Contingent Convertible Bonds in Financial Networks
CContingent Convertible Bonds in Financial Networks ∗ Giovanni Calice † Carlo Sala ‡ Daniele Tantari § September 2, 2020
Abstract
We study the role of contingent convertible bonds (CoCos) in a network of intercon-nected banks. We first confirm the phase transitions documented by Acemoglu et al.(2015) in absence of CoCos, thus revealing that the structure of the interbank network isof fundamental importance for the effectiveness of CoCos as a financial stability enhanc-ing mechanism. Furthermore, we show that in the presence of a moderate financial shocklightly interconnected financial networks are more robust than highly interconnected net-works, and can possibly be the optimal choice for both CoCos issuers and buyers. Finallyour results show that, under some network structures, the presence of CoCos can increase(and not reduce) financial fragility, because of the occurring of unneeded triggers and con-sequential suboptimal conversions that damage CoCos investors.
Keywords : Contingent Convertible Bonds, Financial Networks, Systemic Risk, Conta-gion.
JEL classification : G10, G13, G14, G17. ∗ Financial support from the AGAUR - SGR 2017-640 grant is gratefully acknowledged. † School of Business and Economics, Loughborough University, Loughborough, LE11 3TU, United Kingdom.E-mail:
[email protected]. ‡ Department of Financial Management and Control, Univ. Ramon Llull, ESADE, Avenida de Torreblanca59, Barcelona, Spain; E-mail: [email protected]. § Mathematics Department, University of Bologna, Via Zamboni 33, 40126, Bologna, Italy. E-mail: [email protected]. a r X i v : . [ q -f i n . GN ] A ug Introduction
The 2007-2009 financial crisis has highlighted the critical role of the interbank interconnected-ness for the stability of the global financial system. It also showed the importance of havingaccess to viable short-term funding in periods of crisis, being that liquidity problems canrapidly spillover to other interconnected banks and lead to default propagation. Crucially, anydefault propagation, if not controlled, might then lead to systemic risks and deep economicproblems. Interestingly, although the critical role of banks’ liquidity has been studied prior tothe financial crisis (see, among many others, e.g.: Freixas et al. (2000)), we know comparativelylittle about the causal implications of liquidity on banks default behaviours, and the precisemechanisms through which interbank default propagation operates in the financial system. Inthis regard, Andy Haldane, the Chief Economist of the Bank of England, has noted that “..tosafeguard against systemic risk, the financial system needs to be managed as a system” (Hal-dane (2009)). In another speech, Haldane has stressed how today’s financial system is heavilyinterconnected and raised the question on “...what might be done to close this fault-line, toimprove the resilience of the international monetary system?” (Haldane (2014)). He also ques-tioned the usefulness in the current financial system’s architecture of contingent convertiblebonds (CoCos), an hybrid financial instrument introduced by regulators after the financialcrisis with the aim of enhancing financial stability.Designed to reduce the impact of a lack of short-term liquidity in times of financial distress,CoCos have been extensively issued by financial institutions in the aftermath of the 2008-2009financial crisis, with the aim of providing a buffer in bad times. CoCos are essentially coupon-paying bonds that convert into equity shares, or are fully or partially written-off, when theissuer reaches a pre-specified level of financial distress. Hence, CoCos are regulatory instru- Started with the collapse of the US real estate market and the sub-prime mortgage market, the crisis severelyimpacted the interbank US debt market. Due to a reduced investors’ appetite for risk, overnight bank fundingrates dramatically spiked to unprecedented levels, and the interbank lending market collapsed. Consequently,troubled banks found impossible to refinance their short-term debt liabilities, paving the way to a dramatic creditcrunch. Credit crunch that caused several epic banks failures, such as Bearn Stearns and Lehman Brothersdefaults, subsequently spilled over to the real economy. Additionally, several other systemically importantfinancial institutions (SIFIs) had to rely on governments’ support to prevent bankruptcies, at the expense oftaxpayers. Throughout this paper we refer to CoCo and CoCos for single and multiple contingent convertible bonds,respectively. Firstly introduced by Lloyds in November 2009, CoCos have been promoted by different reg-ulators as a bail-in mechanism to promote banking stability and to reduce the fiscal burdensfor taxpayers. Through the October 2011 reformed Capital Adequacy Rules (Basel Committeeon Banking Supervision (2010)) and the November 2009 Capital Requirements Directive II(CRDII) the Basel Committee on Banking Supervision and the European Commission recog-nized CoCos as Tier 1 bank capital, thus making CoCos even more captivating for banks, asevidenced by the 2011-2015 boom in CoCos issuances. Using as an accounting-based triggerevent like the bank’s Common Equity Tier 1 (CET1), it is possible to differentiate betweenhigh and low trigger CoCos. High-trigger CoCos have a CET1-trigger level above 5.125% and,due to their capacity of reducing bank’s leverage, receive an equity-like treatment by regula-tors. Notably, the assimilation of CoCos to debt from a fiscal viewpoint, and to regulatorycapital (equity) by financial regulators, have paved the way to the emergence (and subsequentrapid growth) of a sizeable and active market for CoCos in both Europe and Asia. Low-triggerCoCos are instead normally considered as Tier 2 capital, and may lead to the orderly resolu-tion of failed banks. Departing from this distinction Goncharenko et al. (2017) documents theconnection between the issuer’s characteristics and the CoCos design.As documented by Avdjiev et al. (2020), since the Lloyd’s first issuance, more than 500 billionUS dollars of CoCos have been issued, with more than 400 issues by different banks in differentcountries. Interestingly, while for fiscal reasons in the US banks use other types of instruments For ease of space and clarity, in this paper we only focus on CoCos with single book-value trigger and equityconversion, and we leave the analysis of other types of CoCos as a future research. According to Moody’s Investors Service (2015), the period 2011-2015 has seen an issuance of approximatelymore than $300 billion CoCos with a boom for the year 2014 with an emission of $93 and $82 billion of Tier Iand Tier II CoCos due in large part to regulatory changes that made CoCos fiscally attractive. While the majority of issuances have been in Europe (inclusive of the UK and Switzerland), other countrieslike China (inclusive of Hong Kong), Australia, Japan, and Canada have also issued a substantial amount ofCoCos.
