Contingent Capital, Tail Risk, and Debt-Induced Collapse

Transcription

1 Contingent Capital, Tail Risk, and Det-Induced Collapse Nan Chen, Paul Glasserman, Behzad Nouri and Markus Pelger This version: January 2017 Astract Contingent capital in the form of det that converts to equity as a ank approaches financial distress offers a potential solution to the prolem of anks that are too ig to fail. This paper studies the design of contingent convertile onds and their incentive effects in a structural model with endogenous default, det rollover, and tail risk in the form of downward jumps in asset value. We show that once a firm issues contingent convertiles, the shareholders optimal ankruptcy oundary can e at one of two levels: a lower level with a lower default risk or a higher level at which default precedes conversion. An increase in the firm s total det load can move the firm from the first regime to the second, a phenomenon we call det-induced collapse ecause it is accompanied y a sharp drop in equity value. We show that setting the contractual trigger for conversion sufficiently high avoids this hazard. With this condition in place, we investigate the effect of contingent capital and det maturity on capital structure, det overhang, and asset sustitution. We also calirate the model to past data on the largest U.S. ank holding companies to see what impact contingent convertile det might have had under the conditions of the financial crisis. Keywords: Contingent convertile det, ail-in det, capital structure, too ig to fail JEL Classification Codes: G12, G13, G32 Chinese University of Hong Kong, nchen@se.cuhk.edu.hk Columia University, pg20@columia.edu Columia University, n2164@columia.edu Stanford University, mpelger@stanford.edu

2 1 Introduction The prolem of anks that are too ig to fail plays out as an unwillingness on the part of governments to impose losses on ank creditors for fear of the disruptive consequences to the financial system and the roader economy. Higher capital requirements and restrictions on usiness practices may reduce the likelihood of a ank ecoming insolvent, ut they do not commit the regulators, managers or investors to a different course of action conditional on a ank approaching insolvency. Contingent capital addresses this prolem through a contractual commitment to have ond holders share some of a ank s downside risk without triggering failure. Contingent convertiles (CoCos) and ail-in det are the two main examples of contingent capital. Both are det that converts to equity under adverse conditions. CoCos provide going concern contingent capital, meaning that they are designed to convert well efore a ank would otherwise default. Bailin det is gone-concern contingent capital and converts when the ank is no longer viale, wiping out the original shareholders and transferring ownership to the ailed-in creditors. These instruments are increasingly important elements of reforms to enhance financial staility. Worldwide issuance of CoCo now exceeds $420 illion, mainly y European and Asian anks. The Swiss anking regulator has increased capital requirements for Swiss anks to 19% of risk-weighted assets, of which 9% can take the form of CoCos. The European Commission s proposed resolution framework relies on ail-in det as one of its primary tools. In the U.S., ail-in is central to the implementation of the FDIC s authority to resolve large complex financial institutions granted y the Dodd-Frank Act. Raising new equity from private investors is difficult for a ank nearing financial distress, which strengthens arguments for government support once a crisis hits; contingent capital seeks to solve this prolem y committing creditors to provide equity through conversion (or a writedown) of their claims. Nevertheless, the relative complexity of these instruments has raised some questions aout whether they can e designed to function as expected and whether they might have unintended consequences. Critics have argued that increasing equity levels would e simpler and more effective than relying on CoCos; see Admati et al. [1]. 1

3 The goal of this paper is to analyze the design of contingent capital and to investigate the incentives these instruments create for shareholders. This work makes several contriutions. First, our analysis reveals a new phenomenon we call det-induced collapse. With CoCos on its alance sheet, a firm operates in one of two regimes: one in which the CoCos function as intended or another in which the equity holders optimally declare ankruptcy efore conversion, effectively reducing the CoCos to straight det. A transition from the first regime to the second is precipitated y an increase in the firm s det load, and its consequences include a sharp increase in the firm s default proaility and a drop in the value of its equity. This is the sense in which det induces a collapse. We show that this hazard is avoided y setting the trigger for conversion at a sufficiently high level. Once det-induced collapse is precluded, we can investigate the incentive effects of CoCos effects that would e lost in the alternative regime in which CoCos degenerate to straight det. We investigate how the value of equity responds to various changes in capital structure and find, perhaps surprisingly, that equity holders often have a positive incentive to issue CoCos. We also find that CoCos can e effective in mitigating the prolem of det overhang the reluctance of equity holders to inject additional capital into an ailing firm when most of the resulting increase in firm value is captured y det holders. CoCos can create a strong positive incentive for shareholders to invest additional equity to stave off conversion. We also examine how CoCos affect the sensitivity of equity value to the riskiness of the firm s assets. This sensitivity is always positive in simple models, creating an incentive for asset sustitution y shareholders once they have issued det. We will see that this is not necessarily the case in a richer setting in which new det is issued as old det matures. We develop our analysis in a structural model of the type introduced in Leland [28] and Leland and Toft [29], as extended y Chen and Kou [10] to include jumps. The key state variale is the value of the firm s underlying assets, and equity and det values are derived as functions of this state variale. CoCo conversion is triggered y a function of this state variale, such as a capital ratio. The model has three particularly important features. First, default is endogeneous and results from the optimal ehavior of equity holders. This feature is essential 2

