Contingent Convertible Obligations and Financial Stability
aa r X i v : . [ q -f i n . R M ] J un Contingent Convertible Obligations and Financial Stability
Zachary Feinstein ∗ T.R. Hurd † June 2, 2020
Abstract
This paper investigates whether a financial system can be made more stable if financial insti-tutions share risk by exchanging contingent convertible (CoCo) debt obligations. The questionis framed in a financial network model of debt and equity interlinkages with the addition ofa variant of the CoCo that converts continuously when a bank’s equity-debt ratio drops to atrigger level. The main theoretical result is a complete characterization of the clearing problemfor the interbank debt and equity at the maturity of the obligations. We then consider a simplesetting in which introducing contingent convertible bonds improves financial stability, as wellas specific networks for which contingent convertible bonds do not provide uniformly improvedsystem performance. To return to the main question, we examine the EU financial networkat the time of the 2011 EBA stress test to do comparative statics to study the implications ofCoCo debt on financial stability. It is found that by replacing all unsecured interbank debt bystandardized CoCo interbank debt securities, systemic risk in the EU will decrease and bankshareholder value will increase.
Systemic risk is the risk of financial contagion – the spread of losses from one bank or institutionto other firms through interconnections in the financial system. This type of risk was exhibitedmost prominently in the global financial crisis of 2007-2009. The deep losses incurred by thereal economy as a result of that crisis show the need for a better understanding of financialcontagion to mitigate the impact of such events in the future. ∗ Stevens Institute of Technology, School of Business, Hoboken, NJ 07030, USA, [email protected] † McMaster University, Department of Mathematics & Statistics, Hamilton, ON L8S 4L8, Canada n the wake of the 2007-2009 financial crisis, contingent convertible bonds were introducedinto European markets. This financial instrument is a derivative contract that converts debt toequity at a pre-defined debt-equity ratio. Since these instruments may improve the solvency ofthe issuing company, they have become prominent in discussions on financial regulation. Werefer to Glasserman and Nouri (2012); Avdjiev et al. (2013); Spiegeleer et al. (2018) for detaileddiscussion of specific kinds of convertible bonds and methodologies for pricing these obligationsin a single institution setting. Gupta et al. (2019) presents a case study of the use of contingentconvertible debts for reducing systemic risk when all such obligations are due outside the financialsystem. The present model of networks with contingent convertible obligations is, fundamentally,a variation of the network clearing model first proposed by Eisenberg and Noe (2001) thataccounts for bankruptcy costs as proposed in Rogers and Veraart (2013), and equity cross-holdings between financial institutions as modeled in Suzuki (2002); Gouriéroux et al. (2012).We wish to highlight the related work of Weber and Weske (2017) which also considers the jointimpacts of bankruptcy costs and equity cross-holdings, assuming fractional recovery of the valueof equity in case it is sold to pay off liabilities.The above cited works consider only fixed banking books where each firm has only “vanilla”external and interbank assets and liabilities. This fixed network assumption no longer holdswhen derivatives such as contingent convertible bonds (CoCos) are included. For networks withcontingent payments based on generalized credit default swap obligations, Schuldenzucker et al.(2017, 2019) show that the clearing payments may not be well-defined and, in fact, may notexist outside of specific network topologies. Klages-Mundt and Minca (2020) studies reinsurancenetworks and is able to determine the existence and uniqueness of the realized liabilities andclearing payments between firms in the system. Banerjee and Feinstein (2019) presents a modelin the vein of Banerjee et al. (2018) that is a time dynamic extension of Eisenberg and Noe(2001) for interbank liabilities with insurance obligations contingent on the state of the financialsystem.The motivation of the present paper is to understand the implications of contingent con-vertible bonds on financial stability within a contingent network model. In contrast to theabove cited papers, we are able to characterize network clearing with contingent convertiblebonds as a standard equilibrium problem in terms of a fixed point condition on a single vectorof firm wealths, see Veraart (2019); Barucca et al. (2016); Banerjee and Feinstein (2019, 2020);Banerjee et al. (2018), rather than joint payment and equity vectors as studied in Suzuki (2002);Gouriéroux et al. (2012); Weber and Weske (2017). This is made possible by adopting a CoCo ariant with fractional conversion, as introduced in Glasserman and Nouri (2012). This frac-tional conversion avoids many of the fixed point problems detailed in, e.g., Kusnetsov and Veraart(2019); Banerjee and Feinstein (2019), in which cycles exist in which either the bond is convertedor not. With these innovations, our first main contribution is to prove general conditions forthe existence of maximal and minimal clearing solutions for the contingent network model withbankruptcy costs.The second main contribution of this paper is to apply this contingent network frameworkto understand the conditions under which CoCo bonds improve financial stability. We provide ageneral discussion of when CoCo liabilities do and do not reduce the number of defaulting banks.Finally, we produce a stress testing framework to study the impacts of contingent convertibledebts on financial stability within the EU financial system as it appeared in the 2011 EBA stresstesting database. This study leads to the observation that replacing unsecured interbank debtin the EU financial system by CoCo securities will usually improve systemic risk.The organization of this paper is as follows. First, in Section 2, we present contingentconvertible bonds for a single bank, with an emphasis on the CoCo variant with fractional con-version. We extend this setting to a network setup of contingent convertible bonds in Section 3where we present the main result of this work – the existence of maximal and minimal clear-ing solutions for the interbank network with contingent convertible bonds. We expound uponthis result in Section 4 with a general discussion of when contingent convertible bonds improvefinancial stability and a provide an example of a small network showing that introducing suchfinancial instruments may increase the number of defaults in the system. We extend this workin Section 5 to networks of contingent convertible bonds in a discrete time framework whereconversion may occur before the maturity date. In Section 6 we utilize the 2011 EBA stresstesting data for the EU financial system in order to study the implications on stability whencontingent convertible bonds are introduced to a real financial network. To conclude, we suggestthat a regulatory intervention that replaces all unsecured interbank debt in the EU financialsystem by a standardized CoCo security will create a win-win situation in which systemic riskis decreased while bank shareholder value increases. Notation:
