Optimal Design and Incentive Effects of ContingentCapital, and Capital Structure Dynamics1

Charles P. Himmelberg2

Goldman Sachs & Co.

Sergey Tsyplakov3

University of South Carolina

December 2, 2016

Abstract

Contingent Capital bonds— known as contingent convertibles (CoCos) — are bondsthat automatically write-down or convert to equity when the financial health ofthe issuer (typically a bank) deteriorates to a pre-defined trigger. Using a dy-namic model of capital structure we show that if conversion terms are dilutive forexisting shareholders, banks will have incentive to reduce likelihood of triggeringby pursuing lower leverage, leading to less defaults and lower borrowing costs.Conversely, if at conversion, bond principal is written down without dilutingshareholders, then banks will have perverse incentives to pursue higher leverageand destroy capital (burn money).

JEL Classification Codes: G12, G13, G32

Key Words: Contingent capital, structural pricing model, credit spreads, marketimperfections, money burning.

1The views in this paper are those of the authors and do not necessarily reflect the views of GoldmanSachs. An earlier title of the paper was "Incentive Effects and Pricing of Contingent Capital". We thankCharles Calomiris, Stephen Figlewski, Paul Glasserman, Tony He (discussant), Amanda Hindlian, SandraLawson, Robert McDonald, Erwan Morellec, Louise Pitt, Steve Strongin, Theo Vermaelen, Andy Winton(WFA discussant), Paul Woolley, Zhenyu Wang (discussant), Alexei Zhdanov and seminar participants atthe Federal Reserve Bank of New York, the Higher School of Economics, the Hong Kong University, theUniversity of Melbourne, the University of Southern Denmark, Swiss Finance Institute, Shanghai AdvancedInstitute of Finance, as well as participants of the 2012 Conference of Paul Woolley Centre for the Studyof Capital Market Dysfunctionality at the University of Technology in Sydney, the 2012 Western FinanceAssociation (WFA) meetings, the 2012 Meetings of European Financial Management Association, 2014 FixedIncome Conference in Charleston, SC, and 2014 Credit Market Research Conference in Shanghai for helpfuldiscussions and comments. The authors are grateful for the Best Paper Award from the Paul Woolley Centrefor the Study of Capital Market Dysfunctionality, and financial support from the Institut Louis Bachelier, theKPMG Global Valuation Institute, and the Center for the Study of Financial Regulation at the Universityof Notre Dame. This work was completed while Sergey Tsyplakov was a visiting scholar in the GlobalInvestment Research division at Goldman Sachs.

2Goldman Sachs & Co., 200 West Street, NY, NY 10282; email: Charles.Himmelberg@gs.com; phone:917-343-3218.

3Moore School of Business, University of South Carolina, Columbia, SC 29208; email:sergey@moore.sc.edu; phone: 803-466-2197.

Optimal Design and Incentive Effects of ContingentCapital, and Capital Structure Dynamics

Abstract

Contingent Capital bonds — known as contingent convertibles (CoCos) — are bonds that

automatically write-down or convert to equity when the financial health of the issuer (typi-

cally a bank) deteriorates to a pre-defined trigger. Using a dynamic model of optimal capital

structure we quantify the extent to which banks facing conversion terms that are more di-

lutive for its shareholders, will have incentive to reduce likelihood of triggering by issuing

equity preemptively, which would lead to less defaults, lower borrowing costs. However,

banks would optimally elect to operate with higher (not lower) leverages when compared to

less dilutive conversion terms. The model quantifies the optimal CoCo design and optimal

capital structure dynamics that maximizes the total value of the bank that faces trade-offs

of tax benefits of debt and transaction costs of recapitalization along with conversion dilu-

tion and default risks as well as various market and contractual frictions. Conversely, if at

conversion, bond principal is written down without diluting shareholders, then banks will

have perverse incentives to pursue higher leverage and destroy capital (burn money).

1 Introduction

This paper proposes a model of dynamic capital structure to quantify the optimal design and

ex-ante incentive effects of Contingent Capital bonds — also known as contingent convertibles

(or CoCos). Cocos are bonds that are automatically written-down or converted to equity

when the issuer’s financial health deteriorates to a pre-defined threshold or trigger. Because

Basel III requires capital-qualifying debt to be loss-absorbing, the most natural issuers of

contingent capital are often assumed to be banks.4 Such bonds, however, could in theory be

issued by non-financial firms as well.

The primary advantage of CoCos, which is well-known, is that they can automatically

recapitalize a bank when it is overleveraged and likely in distress, but still has significant

enterprise value. When conversion is triggered, leverage is reduced, no new external funds

are raised from capital markets, and no government funds are needed for bailouts. For this

reason, CoCos are viewed as a valuable tool for helping bank regulators address the “too big

to fail” problem (Flannery, 2009a, b).5

Researchers and practitioners, however, do not fully understand quantitative determi-

nants of the optimal CoCo design and how various CoCo terms will affect bank management

incentives to manage capital structure. Current understanding has been mostly based on

simplified verbal arguments or one-period static models, which as we seek to show theoreti-

4For banks, the trigger is normally linked to its Tier 1 capital ratio. For example, in April 2013, BarclaysBank sold $1B CoCo bond which can be written-off entirely if the bank’s core Tier 1 capital ratio drops below7%. The Barclays CoCo will count as Tier 2 capital under Pillar 1, the minimum capital requirements set byglobal and European regulators. The trigger mechanisms are either accounting-based, or a hybrid accounting-regulatory-based. In a latter case, regulators are given contractually-specified discretion to trigger. Forexample, in February 2011, Credit Suisse issued around CHF6 billion equivalent of high-trigger Tier 1contingent capital notes. These securities will convert into bank’s equity if 1) the group’s reported Basel3 common equity Tier 1 ratio falls below 7%, or 2) Swiss Financial Markets Authority FINMA determinesthat Credit Swiss “requires public sector support to prevent it from becoming insolvent, bankrupt or unableto pay a material amount of its debts, or other similar circumstances.”

5See also the discussion by Squam Lake Working Group on Financial Regulation (2009) and Duffie (2009).

1

cally, are not entirely correct.

To fill this gap, we emphasize that the incentive effects and optimal CoCo terms are com-

plex and crucially depend on market frictions, and various trade-offs including a trade-off

between the tax benefits of debt and the deadweight costs of bankruptcy. We build the dy-

namic model to quantify the extent to which various contractual terms of CoCos may create

socially constructive or destructive incentives and how these incentives would influence the

capital structure strategy. We concentrate on the design features such as conversion ratios,

tranche sizes, as well as the “trigger uncertainty,” risk of premature regulatory intervention,

and other real-world frictions.

Conversion ratios, in particular, are critically important to understanding the incentives

governing the dynamics of a bank’s capital structure. These ratios determine whether, at the

trigger event, the losses are absorbed by shareholders or by holders of contingent capital. As

first proposed in Strongin, Hindlian, and Lawson (2009), and later discussed in Pitt, Hindlian,

Lawson, and Himmelberg (2011) and Calomiris and Herring (2013), if the conversion terms

are dilutive for the pre-existing shareholders (meaning that one dollar of par value converts to

more than one dollar of equity), then banks have an incentive to recapitalize more frequently.

In response to negative capital shocks, banks would face greater risk of future incremental

dilution from forced conversion of CoCos, and thus would have strong incentives to preempt

this, for example, by raising new equity. Because managers acting on behalf of shareholders

would raise new equity preemptively to move away from the trigger boundary to reduce the

equity dilution costs associated with CoCo conversion, such incentives can in theory mitigate

the ex-post conflict of interest between shareholders and bondholders known as the “debt

overhang problem” and reduce expected social costs from default.6

6Myers (1984) and Myers and Majluf (1984) show that a firm financed with straight debt and managedon behalf of shareholders has no incentive to issue equity during financial distress because the cost of dilutionexceeds the benefit of lower financial distress with the most of the benefits going to bondholders. For a recent

2

The model provides a quantitative measure of the extent to which a tranche of the

“dilutive” CoCo in the capital structure can effectively pre-commit management to build

up capital during distress times and, thus, mitigate the consequences of debt overhang. We

stress, however, that positive incentives effects of CoCo should be analyzed in connection

with other trade-offs and objectives that bank management is facing, including the objective

to minimize taxes, default and transaction costs. Our analysis demonstrates that quantifying

an optimal CoCo contract requires a delicate balance. Its terms should not be too dilutive as

to constrain bank’s activity and ensure that the banks will be willing adopt them, but at the

same time contract terms needs to generate enough incentives to promote a more prudent

behavior.

We also uncover that in contrast to the positive incentives motivated by dilutive conver-

sion terms — accretive conversion terms that write down principal for bondholders can create

perverse incentives for a distressed banks to remain overleveraged and even destroy its own

capital (“burn money”). The logic is the following: During distress times, when the bank

becomes overleveraged and its capital ratio is near the trigger, the bank could find it optimal

to deliberately destroy part of its capital (“burn money”) in order to push its capital ratio

below the trigger. By destroying assets, the bank will breach the trigger and write-down

CoCo principal, and thus can guarantee windfall gains for bank shareholders.7 The losses

generated by bad incentives can negate and potentially even exceed the loss-absorbing ca-

pacity of CoCos per se. To our knowledge, we are the first to discover that such destructive

incentives like “money burning” can arise from CoCo contracts. Prior literature on CoCos

concentrated on well-known incentives such as risk-shifting or incentive to increase dividends,

discussion in the context of bank regulation, see Admati, DeMarzo, Hellwig, and Pfleiderer (2012).7Theoretically, a strategy to burn money or destroy assets would be dominated by a strategy to increase

payouts. In practice, though, a distressed bank is normally under heightened monitoring by regulatorsand debtholders and will likely be constrained to increase payouts (e.g., because of debt covenants). Note,“money burning” strategy cannot be constrained, in part, because it cannot be contracted.

3

that are also common for ordinary debt contracts.8

We quantify the competing incentive effects of contingent capital and debt overhang

using a continuous-time dynamic model of capital structure of a bank that operates with a

CoCo tranche and senior debt. The values of bank assets are stochastic and, in addition, are

subject to a negative “jump” risk or “jump to default” which represent, for example, a risk

of bank runs or a revelation of a large fraud. In the model, the bank chooses initial leverage

and continuously manages its capital in the best interests of its shareholders by trading off

the tax benefits of debt and financial distress, and in addition balancing the cost of issuing

equity (and foregoing the equity benefit of debt overhang) against the risk of accepting the

punitive terms dictated by the CoCo contract. We are particularly interested in answering

such questions as “how dilutive do the conversion terms need to be in order to materially

influence management incentives?” and “what are the quantitative terms for optimal CoCo

design that maximizes the value of the bank?” The model helps to quantify (and price) the

incentive effects of tranche size and conversion ratio by measuring expected future changes

in equity issuance, the initial leverage choice and subsequent recapitalization strategy, and

default risk.

The model, parameterized to 50 largest U.S. bank holding companies endogenously re-

produces capital ratios that consistent with empirically observed values. For our baseline

case, even a relatively small tranche of contingent capital bonds where conversion terms

are dilutive to shareholders (as described above) can go a long way toward neutralizing the

negative incentive effects of debt overhang and, at thus, will increase overall debt capacity.

The material reductions in default risk, and credit spreads can be created with tranche sizes

that are a relatively small fraction of debt outstanding. For example, we show that a CoCo

tranche equal to 5% of total debt, a conversion ratio of 10-15% above par (i.e., $1 of CoCo

8See Berg and Kaserer (2011).

4

converts to $1.10-$1.15 stake of equity), and a trigger set at 6% capital will lower default

probability by about one third when compared to a similar CoCo with a conversion ratio

set at par. Moreover, since CoCos that convert to equity at a more dilutive rate can pre-

commit banks to more prudent capital structure strategy during downturns, borrowing rates

will be lower. Because of lower borrowing costs, the banks will gain higher debt capacity

and will choose to operate with higher leverages (not lower) leading to more effective tax

deductions, when compared to the case with less dilutive conversion terms. For example,

when the conversion rate increases from par-for-par conversion to the conversion with 12.5%

premium, the optimal leverage increases from 88.6% to 90%. This implication of the model

is in a sharp contrast with general argument presented in Strongin, Hindlian, and Lawson

(2009), Pitt, Hindlian, Lawson, and Himmelberg (2011) and Calomiris and Herring (2013),

concluding that a bank with more dilutive conversion terms would maintain higher levels

of capital. The main reason for such a different conclusions is that their predictions were

drawn from simplified “back-on-the envelope” calculations or frictionless static models that

analyze incentives with insulation from other trade-offs and incentives, such as, for example,

incentives to minimize taxable income.

There is another important result which contrasts with predictions of the previous liter-

ature. Our model implies that dilutive CoCos with a trigger set at a higher bank capital will

not necessarily be more effective in generating positive incentives. As the model predicts,

for a given conversion rate, the bank will optimally start with higher capital, but it will

raise less equity in the future when compared to the case with a lower-set trigger, resulting

in higher (not lower) expected default costs. Since the bank faces a combination of capital

structure trade-offs including incentives to reduce taxable income, a higher-set trigger can

diminish flexibility of managing tax shields and the need to retain such flexibility can poten-

tially outweigh the incentives to avoid conversion dilution leading to higher expected default

5

costs. As a regulatory recommendation, trigger should not be set at a very high capital as

to avoid overly constraining bank economic activities.