4 to the analysis of incentive effects and to the emergence of the two default regimes descried aove. Second, the firm s det has finite maturity and must e rolled over as it matures. This, too, is crucial in capturing incentive effects. In a classical single-period model of the type in Merton [32], all the enefits of reducing default risk accrue to ond holders equity holders always prefer riskier assets and are always deterred from further investment y the prolem of det overhang. But in a model with det rollover, reducing default risk allows the firm to issue det at a higher market price, and part of this increase in firm value is captured y equity holders, changing their incentives. This feature also allows us to investigate how det maturity interacts with the efficacy of CoCos. Finally, jumps are also essential to understanding incentive effects. Downward jumps generate a higher asset yield (in the form of an increase in the risk-neutral drift) ut expose the firm to tail risk. CoCos can increase equity holders incentive to take on tail risk ecause equity holders would prefer a dilutive conversion at a low asset value over one at a high asset value. For the same reason, CoCos are more effective in mitigating det overhang when asset value is suject to downward jumps. After demonstrating these implications through a mix of theoretical and numerical results, we calirate the model to data on the largest U.S. ank holding companies for the period Some of the comparative statics in our numerical examples depend on parameter values, so the purpose of the caliration is to investigate the model s implications at parameter values representative of the large financial institutions that would e the main candidates for CoCo issuance. We calculate the model-implied increase in loss asorption that would have resulted from replacing 10% of each firm s det with CoCos, estimate which firm s would have triggered conversion and when, and compare the impact on det overhang costs at three dates during the financial crisis. Overall, this counterfactual exploration suggests that CoCos would have had a eneficial effect, had they een issued in advance of the crisis. Alul, Jaffee, and Tchistyi [2] also develop a structural model for the analysis of contingent capital; their model has neither jumps nor det rollover (they consider only infinite maturity det), and their analysis and conclusions are quite different from ours. Pennacchi s [35] model includes jumps and instantaneous maturity det; he studies the model through simulation, 3

5 taking default as exogenous, and thus does not investigate the structure of shareholders optimal default. Hilscher and Raviv [23], Himmelerg and Tsyplakov [24], and Koziol and Lawrenz [27] investigate other aspects of contingent capital in rather different models. Glasserman and Nouri [17] jointly model capital ratios ased on accounting and asset values; they value det that converts progressively, rather than all at once, as a capital ratio deteriorates. None of the previous literature comines the key features of our analysis endogenous default, det rollover, jumps, and analytical tractaility nor does previous work identify the phenomenon of det-induced collapse. Much of the current interest in contingent capital stems from Flannery [15]. Flannery [15] proposed reverse convertile deentures (called contingent capital certificates in Flannery [16]) that would convert from det to equity ased on a ank s stock price rather than an accounting measure. Sundaresan and Wang [39], Glasserman and Nouri [18], and Pennacchi and Tchistyi [36] study market-ased triggers. Several authors have proposed various alternative security designs; these include Bolton and Samama [6], Calomiris and Herring [7], Duffie [14], Madan and Schoutens [30], McDonald [31], Pennacchi, Vermaelen, and Wolf [37], and Squam Lake Working Group [38]. The rest of this paper is organized as follows. Section 2 details the structural model and derives values for the firm s liailities. Section 3 characterizes the endogenous default arrier and includes our main theoretical results descriing det-induced collapse. Sections 4 6 investigate the impact of det rollover and incentive effects on det overhang and asset sustitution. Section 7 presents the caliration to ank data. Technical details are deferred to an appendix. 2 The Model 2.1 Firm Asset Value Much as in Leland [28], Leland and Toft [29], and Goldstein, Ju, and Leland [19], we consider a firm generating cash through its investments and operations continuously at rate {δ t, t 0}. This income flow is exposed to oth diffusive and jump risk. We assume the existence of a risk-neutral proaility measure Q under which the value of the firm s assets is the present 4

6 value of the future cash flows they generate, [ ] V t = E Q e r(u t) δ u du δ t, t for all t 0. If we adopt the dynamics in Kou [25], then δ := V t /δ t is a constant and V t evolves as a jump-diffusion process under Q, ( dv t = r δ + λ ) dt + σ dw t + d V t 1 + η ( Nt ) (Y i 1), (2.1) where {W t } is a standard Brownian motion, {N t } is a Poisson processes with intensity λ, and the jump sizes {Y i } are i.i.d., independent of {W t } and {N t }, with log Y i having an exponential distriution with mean 1/η. We will value the firm s liailities as contingent claims on the asset value process V, taking expectations under Q and using the dynamics in (2.1). 2.2 The Capital Structure and Endogenous Default The firm finances its assets through straight det, contingent convertile det (CoCos), and equity. We detail these in order of seniority Straight Det i=1 We use the approach of Leland and Toft [29] to model the firm s senior det. The firm continuously issues straight det with par value p 1 dt in (t, t + dt). The maturity of newly issued det is exponentially distriuted with mean 1/m; that is, a portion m exp( ms)ds of the total amount p 1 dt matures during the time interval (t + s, t + s + ds), for each s 0. The det pays a continuous coupon at rate c 1 per unit of par value. In the case of ank deposits with no stated term, the maturity profile reflects the distriution of time until depositors withdraw their funds. The exponential maturity profile and the constant issuance rate keep the total par value of det outstanding constant at P 1 = t ( t ) p 1 me m(s u) du ds = p 1 m. 5