6o boost their Tier 1 capital (e.g.: preferred stocks), US investors are net buyers of CoCos,possibly attracted by the higher yields they offer within a low-interest-rate environment. Ina document for the Chairman of the U.S. House of Representatives Committee on FinancialServices, Greene (2016) discusses the conditions under which the introduction of CoCos in theUS capital markets may augment overall social welfare.Our paper contributes to the broader literature on systemic risk. To the best of our knowledge,this is the first paper, with Gupta et al. (2020), that analyzes the role of CoCos in an interbanknetwork. However, we depart from the framework in Gupta et al. (2020) and we propose atotally different network structure to describe our economy. While the very first idea of aCoCos-like product has been initially proposed by Merton (1991), the literature on CoCos be-gins with Flannery (2002) and Flannery (2005) which propose to use CoCos as a capitalizationbuffer in bad states of the economy and is surveyed in Flannery (2014). Kashyap et al. (2008)and the The Squam Lake Report (2010) discuss the role of CoCos in bank capital regula-tion. A recent strand of the literature discusses how the possible configurations of CoCos canlead to corporate governance problems like debt overhang, or to risk shifting/taking incentivesand possible bank failures due to extreme deleveraging (Koziol and Lawrenz (2012), Hilscherand Raviv (2014), Berg and Kaserer (2011), Chan and Wijnbergen (2017), Goncharenko et al.(2017), Martynova and Perotti (2018), Albul et al. (2013), Chen et al. (2017) and Goncharenko(2019). Notice that the effective modeling and design of CoCos is still a fundamental open andunresolved research question in the academic literature. Calomiris and Herring (2013), Boltonand Samama (2012), McDonald (2013), and Pennacchi et al. (2014) develop models where Co-Cos have trigger linked to accounting values. Sundaresan and Wang (2015), Glasserman andNouri (2016), Pennacchi and Tchistyi (2018), and Pennacchi and Tchistyi (2019) also analyzethe CoCos configuration, but from a market value perspective. The incentive effects of CoCosin individual banks have been investigated by, among others, Hori and Cer´on (2017).Our paper also adds in some respects to the financial network literature. While Allen and Gale(2000) and Freixas et al. (2000) are the first theoretical papers that analyze the stability of In the US, notably due to different accounting and tax rules, there is no marketability of CoCos. The interestedreader is referred to a 2012 Report published by the Financial Stability Oversight Council. As defined in their website, “The Squam Lake Group” is a non-partisan, non-affiliated group of academics whooffer guidance on the reform of financial regulation”. The group, made up of US academics, has been createdduring the 2007-2008 financial crisis, and focuses on long-term financial issues.
The remainder of the paper is organized as follows. Section 2 presents the interbank networkand the repayment system of the model. Section 3 describes, still without the presence ofCoCos, the ring, the complete and the intermediate financial networks. Section 4, introducesCoCos in the different networks and examines their financial stability effects. To better elu-cidate the role of the repayment rule in the network, we discuss the main differences betweenCoCos with and without equity conversion in Sections 4.1 and 4.2, respectively. Section 5concludes. All proofs and supplemental formal results are provided in the Appendix.
In its most simplified set-up (i.e. cash c i = 0) a stylized bank i ’s balance sheet looks like inFigure 1 where x ij represents what bank j returns to bank i , s represents the bank seniorobligations (e.g.: taxes, wages, money market funds) and z i is the remaining fraction of thebalance sheet not explained by the previous elements (e.g.: bank i ’s investments). Given this8igure 1: Bank i ’s balance sheet as defined in Equation (1).framework, the bank i ’s balance sheet identity is: (cid:88) j (cid:54) = i x ij + z i = (cid:88) j (cid:54) = i x ji + s (1)It follows that (cid:80) j (cid:54) = i x ji = x i (cid:54) y i , where y i is the face value of the inter-bank liability. Thus,when x i < y i then bank i defaults on its junior obligations.The presence of CoCos on the balance sheet of a bank and, more in general, any typeof debt securities, creates bilateral obligations among the issuer and the owner. To analyzehow these obligations affect the issuer and the owner we set up an interbank network. Banksobligations are represented as a weighted and directed graph on n nodes. Each bank in thenetwork is represented by a node. The obligations are then represented by directed edgesamong nodes. Specifically, a directed edge from node i to node j exists if bank i is creditorof bank j , such that y ij represents the face value of the contract among the two banks. Theinterbank network is thus identified by the collection of all the interbank liabilities among allbanks in the network, { y ij } . Definition 2.1.
Under this setting, a network is regular if (cid:80) j (cid:54) = i y ij = (cid:80) j (cid:54) = i y ji = y , i.e.everyone owes everyone the same amount. Following Acemoglu et al. (2015), we consider a single good, finite economy with threestates, t = 0 , , n risk-neutral banks, n < ∞ , where each bank i is endowed with aninitial capital k i . Banks cannot invest using their own fund, but can borrow money each otherto finance their investments. The initial capital k can be lent to other banks, kept as cash,or invested in competitive projects. Specifically, in state t the interbank lending takes place,9nd banks use the money borrowed to finance their investments. The investment can producetwo types of returns. A short-term t stochastic return z i , or a long-term t deterministicnon-pleadgeable return A , if the project is held until maturity. Also, at time t banks honortheir senior and interbank obligations. Due to their nature, senior obligations are nonnegative, s > R on the principal, so that the face vale of the j debt to bank i is the product of the amount borrowed and the interest rates: y ij = k ij R ij . Identifying with y i the bank’s i interbank obligations, it follows that the bank i total liabilities at time t are (cid:80) j (cid:54) = i y ji + s = y i + s . In terms of liabilities liquidation, all junior (interbank) debts have equalseniority, and are paid after the senior debts. To lighten the notation we assume s i = s , forany i = 1 , . . . , n and y i = (cid:80) j (cid:54) = i y ji = y , for any i = 1 , . . . , n thus implying a uniform senior andinter-bank liability, respectively. Once senior debts are paid, if the company defaults, juniordebts are repaid proportionally to their face values: x ji = φ i y ji , φ i ∈ [0 ,
1] for any i, j = 1 , . . . , n where parameter φ i is the bank fitness: if the bank i is insolvent to its junior debt then φ i < t , if necessary, banks can liquidate their investments at a cost ζ ∈ [0 , t . Formally, bank i returns to bank j the amount: x ji = y ji , if z i + ζl i + (cid:80) k x ik > s + y i φ i y ji if z i + ζl i + (cid:80) k x ik ∈ ( s, s + y i )0 if z i + ζl i + (cid:80) k x ik ∈ (0 , s ) (2)where l i ∈ [0 , A ] represents the bank’s liquidation decision: l i = (cid:20) min (cid:26) ζ ( s + y i − h i ) , A (cid:27)(cid:21) + (3)10here h i = (cid:80) j (cid:54) = i x ij + z i and [ · ] + = max[ · , t , we assume it to be greater than thesenior obligations in the case of no shock, z i = a > s , while in the case of a shock, z i = a − ε .Combining the liquidation decision, l i with the debt repayment rule, we obtain: x ji = y iji y i [min [ h i + ζl i − s, y i ]] + . (4)It is straightforward to see that when l i (cid:54) A , i.e. the project can be only partially liquidated,then x ji = y ji , thus x ji < y ji always together with a full liquidation l i = A . For this reason wecan substitute in the previous repayment directly l i = A to get: x ji = y ji y i [min [ h i + ζA − s, y i ]] + . (5)which does not depend on the liquidation rule and