The model also helps to design optimal contractual terms for varying bank parameters

and real-life frictions. The optimal contract characterizes the conversion terms that maximize

the market value of the bank taking into account countervailing incentives, initial choice of

capital structure and subsequent recapitalization decisions and balancing the three types of

costs including default costs, transaction costs and dilution costs. The main implication of

the CoCo design is that “one size does not fit all”, where the conversion terms should be

adjusted to reflect different bank characteristics. One interesting prediction is that highly

dilutive conversion terms do not always enhance the efficiency. This is because, facing the

risk of larger dilution, the bank will be recapitalizing unnecessarily too frequently causing

excessive transaction expenses. From a policy perspective, the regulator should promote

CoCo that carry low to moderate dilutive conversion rates to avoid excessively frequent

recapitalizations.

We also show that real-world market frictions can weaken the effect of incentive for banks

to issue equity capital. First, we quantify the extent to which beneficial incentive effects are

obviously diminished by costs of issuing equity (such costs may arise from explicit transaction

costs or from management’s view that its equity is undervalued). Second, we quantify the

magnitude by which incentives are diminished by a negative “jump” risk or “jump to default”.

The jump risk dampens incentives because they reduce the ability of management to alleviate

conversion risk by holding excess capital. This result further challenges the “static” argument

predicting that a bank with riskier assets will be more willing to raise new stock when facing

dilutive conversion than the bank with less risky assets.

Finally, we show that incentives to voluntarily raise new equity are dampened by “trig-

ger uncertainty” (or “trigger ambiguity”). In practice we see trigger uncertainty arising in a

6

number of ways, but two stand out. On the one hand, triggers based on regulatory capital

are often thought to be vulnerable to “regulatory failure” resulting in delayed triggering and

conversion. Delayed trigger can also be due to regulators worrying about creating a panic

or/and “run on the bank” would be reluctant to call the trigger in a timely manner until the

bank becomes deeply in distress.9 Our results suggest that if markets anticipates a possi-

bility of regulators delaying the trigger, the risk of such delays will weaken incentives. This

underscores the value of enhanced disclosure and transparency as mechanisms for increasing

political scrutiny of regulators. On the other hand, triggers based on regulatory discretion

(regulatory intervention) or market values of equity are often thought to be vulnerable in

the other direction — to premature triggering and conversion (including market failures due

to strategic behavior of investors large enough to manipulate equity prices). Again, con-

sistent with intuition, our results show that the risk of early triggering weakens incentives.

The model quantifies that in order to guard against “trigger uncertainty” and/or regulatory

failure, the optimal CoCo design has to include more dilution.10

The discussion of positive incentives raises an important practical question for bank reg-

ulators: whether and when regulators need to mandate the issuance of CoCo that carry

dilutive conversion terms. As the model demonstrates, if the dilutive CoCos are issued when

the bank is already overleveraged, the newly created incentives will immediately become

highly effective (“in the money”) but their benefits will be flowing to debtholders while the

existing shareholders will be experiencing losses. Therefore, shareholders of the undercapi-

talized banks will avoid a voluntary issuance of dilutive CoCo bonds “midstream” because

the subsequent benefits of more conservative leverage will pass disproportionately to existing

bondholders. On the other hand, if the bank is well-capitalized, the newly issued CoCo will

9McDonald (2013) discusses possibility of trigger failure.10See discussions in McDonald (2010) and Skinner and Ioannides (2012).

7

create incremental value for shareholder due to increased future tax shields, while the net

wealth transfer from equityholders to existing debtholders will be minimal. As a policy rec-

ommendation, to extract the maximum incentive benefits of dilutive CoCos, bank regulators

would theoretically need to mandate their issuance during bad times, i.e., when banks lose

capital, and simply encourage well-leveraged banks to use CoCo as recapitalization mecha-

nism, because during good times, banks should issue these bonds voluntarily.

It is important to emphasize that desirable incentive effects of CoCos are reversed for the

case when conversion terms impose losses on CoCo holders rather than on bank shareholders.

If conversion ratio is less than one-to-one — that is, the par value of the CoCo is simply written

down, such conversion terms are therefore accretive (rather than dilutive) to shareholders.

As we discussed earlier, this design creates perverse incentives for increased leverage, or

worse, “money burning” or destruction of assets. When the bank suffers capital losses and

its capital ratio is close to the trigger, it can deliberately destroy part of its capital in order

to breach the trigger, write-down the bond’s principal, and thus capture a windfall gain for

shareholders at the expense of the CoCo investors.

Theoretically, the shareholders will benefit from “money burning” if the contractual

write-down amount (i.e., shareholders’ gain) exceeds the amount of capital that the bank

needs to destroy to activate the trigger.11 Thus, such CoCos can make the debt-overhang

problem even worse than it is for straight debt, because the usual “debt overhang” do not

create incentive to intentionally destroy assets. As a result, these write-down features of

CoCos can increase social costs of distress even more.

Our model helps to formalize the logic of money-burning incentives and also to calibrate

11Our notion of “money burning” is driven by adverse incentives rather than adverse selection, and is thusdistinct from “costly signaling” intuition described by Spence (1973). The “money burning” in our settingresembles the net dividend payout in a tax-disfavored form discussed in Bernheim and Redding (2001) witha significant difference that we do not assume asymmetric information.

8

the magnitude of value destruction in the event of financial distress. Our numerical results

show, for example, that if a bank has issued a tranche of CoCo bonds equal to 5% of

total debt, and with a principal write-downs upon triggering, then as the bond approaches

maturity, shareholder value could be maximized by “burning” as much as 3.5% of assets to

accelerate conversion. For the $13 trillion of assets held by U.S. banks this could imply the

additional value destruction in the event of a future crisis of nearly $400 billion.

The analysis of money burning incentives of the CoCo is not simply a theoretical exercise,

because a money burning rarely emerge as a rational response in a general contract theory.

As demonstrated in Spence (1974) and Milgrom and Roberts (1986), money burning may

serve as a signal to reveal private information in settings with imperfect information, leading

to an efficient equilibrium.12 In contrast, money burning incentives governed by poorly-

designed CoCo may generate inefficient wealth transfer from CoCo holders to the common

stock even without assumption of asymmetric information. Money burning can take many

forms: for example, distressed banks may have incentives to accelerate loan charge-offs,

accelerate write down the value of equity investments, and increase loan-loss provisions in

order to intentionally breach the conversion trigger. This argument further stresses the

design flaw of conversions accretive to equity because they perversely encourages managers

of distressed banks to refrain from issuing equity and destroy assets when they should be

doing exactly the opposite.

This incentive to “burn money” does not appear to be widely appreciated either by

academics or by policymakers. On the contrary, the vast majority of CoCos issued to date

(mostly by European banks) are accretive rather than dilutive to shareholders, evidently

without objection from regulators. This lack of understanding for these value-destroying

incentives of accretive conversion terms has potentially serious consequences for the future

12See Spence (1974), Milgrom and Roberts (1986).

9

behavior of banks in the next financial crisis. The incentives driven by accretive (“write-

down”) conversion terms are obviously objectionable since they give banks the incentive to

destroy value as they spiral into financial distress. And of course, from a policy maker’s

perspective, it is exactly during the crisis that is important to have mechanisms in place

that create rather than destroy incentives to preserve total asset value.

To the best of our knowledge, this is the first dynamic model that quantifies how a

bank would respond to changes in CoCo terms and how to design optimal CoCo terms to

maximize the positive gains from incentives. In addition to research that has noted the

possible incentive effects of contingent capital (e.g., Strongin, Hindlian, and Lawson (2009),

Pitt, Hindlian, Lawson, and Himmelberg (2011), Berg and Kaserer (2011), and Calomiris

and Herring (2013)), our paper relates to existing research that develops pricing models of

contingent capital. Albul, Jaffee, and Tchistyi (2010) derive an analytical solution for the

market value of CoCo debt in an infinite maturity setting with a static capital structure.

Pennacchi (2010) models both the bank’s risky assets and its deposits and solves for the

market price for contingent capital debt and the fair deposit insurance premium. Pennacchi,

Vermaelen and Wolff (2011) propose an alternative form of contingent capital in which the

shareholders have an option to buy shares back at par value from the CoCo holders after

conversion. This option allows shareholders “undo” a potential wealth transfer to CoCo

holders caused by conversion terms or market manipulation. McDonald (2013) introduces

and prices a contingent capital claim with a dual trigger that depends on both the bank’s own

stock price falls as well as a broad financial index of financial distress. Finally, Sundaresan

and Wang (2011) show that conversion triggers based on market value of bank’s stock can

result in multiple equilibrium prices (or no equilibrium price) for contingent capital.13 In

13Glasserman and Nouri (2012) show that, multiple equilibria cannot occur if there is continuous tradingin the bank’s stock and if conversion terms are dilutive.

10

contrast to our model, these models all assume that future capital structure is determined by

the exogenous process for assets, thus precluding the incentive effects of contingent capital

on capital structure choices.

The rest of our paper proceeds as follows. The next section develops the model, section

3 presents numerical results and comparative statics of the model, and section 4 concludes.

Following this, Appendix provides more details on the dynamic stochastic control problem.

2 Model

2.1 Model description and time line

We consider a dynamic continuous-time model of a bank. Bank assets are stochastic (and

subject to negative jumps) and cash flows are proportional to assets. After-tax dividends

are paid as a residual after coupon payments, which are tax deductible.

At time 0, the bank issues both the CoCo debt and senior debt in its capital structure.

We assume that the Regulator requires that CoCo’s size has to be of a predetermined per-

centage of the total debt, but bank can choose initial total debt ratio and a subsequent

recapitalization strategy. Both CoCo bond and senior debt require continuous coupon pay-

ments and both have the same maturity, but the CoCo bond is junior. At maturity, par

values are paid to their claimants respectively, assuming the CoCo has not been converted

in prior periods.14 We also assume that a certain fraction of bank’s senior liabilities is con-

tinuously repaid at par and reissued at market price, such that the principal value remains

unchanged.15 This structure reflects the fact that in practice, senior liabilities of large banks

14For simplicity, we assume that senior liabilities have the same maturity as CoCos. In practice, debtstructure of banks is more complex. Most of retail banks’ senior debt is short-term deposits. Large investmentbanks constantly roll over their senior debt by borrowing overnight federal funds or participating in repomarkets. Theoretically, we can mimic such structure by assuming that a certain fraction of bank’s seniorliabilities is continuously repaid and reissued. See Titman and Tsyplakov (2005).15We follow roll-over structure of Leland (1994) and Titman and Tsyplakov (2005).

11

are the combination of short- and long-term deposits, overnight federal funds, term wholesale

deposits and repo financing instruments, implying that banks constantly roll over part of

their senior debt. The higher the roll-over rate, the shorter will be the effective maturity of

senior liabilities relative to CoCo maturity. For simplicity, we assume that the roll-over rate

is constant.

For the base case, the conversion of the CoCo is triggered as soon as the bank’s capital

ratio drops below a contractually specified accounting-based trigger. This is equivalent to

the assumption that the discretion on conversion is given to a regulator, which calls the

trigger at the first instant when a bank’s capital falls to or below a prespecified ratio.16

For simplicity, we assume that bank capital ratio can be observed by the market and the

regulator. The CoCo bond converts to a fixed market value of shares. The conversion ratio

determines the market value of equity to be owned by CoCo holders (could be below or

above par), where the exact number of new shares is calculated after conversion.17 After

conversion, the bank continues servicing the senior debt until maturity. If asset value falls

further below book value of debt anytime before or at maturity, the regulator will intervene

and classify the bank as in insolvency/default. When in insolvency, the bank’s equity is

wiped out, and debtholders recover the bank’s unlevered assets minus default costs.

After the initial debt ratio choice, the bank can issue equity capital at any time prior

to breaching the trigger, thereby raising its capital ratio and moving away from the trigger.

The bank can also choose to “burn money” (destroy assets or waste capital), which allows

for the possibility that the bank may be willing to “pay” to reduce its capital ratio and thus

force conversion.18 Equity issuance, money burning and default are endogenous and chosen

16Avdjiev, Bolton, Jiang, Kartasheva, and Bogdanova (2015) document that most of the recently issuedCoCos incorporate do not stock-price based triggers.17The model can incorporate the conversion to a fixed number of shares. McDonald (2010) discusses the

advantages and disadvantages of these two cases.18For simplicity, we assume that the bank cannot change its dividend policy or sell assets.

12

to maximize the existing shareholders’ wealth, i.e., the current share price.

We augment the model described above with a number of frictions designed to accom-

modate practical design considerations arising from real-world concerns:

• Transaction costs for equity issuance. When the bank issues equity, it incurs transac-

tion costs that have fixed and variable components. Such frictions are consistent with

the lumpiness and relative infrequency of equity issues observed in the data19.

• Negative “jumps” in the diffusion process for the value of bank unlevered assets. Jumps

in the process for asset value are meant to capture the fact that financial systems are

periodically hit by large crises. Since jumps can more easily swamp precautionary

capital buffers held to increase the risk of undesired (dilutive) CoCo conversion and

default, their inclusion here allows us to calibrate their practical significance.

• Uncertainty in the trigger mechanism. In the later section, we introduce two types of

trigger failure. The first is the “type I” error that the CoCo fails to trigger despite

capital ratios having breached the minimum threshold (this risk might arise due to,

say, poor disclosure or regulatory failure). The second is the “type II” error that the

CoCo accidentally triggers even though the bank’s capital is still above contractually

specified threshold (this risk might arise due to, say, a noisy market trigger or overly-

aggressive regulatory intervention). We calibrate the incentive and pricing effects of

both types of trigger failure.

2.2 Value of the bank unlevered assets

We assume that when there is no injection of capital (or money burning) and no arrival of

jumps, the (after tax) value of the bank’s unlevered assets follows a log-normal stochastic

19Such frictions are commonly thought to reflect information frictions; see Loughran and Ritter (1995).