7 Thus, the firm continuously settles and reissues det at a fixed rate. 1 This det rollover will e important to our analysis through its effect on incentives for equity holders. 2 Tax deductiility of coupon payments lowers the cost of det service to the firm. Bank deposits have special features that can generate additional funding enefits. 3 Deposit insurance creates a funding enefit if the premium charged includes an implicit government susidy. Customers value the safety and ready availaility of ank deposits and are willing to pay (or accept a lower interest rate) for this convenience. DeAngelo and Stulz [11] and Sundaresan and Wang [40] model this effect as a liquidity spread that lowers the net cost of deposits to the ank. In Allen, Carletti, and Marquez [3], the funding enefit of deposits results from market segmentation. We model these funding enefits y introducing a factor κ 1, 0 κ 1 < 1, such that the net cost of coupon payments is (1 κ 1 )c 1 P 1. In the special case of tax enefits, κ 1 would e the firm s marginal tax rate Contingent Convertiles We use the same asic framework to model the issuance and maturity of CoCos as we use for straight det. In oth cases, we would retain tractaility if we replaced the assumption of an exponential maturity profile with consols, ut we would then lose the effect of det rollover. We denote y P 2 the par value of CoCos outstanding, which remains constant prior to conversion or default and pays a continuous coupon at rate c 2. The mean maturity is assumed to e the same as for the straight det, 1/m, and new det is issued at rate p 2 = mp 2. It is straightforward to generalize the model to different maturities for CoCos and straight det, and we use different maturity levels in our numerical examples later in the paper. As in the case of straight det, we introduce a factor κ 2 that captures any funding enefit of CoCos. The tax treatment of CoCo coupons varies internationally. Conversion of CoCos from det to equity is triggered when a function of the state variale 1 Diamond and He [13, p.750] find this mechanism well suited to modeling anks. One alternative would e to have a ank shrink its alance sheet after a negative shock to assets. As stressed y Hanson, Kashyap, and Stein [20], this would run counter to the macroprudential ojective of maintaining the supply of credit in an economic downturn, which, as they further note, is one of the ojectives of contingent capital. 2 We do not model ank runs, ut with det rollover the yield on the ank s det increases as its asset value declines, capturing the idea that the ank needs to offer a higher yield to prevent a run. The extended model of Chen et al. [9] treats insured deposits. 3 See Appendix A for a discussion of the application of Leland [28] and its extensions to anks. 6

8 V t reaches a threshold. As long as the function is invertile, we can model this as conversion the first time V t itself falls elow an exogenously specified threshold V c. Thus, conversion occurs at τ c = inf{t 0 : V t V c }. (2.2) In particular, we can implement a capital ratio trigger y having CoCos convert the first time (V t P 1 P 2 )/V t ρ, with ρ (0, 1) equal to, say, 5%. The numerator on the left is an accounting measure of equity, and dividing y asset value yields a capital ratio. 4 To put this in the form of (2.2), we set V c = (P 1 + P 2 )/(1 ρ). (2.3) Another choice that fits within our framework would e to ase conversion on the level of earnings δv t, as in Koziol and Lawrenz [27]. For the derivations in this section we will keep the value of V c general, except to assume that V 0 > V c so that conversion does not occur at time zero. At the instant of conversion, the CoCo liaility is erased and CoCo investors receive shares of the firm s equity for every dollar of principal, for a total of P 2 shares. We normalize the numer of shares to 1 prior to conversion. Thus, following conversion, the CoCo investors own a fraction P 2 /(1 + P 2 ) of the firm. In the ail-in case, =, so the original shareholders are wiped out and the converted investors take control of the firm Endogenous Default The firm has two types of cash inflows and two types of cash outflows. The inflows are the income stream δ t dt = δv t dt and the proceeds from new ond issuance t dt, where t is the total market value of onds issued at time t. The cash outflows are the net coupon payments and the principal due (p 1 + p 2 )dt on maturing det. 6 The net coupon payments, factoring in tax deductiility and any other funding enefits, are given y A t = (1 κ 1 )c 1 P 1 + (1 κ 2 )c 2 P 2. 4 This approximates a tangile common equity ratio. If CoCos are treated as Tier 1 capital, we could define a trigger ased on a Tier 1 capital ratio through the condition (V t P 1)/V t ρ and thus V c = P 1/(1 ρ). 5 We do not distinguish etween contractual and statutory conversion. Under the former, conversion is an explicit contractual feature of the det. The statutory case refers to conversion imposed on otherwise standard det at the discretion of a regulator granted explicit legal authority to force such a conversion. 6 Our discussion of cash flows is informal and used to provide additional insight into the model. For a rigorous formulation of the Leland-Toft model through cash flows, see Décamps and Villeneuve [12]. 7

9 Let p denote the total rate of issuance (and retirement) of par value of det, just as t denotes the total rate of issuance measured at market value. We have p = p 1 + p 2 prior to conversion of any CoCos and p = p 1 after conversion. Whenever t + δv t > A t + p, (2.4) the firm has a net inflow of cash, which is distriuted to equity holders as a dividend flow. When the inequality is reversed, the firm faces a cash shortfall. The equity holders then face a choice etween making further investments in the firm in which case they invest just enough to make up the shortfall or aandoning the firm and declaring ankruptcy. Bankruptcy then occurs at the first time the asset level is at or elow V, with V chosen optimally y the equity holders. In fact, it would e more accurate to say that V is determined simultaneously with t, ecause the market value of det depends on the timing of default, just as the firm s aility to raise cash through new det influences the timing of default. The equity holders choose a ankruptcy policy to maximize the value of equity. To e feasile, a policy must e consistent with limited liaility, meaning that it ensures that equity value remains positive prior to default. This formulation is standard and follows Leland [28] and Leland and Toft [29] and, in the jump-diffusion case, Chen and Kou [10]. However, the presence of CoCos creates a distinctive new feature, driven y whether default occurs efore or after conversion. Depending on the parameters of the model, the equity holders may find either choice to e optimal. If they choose to default efore conversion, then the CoCos effectively degenerate to junior straight det. Importantly, we will see that positive incentive effects from CoCo issuance are lost in this case. Indeed, the ehavior of the model and, in particular, the value of equity, are discontinuous as we move from a regime in which conversion precedes default to a regime in which the order is reversed. We will see that this change can result from an increase in det either straight det or CoCos so we refer to this phenomenon as det-induced collapse. Upon default, we assume that a fraction (1 α), 0 α 1, of the firm s asset value is lost to ankruptcy and liquidation costs. Letting τ denote the time at which ankruptcy is declared and V τ the value of the firm s assets at that moment leaves the firm with αv τ 8