can be studied independently. Finally, wecan define the interbank social surplus as: u = n (cid:88) i =1 ( π i + T i )= n ( a + A ) − pε − (1 − ζ ) n (cid:88) i =1 l i where T i (cid:54) s is the returns to senior creditors, π i is the bank’s i profit. Denoting with p (cid:54) n the number of banks with shocks, the social surplus of the network changes and depends onthe magnitude of the shocks and the liquidation value. As the liquidations decrease ζ → u = n ( a + A ) − pε − n A (6)Hence, the interbank social surplus is inversely related to the number of defaults.Equation (5) suggests that the project acts simply as an additional asset and, therefore, canbe absorbed in the variable z i . This ensures that we can study the stability properties of thebanking system in the case A = 0 without loss of generality, where the debt repaying rule is11efined as: x ji = y ji y min y, z i + (cid:88) k (cid:54) = i x ik − s + . (7)The previous rule can be interpreted as a propagation rule for financial distress, i.e. in termsof bank’s fitness it reads as: φ i = min (cid:18) , z i − s + (cid:80) k (cid:54) = i y ik φ k y (cid:19) + = f y,s ( h i ( φ )) , (8)where we have defined the activation function f y,s ( h ) = min(1 , h − sy ) + and the bank income h i ( φ ) = z i + (cid:80) k (cid:54) = i y ik φ k , depending on the asset side of the balance sheet and the system’samount of distress. Equation (8) can be thought either as an updating rule for fitness propa-gation, φ t +1 i = f y,s ( h i ( φ t )), or as an equilibrium defining map F : [0 , n → [0 , n : φ = ( φ , . . . , φ n ) → F ( φ ) = ( f y,s ( h i ( φ ))) ni =1 , (9)which, aside from a very specific choice of the model parameters, it generally has a unique fixedpoint, that can be obtained numerically by simply iterating the rule from a starting point φ (e.g.: (1 , . . . , extent of contagion: E ( φ ) = 1 − n n (cid:88) i =1 δ φ i , ; (10)where δ φ i , is the Kronecker delta defined as: δ φ i , = i = j i (cid:54) = j (11)and 2) system’s distress : D ( φ ) = 1 − n n (cid:88) i =1 φ i . (12) More details in Acemoglu et al. (2015). topologicalproperties of the interbank directed and weighted network Y = ( y ij ); and 2) the size and distribution of the shocks.It is worth noticing how under the Acemoglu et al. (2015)’s setting, the number of negativeshocks in the network, and their entity provide a way to evaluate two financial networks,for example { y ij } and { (cid:101) y ij } , in terms of stability and resilience. The former is a max-minclassification, the latter is an expectation. Given p , the financial network { y ij } is more stable than { (cid:101) y ij } if E p u (cid:62) E p (cid:101) u (which when p = 1 and for symmetric regular network implies u (cid:62) (cid:101) u ).Given p , the financial network { y ij } is more resilient than { (cid:101) y ij } if min u (cid:62) min (cid:101) u (whichfor p = 1 and symmetric regular networks implies again that u (cid:62) (cid:101) u ). Note that under oursetting, shocks are deterministic. Hence, the concepts of resilience and stability as previouslydefined would coincide. Interestingly, the extent of contagion proposed in Equation (10) is aproxy for financial stability and resilience under deterministic shocks. Consequently, we departfrom Acemoglu et al. (2015) as we provide a novel and more general framework to analyze thebehaviour of the network structure. In this section, we formally introduce the structure of the networks and examine how exogenous(large and small) shocks impact on the different interbank networks. More precisely, to pro-vide evidence on how different network connectivities may lead to different results under smalland big shocks, we simulate negative shocks of different magnitude in the interbank networksdescribed above. Note that at this stage, we still assume no CoCos in the network. CoCos willbe introduced in Section 4, and the types of interbank networks - without and with CoCos -will be presented and compared. From the obtained results we also suggest some preliminarypolicy implications.For the objectives of our analysis, we consider three types of regular financial networks,namely ring, complete and intermediate networks. Specifically, a financial network is a ringnetwork if y i,i − = y ,n = y and y ij (cid:54) = 0 otherwise, where n is the total number of banks. From13his configuration, bank i is the unique creditor of bank i −
1, and bank 1 is the unique creditorof bank n . As a consequence of the ring structure, a default of a bank spillovers entirely onthe subsequent banks. This property makes the ring network the least interconnected type offinancial network.A financial network is a complete network if y ij = yn − ∀ i (cid:54) = j . Under this setting, a liabilityand thus a possible bank default, is equally divided among all n banks in the financial network,thereby making the complete network the most interconnected type of financial network. Thering network and the fully connected network are just two extreme cases of regular networks, thefirst having the minimum possible connectivity ( k in = k out = 1) the latter the maximal possibleconnectivity ( k in = k out = n − k in = k out = c,
Given ε ∗ = n ( a − s ) and y ∗ = ( n −
1) ( a − s ) , then: • as soon as ε < ε ∗ (small shock regime) or y < y ∗ (low exposure regime) the extent ofcontagion in the ring network is larger than that in the complete network. • as soon as ε > ε ∗ and y > y ∗ default becomes systemic in both the ring and the completenetworks. As shown in Appendix A, for both cases, Theorem 3.1 can be proven analytically by solv-ing the equilibrium Equation 8 for bank fitnesses, and then comparing the extent of financialcontagion. 14igure 2: Simulation on ring and complete network with N = 50, a = 21, s = 20, y = 75 > y (cid:63) .Left panel: extent of contagion; Right panel: system’s distress.Focusing on the extent of financial contagion and on banking distress, the left and right pan-els of Figure 2 indicate the values of E( φ ) and D( φ ) as a function of the shock in the highexposure regime ( y > y (cid:63) ), respectively. Both figures depict with a blue (green) continuousline the ring (complete) network and, dotted in black, where the contagion becomes systemic, ε = ε ∗ . Results are obtained by simulating the equilibrium in a network of N = 50 banks, witha return from the investment equal to a = 21, senior obligations s = 20 and in high exposureregime y = 75 > y (cid:63) . To simplify the simulation we set R = 0, thus assuming the debt tobe already inclusive of any interest and, as in Acemoglu et al. (2015), we set the liquidationcost equal to ζ = 0. Notably, although our focus is not directly on resilience and stabilityas in Acemoglu et al. (2015) we also find, for both our variables of interest, a clear phasetransition between the ring and the complete networks. Therefore, these results are suggestiveof at least two important policy implications. A first ex-ante recommendation, and consistentwith the evidence presented in Acemoglu et al. (2015), is for regulators to enhance the regu-latory oversight and the effective monitoring of the interbank interconnections (connectivity).This activity in turn would require the design and implementation of data-based measures ofinterconnectedness catching the degree of exposures between financial institutions. A second,ex-post recommendation, is to contain, in presence of large negative shocks, the propagation ofcontagion by providing financial support or even bail out the SIFIs and the too interconnectedto fail financial institutions. Yet, regulators should carefully weight in the associated opportu-nity costs of moral hazard and incentives for excessive risk taking by banking intermediaries.15herefore, our results strongly corroborate the view that government interventions might bebeneficial (welfare improving) for the stability and efficiency of the financial network’s struc-ture (especially for a highly interconnected, such as the complete network).The reason for the existence of a phase transition is the presence of two types of shock ab-sorbers in the