13

process:dV

V= (r − α)dt+ σdWV , (1)

where r is a short-term risk-free rate which is assumed to be constant; α is the payout rate;

WV is a Weiner process under the risk-neutral measure; and σ is the instantaneous volatility

coefficient. Given payout rate of α, bank’s assets generate continuous after-tax cash flows of

αV. Equityholders can choose to increase (decrease) the size of the bank’s capital by issuing

equity (burning money) of any size. If the bank decides to issue equity (burn money) and

raise (reduce) capital by 4V , the value of the bank’s assets at time t+ will be:

Vt+ = Vt +4V, (2)

where 4V > 0 is the size of newly raised equity (or money burned, if 4V < 0). We

assume that when the bank issues equity and raises capital, it incurs transaction costs TC,

which have both a fixed component (proportional to the level of its current assets) and a

variable component (proportional to the size of newly raised assets 4V > 0). As such,

if the bank issues equity and raises the size of its assets by 4V , total transaction costs

TC = e1Vt + e24V , and e1 and e2 are positive constants. These transaction costs are paid

by the equityholders of the bank. For simplicity, we assume that there are no transaction

costs if the bank burns money.

We also assume that the bank’s asset value can drop discontinuously due to the random

arrival of negative “jumps.”20 Such jumps allow for the possibility of shocks that cause asset

value falls far enough (and fast enough) to cause CoCo conversion or even default. Such

jumps represent a real-world “friction” in the sense that they preclude the possibility of the

firm issuing equity fast enough to prevent conversion (or default).

20Pennacchi (2010) and Andersen and Buffum (2002) also assume jumps in asset value process.

14

Formally, we let the independent and uniform random variable Y ∈ [0, 1] describe the

magnitude of the shock as a percentage of assets. Arrival times are independent and follow

a Poisson process (Merton (1990)). Specifically, the probability that a jump arrives during

time interval ∆t is λ∆t, where λ is a risk-neutral arrival intensity describing the expected

number of jumps per year. The expected change in assets due to jumps is −λk, where

k = EQ(Y ) and EQ is the expectation under the risk neutral measure Q. The value of bank’s

unlevered assets before taxes is therefore described by:

dV

V= (r − α− λk)dt+ σdWV + dq, (3)

where dq = (Y − 1)dNt describes fluctuations in bank assets due to jumps, where Nt is

a Poisson process. Figure 2 provides an illustration of the two types of sample paths: the

ordinary diffusion process and the jump-diffusion process with asymmetric jumps. Note that

both processes have the same long-run drift.

2.3 Senior debt and contingent capital

A time 0, the bank chooses its initial capital structure and issues both senior debt and

contingent capital bond. Senior debt and contingent capital require continuous coupon

payments at the rate of f and c, respectively, and principal payments of F and C at maturity.

By assumption, the regulator requires that the size of the CoCo is a predetermined percentage

ρ of total debt, i.e., ρ= CC+F

. The senior debt is continuously retired (repaid at par value F )

and reissued at constant rate m. We call parameter m the annual roll-aver rate of the senior

debt. The retired debt is reissued at market price D. With higher retirement rate m, the

effective maturity of senior debt will be shorter than it’s stated maturity T . The annual net

refunding cost is m · (F −D)), D is the market value of the senior debt.21 Given a payout

21We are using the same denotation as in Leland (1998).

15

rate from bank’s assets, α, the bank generates continuous after tax dividend net of interest

payments of (αV − (1 − τ) · (f + c) − m · (F − D)), where parameter τ is the corporate

tax rate. The interest payments for both senior liabilities and CoCo are tax deductible. We

assume that dividend policy is fixed and the bank’s dividend is the residual payments after

interest and taxes. We further assume that the bank cannot sell its assets (for example, due

to debt covenants) and cannot change its dividend policy.22

If at any time before maturity the bank’s capital ratio, (V−F−C)V

, falls below the prede-

termined threshold θ (conversion trigger ratio), the CoCo bond automatically converts to a

fixed dollar amount of shares. Particularly, the terms of conversion pre-specify that each $1

of CoCo bond converts to $π of market value of equity, where π is a conversion ratio.23 If

π > 1, conversion dilutes existing equityholders because it transfers wealth from equityhold-

ers to CoCo investors. At conversion, the existing equityholders transfer shares valued at πC

to CoCo holders and the bank’s debt size declines from F+C to F. In the baseline model, we

assume that there is no trigger uncertainty or trigger failure. Post-conversion, the bank con-

tinues servicing the remaining debt, so after tax dividends are (1−τ) ·(αV −f−m ·(F−D)).

At maturity, the bank repays the par value of F .

We assume that the regulator classifies the bank as insolvent or in default and closes it

any time before maturity when bank’s capital ratio declines to zero, i.e., when the ratio of

(V−F−C)V

= 0. When the bank is insolvent, the equityholders’ value is zero, and debtholders

recover the bank’s unlevered assets V minus proportional default costs (1−DC) · V, where

22These two assumptions are common for dynamic models of firms or Merton-style structual models. Seefor example, Leland (1998), Titman, S., and S. Tsyplakov, (2005), and Albul, Jaffee, and Tchistyi (2010).23These two assumptions differ from assumptions in Sundaresan and Wang (2011). In their model, they

assume a market-based trigger where both the conversion trigger and conversion price themselves dependon the market prices of the bank’s shares. They show that this inter-dependence may result in multipleequilibria or even in an absence of equilibrium price of the bank’s equity. However, Glasserman and Nouri(2012) show that, multiple equilibria cannot occur if there is continuous trading in the bank’s stock and ifconversion terms are dilutive.

16

DC is a positive constant, 0 < DC < 1. CoCos are assumed to be junior in insolvency, so

that CoCo investors are paid only after senior debt recovers 100%.

With jumps in the asset value process, the bank assets can fall below both the trigger

and default boundaries. If the jump reduces the assets value to some level V 0, which is

below the trigger but above the default boundary, then it could not be always possible to

guarantee that CoCos convert to the contracted fixed market value of shares (πC) because

the remaining asset value is less than the sum of senior debt and πC (i.e., πC > V 0−F > 0).

In such cases, existing shareholders will be wiped out leaving remaining residual value to

CoCo investors, who could get less than the conversion ratio assumes (i.e., less than<πC).

In other words, the conversion ratio π will be reduced to π0 (where, π0C = V 0 − F > 0)

reallocating the entire residual equity value to CoCo holders, implying that post-conversion,

the bank becomes wholly-owned by CoCo bondholders.

If the size of the realized jump is large enough, the bank assets can fall through the

conversion boundary and below the insolvency boundary. In such case, the regulator will

immediately classify the bank as insolvent without triggering the conversion. Given that

the CoCo bond is junior, the CoCo holders will have lower priority in recovering value at

default, and for most parameter values, will receive zero value.

2.4 Base Case Calibration of Parameter Values

The purpose of calibration presented in this section is not to mimic characteristics of a certain

bank, but to justify parameter values that crudely describe a large U.S. bank. To do so,

we collect quarterly financial data for the top 50 largest publicly traded U.S. bank holding

companies (BHCs) between 2005 and 2014. We also consider separately the subsample

of BHCs that are globally systemically important banks (GSIBs), with more than $100

billion in assets. There are 8 institutions that are GSIBs in our data set. Most of our

17

information comes from the Federal Reserve’s Y-9C Consolidated Financial Statements for

BHCs. Because of extreme volatility in the banking sector during the crisis, we exclude the

period of crisis (from 2007: Q3 to 2009: Q4) and report summary statistics separately for

the Pre-Crisis Period (2005: Q1-2007: Q2) and the Post-Crisis Period (2010: Q1-2014: Q4).

Table 1 reports several statistical measures that can be used to approximate the volatility

of bank’s assets.24 Table 1 documents averages and standard deviation of accounting return

on assets over the preceding twelve quarters. It varies between 0.2% and 1.3%, and slightly

lower for GSIB and for the Pre-Crisis Period. Another measure of asset risk is a standard

deviation of asset growth. This measure is observed to be varied between 3% and 13%.

Standard deviation tends to be higher in the Post-Crisis period, which is probably explained

by more active bank acquisition activity. Based on these observations, the volatility of bank

unlevered assets is set at σ = 5%.25 The risk-free interest rate is r = 5%, and the asset

payout yield α is 5%, which means that the risk neutral diffusion drift of assets is 0% per

year.

For large BHCs, senior liabilities are comprised of deposits, overnight federal funds,

term wholesale deposits and repo financing instruments and other types of subordinated

and unsubordinated debt. In case of insolvency, insured depositors will likely be able to

recover 100% of their values. However, the other part of debt liabilities will not necessarily

recover 100%. To estimate the deadweight cost of insolvency, we calculate the ratio of

insured deposits to unsubordinated debt liabilities. This ratio is around 40% for the entire

sample and is lower for GSIBs at around 16%. In the related literature on corporate default

costs, these costs are estimated to be around 50%. Due to regulatory intervention during

insolvency, these costs will be lower than for corporations. Given that between 16% and 40%

24All statistics is winsorized at the 5% level.25The volatility parameter is similar in magnitude to the bank assets volatility assumed in Pennacchi

(2010) and Albul, Jaffee, and Tchistyi (2010).

18

of senior liabilities (insured deposits) recover 100% value, we assign the proportional default

costs for senior bank liabilities at 20%., i.e., DC = 20%.

For the base case, we assume that the regulator mandates the CoCo size to be 5% of

the total debt. i.e., ρ = CC+F

= 5%. Both CoCo bond and senior debt mature in six years.

Avdjiev, Bolton, Jiang, Kartasheva and Bogdanova (2015) report that most recent issuances

of CoCos have trigger set at around 6% capital. As such, for our base case we assume that

the trigger is set at θ = 6%, which means conversion occurs when a bank’s capital ratio

satisfies V−F−CV

≤ 6%, where the numerator of this ratio is book value of equity. In order to

study effect of incentives, the conversion ratio is varied from π = 1.0, which is a par-for-par

conversion, to π = 1.25, which is equivalent to 25% above par.

Transaction costs of raising new equity are assumed to be e1 = e2 = 0.025%, reflecting

both fixed and proportional components. Butler, Grullon and Weston (2005) document

investment banking fees for equity issuance around 5%. If we assume equity issuance is 5%

of total capitalization and equity capital is 10% of assets, then total fixed transaction costs

are 5%× 5%× 10% = 0.025% of asset value.

We assume that a negative jump describes a catastrophic-type event like a major crisis, a

big fraud revelation, or a Lehman-like event which is characterized by a very low probability

but significant losses to bank’s assets. Thus, we assume that an annual probability of a

negative jump is λ = 3%, representing the economic environment in which financial crisis

(bank runs) happen on average every 33 years. The jump size Yt is assumed to be uniformly

distributed on [0, 1]. This means that conditional on arrival, a jump leads on average to a

50% loss in asset value (parameter k = −0.5).Due to jumps, the risk neutral diffusion drift

of the value process is adjusted upward by −λk = 1.5%.

Table 1 also reports two capital ratio variables, corresponding to the two commonly-used

measures of regulatory capital: 1) Tier 1 leverage ratio which is Tier 1 capital divided by

19

total unweighted, on-book assets; 2) CAPTIER1 (Tier 1 risk-based capital ratio) is equal to

Tier 1 capital divided by risk-weighted assets. Depending on the measure, capital ratios are

varied between 6% and 14% and overall are slightly lower for the GSIBs. Most of the capital

ratio measures increased for the Post-Crisis Period, which is likely driven by more stringent

regulatory oversight. Finally, Table 1 reports Market-to-book ratios that are varied between

.99 to 1.13. We will compare the model generated optimal capital ratios and Market-to-book

ratio with empirically observed ones.

The coupon rates f and c are set so that initial market prices of the senior debt and

the CoCo bond are approximately equal to their par values of F and C, respectively.26 We

calculate credit spreads at origination as the difference between coupon rates and the risk-

free rate. We also measure the bank’s average cost of debt by calculating its weighted-average

(WA) spread.

We calculate expected default costs as a function of bank’s unlevered assets at time zero,

(Expected Default Costs)V0

. We measure the total expected default costs and expected costs of

recapitalization both as percentage of unlevered assets at time zero. By analyzing the size of

default costs costs as a function of the CoCo contract terms we can gauge the incentive-driven

efficiency of trading off tax benefit of debt and expected default costs. We also calculate

the expected value of future net equity issuances or the expected value of burnt money both

measured ex-ante at time zero calculated as a fraction of the unlevered assets at time 0. We

also calculate the ratio of the market value of the bank as a percentage of assets, E+F+CV0

,

this ratio roughly corresponds to market-to-book ratio. Finally, we calculate the probability

of triggering under the real-world probability measure.27

26We use several numerical iterations to approximately find par coupon rates for senior debt and the CoCo.27For these calculations we assume a 3% premium and thus increase diffusion drift from 5% to 8%. Mapping

the risk-neutral intensity of the jump process to the real world is more complex. Chen, Liuy and Maz (2007)show that for a recursive utility function, there is a linear relation between the risk-neutral arrival intensityand the real world intensity, which has the following form: λ = λRW f(·), where λRW is the real-world

20

We assume complete markets for the bank’s assets, so that the CoCo bond, senior debt

and equity can be regarded as tradeable financial claims for which the usual pricing conditions

must hold. The market values of equity, CoCos, and senior debt are functions of three

variables: asset value, V , face value of total debt, F + C (or F if conversion has already

taken place), and time, t ≤ T . Equity value has to satisfy a stochastic control problem with

fixed and free boundary conditions, where the control variables are the equity issuance (or

money burning). The default/insolvency is called any time at time t as soon as book equity

is negative, i.e., if Vt − F − C < 0. These valuation problems are described in detail in

appendix 3.