The following notation will be consistent for the entirety of this paper. Let x, y ∈ R n for some positive integer n , then x ∧ y = (min( x , y ) , min( x , y ) , . . . , min( x n , y n )) ⊤ , ∨ y = (max( x , y ) , max( x , y ) , . . . , max( x n , y n )) ⊤ ,x + = x ∨ , and x − = − ( x ∧ . Further, to ease notation, we will denote [ x, y ] := [ x , y ] × [ x , y ] × . . . × [ x n , y n ] ⊆ R n to be the n -dimensional compact interval for y − x ∈ R n + . Similarly,we will consider x ≤ y if and only if y − x ∈ R n + . We first consider a single bank whose total debt obligations are structured as a combinationof pure vanilla debt and a single class of contingent convertible (CoCo) debt. To avoid techni-cal difficulties that arise with certain CoCo specifications, we assume a variant of CoCo withfractional conversion, as first introduced by Glasserman and Nouri (2012). This type of CoCo,parametrized by a trigger level τ > and conversion factor η ∈ [0 , , converts a fraction of debtto equity at a set of conversion dates ≤ T < T < · · · < T K , until the maturity of the claimsat T = T K , if on these dates the equity of the issuer drops below τ times its debt. For each $1of CoCo debt converted to equity, the total firm equity increases by $1 and the CoCo investorreceives new equity shares with value η .In this section, we suppose the initial vanilla and CoCo debt have face values ¯ p ≥ and ¯ p c ≥ respectively, and a fraction λ of the CoCo is converted to equity at the final conversiondate T K . Our objective here is to define clearly the relationships between the assets, debt andequity of this bank, in all possible states of the balance sheet at the final conversion time T K .(Earlier conversion dates are considered in Section 5.) We focus on the firm’s wealth V , definedas the difference between total assets and total debt. Immediately after the final conversiondate T K we suppose, before considering bankruptcy, that this is given by ˜ V = x − ¯ p ( λ ) where x ≥ denotes the total external assets and ¯ p ( λ ) := ¯ p + (1 − λ )¯ p c . (1)Wealth V is also called the capital buffer , and equals equity E if V ≥ , in which casethe bank is solvent. Insolvency is defined by the condition V < ; we assume limited liability meaning that insolvency implies the default of the bank and that the value of equity is zero. If bank defaults, we also assume that only a fixed fraction α ∈ [0 , of its assets x are recoveredby the debtholders, in line with Veraart (2019).Under these assumptions, the conversion fraction will be λ = 0 (no conversion) if E = x − ¯ p − ¯ p c ≥ τ (¯ p + ¯ p c ) , and will be λ = 1 (full conversion) if x − ¯ p ≤ τ ¯ p . Partial conversionwith λ ∈ (0 , occurs if x ∈ (1 − τ ) × (¯ p , ¯ p + ¯ p c ) . The following relationships hold in general: λ ( V ) := 1 ∧ (cid:18) τ [¯ p + ¯ p c ] − Vτ ¯ p c (cid:19) + , (2) V = I { V ≥ } x + I { V < } αx − ¯ p ( λ ) . (3)Note that neither λ nor post-conversion equity E = V + depend on η .We now need to determine how the conversion factor η ∈ [0 , influences the fractions c ( λ ) , − c ( λ ) of equity E ( λ ) held by the CoCo holders and original shareholders respectively forany conversion fraction λ ∈ [0 , . If the external asset has value x ≥ (1 + τ )(¯ p + ¯ p c ) then λ = 0 and c (0) = 0 . If x ∈ (1 + τ ) × (¯ p , ¯ p + ¯ p c ) so that partial conversion λ = (1+ τ )(¯ p +¯ p c ) − x (1+ τ )¯ p c ∈ (0 , results, the equity will be E ( λ ) = τ (¯ p + (1 − λ )¯ p c ) since the bank must be at the equity-debtratio τ . To determine the function c ( λ ) , we consider a small shock ∆ x < to the external assetsthat induces a small change ∆ λ = − ∆ x (1+ τ )¯ p c . By the conversion condition for the given η , c ( λ + ∆ λ ) E ( λ + ∆ λ ) = c ( λ ) E λ + η ¯ p c ∆ λ + O (∆ λ ) where E λ denotes the bank’s equity under the shock ∆ x but prior to the additional ∆ λ CoCoconversion, i.e., E ( λ + ∆ λ ) = E λ + ¯ p c ∆ λ + O (∆ λ ) . Therefore, the fractional equity holdings c ( λ + ∆ λ ) can be computed via the relation: c ( λ + ∆ λ ) = c ( λ ) + (cid:18) ( η − c ( λ ))¯ p c τ (¯ p + (1 − λ )¯ p c (cid:19) ∆ λ + O (∆ λ ) . Taking the limit as ∆ x ր (i.e., ∆ λ ց ) results in the differential equation dc ( λ ) dλ = ( η − c ( λ ))¯ p c τ (¯ p + (1 − λ )¯ p c ) . (4)This can be solved with initial condition c (0) = 0 , giving the required formula for the proportion c ( λ ) of equity owned by CoCo holders given the fractional conversion of λ ∈ [0 , : c ( λ ) := η " − (cid:18) ¯ p + (1 − λ )¯ p c ¯ p + ¯ p c (cid:19) τ (5)
10 15 20 − External Asset x W e a l t h V (a) The bank’s wealth V as a function of the valueof the external assets x . External Asset x V a l u e o f E q u i t y Total Equity ( η = 0 )CoCo η = 0 . CoCo η = 1 . (b) The value of the bank’s equity V + ownedby the original shareholders as a function of thevalue of the external assets x . . External Asset x V a l u e o f B o nd s VanillaCoCo η = 0 . CoCo η = 1 . (c) The payoff as a fraction of the face value, fordifferent bonds, as a function of the value of theexternal assets x . Figure 1: Example 2.1: Illustrations of the value of CoCo bonds.
We conclude this section with a simple example to demonstrate the effect of CoCo financingon the balance sheet of a stylized bank.
Example 2.1.
Consider a bank with vanilla liabilities ¯ p = 10 and CoCo liabilities ¯ p c = 4 structured with trigger level τ = 0 . . This means the CoCo bonds convert from debt to equitywhen the debt-equity ratio exceeds /τ = 10 . Suppose also the recovery rate at default is α = 0 . . Notably, from (3) one finds that the wealth V of the bank satisfies a scalar fixed point quation parametrized by x : V = x + 4 λ ( V ) − if V ≥ x − if V < In our simple example, we will demonstrate that this has an explicit solution V ∗ = V ∗ ( x ) where V ∗ ( x ) = x − if x ≥ . x if x ∈ [11 , . x − if x ∈ [10 , x − if x ∈ [0 , . The key to this construction is considering the different scenarios for the conversion of the CoCobonds: • if the original equity-debt ratio overperforms the trigger level τ = 0 . then no conversionoccurs, i.e., V = x − ≥ . × ; • if the original equity-debt ratio is such that the bank can remain at the trigger level τ = 0 . by converting some CoCo bonds to equity then fractional conversion occurs, i.e., V = x + 4 λ ( V ) −