10 after ankruptcy costs. These remaining assets are used first to repay creditors. If default occurs after conversion, only the straight det remains at ankruptcy. If default occurs efore conversion, the CoCos degenerate to junior det and are repaid from any assets that remain after the senior det is repaid. 2.3 Liaility Valuation Our model yields closed-form expressions for the values of the firm s liailities. We proceed y taking the level of the default oundary V as given and valuing each layer of the capital structure. We then derive the optimal level V, leading to the concept of det-induced collapse. We egin y limiting attention to the case V V c, which ensures that the firm does not default efore conversion. 7 With V fixed, the default time τ is the first time the asset value V t is at or elow V. To value a unit of straight det at time t that matures at time t + T, we discount the coupon stream earned over the interval [t, (t + T ) τ ] and the (partial) principal received at (t + T ) τ to get a market value of (V t ; T ; V ) = E Q [ e rt ] 1 {τ >T +t} V t (principal payment if no default) [ +E Q e rτ 1 {τ T +t} αv τ P ] 1 Vt (payment at default) P 1 [ ] τ +E Q (T +t) c 1 e r(u t) du V t. (coupon payments) (2.5) 0 To simplify notation, we will henceforth take t = 0 and omit the conditional expectation given V t, though it should e understood that the value of each liaility is a function of the current value V of the firm s assets. Recall that the det maturity T is exponentially distriuted with density m exp( mt ), and the total par value is P 1. The total market value of straight det outstanding is then B(V ; V ) = P 1 (V t ; T ; V )me mt dt 0 = P 1 ( m + c1 m + r ) [ ] [ ] E Q 1 e (m+r)τ + E Q e (m+r)τ (αv τ P 1 ). (2.6) 7 In a model with jumps, the default time τ and conversion time τ c may coincide, even if V < V c. If CoCos automatically convert to equity at default, the order of events does not matter. If CoCos are treated as junior straight det at default, we adopt the convention that events occur in the order implied y the arrier levels, so if V < V c then the CoCos would e treated as having converted efore default. 9

11 The market value of a CoCo comines the value of its coupons, its principal, and its potential conversion to equity. To distinguish the equity value the CoCo investors receive after conversion from equity value efore conversion or without the possiility of conversion, we adopt the following notation: E BC denotes equity value efore conversion for the original firm, one with P 1 in straight det and P 2 in CoCos; E PC denotes post-conversion equity value and thus refers to a firm with P 1 in straight det and no CoCos; E NC denotes no-conversion equity value, which refers to a firm with P 1 in straight det and P 2 in non-convertile junior det. Each of these is a function of the current asset value V and a default arrier V. We will use the same superscripts to differentiate total firm value and other quantities as needed. With this convention, a CoCo with maturity T and unit face value has market value d(v ; T ; V ) = E Q [ [ e rt ] T τc ] 1 {τc>t } + E Q c 2 e rs ds 0 + [ ] E Q e rτc E PC (V τc ; V )1 1 + P {τc<t }. (conversion value) 2 In writing E PC (V τc ; V ), we are taking the value of post-conversion equity when the underlying asset value is at V τc and the default arrier remains at V. At conversion, the CoCo investors collectively receive P 2 shares of equity, giving them a fraction P 2 /(1 + P 2 ) of the firm; dividing this y P 2 yields the amount that goes to a CoCo with a face value of 1. The total market value of CoCos outstanding is then D(V ; V ) = P 2 d(v ; T ; V ) me mt dt = P 2 ( c2 + m m + r 0 ) ( 1 E Q [ e (r+m)τc ]) + P P 2 E Q [ e (r+m)τc E PC (V τc ; V ) ]. (2.7) To complete the calculation in (2.7), it remains to determine the post-conversion equity value E PC (V τc ; V ). We derive this value y calculating total firm value and sutracting the 10

12 value of det. After conversion, the firm has only one class of det, so E PC (V τc ; V ) = F PC (V τc ; V ) B(V τc ; V ), (2.8) where F PC (V τc ; V ) is the total firm value after conversion: F PC (V τc ; V ) = [ τ V }{{} τc + E Q τ c unleveraged firm value = V τc + κ 1c 1 P 1 r =: V τc + F B 1 BCOST. κ 1 c 1 P 1 e rs ds V τc } {{ } funding enefits ] E Q [ e r(τ τ c) (1 α)v τ V τc ] }{{} ankruptcy costs (1 E Q [ e r(τ τ c) V τc ]) E Q [ e r(τ τ c) (1 α)v τ V τc ] The conversion of the CoCos does not affect the value of the senior det, so the valuation expression in (2.6) applies to B(V τc ; V ) in (2.8). To find the value of equity efore conversion, we again derive the total firm value and sutract the det value. We continue to limit attention to the case V V c. Any funding enefit from CoCos terminates at the conversion time τ c. So, the firm value efore conversion is F BC (V ; V ) = V + κ 1c 1 P 1 ( 1 E Q [ e rτ ]) + κ 2c 2 P 2 ( 1 E Q [ e rτc]) } r {{}} r {{} funding enefits from straight det funding enefits from CoCos E Q [ e rτ ] (1 α)v τ (2.9) =: V + F B 1 + F B 2 BCOST. The market value of the firm s equity is given y E BC (V ; V ) = F BC (V ; V ) B(V ; V ) D(V ; V ). (2.10) A similar calculation leads to closed-form liaility evaluation if conversion does not occur prior to ankruptcy, i.e., V > V c. Here we need to specify the treatment of CoCos at default: either (1) CoCos are automatically converted to equity in ankruptcy, 8 or (2) CoCos are treated as junior det in ankruptcy. We have solved the model under oth formulations, ut we treat 8 Many CoCos include a secondary point of non-viaility trigger, in addition to a capital ratio trigger, which would force conversion at failure. 11