The numerical algorithm used to compute the values of equity and debt is based on

the finite-difference method augmented by policy iteration. Specifically, we approximate

the solution to the dynamic programming on a discretized grid of the state space (V, {F,

F + C}, t). At each node on the grid, the partial derivatives are computed according to

Euler’s method. The backward induction procedure starts at the terminal date T , at which

the values for senior debt, the CoCo bond and equity are determined by payoff to CoCo

holders, holders of the senior debt, and holders of bank equity. The backward recursion

using time steps ∆t takes into account the bank’s optimal strategy to raise capital (or burn

assets). For more detail of the numerical algorithm see Titman and Tsyplakov (2005) and

Tsyplakov (2007).

intensity of jump arrival and f(·) is a function of the average jump size, volatility of the jump, and therisk aversion coefficient. For our parameterization, we assume that f(·) = 10, which corresponds to the riskaversion factor of about 3.We assume that in the real world probability measure, the distribution of jumpsize do not change.

21

3 Results

We use numerical solutions of the model described in the previous section to address the

following general questions:

1. How do the contractual terms of CoCos, including tranche size, conversion ratio and

the trigger, affect management’s propensity to raise equity capital?

2. Given the conversion rate, what is optimal initial leverage ratio at which the market

value of the bank (i.e., market value of equity and total debt) is maximized?

3. What is the optimal conversion ratio (π∗) that maximizes the market value of the bank

taking into account optimal initial leverage and subsequent strategy of rasing equity

capital.

4. To what extent are these incentive effects offset by countervailing frictions, namely

issue costs for new equity, negative jumps in asset value, and uncertainty in the trigger

mechanism?

We begin with an analysis of a base-case parameter range that assume “neutral-to-

dilutive” conversion terms. This analysis will shows how the initial capital structure and its

future dynamics changes as conversion ratios become more dilutive. In the later section, we

consider conversion terms that are accretive for shareholders, and thus create incentives to

burn money.

3.1 The Base Case — No Contractual Frictions

3.1.1 Case with Par-for Par Conversion Rate

Figure 3 illustrates the consequence of CoCo incentives on the initial choice of the capital

structure by comparing two scenarios: the bank is financed with all senior debt, and a

22

baseline case in which the CoCo is 5% portion of total debt. In both scenarios, the bank

chooses the optimal initial leverage at which the total market value of debt plus equity is

maximized. At the origin in Figure 3, one dollar of par converts to one dollar of bank’s

equity (π = 1.0) . The comparison of two capital structures at this point shows that adding

a “par” CoCos to the capital structure will increase the initial optimal leverage ratio (C+FV0)

from 84.5% to 88.6%. The reason is simply the fact that the with CoCo the firm will be able

to create more tax shields without exposing the firm to higher default risk. This observation

is confirmed by the fact that the expected cost of default and weighted-average credit spread

does not change much relative to the case when the bank is financed with all senior debt. At

the same time, the firm starting with higher total level of CoCos plus straight debt will be

paying less corporate taxes which will increase the market valuation of the bank. The ratio

of market value of the bank to its unlevered asset value ,E0+C+FV0

, which corresponds to the

Market-to-Book ratios increases from 1.054 to 1.057. So, if incentives are not enhanced by

dilutive conversion terms, CoCo bonds will not materially reduce overall default risk but can

more effectively decrease taxable income.28 We conclude this section by pointing out that

both the model-generated optimal leverages and Market-to-Book ratio are within empirically

observable values reported in Table 1.

3.1.2 Dilutive Conversion Terms

Figure 3 also displays how the future equity issuance, expected default costs, credit spreads,

and market-to-book ratios are affected as the conversion ratio increases from the par-for-par

level to above par conversion (i.e., made more dilutive of existing shareholders). Panel 2 that

plots the expected rate of future equity issuance shows that a small tranche of CoCo bonds

with moderately dilutive conversion terms can motivate the bank to raise equity and de-lever

28These results confirm earlier studies that covertable debt will increase total debt ratio. See Albul, Jaffee,and Tchistyi, (2010).

23

in the future in order to move from the trigger and that these incentives will be stronger for

more dilutive terms. In anticipation of such incentives, CoCo holders and holders of senior

debt would demand a lower rate of return, and thus, the bank’s cost of borrowing will be

lower. As plotted in Panel 1, the model predicts that because of lower borrowing costs,

the banks will gain higher debt capacity and will choose higher initial leverage ratios (not

lower) leading to even more effective tax deductions. We also calculate and plot the bank’s

average leverage measured over time until maturity or until the bank becomes insolent. As

illustrated in Panel 1, the bank with more dilutive CoCo terms will on average operate with

higher leverage. These implications are in a contrast with general argument presented in

Strongin, Hindlian, and Lawson (2009), Pitt, Hindlian, Lawson, and Himmelberg (2011),

and Calomiris and Herring (2013), predicting that a bank with more dilutive conversion

terms would maintain higher levels of capital.

In quantitative terms, holding the size of CoCo at 5% of bank unlevered assets, an

increase in the conversion ratio from π = 1.0 to 1.12 will lead to an increase in the initial

optimal leverage (C+FV0) from 88.6% to 90% and will incentivize the bank to increase its

future equity issuance rate from zero to 3.5%. The future equity issuance rate is calculated

as the expected equity issuance rate measured over the entire maturity period given that the

bank start with an optimal leverage and remains solvent. As the conversion ratio increases

further from π = 1.22 and higher, the expected rate of equity issuance plateaus at around

4.3% (See Figure 3). For conversion ratios of π = 1.22 and above, the bank will always avoid

drifting through the trigger by preemptively raising enough equity to keep its leverage above

the conversion trigger. Due to the presence of negative jumps, however, the risk of triggering

cannot be reduced to zero. Figure 3 shows that despite the higher initial leverage, the the

weighted-average credit spread and default costs both gradually decrease as the conversion

ratio increases. Specifically, an increase in conversion ratio from 1.0 to 1.12 can meaningfully

24

reduce the deadweight cost of default from 1.6% to 1.13%. However, as illustrated in Figure

4, the total deadweight costs — consisting of expected default costs plus expected transaction

costs of equity issuance— do not decrease significantly for the case with more dilutive CoCos.

One interesting result is a non-monotonic relation between the CoCo spread and the

conversion ratio. The CoCo’s spread narrows as the conversion ratio increases from π = 1.0

to 1.02. However, the spread widens as the conversion ratio increases further from 1.02. The

intuition for this result is the following. Over the range of 1.0 < π < 1.02, the CoCo spread

declines because an increase in the conversion ratio will increase the expected “windfall” to

the holders of the CoCo debt. At the same time, the probability of conversion (shown in

the Figure 3) is does not change. As the conversion ratio increases from 1.02 and higher,

however, the bank’s expected equity issuance increases. This reduces the probability of

conversion (even with higher initial leverage), thus resulting in a lower expected windfall

for CoCo holders and a higher spread. This non-monotonic relationship challenges general

static intuition that a more dilutive conversion will always make the cost of contingent capital

lower.29

3.1.3 The Optimal Conversion Ratio

In this section, we analyze an optimal conversion ratio π∗ for which the total market value

of the combined debt and equity is maximized given that the bank starts with optimal

debt ratio. In order to find the maximum, we use numerical search procedure by varying

parameter π between 1 < π < 1.25, and the initial capital structure between 80% and 100%,

while other parameters stay at the base case level, including the trigger ratio and the CoCo’s

relative size.

Numerical search indicates that the combination of conversion ratio of π∗ = 1.125 and

29See discussion in Strongin, Hindlian, and Lawson (2009).

25

the optimal initial leverage ratio (C+FV0) of 90% maximizes the market value of the bank. See

Figure 4. In other words, given the option, the bank will choose to issue CoCo with the

conversion rate of π∗ = 1.125, because such ratio dominates any other ratios and maximizes

bank’s value. For conversion rates above 1.125, the firm value will be decreasing because

the transaction costs of recapitalization will outweigh the benefits of lesser conversion risk

and lower default costs. This is because, with a risk of even higher dilution, the bank will

be exposed to more frequent recapitalizations resulting in excessive transaction costs. To

confirm this point, Figure 4 depicts total expected deadweight costs that include both default

costs and transaction costs. The figure demonstrates that total deadweight costs increases as

the dilution parameter π increases further beyond the level of π∗ = 1.125. This implication

does not support the opinion expressed in Strongin, Hindlian, and Lawson (2009), Pitt,

Hindlian, Lawson, and Himmelberg (2011), and Calomiris and Herring (2013) implying that

highly dilutive terms are more wellfare-enhancing than less dilutive ones. In contrast, for

the current calibration, the model suggest that very high dilution terms can, in fact, lead

to a lesser efficiency, given a more realistic setting where the bank can manage dynamically

the trade-offs of all deadweight costs and tax benefits of debt.

3.1.4 At What Conditions Will the Bank issue CoCo bonds Voluntarily?

In this section, we explore whether the bank will be willing to issue a new CoCo bond

midstream when it already has the pre-existing senior debt in its capital structure. The

managers will be willing to undergo the issuance if the equity stock will react positively.

There are two factors that will affect the changes in equity value of the bank at the instant

of the CoCo issuance. On one hand, if the conversion terms dilute shareholders, the value

of committing to raising capital in the future could flow mostly to pre-existing senior debt

26

rather than equity.30 As such, the CoCo tranche will create value to holders of existing

bonds at the shareholders’ expense. On the other hand, the tranche of CoCo will create

incremental tax shields that would benefit existence shareholders of the bank. As we will

show, this trade-off between potential wealth transfer and additional tax shields depends

on whether after the issuance of the CoCo, the bank will emerge below, near or above its

optimal leverage.

To assess this trade-off, we calculate the change in market value of equity and the change

in the value of the preexisting senior debt at the instant of CoCo issuance, assuming that the

issuance is unexpected. The calculation of the changes in equity assumes that the proceeds

of the newly issued CoCo are instantly paid out to equityholders. We first consider the bank

that initially operates with the senior debt which is priced without the CoCo tranche. We

consider three cases where the bank’s initial leverage is low at 80%, optimal at 84.5%, and

above optimal at 88%; and after the CoCo is issued, the bank will emerge below optimal,

near the optimal, and above optimal leverage respectively. We assume that the bank issues

a CoCo bond (5% of the total debt) with the same maturity (6 years) as the existing senior

debt, and vary its conversion terms. Figure 5 demonstrates that the impact on the changes

in equity value and debt value is different form case to case. First, an increase in conversion

terms of the newly issued CoCo (π > 1.0) will lead to smaller benefits (or larger losses)

to equity holders and a larger wealth transfer to senior debt for all cases. In comparison,

with a par-to-par conversion terms (π = 1.0), there will be no value transfer because the

equityholders will not commit to de-lever in the future bad state of the world.

If the bank’s initial leverage is below optimal (leverage ratio is 80%), the newly issued

CoCo will produce positive equity returns due to increased future tax shields, while the

30Albul, Jaffee, and Tchistyi, (2010) analyze a similar case when the CoCo bond is issued on top on thepre-existing straight debt. However, they don’t consider the case when the bank can recapitalize.

27

wealth transfer to debtholders will be minimal. In contrast, if the dilutive CoCos are issued

when the bank is already overleveraged, the newly created incentives will immediately be-

come highly effective (“in the money”) but their benefits will be mostly flowing to debtholders

while the existing shareholders will be experiencing an instant stock price loss. As such, dur-

ing bad times, the bank will not be willing to voluntary add a CoCo with punitive conversion

terms. For example, when the bank is initially at 88% (above optimal leverage), and the

conversion ratios of the newly-issued CoCos are 5% above par (π = 1.05), then upon CoCo

issuance, senior debtholders will experience a 1.14% gain, but the shareholders will lose as

much as 1.0% of their stock market value. This result confirms the intuition that at lower

capital, the bank’s commitment to recapitalization may benefit its pre-existing holders of

senior debt rather than holders of equity.

As a policy implication this result suggests that regulators do not necessarily need to

mandate the issuance of CoCos that carry dilutive terms for well-capitalized banks because

the issuance would be in the interests of stock holders to do so. On the other hand, if banks

are undercapitalized, the regulator might be needed to mandate the issuance of CoCos with

dilutive conversion rates.

3.1.5 Optimal Combination of the trigger and conversion ratio

Figure 6.1 show the optimal initial leverage ratios as a function of conversion ratio for three

different triggers, θ, of 5%, 6% (base case) and 7% at which the CoCo converts. With

the higher trigger, there will be a higher likelihood of breaching the trigger, so that if the

bank wants to keep its capital ratio above a higher trigger, the bank has to start with a

higher levels of capital or commit to higher future equity issuance. Holding conversion ratio

constant and a trigger set at θ = 7%, the bank will optimally start with higher capital, but it

will raise less equity when compared to the case with lower trigger (θ = 5%). What is more

28

interesting is that for relatively low conversion ratios (π <1.15), higher-set trigger will result

in higher expected default costs. This prediction is a result of complex trade-offs including

the need to shield taxes and to mitigate potential dilution and reduce financial distress costs.

