− Es[1min{τ,δ,γ}−s−1>0Γs+1]. (24)

By setting the coupon rate as in (24), we have that Ct = ¯C 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>0Γs+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

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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):

dAt = µAtdt + p VtAtdzt+ At(yt− 1)dqt (25) dVt= κ(θ − Vt)dt + σv p VtdWt+ xtdpt,

where zt and Wt are standard Brownian motions, eyt ∼ N (µy, σ2y), xt is exponentially

distributed with mean µx, while qt and ptare 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).

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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 pro-posed 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 = µAtdt + σAtdzt+ At(yt− 1)dqt, (26)

where we recall that zt is a standard Brownian motion, eyt ∼ N (µy, σy2), and qt is a

poisson process with arrival rate λ. We set µy = −

σ2 y

2 in order to make sure that the

mean of the jump is zero.

The asset value will be simulated on a weekly basis, so t grows in steps of size ₅₂1,

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}

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Probability 240 245 250 255 260 265 270 0.05 0.10 0.15 0.20 Asset Value

Figure 1: Base case: σ2 = 0.01, λ =

5.2, σy = 0.07, Au = 255, and 100, 000 iterations. Probability 252 254 256 258 260 0.05 0.10 0.15 Asset Value Figure 2: σ2 _{= 0.0005, λ = 2.6, σ} y = 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.

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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 σ2, σ2_y 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, σ2, 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 σ2 = 0.0005, λ = 2.6, σy = 0.05, and the trigger level is 230. Although the

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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

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(26)

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effect on the conversion loss for this scenario.

In figures 17 to 20 the variance of the asset value, σ2_{, 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.

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Figure 21: Plot with coupon payments for varying levels of the asset value and the volatility.

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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 dAt = µAtdt + p VtAtdzt+ At(yt− 1)dqt (27) dVt= κ(θ − Vt)dt + σv p VtdWt+ xtdpt,

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 value occur simultaneously as well. We,

fur-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 con-sequences 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

Contingent convertible bonds with ﬂoating coupon payments: ﬁxing the equilibrium problem