14 = 0 . × [14 − λ ( V )] implying V = x − V or V = x occurring when . − x ∈ [0 , from the construction of the fractional conversion λ ; • if the original equity-debt ratio after full conversion underperforms the trigger level τ = 0 . then full conversion must occur, i.e., V = x − ≤ . × ; and • if after full conversion the bank is insolvent then the bankruptcy costs must be applied,i.e., x − < .This outcome is plotted in Figure 1a; it shows that in the region of CoCo conversions x ∈ [11 , . the wealth V is made more stable as the external asset value declines. In contrastto the bank which benefits uniformly from the CoCo bonds, holders of bank debt find thatCoCo bonds underperform the payoff of the vanilla debt. The value of the equity for the originalshareholders is displayed in Figure 1b under three conversion factors η = 0 , . , , i.e., in which$1 of CoCo debt is converted to η of new equity at conversion. The original shareholders retainthe full equity of the bank for η = 0 , and the value retained by the original shareholders dropsas η increases (as more of the bank’s equity is held by CoCo bond owners). The CoCo payout,given as the total of remaining debt (1 − λ ( V ))¯ p c and value of the equity c ( λ ( V )) V + , is shown s a fraction of the face value in Figure 1c for η = 0 . , . We see that the higher the conversionfactor η the more value an investor will recover from CoCo bonds; it is also true that the valueof CoCo bonds will always be dominated by the payout of the vanilla debt. Remark 2.2. (i) The example illustrates that although CoCo bonds underperform vanilladebt, their existence stabilizes the health of the bank. If all debts were vanilla in thisexample then default would occur at x = 14 rather than at x = 10 , which demonstratesthe value of CoCo bonds in a crisis scenario.(ii) Although we assume a mechanical rule for conversion of CoCo bonds from debt to equityas was done in, e.g., Glasserman and Nouri (2012), CoCo securities often have an optionalstructure in which the issuer has a right to convert the debt at the trigger level, but notthe obligation to do so. Provided that banks are equity maximizers and the conversionrate η is at most 1, each institution will choose to exercise the CoCo option structure usingthe mechanical rules set out in this work. The remainder of this work will focus on a network of n financial institutions labelled by i ∈{ , , . . . , n } whose initial vanilla and fractional CoCo debt obligations have face values ¯ p i ≥ and ¯ p ci ≥ as in Section 2. Without loss of generality, we will assume ¯ p i + ¯ p ci > for all banks i . The CoCo parameters τ i , η i may vary across banks. These banks hold external assets andcross-holdings of interbank debt and equity. Like network models with vanilla debt and equitycross-holdings such as Eisenberg and Noe (2001); Rogers and Veraart (2013); Gouriéroux et al.(2012), we seek to determine the vector of firm wealths V after network clearing at a time T K .As in these papers, we assume the following stylized rules for clearing:(i) Limited liabilities : the total payment made by any firm will never exceed the total assetsavailable to the bank.(ii)
Priority of debt claims : a firm with V i < cannot pay its debts in full and hence willdefault, in which case the shareholders receive no value.(iii) All debts are of the same seniority : in case a bank has V i < and defaults, debts are paidout in proportion to the size of the nominal claims.Additionally, as in the prior section, a defaulted bank will realize only a fixed fraction α ∈ [0 , of its total assets. As is common in the systemic risk literature, we also add a “fictitious” bank,labelled by i = 0 , to represent the external holders of bank debt and equity. ur aim here is to determine the relationships between the balance sheets of all banks i ∈ { , , ..., n } just after clearing and CoCo conversion takes place at time T K . We suppose thebanks have the vector of wealths V ∈ R n and conversion factors λ ∈ [0 , n . The liabilities ofbank i consist of: • Vanilla liabilities : L i := π i ¯ p i ≥ is owed from bank i to entities outside the financialsystem and L ij := π ij ¯ p i ≥ is owed from bank i to any other bank j ; • CoCo liabilities : (1 − λ i ) L ci is owed from bank i to entities outside the financial systemwhere L ci := π ci ¯ p ci ≥ is the initial face value, and (1 − λ i ) L cij is owed from bank i to anyother bank j where L cij := π cij ¯ p ci ≥ .Its nominal assets with equity cross-holdings are denoted by: • External assets : x i ≥ is held in assets external to the financial network; • Vanilla interbank debt assets : P nj =1 L ji where bank j owes L ji ≥ to bank i ; • CoCo interbank assets : P nj =1 (1 − λ j ) L cji of remaining CoCo debt; • Interbank equity assets : bank i holds a fraction π eji ∈ [0 , of the original equity shares ofbank j plus the fraction π cji of the additional equity from the conversion of CoCo debt ¯ p cj .We assume that L ii = L cii = 0 to eliminate self-dealing and π eii = 0 to eliminate double countingof a firm’s equity. Recall that wealth V i of bank i is its (realized) total assets minus its totalliabilities.To obtain the required balance sheet relationships in this network setting, we extend thediscussion of Section 2 to account for interbank debt and equity assets. First we note that (2)still holds for each bank i : λ i ( V i ) := 1 ∧ (cid:18) τ i ¯ p i (0) − V i τ i [¯ p i (0) − ¯ p i (1)] (cid:19) + . (6)In case i defaults, because all debts have the same seniority, the interbank debts to other banks j will be paid in proportion to the fractions π dij ( λ i ) defined by the relation π dij ( λ i )¯ p i ( λ i ) = π ij ¯ p i + (1 − λ i ) π cij ¯ p ci . (7)Here we introduce again the notation ¯ p i ( λ i ) = ¯ p i + (1 − λ i )¯ p ci .If bank i is solvent, V i ≥ , then as derived in Section 2, its equity is split in the ratio c i ( λ i ) : 1 − c i ( λ i ) between the CoCo holders and the original shareholders, where c i is given by c i ( λ i ) = η i " − (cid:18) ¯ p i ( λ i )¯ p i (0) (cid:19) τ . Since any other bank j holds a fraction π eij of the original shares of i and a fraction π cij of theCoCo bonds of i , j will hold a fraction π eij ( λ i ) of the equity of bank i after conversion, where π eij ( λ i ) = c i ( λ i ) π cij + (1 − c i ( λ i )) π eij . (8)Recall it also holds that when V i ≥ , then the total asset value recovered from i is V i + ¯ p i ( λ ) .Finally, we note that when V i < and i defaults, λ i = 1 and, from the bankruptcy condition, itfollows that the total asset value recovered from i is α ( V i + ¯ p i (1)) .Putting together these formulas for the assets and liabilities of bank i gives the desiredclearing relation for the wealth V i in terms of the wealth vector V and fractional conversionvector λ : V i = x i + n X j =1 π dji ( λ j ) (cid:2) ¯ p j ( λ j ) I { V j ≥ } + α ( V j + ¯ p j (1)) I { V j < } (cid:3) + n X j =1 π eji ( λ j ) V + j − ¯ p i ( λ i ) . (9) The network clearing problem to find solutions of (9) can now be characterized as findingsolutions V ∗ to a vector valued fixed point equation: V ∗ = Φ( V ∗ ) , (10) Φ( V ) := x + Π d ( λ ( V )) ⊤ h diag( I { V ≥ ~ } )¯ p ( λ ( V )) + α diag( I { V <~ } )( V + ¯ p ( ~ i + Π e ( λ ( V )) ⊤ V + − ¯ p ( λ ( V )) . (11)where Π d ( λ ) and Π e ( λ ) are the matrices defined by (7) and (8) respectively.This clearing condition can be compared with the clearing mechanism in Banerjee and Feinstein(2020) for a vanilla interbank market with equity cross-holdings but without contingent convert-ible obligations. In their setting, Tarski’s fixed point theorem can be used to show that thereexists a greatest and least clearing solution to this clearing problem.We now consider the main result of this paper – the existence of a greatest and least clearingsolution in a network model with equity cross-holding and fractional CoCo bonds. Theorem 3.1.