network. The first absorber is the excess liquidity a − s > s of senior creditors of the distressed bank. The firstshock absorber perform efficiently in the complete network, while the latter is least effectivein the complete network. Thus, contagion (default leading to default) is quicker with the ringnetwork for small shocks hence, the ring network is less resilient and stable than the completenetwork. On the other hand, large shocks, which by definition envisage less interconnectionamong banks, lead to senior debts being wiped out to absorb losses. Hence, the ring networkbecomes as stable as the complete network thus reflecting a robust-yet-fragile framework.Finally, we repeat the same analysis but for different intermediate network connectivity, c = 2, 3, 10, 20, 30, 40. Note that it is possible to generate randomly a regular network using thedirected configuration model Bollob´as (1980); Newman (2010), i.e. by considering for each nodea number k in (resp. k out ) of half in-links (resp. half out-links), and then randomly connectingeach half out-link with a half in-link. A practical complication of the configuration model isthat sampled networks may have self-loops and multi-edges (especially for large connectivity).These can be removed manually but, as a result, the network is not exactly regular. To limitthe impact of this heterogeneity, we define the vector z of bank investments in absence ofshocks as: z i = a + ( y in − y outi ) , i = 1 , . . . , n, (14)where y in = (cid:80) j y ji = y is the bank i liability, while y outi = (cid:80) j y ij is the total system liabilityto bank i , that could be different from y . The previously defined z i compensates the imbalancesuch that each bank, in absence of shocks, has a net incoming flow a , as should be in a pureregular network.The results depicted in Figure 3 are obtained by averaging over different network real-izations simulated with the same degree of connectivity c . Once more, the ring and complete16igure 3: Simulation on Random Regular networks with N = 50, a = 21, s = 20, y = 75 > y (cid:63) .Results are averaged over 10 different realizations sampled from a directed configuration model.Left panel: extent of contagion; Right panel: system’s distress.networks are depicted in blue and green with a continuous line, while the intermediate networksare represented with different colors. Interestingly, we can clearly see that with an extent offinancial contagion almost equal to zero, results confirm the higher stability of the completenetwork to small shocks, with respect to all other networks. In fact, while for the ring and allintermediate networks the extent of contagion is increasing with the shock size, the completenetwork experiences a systemic contagion only for ε > ε (cid:63) , and with a jump that determines thepreviously defined phase transition. Focusing on the intermediate networks, it is possible todig deeper into the behavior of networks and their responses to shocks. Notably, our evidenceindicates the existence of three and not only two regimes. More precisely, starting from thering network and increasing the degree of connectivity, the system becomes more and more re-silient, with a jump in the extent of contagion for increasing values of the shock size. The jumpcorresponds to the point in which the shock propagates simultaneously to all the neighbors ofthe stressed bank. In the limit, the network becomes fully connected, and the system is stableup to the transition point where the shock becomes systemic. It is interesting to note thatthere are regimes in the shock size where higher connectivity does not translate into higherstability. This implies that a more connected network is typically more stable only before theoccurrence of the first jump. Afterwards, the system becomes more fragile as opposed to thecase of lower connectivity. 17 Financial Network with CoCos
In this section, we repeat the same experiment of the previous section, except this time weadjust the model by introducing CoCos within the banks balance sheet structure to examinetheir impact on the interbank network. To provide direct evidence on the effects of the conver-sions, we propose the analysis for equity-conversion CoCos. To add CoCos in the bank balancesheet structure, we first finalize the bank i ’s initial balance sheet including the equity E i , asillustrated in Figure 4. Thus, the bank i ’s balance sheet identity is modified accordingly:Figure 4: Bank i ’s balance sheet as defined in Equation (15). (cid:88) j (cid:54) = i x ij + z i = (cid:88) j (cid:54) = i x ji + s + E i . (15)As a consequence of adding equity in the analysis, the social surplus in the network is nowdefined as: u = n (cid:88) i =1 ( E i + T i ) . (16)As a second step, let V i = (cid:80) j (cid:54) = i x ij + z i . Then, under this set-up, if x ji were vanilla bondswith face value y ji , the payoffs to each creditor are: s i = min [ s, V i ] x i = [min [ V i − s, y i ]] + (17) E i = [ V i − s − y i ] + y i = (cid:80) j (cid:54) = i y ji . It is worth noticing that the payoff x i is equivalent to the Acemogluet al. (2015) payoff for the debt repayment. Third, let all x ij be CoCos with vanilla bondsrepresented by s .A crucial and not trivial element in the CoCos configuration is the trigger. In his broadliterature review, Flannery (2014) shows that a suitable and always valid trigger for a CoCois: P t Q t B t (cid:54) N − ( α | σ A t ) (18)where P t Q t is the firm’s share price and the relative quantity, B t the most recent book valueof the assets, α the desired solvency level and σ A t the volatility of the assets portfolio. Whilethe left hand side is a common indicator for most CoCos with a book-value trigger, the righthand side is fully discretionary. In our case, we generalize Equation 18 and consider a CoCosto be triggered whenever: E i V i (cid:54) τ ⇔ V i (cid:54) s + y i − τ where τ is the trigger capital ratio.Having introduced CoCos in the bank i ’s balance sheet, we next examine their role within anetwork of banks in two steps. In Section 4.1, we first consider a network of banks with CoCosand study how and when shocks lead to a systemic CoCos triggering. Then, in Section 4.2,we explore the real effects of CoCos as a regulatory tool on the banks’ balance sheets bymodelling the dynamics of the actual amount of money (i.e. bonds) converted into equity incase of trigger. For CoCos with equity-conversion, in the case of trigger, bonds are partially or wholly convertedinto equity to make up (if possible) the shortfall up to E i = τ V i . Suppose that before conversion E i = ¯ EV i , with ¯ E (cid:54) τ . The amount of bond ∆ y to be converted into equity is thus ∆ y =( τ − ¯ E ) V i . If y i < ∆ y the entire CoCo is converted. Note that: y i < ∆ y ⇐⇒ V i − s − ¯ EV i < ( τ − ¯ E ) V i = ⇒ V i < s − τ . (19)19e can therefore distinguish between the following cases. First, if s − τ (cid:54) V i (cid:54) s + y i − τ , the bond ispartially converted into equity, hence this implies that the CoCos holders hold ∆ y = ( τ − ¯ E ) V i of converted equity and x i = y i − ∆ y = (1 − τ ) V i − s of unconverted CoCos. Second, if s (cid:54) V i (cid:54) s − τ , the entire CoCo is converted. Hence, the CoCo holders hold ∆ y = y i ofconverted equity and no unconverted CoCo. Third, if V i (cid:54) s , the equity holders bear the lossesbefore senior creditors. Combining all these cases together we obtain: x i = min[ y i , (1 − τ ) V i − s ] + . (20)Equivalently, the trigger propagation rule in terms of the banks fitness φ i = x i /y i is: φ i = f y,s ((1 − τ ) h i ( φ )) , (21)with the same propagation function of the case with no CoCos. The main difference betweenEquation 20 and Equation 21 is in the interpretation of the fitness parameter φ i . In presence ofCoCos, φ i <
1, does not suggest a default, but just the triggering of the CoCo. Consequently,the distress propagation in this system indicates a triggering propagation. It is worth stressingthat there is a unique equilibrium which can be derived analytically and whose property issummarized in the following theorem:
Theorem 4.1.