13 (1) in the paper and leave (2) to a supplementary appendix. The value of equity is unaffected y the treatment of CoCos in ankruptcy and is therefore the same in the two cases. 9 Before default, the total market value of straight det is B(V ; V ) = P 1 ( m + c1 m + r ) [ ] [ ] E Q 1 e (m+r)τ + E Q e (m+r)τ (αv τ P 1 ) (2.11) Under the condition that CoCos are converted to equity at default and wiped out, the total market value of CoCos efore default is D(V ; V ) = P 2 ( m + c2 m + r ) [ ] E Q 1 e (m+r)τ. (2.12) Total firm value in this case is given y 10 ( F BC κ1 c 1 P 1 (V ; V ) = V + + κ ) 2c 2 P 2 (1 E Q [ e rτ ]) E Q [ e rτ ] (1 α)v τ r r E Q [ e (r+m)τ (αv τ P 1 ) +]. Equity value in the case V > V c now follows from (2.10) using these expressions. All pieces (2.6) (2.12) of the capital structure of the firm can e explicitly evaluated through expressions for the joint transforms of hitting times τ or τ c and asset value V given explicitly y Kou [25] and Kou and Wang [26]. The appendix contains additional details. 2.4 The Bail-In Case In the ail-in case, conversion of det to equity occurs when the firm would not otherwise e viale, rather than at an exogenously specified trigger. We model this y taking V c = V, with the understanding that conversion occurs just efore what would otherwise e ankruptcy. We set = so the original shareholders are wiped out, and the firm is taken over y the ail-in investors. As ankruptcy is avoided, we assume that no ankruptcy costs are incurred, so α = 1. Just after conversion, the firm continues to operate, now with just P 1 in det outstanding. 9 In a model with endogenous default with only straight det, the equity value at default will e exactly zero. If CoCos are added to the capital structure it is possile that shareholders may choose an optimal default arrier at which the recovery value is larger than the det value at default. We make the economically sound assumption that equity holders are nevertheless wiped out at default. In a supplementary appendix we have also solved the alternative model, in which the difference etween recovery and det value is paid out to the shareholders. In that formulation, equity holders would have a stronger incentive to default earlier, and the prolem of det-induced collapse introduced in Section 3 would e more severe. 10 Based on the results in Chen and Kou [10] we can show that for the optimal default arrier derived in Section 3 there will not e any residual value left at ankruptcy, i.e. (αv τ P 1) + = 0 and the last term can e dropped. 12

14 3 Optimal Default and Det-Induced Collapse Having valued the firm s equity at an aritrary default arrier V, we now proceed to derive the equity holder s optimal default arrier V 3.1 Endogenous Default Boundary and to investigate its implications. As in Section 2, we denote y E PC (V ; V ) the post-conversion equity value for a firm with asset value V and default arrier V. After conversion, we are dealing with a conventional firm, meaning one without CoCos. In such a firm, the equity holders choose the default arrier V to maximize the value of equity suject to the constraint that equity value can never e negative; that is, they solve suject to the limited liaility constraint max V E PC (V ; V ) (3.1) E PC (V ; V ) 0, for all V V. The limited liaility constraint ensures that the chosen V is feasile. Without this condition, a choice of V that maximizes E PC (V ; V ) at the current asset level V might entail sustaining a negative value of equity at some asset level etween V and V, which is infeasile. Denote the solution to this prolem y V PC. Before conversion, when the firm s liailities include CoCos, equity value is given y E BC (V ; V ), and the shareholders would like to choose V to maximize this value. If they choose V < V c, conversion will precede ankruptcy, and following conversion they and the new shareholders who were formerly CoCo holders will face an equity maximization prolem of the type in (3.1). Hence, efore conversion the equity holders face a commitment prolem, in the sense that they cannot necessarily commit to holding V at the same level after conversion that they would have chosen efore conversion. Anticipating this effect, they will choose V = V PC choose V < V c. Thus, efore conversion, equity holders will choose V to solve if they max V E BC (V ; V ) 13

15 suject to the limited liaility constraint E BC (V ; V ) 0, for all V V and the commitment condition that V = V PC prolem. if V < V c. Let V denote the solution to this Chen and Kou [10] have solved the optimal default arrier prolem with only straight det, and this provides the solution for the post-conversion firm: V PC = P 1 ɛ 1, where ɛ 1 depends on c 1, m, κ 1 and α ut is independent of the capital structure and V. See equation (B.1) in the appendix for an explicit expression. Recall that E NC (V ; V ) denotes the value of equity if the P 2 in CoCos is replaced with non-convertile junior det in the original firm. Extending Chen and Kou [10], we can express the optimal default arrier for this firm as V NC = P 1 ɛ 1 + P 2 ɛ 2, where ɛ 2 is defined analogously to ɛ 1 using c 2 instead of c 1 ; see (B.2). We always have V PC V NC ecause increasing the amount of non-convertile det while holding everything else fixed raises the default arrier. We can now characterize the optimal default arrier with CoCos. Theorem 1. For a firm with straight det and with CoCos that convert at V c, the optimal default arrier V has the following property: Either V = V PC V c or V = V NC V c. (3.2) Moreover, V PC is optimal whenever it is feasile, meaning that it preserves the limited liaility of equity. This result reduces the possile default arriers for a firm with CoCos to two candidates, each of which corresponds to the default arrier for a firm without CoCos. The second case is a candidate only if, without the conversion feature, it would e optimal to default at an asset level higher than the trigger V c. This can occur only if the first case does not yield a feasile solution. We will see that a firm can sometimes move from the first case in (3.2) to the second case y increasing its det load. The transition is discontinuous, creating a jump up in the default arrier and a drop in equity value. We refer to this phenomenon as det-induced collapse. This 14