There exist greatest and least clearing wealth solutions V ∗ + , V ∗− of V = Φ( V ) in he domain D := [ − ¯ p ( ~ , ( I − Π e ( ~ ⊤ ) − (cid:16) ( x + Π d ( ~ ⊤ ¯ p ( ~ − ¯ p ( ~ ∨ (diag( τ )¯ p ( ~ (cid:17) ] if the conversion factors of all CoCo bonds are bounded by 1, i.e., η ∈ [0 , n .Proof. As in the Example 2.1, we consider the four possible kinds of outcome that can occurfor bank i : no conversion of CoCo bonds when V i ∈ I := [ τ ¯ p i (0) , ∞ ) and λ i = 0 , strictlyfractional conversion when V i ∈ I := ( τ ¯ p i (1) , τ ¯ p i (0)) and λ i ∈ (0 , , full conversion but solvent V i ∈ I := [0 , τ ¯ p i (1)] and λ i = 1 , and default when V i ∈ I := ( −∞ , and λ i = 1 .Next, for each i = j we introduce functions A ji ( V j ) representing the assets that bank i recovers from bank j , and A i ( V ) representing the total assets that bank i recovers A ji ( V j ) = π dji ( λ j ( V j )) (cid:2) ¯ p j ( λ j ( V j )) I { V j ≥ } + α ( V j + ¯ p j (1)) I { V j < } (cid:3) + π eji ( λ j ( V j )) V + j , (12) A i ( V ) = x i + n X j =1 A ji ( V j ) . (13)(Note that A ii ≡ .) Then (9) becomes V i = Φ i ( V ) = A i ( V ) − ¯ p i ( λ i ( V i )) . (14)This function Φ i is nonincreasing in V i but nondecreasing in V j , j = i which means Tarski’sfixed point theorem does not apply to V = Φ( V ) . To overcome this obstacle, we use the factthat at a fixed point V ∗ , in the case of strictly fractional conversion V ∗ i ∈ I , the wealth can besimplified because bank i must be at the trigger level, i.e., V ∗ i = τ p i ( λ i ( V ∗ i )) . Combining thiswith (14), we find that if λ i ( V ∗ i ) ∈ (0 , then at the fixed point, ¯ p i ( λ i ( V ∗ i )) = (1 + τ i ) − A i ( V ∗ ) .This suggests that we can rewrite the clearing condition (10) in an equivalent way, as the fixedpoint condition V = ˆΦ( V ) where the function ˆΦ i ( V ) is defined by ˆΦ i ( V ) = Φ i ( V ) + I { V i ∈ I } ¯ p i ( λ i ( V i )) − (1 + τ i ) − ( x i + n X j =1 A ji ( V j )) . (15) ne can show that ˆΦ i ( V ) = F i ( A i ( V )) where the univariate functions F i are given by F i ( A i ) = A i − ¯ p i (0) if A i ∈ (1 + τ i ) × [¯ p i (0) , ∞ ) τ i (1 + τ i ) − A i if A i ∈ (1 + τ i ) × [¯ p i (1) , ¯ p i (0)) A i − ¯ p i (1) if A i ∈ [1 , τ i ] × ¯ p i (1) αA i − ¯ p i (1) if A i < ¯ p i (1) . From this formulation, one can prove two lemmas.
Lemma 3.2.
The fixed point sets of V = Φ( V ) coincide with the fixed point sets of V = ˆΦ( V ) . Lemma 3.3.
The univariate functions F i ( A i ) and A ji ( V j ) satisfy the following properties forall i = j :(i) F i is upper semicontinuous and nondecreasing in A i , and bounded from below by − ¯ p i (1) for A i ≥ .(ii) A ji is upper semicontinuous and nondecreasing in V j . This lemma immediately implies that each A i ( V ) and hence ˆΦ itself is upper semicontinuousand nondecreasing in V . Finally we need to show that ˆΦ maps the domain D into itself:(i) Let V = − ¯ p ( ~ ≤ ~ . Then λ ( V ) = ~ and hence ˆΦ( V ) = αA ( V ) − ¯ p ( ~ ≥ V . (ii) Let V = ( I − Π e ( ~ ⊤ ) − (cid:16) ( x + Π d ( ~ ⊤ ¯ p ( ~ − ¯ p ( ~ ∨ (diag( τ )¯ p ( ~ (cid:17) . By construction, ¯ V ≥ ( I − Π e ( ~ ⊤ ) V ≥ diag( τ )¯ p ( ~ which implies λ ( V ) = ~ and hence ˆΦ( V ) = Φ( V ) = A ( V ) − ¯ p ( ~ . Since A ( V ) = x + Π( ~ ⊤ ¯ p ( ~
0) + π e ( ~ ⊤ V we have ˆΦ( V ) − V = x + Π( ~ ⊤ ¯ p ( ~ − ¯ p ( ~ − ( I − π e ( ~ ⊤ ) V ≤ ~ (iii) By the monotonicity of ˆΦ , for any V ∈ D , V ≤ ˆΦ( V ) ≤ ˆΦ( V ) ≤ ˆΦ( V ) ≤ V , hence ˆΦ( V ) ∈ D .Given the two lemmas, we have shown that ˆΦ is a monotone nondecreasing function from thecomplete lattice D onto itself, and from Tarski’s fixed point theorem we draw the standardconclusion that it has a greatest and least fixed point in D .The proof of Lemma 3.2 is left to the reader. roof of Lemma 3.3. The statement for F i is trivial to verify. Similarly, the only non-trivialstatement for A ji ( V j ) to verify is monotonicity in the conversion region where λ j ( V j ) ∈ (0 , .For this, we can write A ji ( V j ( λ j )) = x j + π dji ( λ )¯ p j ( λ ) + τ ¯ p j ( λ ) π eji ( λ ) and show by differentiation that it is non-increasing with respect to λ . We use (7), (8) and thedifferential equation (4) for c j and find dA ji dλ j = − π cji ¯ p cj [1 + ( η j − c j ) + τ j c j ] − π eji ¯ p cj [( η j − c j ) + (1 − λ ) τ j c j ] (16)for which every term is explicitly non-positive.We conclude this section by remarking on the computation of the greatest clearing solution V ∗ = Φ( V ∗ ) . This can be found via Picard iterations of ˆΦ , defined in (15), beginning at V (0) = ( I − Π e ( ~ ⊤ ) − h ( x + Π d ( ~ ⊤ ¯ p ( ~ − ¯ p ( ~ ∨ (diag( τ )¯ p ( ~ i . By upper semicontinuity,these fixed point iterations will converge to the maximal clearing wealths. In this section, we will compare interbank networks that include CoCo bonds to similar in-terbank network models of equity cross-holdings and vanilla debt such as those presentedin Rogers and Veraart (2013). In this section, all CoCo bonds are assumed to have friction-less conversion η = 1 , as well as assumptions ensuring that the results of Theorem 3.1 hold.The comparisons we make in this section will be helpful in determining settings in whichintroducing CoCo bonds, or “CoCo-izing” debt obligations, may improve system behavior. Wemeasure system performance by the set of defaulting banks: We say system A outperformssystem B if D A ⊆ D B where D • denotes the set of defaulting institutions under the maximalclearing solution in a given system. Note that it is entirely possible that two systems are notcomparable in such a way. We now provide an extreme setting in which the CoCo-ized systemoutperforms (consistent) strict interbank networks, and then provide a simple counterexampleshowing that this ordering does not hold in general. Proposition 4.1.