First, with less capital being raised for a higher trigger, it can lead to higher expected default

costs. Second, with a higher-set trigger, the relative dilution for the preexisting shareholders

is smaller when CoCo is converted, which in turn, would imply weaker incentives to raise

capital. This result contrasts with the predictions derived from a static argument presented

in the prior literature that higher triggers will create stronger commitment to preemptively

raise equity capital and reduce default costs when compared to lower triggers.31 In general,

the bank will always prefer the CoCo design with lower trigger, because with lower trigger,

the bank retains more flexibility to manage its capital structure and reduce more effectively

its taxable income.32

Figure 6.2 shows the optimal conversion ratio for 5 different trigger ratios varied between

5% and 7%. For a given trigger ratio, the bank chooses the conversion ratio and the optimal

initial leverage ratio taking into account subsequent recapitalization decisions, such that

the market value of equity plus debt is maximized.33 As the trigger increases, the optimal

conversion ratio π∗ increases too, and the bank will respond by reducing its initial leverage,

and will raise less equity in the future. Expected default costs will decrease too. This

comparative statics shows a complex nature of trade-offs that the bank faces. This analysis

also suggests that the approach of “one size fit all” does not work in designing an optimal

CoCo contract. Instead, the optimal contract terms (conversion ratio and trigger) should be

designed based on a comprehensive analysis of bank characteristics and other incentives of

31Interestingly, the result changes for the case with no taxes. At the higher trigger, the commitment topreemptively issue equity and avoid conversion will always be stronger even for less dilutive conversion ratios.32Unreported calculations show that the bank’s market value generally increases with lower trigger.33The numerical algorithm searches for the optimal combination of conversion ratio and initial leverage

ratio at which the market value of equity and total debt is maximized.

29

debt, and not only the incentives to avoid dilutive conversion.

3.1.6 Optimal Combination of the CoCo Size and Conversion Ratio

Figure 7.1 depicts the optimal initial leverage ratio and other model-generated values for

three different sizes of CoCo bond, ranging from 4% to 6%, where the size is measured as

a percentage of total debt, CC+F

. For a larger size CoCo tranche (6%) and relatively low

dilution terms (π <1.075), the bank will optimally start with slightly higher leverage, and

will raise less equity capital in the future leading to higher (not lower) expected defaults costs

when compared to the case with a smaller tranche size (4%). This prediction contradicts

a general argument presented in the earlier literature that larger CoCo are always more

efficient in reducing future default costs.

Figure 7.2 depicts the optimal conversion ratio for five different CoCo sizes varied from

5% to 7% of total debt, given that the bank starts with the optimal initial capital structure

and employs optimal de-leveraging strategy.34 The model implies that for the smaller size

CoCos, the optimal dilution parameter (π∗) has to be higher and the initial leverage is lower

compared to the case with larger size CoCo. For example, given the size of CoCo at 4%, the

optimal conversion term π∗ =1.175. As the size of the CoCo increases to 6%, the optimal

conversion ratio declines to π∗ =1.1.

3.1.7 Optimal Conversion Ratio and Asset Risk Parameters

Table 9.1 presents comparative statics for parameters of the model that describe the risk

of bank assets including drift volatility and probability of negative jumps. The parameter

values are varied around base case reported in Table 2. For almost all dilutive terms, a bank

with higher volatility of the drift (σ) will optimally select lower initial leverage and will issue

more equity in the future. In order to design an optimal CoCo contract for the bank with

34In this comparative statics, the trigger remains at 6% capital as in the base case.

30

more volatile assets, one needs to make conversion more dilutive. For example as depicted

in Figure 9.2, the bank with the volatility of σ = 7% should take on the CoCo with the

optimal dilution parameter of π∗ = 1.17. In comparison, for the banks with lower volatility

of σ = 3%, the optimal conversion rate is π∗ = 1.08. The reason behind these results

is that higher volatility increases the equity option value of debt overhang in the post-

conversion, producing weaker incentives to preempt triggering. Thus, in order to strengthen

the incentives, the CoCo should be more dilutive. As a policy suggestion: if regulators can

mandate CoCo terms, conversion rates should be more dilutive for banks that hold riskier

assets.

The effect of jumps on the bank’s incentives is more complicated. Jumps introduce

skewness in the asset return distribution and a “fat” tail for negative returns. The higher

jump probability (i.e., higher value for parameter λ), the higher will be the probability that

the trigger is breached and the probability that default occurs.35 As reported in Table 8.1,

holding the conversion term constant, when jumps are more frequent, the bank will be less

committed to issue equity and recapitalize.36 This is because with higher jump risk, the

equity issuance strategy has a diminishing marginal impact on the bank’ ability to reduce

risk of triggering. In contrast, with less frequent jumps, the bank can avoid triggering with

higher degree of certainty when it raises capital. These results suggest that in the economic

environment in which changes are driven by sudden negative changes (i.e., financial crisis),

the effectiveness of CoCo as a self-disciplining mechanism will weaken, and an increase in

conversion ratio will have a diminished impact on the bank’s incentives to manage its capital

structure.

35There is a countervailing factor in play here: with higher negative jumps, the risk neutral drift is adjustedupward increasing the upside potential for the equity.36For example, given the conversion rate of 1.1 and the probability of jumps of 2.25% per year, the expected

equity issuance rate reaches the plateau at the level of 5.62%. In comparison, with lower jump probabilityof 1.75%, equity issuance rate plateaus at the level of 6.67%.

31

The above argument explains why the optimal conversion ratio is lower for the bank that

holds assets subject to more frequent jumps, and why its management will be less incentivized

to raise capital. For such banks, an increase in conversion ratio will be counterproductive as

it will exert unnecessary frequent recapitalizations and excessive transaction costs without

an effective impact on incentives. Figure 8.2 confirms this argument: the optimal conversion

for asset with jump probability of 2% is π∗ =1.15, compared to π∗ =1.11 for the case with

jump probability of 5%.

As pointed out in this section, both types of asset risk dampen incentives because they

reduce the ability of management to reduce conversion risk by holding excess capital. This

result further challenges the “static” argument predicting that a bank with riskier assets

will be more willing to issue new stock to preempt conversion than the bank with less risky

assets.37

3.1.8 Optimal Conversion Ratio and Other Parameters

We also varied parameter m which is the annual roll-over rate of the senior debt. The higher

parameterm is the shorter will be the effective maturity of the senior liabilities. Comparative

statics presented in Figure 10.1 demonstrate that for a given conversion rate, the optimal

leverage and incentives to raise preemptive capital are higher for higher roll-over parameter,

which leads to lower default costs. This is because higher roll-over rate will reduce the

“debt-overhang” problem due to the fact that the larger fraction of debt is being repaid and

reissued at market value. As to the optimal contract: Figure 10.2 illustrates that when the

senior debt is rolled-over more frequently, the bank can optimally operate with higher debt

and take advantage of higher tax shields more effectively without exposing bank to higher

default risks. The conversion ratio can be less dilutive when compared to the case where

37See Calomiris and Herring (2011).

32

the debt is rolled-over rate less frequently. For example, if the bank roles over 10% of its

senior debt annually (m = 10%), the optimal contract should have the conversion ratio of

π∗ = 1.16. In contrast, as the rollover rate increases to m = 30%, the optimal conversion

ratio declines to π∗ = 1.1.

The affect of transaction costs of raising equity (parameter e1 and e2) on the incentives

and capital structure strategy is as predicted. Table 11.1 shows that with larger transaction

costs, management will be less willing to issue equity leading to higher default costs. Also,

the bank will be selecting initially lower leverage. This prediction reiterates the point made

in the previous paragraph that market frictions will reduce the ex-ante commitment of a bank

to manage its capital ratios. To compensate for weaker incentives, the one needs to design

the contract with more aggressive dilution terms. As the model predicts, when transaction

costs increase from e1=e2 = 0.3% to e1=e2 =0.5%, the optimal conversion ratio increases

from π∗ = 1.075 to 1.2. See Figure 11.2.

3.2 Trigger uncertainty and covenant considerations

In the following sections we will consider a number of contractual frictions that may affect the

bank’s capital structure strategy, incentives, and the optimal CoCo design. In the baseline

model, we assume a “perfect enforcement” of the contract, meaning that the conversion

triggers at the first instant when the conversion ratio it tripped (either through a drift or a

jump). In reality, there can be a failure in the trigger mechanism. We will discuss two types

of trigger failure. The first is the “type I” error that the CoCo fails to trigger even if the

bank capital ratio breaches the contractually specified threshold. The second is the “type

II” error that the CoCo prematurely triggers even though the bank’s capital is still above

the threshold.

33

3.2.1 Delayed triggering weakens ex-ante incentives

There is a number of market frictions that allow the bank to temporarily operate below the

trigger without conversion. For example, if the regulatory oversight is poor, they could call

the trigger with a delay. Alternatively, if regulators are too worried about creating a panic

or/and “run on the bank”, they would be reluctant to call the trigger in a timely manner

and may wait until the bank becomes deeply in distress.

In order to quantify the effect of this friction, we modify the model by introducing the

so-called “delayed trigger”. We assume that given that the bank’s capital ratio is below the

trigger, the conversion event can happen with some probability described by a Poisson-arrival

process with the arrival intensity μ per year. Specifically, if the bank’s ratio declines and

remains below the trigger during any short time interval between t and t+ dt ( dt→ 0), the

probability that the conversion occurs during this period is μ · dt, and the probability that

the trigger does not occur is 1 − μ · dt. In this setting, the bank’s capital ratio can breach

the trigger multiple times and stay below the trigger for some duration of time without

conversion. With the assumption of the Poisson-arrival process, the probability that the

conversion occurs next period does not depend on the duration of time the bank’s capital

ratio have already stayed below the trigger in the prior periods.38 The average length of

time the bank stays below the trigger ratio without converting is 1μ, and the probability that

the conversion happens within one year period is 1 − e−μ. We assume that markets fully

anticipates a possibility of delaying the trigger, and will factor in this possibility into pricing

of debt, equity, CoCos and incentives. Note, in the base case model, the conversion arrival

intensity is μ =∞, meaning no delays.

Figure 12.1 shows that if the probability (the arrival intensity) of the triggering declines,

38The arrival of the trigger event is independent from theWeiner process WV in the asset value returns.

34

the managerial incentive to issue equity preemptively weakens for all conversion ratios. For

example, if the annualized probability of conversion is 50% (i.e., significant delay), manage-

ment is expected to issue equity only if the conversion rate is above 1.19. In comparison, for

the base case (no delays), the bank will start issuing equity for conversion rate of 1.02 and

above. Holding conversion rate constant, the optimal initial leverage is lower and default

costs are higher when delays are longer, i.e., when the conversion probability is lower. Weaker

incentives are explained by the fact that with the “delayed trigger”, the likelihood of dilution

will be lower, and the longer the bank stays below the trigger ratio without activating the

conversion, the lesser will be the need for preemptive recapitalizations.

In order to design the optimal contract in the environment with delayed trigger, dilution

rates have to be higher to compensate for weaker incentives. As Figure 12.2 demonstrates,

when the annualized probability of conversion is 50% (i.e., significant delay), the optimal

conversion rate is π∗ = 1.25. In contrast, for the case with no delays (base case), the optimal

dilution parameter is π∗ = 1.125. Overall, these insights imply that a weaker enforcement of

trigger mechanism diminishes positive effects of CoCo on bank incentives, thereby suggesting

the value of enhanced bank disclosure and greater transparency.

3.2.2 Premature triggering weakens ex-ante incentives

There are several reasons for the “type II” trigger error, i.e., the situation when the conver-

sion is activated even though the bank’s capital ratio is still above the pre-defined trigger.

When banking regulators have a discretion to trigger the conversion, there could be a prema-

ture intervention if the regulator becomes overly risk-averse. Another reason for premature

triggering could be due to noisy market fluctuations or market manipulations.

In order to quantify the effect of premature triggering, we modify the model by assuming

that— in addition to the “hard trigger” at capital ratio of θ = 6%, the CoCo conversion

35

can be also triggered by regulators (or due to market manipulation) at capital ratios above

θ. It is plausible to assume that, everything being equal, regulators will more likely trigger

prematurely if a bank has low capital (but still above the pre-defined threshold). As such,

we assume that at any period t, the probability of premature triggering is negatively related

to the distance between bank’s capital ratio at time t (Vt−F−CVt

) and the prespecified trigger

ratio θ. In the model, the annual probability of the premature triggering is described by

exponential probability function e(θ−Vt−F−C

Vt)∗ϕ, where ϕ is a positive constant, given that the

capital ratio is still above the trigger, i.e., Vt−F−CVt

> θ. 39

Figure 13.1 depicts the analysis for thee cases: two cases with more frequent and less

frequent premature triggering described by parameter ϕ=20 and ϕ=60 respectively, as well

as the base case which has no premature triggering (base case). Panel 1 of Figure 13.1 depicts

the annual probability of triggering as a function of bank capital. The model predicts that

the bank that faces higher odds of premature trigger will be less incentivized to raise capital

preemptively. For example, for the case with higher premature triggering (ϕ=20), the bank

is not raising equity for any conversion ratios π less than 1.175. With no possibility of

premature triggering, the bank is expected to raise equity for dilution ratio of 1.03. This is

because the risk of a premature conversion weakens shareholders’ ability to control the risk

of dilution.

As depicted in Figure 13.2, in order to design the optimal CoCo contract in the economy

where premature triggering is more likely, the conversion terms should be more dilutive.

With such risk, the bank will optimally respond by selecting higher capital initially. Despite

the risk of higher dilution, the bank is expected to raise less equity, which will lead to more

frequent defaults. This prediction resonates with the argument offered in the previous section

39Similarly to the setting in the previous section, the value of (cap.ratio) ∗ϕ is a Poisson arrival intensityper year. The arrival of triggering event is independent from the random term WV .

36

that with more uncertainty (more noise) in the trigger mechanism, the less valuable (from

shareholders’ standpoint) will be the strategy of preemptive equity issuances. From the

regulatory perspective, this prediction also re-emphasizes the design of trigger mechanisms

that reduce such uncertainty.