There exist exposure thresholds y ∗ r ( τ ) , y ∗ c ( τ ) given by Equations (38) , (44) andshock thresholds ε ∗ r ( y, τ ) , ε ∗ c ( y, τ ) given by Equations (39) , (45) such that: • when ( y, τ ) ∈ S r = { ε > ε ∗ r ( y, τ ) , y > y ∗ r ( τ ) } the shock triggers a systemic CoCo triggeringin the ring network. • when ( y, τ ) ∈ S c = { ε > ε ∗ c ( y, τ ) , y > y ∗ c ( τ ) } the shock triggers a systemic CoCo triggeringin the complete connected network. S r and S c identify the systemic unstable regions for the ring and complete networks, respec-tively. It follows that their complement S cr and S cc identify the safe regions for the ring andcomplete networks, respectively. Moreover S r ⊂ S c and • when ( y, τ ) (cid:54)∈ S c (implying ( y, τ ) (cid:54)∈ S r ) the ring network is the least (and the completenetwork is the most stable) financial network. τ . • there exist a region where ( y, τ ) (cid:54)∈ S r but ( y, τ ) ∈ S c where the ring network is the most(and the complete network is the least) stable financial network. The proof is reported in Appendix B.2 and is based again on the analytical solution of thefixed point Equation (21). The first part of the theorem states that, in presence of CoCos,the unstable region (large shock and high exposure) is not universal, as in the case of vanillabond, but strongly depends on the network structure. As a consequence of the dependence onthe network structure, there are different thresholds for different levels of shock size ( ε ) andexposure ( y ). Moreover, as shown in Figure 5, the unstable regions (or equivalently the saferegions) are strongly affected by the triggering parameter τ . When τ tends to zero, CoCosbecome vanilla bonds, and the two regions tend to coincide. When instead τ increases, the tworegions tend to diverge more and more significantly. In the second part of the theorem, the twounstable regions (for the ring and complete networks) are compared. Remarkably, it appearsthat the safe region for the complete network is the smallest one. This suggests that if the size21f the shock is such that it does not determine a systemic triggering in the complete network,the latter is the most stable topology (first point). However (second point), the triggering inthe complete network becomes systemic for a shock size that is smaller than the one necessaryfor a systemic triggering in the ring network. As such, as illustrated in Figure 5 with the orangeregion, there exists a medium shock size region, in which the ring network is more stable thanthe complete one. For larger shocks, the triggering becomes systemic also in the ring network,and the two topologies are equivalently sub-optimal.Figure 6 shows the extent of contagion and system’s distress as a function of the shock size ina high exposure regime, for different network connectivity. For consistency with the previousanalysis, we repeat the simulation exercise on random regular networks with N = 50, a = 21, s = 20, y = 75, and all results are averaged over 10 different realizations sampled from adirected configuration model. Again, the degrees of connectivity are equal to c =2, 3, 10,20, 30, 40 and maintained constant throughout the simulations. As we can observe from theFigure 6: Simulation on random regular networks with N = 50, a = 21, s = 20, y = 75. Resultsare averaged over 10 different realizations sampled from a directed configuration model.figure the system behaves as in the vanilla bond case for a shock of relatively small size. Thering network is the most resilient, the complete network the least robust and for intermediateconnectivity jumps occur again, delimiting the region where the higher connectivity enablesthe system’s robustness. Note that immediately after a jump occurs, the network enters afragility phase where systems with lower connectivity are more robust. For medium shockssizes, there is an inversion point, where all the lines in Figure 6 cross, after which the contagionbecomes systemic in the complete network, whereas the ring network is still relatively stable.22inally, for larger shocks sizes, all the network structures are equivalent, since they cannotprevent the triggering to become systemic. A final remark on the difference between CoCosand vanilla bonds concerns the shape of the shock size threshold ε ( y, τ ) which now explicitlydepends on the bank exposure y (instead of τ ). In particular, for both network topologiesthe critical threshold decreases with the exposure (the higher the exposure the smaller thesafe regions) while in the presence of vanilla bond ( τ = 0) it is independent. An importantfinancial stability consequence is the existence of a maximal exposure, beyond which any (evensmall) shock can determine the triggering of the distressed bank and can potentially lead to asystemic crisis. As a final step, we proceed to analyze the scenario of a trigger event in which the CoCos holderbank uses the equity received from the CoCo issuer bank. Note that neglecting this monetaryvalue would result in an overestimation of both the shock size and the degree of systemicdistress or, equivalently, to underestimate the systemic crisis threshold. More specifically, inthis section we document the possibility for the CoCos holder to liquidate, at a liquidationcost, the dollar amount of shares of the CoCos issuer bank received as a consequence of thetrigger. Formally, if a shock in the economy leads to a trigger, the CoCo issuer bank i repaysan amount x nci of unconverted CoCo: x nci = min[ y i , (1 − τ ) V i − s ] + . (22)while it converts and uses the remaining amount x ci : x ci = y i − x i . (23)If we denote with η ∈ [0 ,
1] the effective market value of the issuer bank’s share after thetrigger, we can generalize the debt repaying rule as: x i = x nci + ηx c = ηy i + (1 − η ) min[ y i , (1 − τ ) V i − s ] + (24)23nd, equivalently, the propagation rule in terms of the bank fitness as: φ i = η + (1 − η ) f y,s ((1 − τ ) h i ( φ )) . (25)The value of η is bounded between 0 and 1, and denotes the dollar value of the convertedshares. Indeed, such a value depends by many internal and external factors, like the overallquality of the bank, of the banking system and the size of the shock that affects an individualbank. Moreover, a CoCo conversion might depress the bank’s equity, being a trigger usuallyseen as a bad signal for the bank issuer by market participants. In this paper, we treat η asexogenously given, and analyze the effects on the level of bank fitness for different values of η .First, for η = 0 (shares have zero value) the bank cannot use any money from the conver-sion and we indeed retrieve the results of the previous section. Second, for 0 < η (cid:54) η becomesthe minimum possible bank fitness. Figure 7 shows the extent of contagion as a function ofthe shock obtained by simulating the equilibrium in a network of N = 50, a = 21, s = 20, y = 75. Given this equilibrium and a fixed value τ = 0 . η = 0 .