Consider a contingent system of obligations that is fully CoCo-ized, i.e., ¯ p = ~ . Such a contingent system will have no defaults. Therefore, the contingent systemoutperforms all other financial systems. x x L = ¯ p L L Figure 2: Example 4.3: Network topology of interbank obligations.
Proof.
Let D denote the set of defaulting banks in the CoCo-ized network. Assume D 6 = ∅ . Take i ∈ D , i.e., V ∗ i < and note that ¯ p i (1) = ¯ p i = 0 . By construction of the fractional conversion, λ i ( V ∗ i ) = 1 . As such, by the construction of the clearing mechanism > V ∗ i = αA i ( V ∗ ) − ¯ p i (1) = αA i ( V ∗ ) ≥ since the assets A i ( V ∗ ) ≥ by construction. Thus we have a contradiction and it must hold that D = ∅ . As an immediate consequence, it must follow that the contingent system outperformsany other network. Remark 4.2.
As we have shown that the fully CoCo-ized system will outperform any othernetwork, this is true for any system with consistent parameters. That is, consider a vanillainterbank network with fixed obligations ˆ L := diag( ~ − λ ) L and interbank equity assets ˆΠ e :=Π e ( λ ) for any fractional conversion λ ∈ [0 , n ; allow all other parameters to keep their constantvalues. This partially CoCo-ized system will be outperformed by the fully CoCo-ized network.We now provide a simple counterexample to show that CoCo bonds do not uniformly improvesystem performance. This is undertaken by making such comparisons over consistent networksas considered in Remark 4.2. Example 4.3.
Consider a 2 bank and societal node system as depicted in Figure 2. We willfirst consider two networks – a purely vanilla interbank network and one where some obligationshave been CoCo-ized – with zero recovery at default ( α = 0 ) for which we can show analyticallythat the introduction of CoCo bonds can make a financial system worse. We will then generalizeparameters in this system in order to understand heuristics for when CoCo bonds may be helpfulfor financial stability.(i) Consider a standard interbank network in the vein of Rogers and Veraart (2013) with novanilla interbank equity assets (i.e., π eij = 0 for i, j ∈ { , } ) and no CoCo bonds. Letthe external assets be x = 6 and x = 1 , the interbank liabilities be L = 10 , L = L = 5 , and all other liabilities be identically . As such, ¯ p = L and ¯ p = L + L ith associated relative liability matrix Π . By construction, a simple clearing solution isprovided by: V ∗ = x + L − ¯ p = 1 V ∗ = x + ¯ p − ¯ p = 1 . It is clear that V ∗ ∈ R and thus neither bank defaults.(ii) Consider the same interbank network, but now where β ∈ [0 , of bank ’s liabilitiesare CoCo-ized with trigger level τ = 1 and with full conversion η = 1 . This choice ofparameters leads to the construction π e ( λ ) = β λ . By the construction of the fractionalconversion there are two cases to consider:(a) If β ∈ [ , then bank 2 will be in default with exactly β × CoCo bondstriggered. The resultant maximal clearing wealths are given by: V ∗ = x − ¯ p ( 710 β ) = 3 > V ∗ = − ¯ p = − < . We wish to note that, in this setting, the CoCo bonds improve the wealth of bank1, but cause bank 2 to default. To prove that this is the maximal clearing solution,consider this problem in the vein of a fictitious default algorithm. Assuming neitherbank defaults the wealths would be V (0)1 = > and V (0)2 = − < . This thenimplies bank 2 must default, and the above clearing solution results.(b) If β ∈ [0 , ] then all CoCo bonds will be triggered and there are three sub-settingsto consider:i. If β ∈ [0 , −√ ] then both banks are solvent with maximal clearing wealths V ∗ = x + L − ¯ p (1) = 1 + 10 β > V ∗ = x + ¯ p (1) + c (1) V ∗ − ¯ p = 10 β − β + 1 ≥ . We wish to note that, in this setting, the CoCo bonds improve the wealth ofbank 1, but decreases the wealth of bank 2. With these alterations, the total(system-wide) wealth improves with increased CoCo-ization β . i. If β ∈ ( −√ , ) then both banks are insolvent with maximal clearing vector V ∗ = − ¯ p (1) = − − β ) < V ∗ = − ¯ p = − < . This can be proven via a fictitious default algorithm. First, assuming neitherbank defaults then the wealths would be V (0)1 = 1 + 10 β > and V (0)2 =10 β − β + 1 < . Therefore, bank must default. This adjusts the value ofbank to V (1)1 = − β < and therefore both banks must default.iii. If β ∈ [ , ) then bank is insolvent with maximal clearing wealths V ∗ = x − ¯ p (1) = − β > V ∗ = − ¯ p = − < . This can again be proven via a fictitious default algorithm. First, assumingneither bank defaults implies the wealths would be V (0)1 = 1 + 10 β > and V (0)2 = 10 β − β + 1 < . Therefore, bank must default, and the aboveclearing solution results.These analytical results show that the conditions for comparing a CoCo-ized system to a strictinterbank network are non-trivial and non-monotonic.We now consider more broadly how the recovery rate α ∈ [0 , , trigger levels τ ∈ R , andfraction of liabilities made contingent β ∈ [0 , interact in this example. To do so we specifythree recovery rates: no recovery, fractional recovery, and full recovery ( α ∈ { , , } ) and threetrigger levels: low, medium, and high ( τ = τ ∈ { , , } ). These nine scenarios are then com-pared over the range of CoCo-ization fractions β ∈ [0 , to determine the default scenarios.Figure 3 provides images of the varying default scenarios under these varying network param-eters. Notably, changing the recovery rate does not affect the general shape of the defaultingregions but only the sum total of defaulting institutions. However, the trigger level causes sig-nificant impacts to the defaulting regions. This indicates that for CoCo bonds to be utilized forfinancial stability, they should be implemented with low trigger levels wherever possible. Remark 4.4.
Proposition 4.1 and Example 4.3 present an intriguing relationship for the useof CoCo bonds in regulation. If a regulatory agency dictated that all bonds must be CoCo-izeddependent on leverage and capital adequacy requirements, then these requirements will always a) α = 0 and τ = 1 (b) α = 0 and τ = 2 (c) α = 0 and τ = 5 (d) α = and τ = 1 (e) α = and τ = 2 (f) α = and τ = 5 (g) α = 1 and τ = 1 (h) α = 1 and τ = 2 (i) α = 1 and τ = 5 Figure 3: Example 4.3: Impact of varying network parameters on the set of defaulting firms for a2 bank system. be satisfied and no forced deleveraging will occur. However, this would occur at the expenseof other investors and funds, which may ultimately trigger greater systemic problems than itprevented.