3.3 Accretive Conversion Terms and Incentives to “Burn Money”

3.3.1 The effect of accretive conversion terms on incentives to burn money

The bank will have incentive to deliberately burn money (burn assets or destroy capital)

in order to breach the trigger only if at conversion, the bond’s principal will be (fully or

partially) written down. When the bank is overleveraged and its capital ratio is close to the

trigger, it can deliberately destroy part of its capital in order to accelerate breaching the

trigger, write-down the bond’s principal, and thus capture a windfall gain for equity holders

at the loss to the CoCo holders.

The incentive to “burn money” will depend on the remaining maturity of the bond

and the level of bank capital. Theoretically, at the instant the bond comes due, the bank

may be willing to burn as much capital as the amount of the write-down itself, conditional

that the “burnt” amount will be enough to push its capital ratio below the trigger. For

longer maturity, these incentives are more complex. When the remaining maturity is long,

bank managers may find it optimal to wait, because burning money eliminates a valuable

optionality that bank assets can increase in value and generate positive gains for the bank’s

shareholders.

Figure 14 presents the optimal initial leverage ratios for accretive conversion terms. Con-

version ratios vary from π =0 (total loss for CoCo holders) to π =1 (a par-for-par conversion).

Panel 2 show that the expected money burning, (negative of equity issuance) increases as the

conversion rates decline from 1 to 0, i.e., becomes more accretive to equity. The spread of

37

the CoCo also increases significantly from 41b.p. to 409 b.p. Panel 4 shows total deadweight

costs calculated as expected default costs plus expected money burning increases as well.

Interestingly, as the conversion term become more accretive (i.e., a larger write-down), the

bank will actually be selecting slightly lower leverage initially.

Figure 15 shows that, one day before the CoCo comes due, the bank is expected to

burn as much as 4.3% of its assets to trigger the conversion. The intuition of this result is

straightforward: An instant before maturity, the shareholders face two choices: 1) pay a full

amount of principal, or 2) burn 4.3% of its assets and reduce its capital ratio from 10.3% to

6% which will breach the trigger and capture gains coming from writing-off the principal.

The 100% write-down of principal generates gains to the shareholders valued at 4.75% of

the banks assets.40 These gains exceed the amount of capital that the bank needs to burn.

Figure 11 demonstrates that for a short maturity case, a two percentage point “spike” in

expected deadweight costs reflects the expected amount that will be “burnt”.

3.3.2 Bank’s capital ratio remaining maturity and money burning

As we pointed earlier, when the remaining maturity of the CoCo is short, the ”burn money”

incentives are likely to be in the money. To quantify the maturity effect, we use the example

of the contingent convertible bond issued in 2013 by Barclays Bank which can be written-

off entirely upon triggering. Footnote 1 provides contractual details of the bond. A 100%

write-down of principal implies that $1 of par value of CoCo will convert to 0 cents of equity,

and parameter π = 0.0. The remaining parameter values are at the base case level. Figure

15 plots the relation between bank’s capital ratio and the expected value of burnt money for

three (remaining) maturity cases: one day, a half year, and one year. In these tests, we vary

the bank’s capital ratio and keep the size of CoCo and of the senior debt constant at the base

40These gains are calculated as the product of 75% · 5% = 3.75%, given that CoCo’s principal is 5% ofassets.

38

case values. The expected amount of “money burning ” exhibits an inverse U-shaped relation

with the bank capital ratio. Specifically, the expected size of burnt money will increase for

the capital ratio for the range between 4% and around 10% and declines for capital ratio

of around 10% and larger. The explanation of this non-monotonic relation is the following:

When the banks capital ratio declines to a level very close to the conversion trigger, there

is a high probability that the trigger will be breached during the remaining maturity simply

because value of bank assets can decline even more. In such cases, the bank has a lesser

need to burn money to accelerate triggering, or if needed, it does not have to burn a large

amount. Conversely, if the bank operates at a capital ratio that is significantly higher than

the trigger of 6%, the option to burn money will be farther out of the money, because it is

unlikely that the bank’s capital ratio will drift close to the conversion trigger. In such cases,

in order to breach the trigger, the bank will need to burn the amount that exceeds the value

of expected gains. Therefore, at the capital ratio well above the conversion trigger, the bank

is not expected to burn much money. In the middle range, at the capital ratio around 10%,

the banks is expected to burn the largest amount of money. The expected deadweight costs

exhibits a similar inverse U-shaped relation with the capital ratio.

The model implies that the option to burn money will be deeper “in-the-money” if the

remaining maturity is short. One day before maturity, the bank will be willing to burn up

to 4.7% of its capital (i.e., reduce its capital from up to 10.7% down to 6% trigger) in order

to breach the trigger and produce gains from a 100% write-down of CoCos. The intuition

of this result is straightforward: A day before maturity, the shareholders face two choices:

1) pay a full amount of principal, or 2) burn 4.7% of its assets and reduce its capital ratio

from 10.7% to 6% which will breach the trigger and capture positive net gains coming from

writing-off the principal. 41 These gains slightly exceed the amount of capital that the bank

41These gains are calculated as the product of 75% · 5% = 3.75%, given that CoCo’s principal is 5% of

39

needs to burn. Figure 15 demonstrates that for a short maturity case, a 4.7 percentage point

“spike” in expected deadweight costs reflects the expected amount that will be “burnt”. At

capital ratios above 10.7%, the bank does not burn assets because the amount it needs to

burn will exceed the gains from the principal write-down.

3.3.3 Discussion of the write-down provision of the CoCo contract

As the model implies, the option to “burn money” is likely be in-the-money in times of crisis,

when the bank has low capital. This write-down feature is thus highly undesirable, because

it is exactly during the crisis that is important to have incentive structures in place that

discourage banks to exercise value-destroying strategies. The possibility that the trigger will

write-down the bond’s principal can also exacerbate other inefficiencies and value-reducing

incentives associated with a typical debt overhang problem. For example, the incentive

to accelerate the trigger will encourage the bank to increase risk even more, especially its

downside risk. Also, the bank will be more incentivized to underinvest in positive NPV

projects. These value-reducing incentives created by the write-down feature will be even

worse than those generated by ordinary senior debt. Finally, due to the impact of money

burning incentives, the bank’s stock price and the price of CoCo bond will likely experience

higher uncertainty near maturity date. This price uncertainty can create liquidity problems

for the bank because near maturity, it will need to raise funds to pay off its maturing debt.

4 Conclusion

Contingent capital will likely play a role in the future regulatory landscape for banks and

financial firms as it designed to help absorb losses during a financial crisis. This paper

demonstrates that CoCos have consequences that go beyond mere loss absorbancy. We show

assets.

40

that even a small tranche of CoCos can generate sizable incentives for banks to choose and

manage capital structure, and these incentives can either create or destroy value, depending

on their design.

Our main finding is that CoCos that carry dilutive conversion terms for the pre-existing

shareholders will commit banks to raising equity capital preemptively during bad times, but

incentivize banks to operate with higher leverages during normal times. These “incentive

dividend” can materially increase valuation, reduce the costs of bank debt, and reduce the

social welfare costs of financial distress. However, optimal contract design implies that

conversion terms should not be too dilutive as to constrain bank’s activity, but at the same

time, contract terms should be punitive enough to promote a more prudent behavior.

The model also predicts that “trigger uncertainty” weakens the bank’s incentive to pur-

sue recapitalization during bad times. Trigger uncertainty can arise either because triggering

is delayed (e.g., inattentive or poorly incentivized regulators), or because triggering is pre-

mature (e.g., because of premature regulatory intervention or strategic market manipulation

by speculators). Either way, we show that trigger uncertainty weakens the bank’s incen-

tives because it weakens the management’s ability to control dilution risk. To compensate

for weaker incentives in the environment with “trigger uncertainty”, an optimally designed

contract has to include more punitive dilutions.

The model shows that — in contrast to the positive incentives created by dilutive con-

version terms — conversion terms that write down principal for bondholders create perverse

incentives for the banks to pursue higher leverage and “money burning.” When the bank

becomes overleveraged and its capital ratio is near the trigger, the bank will find it optimal

to burn capital to push its capital ratio below the trigger. By burning assets, the bank can

breach the trigger, write-down CoCo’s principal and guarantee a windfall gain for share-

holders. So far financial regulations (including Basel-III) do not appear to recognize that

41

this loss-absorbing feature can have value-destroying incentive effects on bank behavior. Our

results underscore the urgency of expanding the policy discussion to emphasize not just “loss

absorbancy” but also incentives.

The main implication of the CoCo design is that “one size fit all” approach does not work.

Instead, conversion terms should be adjusted to reflect banks’ individual characteristics, and

take into account countervailing incentives, an initial capital structure choice and subsequent

recapitalization decisions along with various costs including expected taxes, default costs,

transaction costs and dilution costs.

Substantial room for future research remains. We restricted our attention to common

equity conversions, but the model can easily be extended to analyze conversion to preferred

stock. There are at least two reasons such conversion design might be desirable. First, when

market triggers are used and the stock price is near the trigger, the stock price is vulnerable

to a “death spiral” caused by market manipulation. Converting to preferred stock could

mitigate this risk. Second, fixed income investors may be reluctant to own CoCos because

their mandate prohibits them from owning common equity. Converting to more debt-like

securities in the capital structure might help to address this concern, too.

42

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45

A Appendix: Analytics of the model

A.1 Valuation of the bank’s equity

In this section, we present valuation of the bank’s equity for the case before E(V, F +C, t),

and after the conversion E(V, F, t). The value of equity is a function of its asset value V ; its

principal F + C, if no conversion took place in prior periods, or F , if the CoCo bond was

converted before; and time t (≤ T ). The CoCo bond is described by parameters π, θ and

the size of CoCo bond C. First, we describe the valuation of equity for the case in which

the conversion already have taken place in prior periods. For this case, the bank continues

paying interest of its senior debt f and its par value F at maturity. At maturity date T , the

value of the bank’s equity is

E(V, F, T ) = max(0, V − F ). (4)

Any time prior to maturity, using standard arbitrage arguments, the value of the eq-

uity E(V, F, t) is given by the solution to the following partial integro-differential equation

(PIDE):

(5)

where subscripts t and V denote partial derivatives and EQ is the expectation operator under

the risk neutral measure Q. In this equation, αV is the after-tax cash flow generated by the

bank’s assets and αV − (1− τ) · f −m · (F −D) is the dividend payout to the shareholders;

and the last term represents the expected change in equity value due to jumps. Note that

after the CoCo bond converts, the bank will not elect to raise equity capital because such

transaction would dilute the shareholders’ value and will benefit holders of the senior debt

at the shareholders’ expense. Also, without CoCo bond the bank will not choose to burn its

assets.

46

Now, consider the valuation of the bank’s equity for the case if the CoCo was not con-

verted in the prior periods. At maturity t = T :⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩E(V, F + C, T ) = V − C − F, V−F−C

V> θ, no conversion,

E(V, F + C, T ) = max{0, (V − F − (π) · C}, if V−F−CV

< θ and V ≥ F,

E(V, F + C, T ) = 0, otherwise.

Before maturity, t < T, the bank makes its equity issuance as well as default choices. These

choices maximize the market value of the bank’s equity, which is the present value of net cash

flow to the equityholders. The solution involves determining the free boundary conditions

that divide the state space (V, {F + C or F}, t) into the three regions that characterize the

bank’s choices: the no equity issuance/no money burning region, the equity issuance region,

the money burning region, the conversion region, and the default region.42

In the no equity issuance/ no money burning region, it is not optimal for the bank to

issue equity capital (or burn money). In this region, the equity value E(V, F + C, t) equals

the instantaneous cash flow net of coupon payment plus the expected value of the equity at

time t+∆t calculated under the risk neutral measure Q:

E(V, F+C, t) = [αV−(1−τ)·(f+c)−m·(F−D)]dt+e−rdtEQ{E(Vt+dt, F+C, t+dt)}, t+dt ≤ T.

(6)

In this region, the equityholders will not choose to issue equity (or burn money) and the

following inequalities must hold for any ∆V :

42For brevity, we omit the discussion of the technical detail of boundary and “high contact” conditionsthat applied to the value function E. For details see Oksendal and Sulem (2007).

47

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

[αV − (1− τ) · (f + c)−m · (F −D)]dt+ e−rdtEQ(E(Vt+dt +4V, F + C, t+ dt) )−4V − TC <

[αV − (1− τ) · (f + c)−m · (F −D)]dt+ e−rdtEQ(E(Vt+dt, F + C, t+ dt)),

4V > 0 , i.e., equity issuance is not profitable.

E(V −4V, F, t)− (π) · C < [αV − (1− τ) · (f + c)−m · (F −D)]dt+

e−rdtEQ(E(Vt+dt, F + C, t+ dt))

s.t., V−∆V−F−CV−∆V < θ, 4V > 0 , i.e., ”money burning” at the amount of 4V

that leads to an immediate conversion is not profitable.