03 (left)and η = 0 . η = 0 .
03 (left panel) the ring network is more stableFigure 7: Simulation on Random Regular networks with N = 50, a = 21, s = 20, y = 75.Results are averaged over 10 different realizations sampled from a directed configuration model.24han both the complete network and all the intermediate ones. In fact, while the extent ofcontagion is systemic (=1) in the ring network for big shocks only (from ε (cid:62)
35 on), thecomplete network is the first type of network that achieves maximal instability (for ε (cid:62) η increases, but in a non-linearway. As we can see from the right panel of Figure 7, for η = 0 .
3, the ring network is notonly again the most stable network, but a) it never achieves the full extent of contagion andb) reaches the maximal extent of contagion at around E( φ )=20% also for the largest shocks.Moreover, although the extent of contagion increases with the degree of interconnections, thechange is non-linear. Therefore, the lightly interconnected networks ( c = 2 and c = 3) arenever susceptible to systemic contagion and reach their maximum value at E( φ ) ≈ ε = 17). Hence, our results clearly demonstrate that the equity conversion is themost beneficial for no or lowly interconnected networks and the most detrimental for highlyinterconnected networks. In fact, highly interconnected networks are more prone to financialcontagion in presence of a CoCos trigger. Note that the estimated results depend on the choiceof η , to circumvent discretionary choices and to provide a general overview of the role of η forthe complete and ring networks. Figure 8 summarizes the results. Plotting the level of thecritical size of the shock ε ∗ ( η, τ ) ∈ [0 , η ∈ [0 , η ≈ . η , see AppendixC for explicit expressions.systemic financial risk. By contrast, unnecessary conversions might actually penalize (and hit)the bond (CoCo) holders. We define the conversions to be unnecessary whenever a shock leadsto a systemic conversion without creating systemic risk in the economy. To better elucidatethis result, we now compare Figure 2 with Figure 8. Figure 2 shows that in absence of CoCosthe risk is systemic only from ε (cid:62)
50, while Figure 8 illustrates that the ring and many lightlyinterconnected networks experience a systemic trigger for much smaller shocks in the economy.From a market perspective, to avoid the automatic triggering below a certain threshold, CoCosissuers could consider to issue CoCos with a dual trigger conversion (e.g.: McDonald (2013)),one endogenous trigger, linked to the balance-sheet or firm’s equity value, and one endogenous,controlled by the regulatory authority.Our results have important implications for financial stability, policy analysis and wel-fare. A first notable implication is for policymakers and regulators to be vigilant and pursuea targeted nuanced monitoring approach on the issuance of the contracts and the way theinstruments are actually managed by financial institutions, primarily by the systemically im-portant financial institutions (SIFIs). We advocate that regulators should consider an adequatemonitoring of the degree of exposure to these instruments of the SIFIs in order to mitigate pos-sible systemic risk concerns especially in the case of highly interconnected networks of banksand/versus other financial institutions. A first specific policy recommendation is for the rele-vant supervisory authorities to enhance the disclosure requirements of the typology, featuresand amount of CoCo held by each individual bank as well as their types of interdependencies26counterparty risk) with other banking intermediaries and non-banking financial institutions(shadow banking). A second policy recommendation is for regulators to carefully reconsiderthe design and the structuring of CoCos potentially incentivizing the issuance of going-concernCoCos with market-based triggers which may reduce inherent agency costs problems betweenbanks’ managers and shareholders. Nonetheless, we take an agnostic view as for the welfareeffects of these instruments.
Contingent convertible bonds (CoCos) are regulatory financial instruments introduced in theaftermath of the 2008-2009 financial crisis with the aim of containing the build-up of systemicrisk in the financial system in bad times. In this paper we propose different balance-sheetbased interbank financial networks, with and without CoCos, and we show that the structureof the network is of great importance for the effectiveness of CoCos as risk mitigating securities.Specifically, we demonstrate that for ring and complete networks the state (phase) transitionin a network without CoCos is also naturally operational in a network with CoCos, thusconfirming the robust-yet-fragile result documented by Acemoglu et al. (2015). We also showthat, in presence of moderate shocks, lowly interconnected networks enhance the stability of thefinancial system more than highly interconnected ones. Finally, we show that to maximize theeffectiveness of CoCos both issuers and investors should consider the type of interbank financialnetwork where they live. Overall, policymakers and regulators should carefully consider therole of the interbank network to assess the potential financial contagion dynamics of CoCos.27