16 phenomenon is not present without CoCos (or with ail-in det). Moreover, we will see that the positive incentive effects that result from CoCos under the first case in (3.2) disappear following the collapse. To illustrate, we consider an example. The heavy solid line in Figure 1 shows equity value as a function of asset value for the NC firm, in which the CoCos are replaced y junior det. The optimal default arrier V NC is at 93, and the NC equity value and its derivative are equal to zero at this point. If the conversion trigger V c is elow 93 (two cases are considered in the figure), then V = V NC = 93 is a feasile default level for the original firm ecause the resulting equity values are consistent with limited liaility. The optimal post-conversion default arrier is V PC = 58. Suppose the conversion trigger is at V c = 65, and suppose the original shareholders of the original firm with CoCos attempt to set the default arrier at V = 58. The dashed line shows the resulting equity value. At higher asset values, the dashed lines is aove the solid line, suggesting that equity holders would prefer to set the default arrier at 58 than at 93. However, the dashed line is not a feasile choice ecause it creates negative equity values at lower asset levels; the est the shareholders can do in this case is to set V = 93. If the conversion trigger were at V c = 75, a default arrier of V = V PC = 58 would e feasile ecause the resulting equity values (the dash-dot line) remain positive; in fact, this choice would then e optimal. If we imagine starting with the conversion trigger at 75 and gradually decreasing it toward 65, at some level of V c in etween the default arrier jumps up from 58 to 93, and the equity curve collapses down to the heavy solid line showing the equity curve for the NC firm. In the ail-in case, the original equity holders are effectively choosing V c ecause their default is a conversion that transfers ownership to the new shareholders. After conversion, the new shareholders will choose default arrier V PC. Before conversion, the original equity value is given y E BC, evaluated with =. In maximizing the value of their claim, the original equity holders will choose a level of V c consistent with limited liaility, E BC (V ; V PC ) 0, for all V V c. The value of equity changes continuously with V c and with the det levels P 1 and P 2 (this can e seen from the expression (B.3) given in the appendix) so there is no phenomenon of det-induced collapse. 15

17 Det induced collapse PC V =V =58; Vc =65 PC V =V =58; Vc =75 NC V =V =93; Vc =65 25 Equity value V B PC V C =65 V C =75 V B NC V t Figure 1: Candidate equity value as a function of asset value in three scenarios. The heavy solid (green) line reflects default at V NC = 93, prior to conversion. The other two lines reflect default at V PC = 58 with two different conversion triggers. With V c = 65, equity ecomes negative so V PC is infeasile and default occurs at V NC. With V c = 75, default at V PC is feasile, and it is optimal ecause it yields higher equity than V NC. 3.2 Constraints on Det Levels We now analyze the effect of changing the det levels P 1 and P 2 and the limits imposed y Theorem 1. For purposes of illustration, we start with a simple case in which V c is held fixed as we vary P 1 and P 2. This allows us to isolate individual effects of changes in capital structure. Theorem 1 shows that either of two conditions leads to det-induced collapse: The optimal default arrier for the post-conversion firm is too high: V PC > V c. No default arrier lower than V c is feasile: for any V < V c we can find some V > V c such that E BC (V ; V ) < 0, violating the limited liaility of equity. These two conditions provide guidance in examining when changes in capital structure result in det-induced collapse. Theorem 2 makes this precise. In the theorem, we estalish two critical amounts P 1 and P 2 such that the condition P 1 P 1 is equivalent to V PC holds, P 2 P 2 is equivalent to E BC (V ; V PC ) 0 for all V V c. V c, and, when this 16

18 Theorem 2. Suppose V c is fixed. There exist upper ounds on the amount of straight det and CoCos aove which det-induced collapse ensues. Formally, there exist P 1 and P 2, where P 2 depends on P 1, such that the following holds: If either P 1 > P 1 or P 2 > P 2, then we have det-induced collapse. If 0 P 1 P 1 and 0 P 2 P 2, then det-induced collapse does not occur. The critical levels P 1 and P 2 are derived in the appendix. We illustrate these det limits through numerical examples. We fix the parameters in Tale 1, which are in line with our caliration results, and vary the average maturity 1/m, and the amount of straight det P 1. Parameter Value initial asset value V risk free rate r 6% volatility σ 8% payout rate δ 1% funding enefit κ 1, κ 2 35% jump intensity λ f 0.3 firm specific jump exponent η 4 coupon rates (c 1, c 2 ) (r + 3%, r + 3%) ankruptcy loss (1 α) 50% Tale 1: Base case parameters. Asset returns have a total volatility (comining jumps and diffusion) of 21%. On average every 3 years a jump costs the firm a fifth of its value. The numer of shares issued at conversion is set such that the market value of shares delivered is the same as the face value of the converted det if conversion happens at exactly V c. 17