So far in this work, we have considered conversion of CoCo bonds at maturity only. In thissection we introduce a multi-period discrete time framework and consider a network of vanillaand contingent obligations due on some day T ≥ in which the trigger level for the CoCo onds are checked on days t ∈ { T , T , ..., T K := T } . Due to these checks, even if a bank is wellcapitalized at the maturity of the obligations, conversion of debt for equity can occur at earliertimes.For this setting, we will consider a minimal modification from the single period setup pre-sented in Section 3. Let L ij ≥ be the vanilla interbank debt from firm i to j fixed at time t = 0 and due at maturity T . Let L c, ij ≥ be the CoCo interbank debt from firm i to j fixed at time t = 0 and due at maturity T . Let π e, ij denote the initial fractional holdings offirm i ’s equity held by bank j fixed at time t = 0 . To separate the effects of contingent con-vertible bonds from those of dynamic network models (as in, e.g., Capponi and Chen (2015);Kusnetsov and Veraart (2019); Banerjee et al. (2018)), all obligations in this section will bezero-coupon. Additionally, at time t ∈ { T , T , ..., T K } consider the value of the external assets x i ( t ) ≥ of bank i ∈ { , , ..., n } following some stochastic process. Finally for simplicity letthe risk-free rate r = 0 be zero.At time t ∈ { T , T , ..., T K } , some of the CoCo assets and liabilities may have already con-verted to equity assets and liabilities previously. To keep track of these modifications to thebalance sheet over time, let λ ti ∈ [0 , denote the fractional conversion of remaining CoCo bondsat time t issued by bank i . Then let L c,tij := (1 − λ ti ) L c,t − ij denote the not yet converted CoCodebt issued by bank i and held by bank j at the end of time t , and let ¯ p c,ti := P nj =0 L c,tij denotethe total remaining CoCo debt issued by bank i after time t clearing occurs. With these con-versions, the fractional holdings of equity will change over time as well. Denote the fractionalholdings by the CoCo bond holders c ti of bank i by c ti ( λ ti ) = c t − i " ¯ p i + (1 − λ ti )¯ p c,t − i ¯ p i + ¯ p c,t − i τi + η i − " ¯ p i + (1 − λ ti )¯ p c,t − i ¯ p i + ¯ p c,t − i τi = η − " ¯ p i + (1 − λ ti )¯ p c,t − i ¯ p i + ¯ p c, i τi . As in Section 3, the equity cross-holdings matrix is defined for each pair of banks i, j : π e,tij ( λ ti ) = π cij c ti ( λ ti ) + π e, ij (1 − c ti ( λ ti )) . It remains to determine the fractional conversion λ t that occurs at time t ∈ { T , T , ..., T K } based on trigger levels τ . As in the one period setting, this is dependent on the value of equity ( t ) at time t and the other system parameters, i.e., λ ti ( E ( t )) := 1 ∧ τ i [¯ p i + ¯ p c,t − i ] − E i ( t ) τ i ¯ p c,t − i ! + (17)for every bank i and any time t ∈ { T , T , ..., T K } . If all CoCo bonds have previously beenconverted to equity (i.e., ¯ p c,t − i = 0 ) then this fractional conversion is irrelevant and can bearbitrarily chosen.Intrinsic to (17) is the value of equity (at time t ∈ { T , T , ..., T K } ) for each firm. In orderto proceed, we assume, as in Glasserman and Nouri (2012), that the book value of equity istaken at all times t . In order to accomplish this, at any time t < T interbank assets will bemarked in full due to historical price accounting. Additionally, we will assume that, when pricingequity, only the conversion from debt to equity from the CoCo bonds at time t will be takeninto account. For notational simplicity and for comparison to the notation of Section 3, let ¯ p ti (0) = ¯ p i + ¯ p c,t − i and ¯ p ti (1) = ¯ p i for every bank i and at time t . With this construction, andthe same logic as in the construction of ˆΦ from (15), we find, at time t < T , that the wealthscan be valued by: V i ( t ) = A ti ( V ( t )) − ¯ p ti (0) if A ti ( V ( t )) ≥ (1 + τ i )¯ p ti (0) τ i τ i A ti ( V ( t )) if A ti ( V ( t )) ∈ (1 + τ i ) × [¯ p ti (1) , ¯ p ti (0)) A ti ( V ( t )) − ¯ p ti (1) if A ti ( V ( t )) < (1 + τ i )¯ p ti (1) (18) A ti ( V ( t )) = x i ( t ) + n X j =1 h L ji + (1 − λ tj ( V j ( t ) + )) L c,t − ji i + n X j =1 Π e,tji ( λ tj ( V j ( t ) + )) V j ( t ) + and equity E ( t ) := V ( t ) + is defined as the positive book wealth. Though a bank’s book equitymay be 0, in this setup we assume that no default can occur until maturity T . Remark 5.1.
In this work we consider the book value of equity prior to maturity. This is in con-trast to recent work on network valuation adjustments Barucca et al. (2016); Banerjee and Feinstein(2020) in which the market value is undertaken endogenously from the clearing system. Marketequity valuation with contingent debts is beyond the scope of this work.With this setup, the existence of a clearing procedure over time can be addressed. At alltimes prior to maturity, this is taken as the clearing procedure of (18) in the book value ofwealth; at maturity the problem reduces to that considered in Theorem 3.1 in the actualizedwealth. Algorithm 5.3 presents a method for constructing the clearing solution forward in time. orollary 5.2. Fix time t ∈ { T , T , ..., T K − } . There exists a greatest and least clearing bookvalue of wealth satisfying (18) in D t := [ − ¯ p t ( ~ , ( I − (Π e,t − ( λ t − )) ⊤ ) − (cid:16) ( x ( t ) + Π d,t ( ~ ⊤ ¯ p t ( ~ − ¯ p t ( ~ ∨ (diag( τ )¯ p t ( ~ (cid:17) ] ⊆ L ∞ ( σ ( { x (0) , x (1) , ..., x ( t ) } )) . Proof.