Proof of Theorem 3.1
A.1 Ring Network
In this case y ij = y ( δ j,i +1 + δ j,i − ). We can study analytically the equilibrium of the systemafter an idiosyncratic shock of size ε on one bank: z = ( a − ε, a, . . . , a ), a > s and ε > ( a − s ).Denoting φ (cid:63) the equilibrium we have that φ (cid:63) (cid:54) φ (cid:63) (cid:54) . . . (cid:54) φ (cid:63)n . Let¯ n ( φ (cid:63) ) = max { i : φ (cid:63)i < } (26)the number of insolvent banks. As soon as ¯ n < n , it should be φ n = 1 and thus: φ = f y,s ( y + a − ε ) φ i = φ i − + a − sy = φ + ( i − a − sy , i = 2 , . . . , ¯ nφ i = 1 , i > ¯ n. It holds as soon as ¯ n < n , i.e. f y,s ( y + a − ε ) + ( n − a − sy >
1, whose solution is ε < ε (cid:63) = n ( a − s ) or y < y (cid:63) = ( n − a − s ) , (27)exactly as in Acemoglu et al. (2015). This is the equilibrium in the small shock ( ε < ε (cid:63) ) orlow exposure ( y < y (cid:63) ) regime, where ε (cid:63) is the total system’ liquidity in absence of shocks. Inthis regime the extension of the contagion is, from Equation 27, E ( φ (cid:63) ) = ¯ n ( φ (cid:63) ) /n = (cid:98) ya − s (1 − f y,s ( y + a − ε )) (cid:99) /n, (28)which is linearly stepwise increasing with ε until min( ε (cid:63) , y + ( a − s )). Analogously the totaldistress is D ( φ ) = (cid:107) − φ (cid:63) (cid:107) n ≈ ¯ n ( φ )(1 − φ (cid:63) )2 n (29)which grows quadratically in ε up to the transition. On the contrary, when ε > ε (cid:63) and y > y (cid:63) , it is ¯ n = n . In this regime it should be at the same time φ n = φ + ( n − a − sy and28 = ( φ n + a − s − εy ) + . The only possibility is that: φ = 0 φ i = ( i − a − sy , i > . This solution is independent from ε since the distress propagation is already maximal: E ( φ (cid:63) ) =1 and D ( φ (cid:63) ) = 1 − ( a − s )( n − y = 1 − y (cid:63) y . A.2 Fully connected network
In this case, every pair of banks is connected and the junior liability of a bank is equallydistributed among its neighbors, i.e. y ij = yn − , ∀ i (cid:54) = j . Again, we can study analytically theequilibrium of the system after an idiosyncratic shock of size ε on one bank: z = ( a − ε, a, . . . , a ), a > s and ε > ( a − s ). In fact, in this case all the updating rules Eq. (8) for non shockedbanks have the same structure. Thus, we can make an ansatz for the equilibrium and searchfor a solution of the form: φ i = φ s if i = 1 , ( shocked bank) φ ns if i > , ( non shocked bank) , (30)where ( φ s , φ ns ) satisfying: φ s = f ys ( a − ε + yφ ns ) φ ns = f ys ( a + yn − φ s + y n − n − φ ns ) . (31)Equation (31) do not admit solutions if both φ s , φ ns ∈ (0 , φ ns = 1 = ⇒ φ s = (cid:18) − ε − ( a − s ) y (cid:19) + φ s = 0 = ⇒ φ ns = ( n −
1) ( a − s ) y = y (cid:63) y . (32)The latter is admissible as soon as y > y (cid:63) and φ s = 0, implying ε > n ( a − s ) = ε (cid:63) , thuscorresponding to the large shock large exposure regime. On the contrary, y < y (cid:63) or ε < ε (cid:63) , the29rst solution holds, corresponding to an equilibrium where the shocked bank is insolvent whilethe rest of the system has absorbed the distress. In this second case, the solution is independentfrom ε for ε > y + ( a − s ), corresponding to the maximum distress that bank 1 can propagategiven its (limited) debt. The extension of contagion and overall financial distress can be easilycomputed starting from the analytical solutions 32. In particular, their maximum values inthe large shock regime are E ( φ (cid:63) ) = 1 and D ( φ (cid:63) ) = 1 − ( n − y (cid:63) /yn ) ≈ − y (cid:63) /y < D ring . B Proof of Theorem 4.1
First, we need the following propositions:
Proposition B.1.
In a ring network with z = ( a − ε, a, . . . , a ) it holds ∀ i = 1 , . . . , n − φ (cid:63)i = 1 = ⇒ φ (cid:63)i +1 = 1 . Proof.
For any i = 1 , . . . , n − φ (cid:63)i = 1 means that ((1 − τ ) a − s ) /y + (1 − τ ) φ (cid:63)i − − δ i (1 − τ ) ε/y (cid:62)
1, i.e. ((1 − τ ) a − s ) /y (cid:62) − (1 − τ ) φ (cid:63)i − (cid:62) − (1 − τ ). ]= Thus:(1 − τ ) φ (cid:63)i + (1 − τ ) a − sy = (1 − τ ) + (1 − τ ) a − sy (cid:62) , implying φ (cid:63)i +1 = 1. At τ = 0 the proposition is true simply because the sequence φ (cid:63)i is weaklyincreasing. Proposition B.2.
For any b ∈ R , ε (cid:62) and λ ∈ [0 , , given φ ( y, ε ) = (1 − τ )(1 − ε/y ) + b/y ,then the portion of the positive ( y, ε ) plane defined by the stability condition: (1 − λ ) min( φ ( y, ε ) , + + b λτ y (cid:62) s the intersection of the regions defined by y (cid:54) y (cid:63)λ , ε (cid:54) ε (cid:63)λ ( y ) and ε (cid:54) ε (cid:63) ( y ) with: y (cid:63)λ = bλτε (cid:63)λ ( y ) = b (cid:18) λτ + 11 − τ (cid:19) − ( y − y (cid:63)λ ) (cid:18) − τ )(1 − λ ) − (cid:19) ε (cid:63) ( y ) = b − τ y − τ . The low exposure regime is identified by y (cid:63)λ , the small shock regime by ε (cid:63)λ ( y ) and the no-shockregime by ε (cid:63) ( y ) , which is correctly independent from λ , encoding the network structure.Proof. (1) If φ ( y, ε ) (cid:62)
1, i.e. ε (cid:63) ( y ), then the condition is fulfilled, because ε (cid:62) b (cid:62) τ y and the l.h.s. is a convex combination of two numbers larger than 1. (2) If bλ/τ y (cid:62) y (cid:54) y (cid:63)λ the condition is fulfilled simply because (1 − λ ) φ ( y, ε ) is always positive. This isalso a necessary and sufficient condition when φ ( y, ε ) (cid:54) φ ( y, ε ) ∈ [0 ,