19 maximal amount of CoCos P Critical values for CoCos 1/m=0.1 1/m=1 1/m=4 1/m= Straight det P 1 maximal leverage in CoCos P 2 /F Critical leverage ratios for CoCos 1/m=0.1 1/m=1 1/m=4 1/m= Straight det leverage P 1 /F 1 Critical total leverage maximal total leverage /m= /m= /m=4 1/m= Straight det leverage P 1 /F Figure 2: Top: Critical values of CoCos P 2 as a function of straight det P 1 for different mean maturities and V c = 75. Middle: Critical leverage ratios of CoCos P 2 /F as a function of straight det leverage P 1 /F. Bottom: Critical leverage (P 1 + P 2 )/F as a function of straight det leverage P 1 /F. Figure 2 shows the maximum amount of CoCos and the maximum leverage ratio that can e sustained without det-induced collapse, with a conversion arrier V c = 75. The mean maturity ranges from 1/m = 0.1 years to 1/m = 10 years. In the first plot we show P 2 as a function of P 1. The intersection of each curve with the x-axis represents P 1. For example a firm with a mean det maturity of 1/m = 4 years and face value P 1 = 90 can only add P 2 = 15 CoCos to the capital structure. If the firm adds more CoCos, det-induced collapse occurs. The second plot shows the same relationship, ut now in terms of leverage. For a firm that chooses det 18

20 levels P 1 and P 2, we calculate the resulting total value of the firm F. The ratios P 1 /F and P 2 /F are the leverage ratios for straight det and CoCos. A firm with det maturity of 10 years and a straight det leverage of 80% can increase the CoCo leverage only up to 5%. Finally, in the third plot we show the total leverage (P 1 + P 2 )/F as a function of straight det leverage. A firm with a det maturity of 1 year cannot lever up to more than 78% without triggering det-induced collapse, regardless of how it chooses its capital structure. As we have noted efore, the optimal default arrier V PC = P 1 ɛ 1 is proportional to the amount of straight det. If V c is far aove V PC, a large amount of CoCos can e issued. A short mean maturity 1/m results in a higher default arrier V PC critical level P2. If the amount of straight det is high, this also increases V PC effect takes place. and hence also in a lower and the same Figure 3 shows how the critical values in the top panel of Figure 2 change when we remove any funding enefit for CoCos y setting κ 2 = 0. The figure shows that the upper limit on CoCo issuance to avoid det-induced collapse decreases. We interpret this effect as follows. With the funding enefit reduced, shareholders have less incentive to keep the firm operating and will therefore raise the default arrier; raising the default arrier expands the scope of det-induced collapse. maximal amount of CoCos P Critical values for CoCos 1/m=0.1 1/m=1 1/m=4 1/m= Straight det P 1 Critical leverage ratios for CoCos 1 Figure 3: Critical values of CoCos P 2 as a function of straight det P 1 for different mean 1/m=0.1 maturities 0.8 and V c = 75, when CoCos have no funding enefit (κ 2 1/m=1 = 0). 1/m= /m=10 maximal leverage in CoCos P 2 /F 3.3 Implications for Optimal Capital Structure and Regulation 0.4 In the previous 0.2 section, we showed that an increase in either straight det or CoCos can precipitate det-induced collapse, viewing the det levels P 1 and P 2 as exogenous Straight det leverage P /F 1 In this 1 19 Critical total leverage verage

21 section, we examine the implications of det-induced collapse for the firm s optimal choice of det levels and for the effect of regulatory constraints on these det levels. As in Leland [28] and much of the susequent capital structure literature, we have taken the default arrier to maximize the value of equity, and we now take the det levels to maximize total firm value. The coupon rates c 1, c 2 are set exogenously. A firm that issues only straight det (which we may think of as a post-conversion firm), chooses P 1 to maximize the firm value F PC, solving max F PC = max(v + F B 1 BCOST ). (3.3) P 1 P 1 In the asence of CoCos, this reduces to the prolem studied in Chen and Kou [10], who showed that firm value is strictly concave in P 1 ; for each V there is a unique det level solving (3.3), which we denote y P PC 1. This det level optimally alances the tradeoff etween funding enefits F B 1 and ankruptcy costs BCOST. If we allow the firm to issue CoCos in addition to straight det, the optimization prolem changes to maximizing the efore-conversion firm value with F B 2 the funding enefits from CoCos. max F BC = max(v + F B 1 + F B 2 BCOST ), P 1,P 2 P 1,P 2 We first consider the ehavior of the firm value F BC over the region in which P 1 P 1 and P 2 P 2, recalling that P 1 and P 2 are the upper limits on the two types of det that preclude det-induced collapse. For det levels in this region, we know from Theorem 2 that the optimal default arrier V equals V PC and is thus independent of P 2, the level of CoCo issuance. Within this region, we may write the firm value as F BC = V + F B 1 (P 1 ) + F B 2 (P 2 ) BCOST (P 1 ) = F PC (P 1 ) + F B 2 (P 2 ). We know that the funding enefit from CoCo issuance is strictly increasing in P 2, so for any choice of P 1, the optimal P 2 is P 2 = P 2 (P 1 ). Now suppose that P PC 1 P 1, with P PC 1 the level maximizing the value of the post-conversion firm in (3.3). In this case, we claim that the optimal level of senior det for the efore-conversion firm satisfies P 1 P PC 1. Within the 20