This is a direct extension of the proof of Theorem 3.1 and an application of Tarski’s fixedpoint theorem on the lattice D t .We conclude this section by constructing an algorithm in order to present the “optimal”clearing solution over time for some realization of the external market x . Algorithm 5.3. (i) Initiate k = 0 and network parameters L , L c, , Π e, ;(ii) Increment k = k + 1 ;(iii) If k = K then find the maximal clearing wealths V ( T K ) = Φ( V ( T K )) from (11) andterminate;(iv) Let V ( T k ) be the maximal clearing book wealths satisfying (18) and λ t := λ t ( V ( T k ) + ) bethe associated fractional conversions satisfying (17);(v) Update network parameters L c,T k := (1 − λ T k ) L c,T k − and Π e,T k := Π e,T k ( λ T k ) ;(vi) Return to step (ii).By an application of Corollary 5.2 and Theorem 3.1, this algorithm is guaranteed to provide aunique clearing solution. Additionally, this is called optimal as, at every time t ∈ { T , T , ..., T K } ,the equilibrium is chosen so as to maximize the equity at that specific time. It is, however,possible that a different solution early on may ultimately lead to higher potential equity at theterminal time. In this section we undertake a detailed case study for the implications of CoCo bonds on sys-temic risk in the EU system, using the data from the 2011 EBA stress test to calibrate thenetwork of obligations. This data set has been studied in, e.g., Gandy and Veraart (2016);Chen et al. (2016); Feinstein (2019); Feinstein et al. (2018) under the default contagion modelof Eisenberg and Noe (2001). Due to complications with the calibration methodology, we only consider 87 of the 90 institutions. DE029, LU45,and SI058 were not included in this analysis. he EBA stress test data set provides information on the total assets T A i , capital C i , andinterbank liabilities P nj =1 L ij for all banks i for a date in 2011. We construct stylized balancesheets consistent with this data, by making additional assumptions. First, since equity cross-holdings between systemically important financial institutions are typically small, we assume theoriginal equity cross-holdings are given by π eij = 0 for every pair of firms i, j . Next, we assumethat all assets that are not interbank must be external and all liabilities that are not capitalor interbank must be external, and moreover, as in Chen et al. (2016); Glasserman and Young(2015), we assume that interbank liabilities are equal to interbank assets P nj =1 L ij = P nj =1 L ji . Under these assumptions, the external assets and liabilities are given by x i := T A i − n X j =1 L ji and L i := T A i − n X j =1 L ij − C i . Having calibrated the external assets and liabilities, as well as the total asset and liabili-ties, we still require the full nominal liabilities matrix L . To find a single realization of theliability matrix consistent with the calibrated row and column sums, we utilize the methodof Gandy and Veraart (2016) with parameters p = 0 . , thinning = 10 , n burn-in = 10 , and λ = pn ( n − P ni =1 P nj =1 L ij ≈ . .In these case studies we adopt both the static time framework from Section 3 and thedynamic time framework from Section 5 and, in a similar approach to Example 4.3, considerthe implications on systemic risk through CoCo-izing the debts L at varying trigger levels.Throughout this section we take the recovery rate α = and conversion rate η = 1 . Incontrast to Section 4, herein we define the systemic risk measure as the fraction of total externalliabilities that are paid at maturity T , as debt, or if CoCo-ized, as equity. A histogram of thesize of these external obligations, as a fraction of the total external liabilities of e trillion across the system of banks, is displayed in Figure 4. Summary statistics of the system-widebalance sheet are provided in Table 1. We note that the obligations described by this data setare, predominantly, unsecured. This is due to the nature of interbank lending in Europe and,as such, fits within the clearing payment system described within this work. Example 6.1.
In this example we will undertake a stress-testing study of the contingent networkmodel, calibrated as discussed above to the EBA data, within the static framework of Section 3.As with Example 4.3, let β ∈ [0 , denote the fraction of liabilities that have been made intocontingent convertible bonds. For this study we assume that all banks follow the same CoCo- In actuality, as done in Gandy and Veraart (2016), we perturb the interbank assets slightly so as to satisfy sometechnical conditions.
Total Assets ( e trillion ) Total Liabilities ( e trillion )External: Interbank:
Capital: – 1.002Table 1: Summary statistics of the banking system from the 2011 EBA stress test data set. ization level β with the same trigger level τ ∈ (0 , . Herein we consider two different CoCo-izingschemes: • full CoCo-ization : β fraction of all debts, both interbank and external, are CoCo-ized; and • interbank CoCo-ization : β fraction of interbank debts only are CoCo-ized with all externaldebts remaining vanilla.We do not consider external CoCo-ization where all external debts are CoCo-ized with allinterbank debts remaining vanilla: This scheme appears approximately equivalent to the fullCoCo-ization scheme. For the purposes of this case study, we consider a stress-test scenariounder which the external assets of all banks are decreased by 3%. At β = 0 both schemescorrespond with the plain Eisenberg-Noe system (i.e., all debts are vanilla); under the stressscenario 72 of the 87 banks are in default and approximately 51.48% of debt owed to society isrepaid. The results of varying β and τ are displayed in Figures 5 and 6.In Figures 5a and 5b the fractional repayment of external liabilities is displayed under fullCoCo-ization and interbank CoCo-ization respectively. First we note the similarities between hese two schemes, looking for heuristics that may generally hold. In particular, the behavior ofthe system becomes worse as the trigger level τ increases, but there is a non-monotonic responseto the level of CoCo-ization β . Unlike in the dynamic time framework considered in the finalexample, in this static setting the best case scenario is for full repayment. For this reason thehealth of the system is monotonic in the trigger level τ : the greater the trigger level τ the moreoften debts are converted and, as the equity has no time to grow, a write-down occurs. Nowconsider the full CoCo-ization scheme depicted in Figure 5a. For low levels of CoCo-ization,there is a significant number of defaults (as is the case in the purely Eisenberg-Noe settingwith β = 0 ), but at a threshold level of β ≈ . the cascade of defaults is eliminated and allbanks become solvent with the external system (i.e. the real economy) recovering approximately . of the face value of its initial holdings. However, as more debts are CoCo-ized, eventhough no additional defaults occur, the fractional repayment to the external system decreases;this is especially noticeable for higher trigger levels τ . In contrast, for the interbank CoCo-ization scheme depicted in Figure 5b, the interaction between level of CoCo-ization β and thetrigger level τ is more complicated. For relatively low trigger levels τ , the external systemrecovers as much as . repayment at β ≈ . CoCo-ization. However, for higher triggerlevels, significant defaults may still be exhibited. We conclude the discussion of payments bydirectly comparing the full CoCo-ization and interbank CoCo-ization schemes. Clearly, for muchof the region, full CoCo-ization outperforms the interbank CoCo-ization by significant margins.However, for high CoCo-ization levels β the two schemes are comparable or, even better underinterbank CoCo-ization for high trigger levels τ .In Figures 6a and 6b the value for the original shareholders is displayed under full CoCo-ization and interbank CoCo-ization respectively. Notably, under full CoCo-ization, the originalshareholders benefit under higher trigger levels τ and greater CoCo-ization. In fact, by com-paring Figures 6a and 5a, it becomes clear that for CoCo-ization with high enough β so thatdefaults are avoided, the original shareholders benefit at the expense of the bond holders andvice versa. In contrast, interbank CoCo-ization can increase the original shareholder value onlyto a minimal degree, and does so in tandem with increasing benefits for the external debthold-ers as shown in Figures 6b and 5b. Clearly, the full CoCo-ization scheme benefits the originalshareholders more than interbank CoCo-ization, but the original shareholders would prefer bothschemes over the no CoCo-ization scheme ( β = 0 ). Example 6.2.