1] then we havestraightforwardly the condition ε (cid:54) ε (cid:63)λ ( y ), since(1 − λ ) φ ( y, ε ) + b λτ y (cid:62) ⇒ (1 − λ ) (cid:18) (1 − τ )(1 − εy ) + by (cid:19) + b λτ y (cid:62) ⇒ (1 − λ ) ((1 − τ )( y − ε ) + b ) + bλτ (cid:62) y = ⇒ y (1 − (1 − λ )(1 − τ )) + ε (1 − λ )(1 − τ ) (cid:54) b (cid:18) (1 − λ ) + λτ (cid:19) = ⇒ ( y − y (cid:63)λ )(1 − (1 − λ )(1 − τ )) + ε (1 − λ )(1 − τ ) (cid:54) b (cid:18) (1 − λ ) + λτ (1 − λ )(1 − τ ) (cid:19) This must be a stronger condition if instead φ ( y, ε ) (cid:62) φ ( y, ε ) (cid:54)
0, since in this case(1 − λ ) min( φ ( y, ε ) , + + b λτ y (cid:62) (1 − λ ) φ ( y, ε ) + b λτ y (cid:62) . For this reason, the desired region is just the intersection of (1), (2) and (3), together with theconditions of positive ε and y . 31 .1 Ring Network We follow the same analysis of the previous section (i.e. in absence of CoCo). Again, y ij = y ( δ j,i +1 + δ j,i − ) and z = ( a − ε, a, . . . , a ). We assume no triggers in absence of shocks, i.e.: E i V i = a − sa + y (cid:62) τ = ⇒ (1 − τ ) a − s (cid:62) τ y (cid:62) . (33)Moreover, we consider the situation where the size of the shock is at least responsible of thefirst bank triggering, i.e. a − ε − sa + y (cid:62) τ = ⇒ ε (cid:62) ε (cid:63) ( y, τ ) = ( a + y ) − s + y − τ , (34)where ε (cid:63) ( y, τ coincides with the no-shock threshold of Proposition B.2. Since φ (cid:63)i = 1 = ⇒ φ (cid:63)i +1 = 1, see Proposition B.1, we can define as in the previous case:¯ n ( φ (cid:63) ) = max { i : φ (cid:63)i < } (35)as the number of non-triggered banks. As soon as ¯ n < n , it should be φ n = 1 and thus: φ = f y,s ((1 − τ )( y + a − ε )) φ i = (1 − τ ) φ i − + (1 − τ ) a − sy = (1 − τ ) i − φ + (1 − τ ) a − sy i − (cid:88) k =0 (1 − τ ) k i = 2 , . . . , ¯ nφ i = 1 , i > ¯ n. (36)Defining λ rτ = 1 − (1 − τ ) n − , this solution holds as soon as: φ n = (1 − λ rτ ) φ + λ rτ (1 − τ ) a − sτ y (cid:62) . (37)32he previous condition is satisfied if y (cid:54) y (cid:63)r ( τ ) or ε (cid:54) ε (cid:63)r ( y, τ ), where (Proposition B.
2, with b = (1 − τ ) a − s and λ = λ rτ ), y (cid:63)r ( τ ) = λ rτ τ [(1 − τ ) a − s ] (38) ε (cid:63)r ( y, τ ) = (cid:18) λ rτ τ + 11 − τ (cid:19) [(1 − τ ) a − s ] − ( y − y (cid:63)r ) (cid:18) − τ )(1 − λ rτ ) − (cid:19) (39)In the limit τ →
0, since λ rτ τ → ( n − y (cid:63)r ( τ ) → y (cid:63) = ( n − a − s ) and ε (cid:63)r ( y, τ ) → ε (cid:63) = n ( a − s ), Figure 5. We can compute the extension ofthe triggering contagion as φ ¯ n = 1, bringing to: E ( φ (cid:63) ) = ¯ n ( φ (cid:63) ) /n = log (( φ ∞ − / ( φ ∞ − φ )) n log(1 − τ ) , (40)where φ ∞ = (1 − τ ) a − sτy and φ as in 36. B.2 Complete network
In this case, we can again make the ansatz φ (cid:63)i = φ (cid:63)s if i = 1 , ( shocked bank) φ (cid:63)ns if i > , ( non shocked bank) , (41)where ( φ (cid:63)s , φ (cid:63)ns ) the solution of φ s = f ys ((1 − τ )( a − ε + yφ ns )) φ ns = f ys ((1 − τ )( a + yn − φ s + y n − n − φ ns ) .
33n the safe case, where the trigger doesn’t propagate through the entire network, the solutionmust be φ (cid:63)s = f ys ((1 − τ )( a − ε + y )) φ (cid:63)ns = min(1 , φ ns ) + = 1 φ ns = (1 − τ ) a − sy + 1 − τn − φ s + (1 − τ )( n − n − φ ns = (1 − λ cτ ) φ s + λ cτ τ [(1 − τ ) a − s ] , (42)where we have defined λ cτ = τ ( n − τ ( n − . (43)This solution exists if φ ns (cid:62)
1, i.e. again using Proposition B.2 with b = (1 − τ ) a − s and λ = λ cτ , if and only if y (cid:54) y (cid:63)c ( τ ) or ε (cid:54) ε (cid:63)c ( y, τ ), where y (cid:63)c ( τ ) = λ cτ τ [(1 − τ ) a − s ] (44) ε (cid:63)c ( y, τ ) = (cid:18) λ cτ τ + 11 − τ (cid:19) [(1 − τ ) a − s ] − ( y − y (cid:63)c ) (cid:32) − τ )(1 − λ fτ ) − (cid:33) . (45)We have lim τ → y (cid:63)c ( τ ) = lim τ → y (cid:63)r ( τ ) = y (cid:63) = ( n − a − s ) (46)and lim τ → ε (cid:63)c ( y, τ ) = lim τ → ε (cid:63)r ( y, τ ) = ε (cid:63) = n ( a − s ) . (47) C Cocos with equity liquidation
C.1 Ring Network
Computations similar to those in the previous section bring to ε (cid:63)r ( τ, η ) = [(1 − τ )( a + y ) − ( s + y )] (cid:18) (1 − η )(1 − C ( η, τ ) n ) C n ( η, τ )(1 − C ( η, τ )) (cid:19) (48)34ith C ( η, τ ) = (1 − η )(1 − τ ). In fact as soon as φ n = 1 we have ∀ i (cid:54) ¯ nφ i = η + (1 − η ) f y,s ((1 − τ ) h ( φ i − )) = A + Cφ i − = C i − φ + A i − (cid:88) k =0 C k = C i − φ + A − C i − − C with A = η + (1 − η ) b/y and C = C ( η, τ ) = (1 − η )(1 − τ ). The critical threshold in the ringnetwork is the solution of the equation φ n ( ε ) = 1, i.e. C n − φ ( ε ) + A − C n − − C = 1 (49)where φ ( ε ) = A + C (1 − ε/y ). Straightforward manipulations bring equation (48) C.2 Complete Network
In the case of the complete network we have that ε (cid:63)c ( τ, η ) = [(1 − τ )( a + y ) − ( s + y )] (cid:18) n − τ − η (1 − τ )(1 − η )(1 − τ ) (cid:19) . (50)In fact in this case the critical shock is the solution of the equation φ ns ( ε ) = 1, i.e. φ ns ( ε ) = η + (1 − η ) f ys ((1 − τ )( a + yn − φ s ( ε ) + y n − n − , (51)where φ s ( ε ) = η + (1 − η ) (1 − τ )( a − ε + y ) − sy . (52)Solving in ε equation (50) follows straightforwardly.35 Bibliography
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