(7)

where TC are transaction costs of raising equity, TC = e1 · Vt + e2 · 4V. The value of the

equity E(V, F + C, t) in the no equity issuance/no money burning region is given by the

solution to the following PIDE:

σ2V 2

2EV V + (r − α− λ · k)EV +Et + (f + c)− rE−

λ · EQ{E(Y · V, F, t)−E(V, F + C, t)} = 0, for any V−F−CV

> θ, E > 0.43(8)

In the equity issuance region, the value of the bank’s equity E(V, F + C, t), t < T, can be

determined by maximizing the expected value of equity, over all equity issuance policies

4V :

E(V, F + C, t) = [αV − (1− τ) · (f + c)−m · (F −D)]d

(9)

e−rdtEQ(max4V >0

[E(Vt+dt +4V, F + C, t+ dt) −4V − TC]), for t < T. (10)

The bank will raise new capital if the net benefit of raising capital exceeds the transaction

costs TC. This condition characterizes a region where equityholders raise equity and

increase the size of the bank capital. In the money burning region, there has to be some

48

non-zero amount of assets 4V ∗ > 0 to be burn that satisfies the following condition:

E(V−4V ∗, F, t)− (π)·C > E(V, F+C, t), s.t., V −∆V ∗ − F − CV −∆V ∗

= θ, 4V ∗ > 0 , for t < T.44

(11)

In this region, the equity value is set at: E(V, F +C, t) = E(V −4V ∗, F, t)− (π) ·C. In the

CoCo conversion region, the banks capital ratio is below the trigger θ, and the following

condition is held V−F−CV

≤ θ. In this region, the CoCo converts to equity at the dollar

amount of shares (π) ·C. In this region, the equity value for the existing stockholders is set:

E(V, F + C, t) = E(V, F, t)− (π) · C, if V > F, and E(V, F, t)− (π) · C > 0. (12)

where (π) · C is the equity value that the CoCo holders take over after the conversion. The

remaining fraction of equity value E(V, F, t)− (π) · C belongs to pre-existing shareholders

of the bank. The first inequality reflects the condition that the CoCo conversion can take

place only if the bank’s assets are larger than the size of its debt. As we pointed out

earlier, a large negative jump can instantly reduce the value of bank’s assets to some level

V 0 well below the conversion trigger so that V0−F−CV 0 < θ, but above the default level

V 0 > F . If the conversion ratio π is high, the conversion ratio is reduced from π to π0(>0)

such that the value of the existing shareholders is wiped out, i.e., E(V 0, F + C, t) =0, and

the bank is taken over by the holders of contingent capital, where π0 satisfies

E(V 0, F, t) = (π0) · C. In this region, the value of the CoCo holders will be E(V 0, F, t). In

default region, the equity value E = 0, if

E(V, F, t) = [αV −(1−τ) ·f−m ·(F−D)]dt+e−rdtEQ{E(Vt+dt, F, t+dt)} < 0, t < T. (13)

49

A.2 Valuation of the senior debt

To calculate the value of the senior debt, we need to consider the bank’s default strategy

and the shareholders’ optimal strategy to raise capital. At maturity date T , the value of the

bank’s debt is⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

D(V, F, T ) = F, if V ≥ F and D(V, F, T ) = (1−DC) · V, otherwise,

CoCo bond was converted in prior periods.

D(V, F + C, T ) = F, if V ≥ F and D(V, F + C, T ) = (1−DC) · V, otherwise,

no prior CoCo conversion,

(14)

where DC represents proportional default costs (0 ≤ DC ≤ 1). Any time prior to maturity

t < T , the value of debt satisfies:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

σ2V 2

2DV V + (r − α− λ · k)DV +Dt + f +m · (F −D)− rD−

λ · EQt {D(Y · V, F + C, t)−D(V, F + C, t)} = 0,

if E(V, F + C, t) ≥ 0, no prior CoCo conversion.σ2V 2

2DV V + (r − α− λ · k)DV +Dt + f +m · (F −D)− rD−

λ · EQt {D(Y · V, F, t)−D(V, F, t)} = 0,

if E(V, F, t) ≥ 0, the CoCo converted in prior periods.

(15)

In the conversion region, the value of the senior debt is

D(V, F + C, t) = D(V, F, t), ifV − F − C

V≤ θ. (16)

In the region where the bank issues equity and increases its capital by 4V , the value of

senior debt satisfies:

D(V, F + C, t+) = D(V +4V, F + C, t). (17)

50

In the region where the bank ”burns” assets by the amount of4V ∗ and forces the conversion,

the value of senior debt satisfies:

D(V, F + C, t+) = D(V −4V ∗, F, t), s.t.,V −4V ∗ − F − C

V −4V ∗ = θ. (18)

In default region, the assets are transferred to the holders of the senior debt:

D(V, F, t) = (1−DC) · V, if E(V, F, t) = 0. (19)

A.3 Valuation of the CoCo Bond

Because of lower priority, the CoCo holders recover zero value at default. To calculate the

value of CoCo debt, we need to consider the bank’s default strategy and strategy to raise

capital as well the possibility of conversion. At the debt maturity date T , if conversion did

not take place in prior periods, the value of the CoCo bond, Z(V, F + C, T ) is⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩Z(V, F + C, T ) = C, V−F−C

V> θ,

Z(V, F + C, T ) = min{(V − F ), (π) · C}, if V−F−CV

< θ and V ≥ F,

Z(V, F + C, T ) = 0, otherwise.

(20)

Any time prior to maturity t < T, the value of CoCo bond satisfies:

σ2V 2

2ZV V + (r − α− λ · k)ZV + Zt + c− rZ − λ · EQt {Z(Yt · V, F + C, t)− Z(V, F + C, t)} = 0,

if E(V, F + C, t) ≥ 0.(21)

At the conversion boundary, the value of the CoCo bond is

Z(V, F + C, t) = min{E(V, F, t), (π) · C} ifV − F − C

V≤ θ. (22)

For V 0, s.t., V0−F−CV 0 < θ and V 0 > F, existing equity investors are wiped out, and the residual

value is allocated to CoCo investors. The bank remains operational and is taken over by the

51

holders of the CoCo bond. For such values of assets, the value of the CoCo holders will be

Z(V 0, F + C, t) = E(V 0, F, t). In the region where the bank burns money by the amount of

4V 0 > 0 and forces the conversion, the value of CoCo debt satisfies:

Z(V, F + C, t+) = Z(V −4V 0, F + C, t+) = (π) · C, s.t.,V −4V 0 − F − C

V −4V 0 = θ. (23)

In the region where the bank issues equity and increases its capital by 4V at t, the value of

CoCo bond satisfies:

Z(V, F + C, t+) = Z(V +4V, F + C, t). (24)

In the default region:

Z(V, F + C, t+) = 0. (25)

52

Panel 1: Pre-Crisis Period (2005: Q1-2007: Q2)

All banks GSIB only

Variable mean sd N mean sd NAsset Growth, % 2.5% 7.2% 602 2.8% 3.9% 45

Ratio of Tier 1 capital to total (unweighted) assets. 7.7% 1.4% 602 6.2% 0.5% 45

Ratio of Tier 1 capital to risk-weighted assets. 10.4% 2.3% 602 9.2% 1.4% 45

Standard deviation of (net current operating earnings/TA) 0.4% 0.5% 602 0.3% 0.2% 45

Table 1: Summary Statistics for the top 50 largest publicly traded U.S. bank holding companies (BHCs) and globally systemically important banks (GSIBs)

Table presents a summary statistics of quarterly financial data for the top 50 largest publicly traded U.S. bank holding companies (BHCs) between 2005 and 2014. The table also consider separately the subsample of BHCs that are globally systemically important banks (GSIBs), with more than $100 billion in assets. There are 8 institutions that are GSIBS in our data set. Tables report summary statistics separately for the Pre-Crisis Period (2005: Q1-2007: Q2) and the Post-Crisis Period (2010: Q1-2014: Q4). All statistics is winsorized at the 5% level.

Market to book ratio. 1.13 23.0% 602 108.6% 4.3% 45

Ratio of insured deposits to unsubordinated debt liabilities. 0.364 0.201 602 0.18 0.149 45

Panel 2: Post-Crisis Period (2010: Q1-2014: Q4)

All banks GSIB only

Variable mean sd N mean sd NAsset Growth, % 2.7% 12.4% 1,090     2.9% 13.6% 135

Ratio of Tier 1 capital to total (unweighted) assets. 9.2% 1.9% 1,090 7.1% 1.0% 138

Ratio of Tier 1 capital to risk-weighted assets. 13.4% 2.5% 1,090 14.0% 2.3% 138

Standard deviation of (net current operating earnings/TA) 0.9% 1.3% 1,090 0.7% 0.7% 138

Market to book ratio. 1.03 17.5% 1,090 99.6% 2.9% 138

Ratio of insured deposits to unsubordinated debt liabilities. 0.43 0.239 1,090 0.142 0.16 138

Table 2: Base Case Parameter Values

Parameter Value

The volatility of bank assets σ = 5%

Asset cash flow yield α = 5%

The risk-free rate r = 5%

The proportional default costs DC = 20%

Maturity of CoCo and senior debt 6 years

A conversion trigger θ = 6% of capital

The conversion ratio π Varied from 1.0 (conversion at par) to 1.25

(25% above par)

Tax rate 35%

Roll-Over rate of Senior debt m =20%

Transaction costs of raising equity e1 = e2 = 0.025%

The jump size k = - 0.5, i.e., the jump size is uniformly distributed

between 1 and 0, implying an average loss of 50%

The annual probability of jump λ = 2% per year

Figure 3: Base case and the case with a straight debt onlyPanel 1 show the model-implied optimal initial leverage (i.e., the leverage at which the market value of debt and equity is maximized) as a function of conversion ratio. Panel 1 also depicts average bank leverage given optimal initial choice. Panel 2, 3 and 4 depict other model-generated values measured at the optimal initial leverage as a function of the CoCo conversion ratio. The credit spread is the difference between the yield of the bond and the risk free rate. The expected value of total future equity issuances is measured ex-ante at time zero as % of the bank's assets. Total deadweight costs include expected defaults and transaction costs of issuing equity. For comparison, the graphs also present the case where the bank have straight debt only (no CoCo) in its capital structure. In the base case, the size of CoCo bond is 5% of the bank's total debt. The CoCo conversion is triggered when the bank's capital ratio declines to 6% or below. The conversion ratio is the market value of equity to which $1 of par value of the CoCo bond will convert when the trigger is breached. The size of the CoCo bond is 5% of the bank's total debt. The remaining parameter values are as in the base case reported in Table 1.

81%82%83%84%85%86%87%88%89%90%91%

1.00 1.05 1.10 1.15 1.20 1.25

Conversion Ratio

Panel 1: Initial Optimal Leverage and Average Leverage 

Optimal Initial Leverage for Base Case with CoCo

O ti l I iti l L ith t i ht d bt C C

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

4.0%

4.5%

5.0%

1.00 1.05 1.10 1.15 1.20 1.25

Panel 2: Expected size of equity issuance as % of initial asset value

Conversion Ratio

Optimal Initial Leverage for Base Case with CoCo

Optimal Initial Leverage with straight debt, no CoCo

Average Leverage base case with CoCoAverage Leverage with straight debt, no CoCo

15202530354045505560

1.00 1.05 1.10 1.15 1.20 1.25Conversion Ratio

Panel 4: Credit Spreads, bp

Weighted‐Average Spread of CoCo and senior Debt

Spread of senior debt

Spread of CoCo bond

0.0%

0.5%

1.00 1.05 1.10 1.15 1.20 1.25

Conversion Ratio

0.0%0.2%0.4%0.6%0.8%1.0%1.2%1.4%1.6%1.8%

1 1.05 1.1 1.15 1.2 1.25

Conversion Ratio

Panel 3: Expected default costs and transaction costs

Bank has only senior debt in capital structure

Base Case: Expected Default Costs

Base Case: Expected Transaction Costs

1.55%

1.56%

1.57%

1.58%

1.59%

1.60%

1.61%

1.62%

1.00 1.05 1.10 1.15 1.20 1.25Conversion Ratio

Total Expected Default Costs plus Transaction Costs 

1.0570

1.0575

1.0580

1.0585

1.0590

1.0595

1.0600

1.00 1.05 1.10 1.15 1.20 1.25Conversion Ratio

Market Value of the Bank divided by Value of Banks Assets

Figure 4: Optimal Conversion RatioThese figures show the expected total deadweight costs measured at the optimal initial leverage as a function of conversion ratio. Deadweight costs is the sum of expected default costs plus expected transaction costs of issuing equity. These values are measured at the optimal initial leverage as a function of the CoCo conversion ratio. The conversion ratio is the market value of equity to which $1 of par value of CoCo bond will convert when the conversion trigger is breached. The size of CoCo bond is 5% of the bank's total debt. The CoCo conversion is triggered when the bank's capital ratio declines to 6% or below. The remaining parameter values are as in the base case reported in Table 1. Expected default costs and expected value of transaction costs are measured as % of the bank's unlevered assets.

1.55%1.00 1.05 1.10 1.15 1.20 1.25

Conversion Ratio

1.05701.00 1.05 1.10 1.15 1.20 1.25

Conversion Ratio

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

1 1.05 1.1 1.15 1.2 1.25

Panel 3: Trigger probability

192021222324252627

1.00 1.05 1.10 1.15 1.20 1.25Conversion Ratio

Panel 4: Credit Spreads, bp

Weighted‐Average Spread of CoCo and senior Debt

Spread of senior debt

Bank has only senior debt in capital structure

Figure 5: Change in equity value and the  value of senior debt at the instant the CoCo tranche is issued 

Panel 1 and 2 depicts the change in market value of equity and of the preexisting senior debt at the instant of CoCo issuance with different conversion terms, assuming that the issuance is unexpected. The value change calculationassumes that the proceeds of the newly issued CoCo are instantly paid out to equity holders. The figure depicts three cases when the bank's initial leverage prior to Coco the issuance is low at 80%; optimal at 84.5%; and above optimal at 88%. The newly issued CoCo bond is 5% of the total debt with the same maturity of 6 years as in the base case. The trigger is at 6% (base case). The conversion ratio is the market value of equity to which $1 of par value of CoCo bond will convert when the conversion trigger is breached. The CoCo conversion is triggered when the bank's capital ratio declines below the trigger. The remaining parameter values are as in the base case reported in Table 1.