In this example we continue to study the EU network in the static frameworkof Section 3, now with a variable stress testing scenario and 4 distinct CoCo-ization schemes: a) Full CoCo-ization (b) Interbank CoCo-ization Figure 5: Example 6.1: Impact of varying fraction of contingent convertible bonds ( β ) and theirtrigger level ( τ ) on payments to the real economy using the 2011 EBA stress testing data. (a) Full CoCo-ization (b) Interbank CoCo-ization Figure 6: Example 6.1: Impact of varying fraction of contingent convertible bonds ( β ) and theirtrigger level ( τ ) on the value for the original shareholders using the 2011 EBA stress testing data. • no CoCo-ization : all debts, both interbank and external, remain vanilla; • full CoCo-ization : all debts, both interbank and external, are CoCo-ized with trigger level τ = 0 . ; • interbank CoCo-ization : all interbank debts are CoCo-ized with trigger level τ = 0 . withall external debts remaining vanilla; and • external CoCo-ization : all external debts are CoCo-ized with trigger level τ = 0 . withall interbank debts remaining vanilla.The no CoCo-ization scheme corresponds with the plain Eisenberg-Noe system. For the purposesof this case study, we stress the external assets of all banks by a variable fraction ξ , so that
0% 2% 4% 6% 8% 10%0.50.60.70.80.91 F r a c t i ona l R epa y m en t o f E x t e r na l L i ab ili t i e s No CoCo-izationFull CoCo-izationInterbank CoCo-izationExternal CoCo-ization (a) Payments to the real econ-omy
0% 2% 4% 6% 8% 10%00.10.20.30.40.50.60.70.80.91
No CoCo-izationFull CoCo-izationInterbank CoCo-izationExternal CoCo-ization (b) Value for the original share-holders
0% 2% 4% 6% 8% 10%01020304050607080 N u m be r o f D e f au l t s No CoCo-izationFull CoCo-izationInterbank CoCo-izationExternal CoCo-ization (c) Number of defaults
Figure 7: Example 6.2: Impact of varying stress levels ( ξ ) on payments to the real economy, valuefor the original shareholders, and banking defaults using the 2011 EBA stress testing data. x ∗ = (1 − ξ ) x external assets are available to the banks. The results of varying ξ ∈ [0% , aredisplayed in Figure 7. In Figure 7a, the value of debts owed external to the system is displayed.In Figure 7b, the value for the original shareholders is displayed. In Figure 7c, the numberof defaulting banks is plotted as a function of the stress level ξ . Notably, the system withoutCoCo-ization performs poorly, quickly encountering large number of defaults, low repaymentunder low stress scenarios, and large losses for the original shareholders. The full CoCo-izationand external CoCo-ization schemes are very similar; under both schemes, all banks remainsolvent and the losses to the external system are solely caused by the fractional conversion ofCoCo debts (see Example 2.1 for a simple construction for such losses). In fact, the no CoCo-ization scheme, for debtholders, outperforms both the full and external CoCo-ization schemesat low stress levels ( ξ ≤ . ). In constrast, the original shareholders strictly prefer eitherfull and external CoCo-ization over the scheme without CoCo-ization. Finally, the interbankCoCo-ization scheme outperforms the payments for all other schemes for much more sizeablestresses ( ξ ≤ . ), albeit at the expense of a single bank defaulting. Beyond ξ = 3 . the interbank CoCo-ization exhibits significant default contagion, though it always outperformsthe no CoCo-ization scheme in repayments of debts, equity for the original shareholders, anddefaults. Example 6.3.
In this example we study the unstressed ( ξ = 0 ) EU network in the dynamicframework of Section 5 under the full CoCo-ization scheme. Consider now the dynamic frame-work of Section 5. In this example we will undertake a numerical study of the EBA datacalibrated as discussed above, but without any stress scenario. As with the prior example, let β ∈ [0 , denote the fraction of liabilities that have been made into contingent convertible bonds;this CoCo-ization takes place for all liabilities (the full CoCo-ization scheme from Example 6.1). a) 1 time step (b) 2 time steps (c) 4 time steps Figure 8: Example 6.3: Impact on fractional repayment of external obligations by varying thenumber of time steps in a dynamic model of contingent convertible bonds.
For this study we assume that all banks follow the same level β and the same trigger level τ ∈ (0 , . The dynamic setting requires the external assets to evolve over time, and thus wewill assume that external assets follow correlated geometric Brownian motions. We take theinitial values x (0) = x and simulate the geometric Brownian motion forwards in time until time year. The remaining parameters for the external asset processes are fixed as follows: the risk-free rate r = 0 is fixed, the physical and risk-neutral measures are assumed to be identical, eachbank has volatility σ = 20% (from comparisons to annualized historical volatility of Europeanmarkets in 2011), and all banks have pairwise correlation.A single realization of the external asset process is simulated, leading to the results displayedin Figures 8 and 9 for the cases of , , and equally spaced time steps. Note that the singletime step setting corresponds exactly with the static model of Example 6.1 under full CoCo-ization, except now the value of the external assets is random. In contrast to the static setting,increasing the number of time steps evaluated can now produce gains for the debt holders inthe system. As indicated by the red areas of Figures 8b and 8c, there are regions of gains aslarge as 13% for the real economy due to contingent convertible bonds. However, these gainsfor CoCo debt holders are at the expense of the original shareholders as indicated by the blueareas of Figures 9b and 9c. Finally, we note that the network parameters that cause losses tothe debt holders in the static single time step framework cause comparable losses in the discretetime systems. Thus, for the purposes of stress testing, a static model may be appropriate andalleviate the complexity involved in the discrete time framework. We could, instead, consider a backwards evolution of the external assets so that x (1) = x , but such a setting isdirectly comparable to the static setting depicted in Example 6.1 above. a) 1 time step (b) 2 time steps (c) 4 time steps Figure 9: Example 6.3: Impact on the value for the original shareholders by varying the numberof time steps in a dynamic model of contingent convertible bonds.
In this work we studied a network model of interbank obligations and equity that includes CoCobond obligations between firms. We proved general conditions for the existence of maximal andminimal clearing solutions under this setting with bankruptcy costs. From the perspective offinancial stability, we proved that if all liabilities are rewritten as contingent convertible bonds,then the financial system will never experience any defaults. However, if only partial obligationsare CoCo bonds, then this can cause increased defaults in the interbank system. Heuristically,we come to the conclusion that lower trigger levels for the CoCo bonds in the system tend toimprove the financial stability. Countering this instinct, which would be a worthwhile follow-upstudy, the lower the trigger level the higher the interest rate that would need to be offered soas to make the CoCo bond attractive for investors. This would, ultimately, increase the totalliabilities of the issuer which can counteract the improvements in financial stability that mayresult from the CoCo bonds.Additionally, we consider a network of CoCo bonds in a discrete time setting so that con-version from debt to equity may occur prior to maturity. This is presented in a setting with thebook value of equity over time. An interesting extension of this work, along with further workon network valuation adjustments, is to endogenize the market value of equity in this system.Based on the theoretical and numerical results, especially in our EBA case studies fromSection 6, we propose that CoCo bonds should be used for interbank debts only. Heuristically,at the low trigger level τ = 0 . with full conversion η = 1 , CoCo-izing all interbank debtsstabilizes the system above the no CoCo setting with a higher value for the original shareholdersas well. This interbank CoCo-ization, empirically, reduces system defaults without the externalsystem realizing any writedowns due to the conversion of CoCo debt into equity. In addition, theoriginal shareholders in the banks realize at least as high a value as the no CoCo setting without he risk of large losses due to the early conversion of CoCo debts as exhibited by the full orexternal CoCo-ization schemes. In fact, we caution against external debts being redenominatedsince, beyond the threshold at which CoCo bonds rescue the system from defaults, they cancause significant writedowns without any defaults occurring. Due to these writedowns, CoCobonds will likely be less expensive than the vanilla instruments, i.e., higher interest rates willneed to be offered to raise the same amount of cash which can cause greater total liabilitiesand interlinkages, as such we propose the full conversion factor η = 1 so that the purchasersof these instruments can be appropriately renumerated for the potential drop in value of theCoCo bonds at or below the trigger level τ . Further work on network valuation and networkformation are necessary to adequately test this heuristic belief on where CoCo bonds should beimplemented in the financial system. References
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