22 no-collapse region, increasing P 1 eyond P PC 1 would decrease oth F PC (P 1 ) (y the optimality of P PC 1 ) and F B 2 ( P 2 (P 1 )) (ecause P 2 is decreasing in P 1 ). The option to issue CoCos lead the firm to reduce (or at least not to increase) its level of senior det. The other region to consider has at least one of the two det levels P 1, P 2 aove the threshold for det-induced collapse. In this case the default arrier is equal to V NC = P 1 ɛ 1 + P 2 ɛ 2 and hence F NC = V + F B 1 (P 1, P 2 ) + F B 2 (P 1, P 2 ) BCOST (P 1, P 2 ) The optimal capital choice for an NC firm depends crucially on the funding enefits. Within this region, CoCos degenerate to ordinary det, and with c 1 = c 2 and κ 1 = κ 2 the two types of det are perfect sustitutes, so the det levels affect firm value only through their total P 1 + P 2. The general case is more complex, ut for example for c 1 > c 2 and κ 1 > κ 2 straight det provides a etter trade-off etween funding enefits and ankruptcy costs and is preferred to CoCos. 11 To formalize the result, we denote y P P C 2 the optimal amount of straight det for a hypothetical firm that issues only straight det, ut with the coupon rate c 2 of the CoCos. Proposition 1. The optimal capital structure (P1, P 2 ) has the following properties: 1. If the firm s optimal choice does not produce det-induced collapse, then it has the form P1 P 1 PC and P2 = P 2 (P1 ). (3.4) 2. If κ 1c 1 ɛ 1 κ 2c 2 ɛ 2 det-induced collapse. 3. If κ 1c 1 ɛ 1 < κ 2c 2 ɛ 2 and V c > ɛ 1 P P C 1, then the firm s optimal capital structure does not produce there exists a critical level V c > max ( ɛ 1 P1 P C, ɛ 2 P2 P C ), such that for Vc > V c the firm s optimal capital structure does not produce det-induced collapse. The marginal gain in the funding enefits κ 1 c 1 is the marginal enefit of increasing straight det. The marginal increase in the default proaility is proportional to ɛ 1 and represents the marginal cost of raising straight det. Hence, κ 1c 1 ɛ 1 captures the marginal tradeoff of increasing 11 Note that total firm value does not depend on how the remaining assets at ankruptcy are shared etween det holders and CoCo holders. If only degenerated CoCos are issued we assume that at ankruptcy either the remaining assets are given to the CoCo holders or are completely lost. 21

23 P 1. If κ 1c 1 ɛ 1 κ 2c 2 ɛ 2 then straight det has etter enefit-cost tradeoff than degenerated CoCos and hence is preferred to CoCos. Analogous arguments apply to the other case. The results in Proposition 1 have implications for regulation. In case 1, a regulator can control the firm s default proaility y controlling the level of senior det. So long as the firm s det levels are in the no-collapse region, the default arrier is strictly increasing in P 1, so reducing P 1 makes default less likely. If a regulator limits senior det to a level lower than the firms optimum, so that P 1 < P 1 P PC 1, the firm will increase its CoCo issuance to P 2 (P 1 ). In so doing, it recovers at least part of the firm value lost through the reduction in senior det without changing the proaility of default, ecause the additional CoCos do not move the default arrier. The situation is quite different if the firm s optimal choice (P1, P 2 ) puts it in the region of det-induced collapse. In this case, limiting P 1 is ineffective as any increase in P 2 without an offsetting decrease in P 1 increases the proaility of default. The regulator can ensure that the firm will optimally choose det levels in the no-collapse region y requiring that the conversion trigger e set sufficiently high. If the funding enefits of straight det are higher than for CoCos, then the minimum level for the conversion arrier depends only on the optimal default level chosen y hypothetical firms that have only straight det. The main takeaway is that a sufficiently high conversion arrier avoids det-induced collapse and allows the regulator to control the default proaility through capital requirements. 3.4 Constraints with a Capital Ratio Trigger In the analysis of Theorem 2, we held V c fixed while varying P 1 and P 2 to isolate individual effects. We now let V c vary with the det levels y setting V c = (P 1 + P 2 )/(1 ρ), following the conversion rule in (2.3) ased on a minimum capital ratio ρ. According to Theorem 1, we have det-induced collapse if V PC > V c, which now reduces to the condition (ɛ 1 (1 ρ) 1)P 1 > P 2. (3.5) As efore, ɛ 1 is given explicitly y equation (B.1) in the appendix. It follows from (3.5) that a sufficiently large P 1 will produce det-induced collapse if ɛ 1 (1 ρ) > 1. We explore when this 22

24 100 Critical P Critical P 1 Critical P c=0.01 c=0.03 c=0.05 c=0.07 c=0.09 c=0.11 c=0.13 c=0.15 c=0.17 c=0.19 Critical P c=0.01 c=0.03 c=0.05 c=0.07 c=0.09 c=0.11 c=0.13 c=0.15 c=0.17 c= mean maturity 1/m mean maturity 1/m Figure 4: Critical values of straight det P 1, that lead to V PC > V c and hence det-induced collapse for a capital ratio trigger. We set ρ = 0.05 and P 2 = 5. Figure 5: Limited Funding Benefits: Critical values of straight det P 1, that lead to V PC > V c and hence det-induced collapse for a capital ratio trigger. We set ρ = 0.05 and P 2 = 5. condition 12 holds and its implications through numerical examples varying the det rollover frequency m and coupon c. For the numerical examples we take the aseline values in Tale 1 and set the capital ratio and the amount of CoCos to e ρ = 5% and P 2 = 5, respectively. Figure 4 plots the critical levels of P 1 that lead to (3.5). For example, a firm with mean maturity of four months (1/m = 0.3) and coupon of c = 0.11 on its straight det will experience det-induced collapse at any P 1 larger than 80. Note that this condition is completely independent of the parameters of the CoCos other than the amount P 2 (in particular CoCos can have different maturity than straight det). Figure 4 reveals an important interaction etween det maturity and detinduced collapse: rolling over det more frequently lowers the threshold of P 1 for det-induced collapse. Directly from (3.5), it is also clear that lowering the required capital ratio ρ also widens the scope of parameters leading to det-induced collapse. The threshold for P 1 depicted in Figure 4 gives a sufficient condition ased on (3.5): setting P 1 aove the critical value guarantees det-induced collapse. It is actually possile to have det- 12 This condition also yields det-induced collapse with the Tier 1 trigger in the footnote just efore (2.3). 23