‐0.50%

0.00%

0.50%

1.00%

1.50%

1 1.05 1.1 1.15 1.2 1.25

Change in Equity Value at the instant the CoCo is issued

0.40%

0.60%

0.80%

1.00%

1.20%

1.40%

Change in Debt Value at the instant the CoCo is issued

‐2.00%

‐1.50%

‐1.00%

‐0.50%

0.00%1 1.05 1.1 1.15 1.2 1.25

Initial leverage prior to CoCo issuance is 80%

Initial leverage prior to CoCo issuance is 84.5%

Initial leverage prior to CoCo issuance is 88%

‐0.20%

0.00%

0.20%

0.40%

0.60%

1 1.05 1.1 1.15 1.2 1.25

Initial leverage prior to CoCo issuance is 80%

Initial leverage prior to CoCo issuance is 84.5%

Initial leverage prior to CoCo issuance is 88%

Figure 6.1: Optimal CoCo design: Triggers Ratios and Conversion RatiosPanel 1 show the optimal initial leverage ratios as a function of conversion ratio for three different triggers of 5%, 6% (base case) and 7%. Panel 2, 3, and 4 show the expected equity issuance, expected default costs, and market value of debt and equity (as % of unlevered assets) calculated at the optimal initial leverage ratios. The conversion ratio is the market value of equity to which $1 of par value of CoCo bond will convert when the conversion trigger is breached. The CoCo conversion is triggered when the bank's capital ratio declines below the trigger. The remaining parameter values are as in the base case reported in Table 1. Expected equity issuance costs and expected default costs are measured ex-ante at time zero as percentage of the bank's unlevered assets.

0.885

0.89

0.895

0.9

0.905

0.91

1 1.05 1.1 1.15 1.2 1.25

Panel 1: Optimal Initial Leverage

Trigger is 5% of Capital

Trigger is 6% of Capital

Trigger is 7% of Capital

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

1 1.05 1.1 1.15 1.2 1.25

Panel 2: Total Expected Equity Issuance

Trigger is 5% of Capital

Trigger is 6% of Capital

Trigger is 7% of Capital

Trigger is 6% of Capital

Trigger is 7% of Capital

Trigger is 6% of Capital

Trigger is 7% of Capital

0.0%

0.2%

0.4%

0.6%

0.8%

1.0%

1.2%

1.4%

1.6%

1.8%

1 1.05 1.1 1.15 1.2 1.25

Panel 3: Expected Default Costs

Trigger is 5% of Capital

Trigger is 6% of Capital

Trigger is 7% of Capital

1.056

1.057

1.058

1.059

1.060

1.061

1.062

1.00 1.05 1.10 1.15 1.20 1.25Conversion Ratio

Panel 4: Market Value of the Bank divided by Value of Banks Assets

Trigger is 5% of Capital

Trigger is 6% of Capital

Trigger is 7% of Capital

Figure 6.2: Optimal CoCo design: Optimal Combination of different Trigger Ratios and Conversion RatiosPanel 1 shows the optimal conversion ratio for 5 different trigger ratios varied between 5% and 7%. The five combinations are labeled from #1 to #5. The size of the CoCo remains as 5% of the total debt. For a given trigger ratio, the firm chooses the conversion ratio and the optimal initial leverage ratio taking into account subsequent recapitalization decisions, in such a way that the initial market value of equity plus debt is maximized. Panel 2-4 show the optimal initial leverage, expected equity issuance rate, and expected default costs for the 5 combinations of conversion ratio and trigger ratios labeled #1-#5. The remaining parameter values are as in the base case reported in Table 1. Expected default costs are measured as the difference of bank's unlevered assets minus total market value of debt and equity as % of the bank unlevered assets.

#1#2

#3

#4

#5

1.115

1.12

1.125

1.13

1.135

1.14

1.145

1.15

1.155

5.00% 5.50% 6.00% 6.50% 7.00%

Conversion

 Ratio

Trigger

Panel 1: Optimal combination of conversion rate 

and trigger ratio  #1

#2

#3

#4

#5

88.5%

89.0%

89.5%

90.0%

90.5%

91.0%

5.0% 6.0% 7.0%

Trigger

Panel 2: Optimal Initial Leverage

1.1155.00% 5.50% 6.00% 6.50% 7.00%

Trigger

88.5%5.0% 6.0% 7.0%

Trigger

#1 #2 #3#4

#5

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

4.0%

5.0% 5.5% 6.0% 6.5% 7.0%

Trigger Ratio

Panel 4: Expected Equity Issuance

#1

#2#3

#4

#5

1.00%

1.01%

1.02%

1.03%

1.04%

1.05%

1.06%

5.0% 5.5% 6.0% 6.5% 7.0%

Trigger Ratio

Panel 3: Default Costs

0.882

0.884

0.886

0.888

0.89

0.892

0.894

0.896

0.898

0.9

0.902

1 1.05 1.1 1.15 1.2 1.25

Optimal Initial Leverage

CoCo is 4% of total debt

CoCo is 5% of total debt

00.0050.010.0150.020.0250.030.0350.040.0450.05

1 1.05 1.1 1.15 1.2 1.25

Expected Equity Issuance

CoCo is 4% of total debt

CoCo is 5% of total debt

Figure 7.1: Optimal CoCo design: CoCo size and Conversion RatiosPanel 1 show the optimal initial leverage ratios as a function of conversion ratio for three different sizes of CoCo 4%, 5% (base case) and 6% of the bank's total debt. Panel 2, 3, and 4 show the expected equity issuance, expected default costs, and market value of debt and equity calculated at the optimal initial leverage ratios. These values are measured ex-ante at time zero as percentage of the bank's unlevered assets. conversion ratio is the market value of equity to which $1 of par value of CoCo bond will convert when the conversion trigger is breached. The CoCo conversion is triggered when the bank's capital ratio declines below the trigger. The

CoCo is 4% of total debt

CoCo is 5% of total debt

CoCo is 6% of total debt

CoCo is 4% of total debt

CoCo is 5% of total debt

CoCo is 6% of total debt

0.008

0.009

0.01

0.011

0.012

0.013

0.014

0.015

0.016

0.017

0.018

1 1.05 1.1 1.15 1.2 1.25

Expected Default Costs

CoCo is 4% of total debt

CoCo is 5% of total debt

CoCo is 6% of total debt

1.055

1.056

1.057

1.058

1.059

1.06

1.061

1 1.05 1.1 1.15 1.2 1.25

Market Value of the Bank's debt and equity divided by Value of Banks 

Assets

CoCo is 4% of total debt

CoCo is 5% of total debt

CoCo is 6% of total debt

Figure 7.2: Optimal CoCo design: Combination of different CoCo sizes and Conversion RatiosPanel 1 shows the optimal conversion ratio for 5 different sizes of CoCos varied between 4% and 6%. as percentage of total debt The five combinations are labeled from #1 to #5. The trigger remains as 6% capital. For a given CoCo size, the firm chooses the conversion ratio and the optimal initial leverage ratio taking into account subsequent recapitalization decisions, in such a way that the initial market value of equity plus debt is maximized. Panel 2-4 show the optimal initial leverage, expected equity issuance rate, and expected default costs for the 5 combinations of conversion ratio and trigger ratios labeled #1-#5. The remaining parameter values are as in the base case reported in Table 1. Expected default costs are measured as the difference of bank's unlevered assets minus total market value of debt and equity as % of the bank unlevered assets.

#1

#2

#3#4

#5

1.091.11.111.121.131.141.151.161.171.18

4.00% 4.50% 5.00% 5.50% 6.00%

Panel 1: Optimal combination of conversion rate 

and trigger ratio 

#1

#2

#3

#4

#5

88.8%89.0%89.2%89.4%89.6%89.8%90.0%90.2%90.4%90.6%90.8%

4 00% 5 00% 6 00%

Panel 2: Optimal Initial Leverage

1.091.11.11

4.00% 4.50% 5.00% 5.50% 6.00%

CoCo size as % of total debt

#1

88.8%89.0%89.2%89.4%

4.00% 5.00% 6.00%

CoCo size as % of total debt

#1

#2

#3

#4

#5

3.3%

3.4%

3.5%

3.6%

3.7%

3.8%

4.00% 4.50% 5.00% 5.50% 6.00%

CoCo size as % of total debt

Panel 4: Expected equity Issuance#1 #2 #3

#4#5

1.00%

1.01%

1.01%

1.02%

1.02%

1.03%

1.03%

1.04%

1.04%

1.05%

4.00% 5.00% 6.00%

CoCo size as % of total debt

Panel 3: Default Costs

0.87

0.875

0.88

0.885

0.89

0.895

0.9

0.905

1 1.05 1.1 1.15 1.2 1.25

Optimal Initial Leverage

Jump Probability = 2%

Jump Probability = 3%

Jump Probability = 4%

0

0.01

0.02

0.03

0.04

0.05

0.06

1 1.05 1.1 1.15 1.2 1.25

Expected Equity Issuance

Jump Probability = 2%

Jump Probability = 3%

Jump Probability = 4%

Figure 8.1: Optimal CoCo design: Jump probability and Conversion RatiosPanel 1 show the optimal initial leverage ratios as a function of conversion ratio for three different jump probabilities 2%, 3% (base case) and 4%. Panel 2, 3, and 4 show the expected equity issuance, expected default costs, and market value of debt and equity calculated at the optimal initial leverage ratios. These values are measured ex-ante at time zero as percentage of the bank's unlevered assets. conversion ratio is the market value of equity to which $1 of par value of CoCo bond will convert when the conversion trigger is breached. The CoCo conversion is triggered when the bank's capital ratio declines below the trigger. The remainingparameter values are as in the base case reported in Table 1.

Jump Probability = 2%

Jump Probability = 3%

Jump Probability = 4%

Jump Probability = 2%

Jump Probability = 3%

Jump Probability = 4%

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

1 1.05 1.1 1.15 1.2 1.25

Expected Default Costs

Jump Probability = 2%

Jump Probability = 3%

Jump Probability = 4%

1.055

1.056

1.057

1.058

1.059

1.06

1.061

1.062

1.063

1.064

1 1.05 1.1 1.15 1.2 1.25

Market Value of the Bank's debt and equity divided by Value of Banks Assets

Jump Probability = 2% Jump Probability = 3%

Jump Probability = 4%

Figure 8.2: Optimal CoCo design: Jump probability and Conversion RatiosPanel 1 shows the optimal conversion ratio for 5 different jump probability rates varied between 1% and 5%. The five combinations are labeled from #1 to #5. The trigger remains as 6% capital and the size of the CoCo is 5% of the total debt. For a each case, the firm chooses the conversion ratio and the optimal initial leverage ratio taking into account subsequent recapitalization decisions, in such a way that the initial market value of equity plus debt is maximized. Panel 2-4 show the optimal initial leverage, expected equity issuance rate, and expected default costs for the 5 cases . The remaining parameter values are as in the base case reported in Table 1. Expected default costs are measured as the difference of bank's unlevered assets minus total market value of debt and equity as % of the bank unlevered assets.

#1

#2

#3#4

#51.115

1.12

1.125

1.13

1.135

1.14

1.145

1.15

1.155

Panel 1: Optimal combination of conversion rate 

and trigger ratio 

#1

#2

#3#4

#5

89.4%

89.6%

89.8%

90.0%

90.2%

90.4%

90.6%

Panel 2: Optimal Initial Leverage

#4

#5

1.11

1.115

1.12

1.125

1.00% 2.00% 3.00% 4.00% 5.00%

Jump Probability

#1

89.2%

89.4%

89.6%

0.01 0.03 0.05

Jump Probability

#1

#2

#3

#4

#5

0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

6.0%

0.01 0.02 0.03 0.04 0.05

Jump Probability

Pane 4: Expected Equity Issuance

#1

#2

#3

#4

#5

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

0.01 0.02 0.03 0.04 0.05

Jump Probability

Panel 3: Default Costs

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

1 1.05 1.1 1.15 1.2 1.25

Expected Equity Issuance

0.84

0.85

0.86

0.87

0.88

0.89

0.9

0.91

0.92

0.93

1 1.05 1.1 1.15 1.2 1.25

Optimal Initial Leverage

Figure 9.1: Optimal CoCo design: Asset Volatility and Conversion RatiosPanel 1 show the optimal initial leverage ratios as a function of conversion ratio for three different asset volatilities of 3%, 5% (base case) and 7%. Panel 2, 3, and 4 show the expected equity issuance, expected default costs, and market value of debt and equity calculated at the optimal initial leverage ratios. These values are measured ex-ante at time zero as percentage of the bank's unlevered assets. conversion ratio is the market value of equity to which $1 of par value of CoCo bond will convert when the conversion trigger is breached. The CoCo conversion is triggered when the bank's capital ratio declines below the trigger. The remainingparameter values are as in the base case reported in Table 1.

0

0.01

1 1.05 1.1 1.15 1.2 1.25

Asset Vol=3% Asset Vol=5%

Asset Vol=7%

0.84

0.85

1 1.05 1.1 1.15 1.2 1.25

Asset Vol=3% Asset Vol=5%

Asset Vol=7%

1.03

1.035

1.04

1.045

1.05

1.055

1.06

1.065

1.07

1.075

1 1.05 1.1 1.15 1.2 1.25

Market Value of the Bank's debt and equity divided by Value of Banks Assets

Asset Vol=3%Asset Vol=5%Asset Vol=7%

0.0075

0.0095

0.0115

0.0135

0.0155

0.0175

0.0195

0.0215

0.0235

1 1.1 1.2

Expected Default Costs

Asset Vol=3%

Asset Vol=5%

Asset Vol=7%

Figure 9.2: Optimal CoCo design: Volatility of Assets and Conversion RatiosPanel 1 shows the optimal conversion ratio for 5 different roll-over rates of the straight debt varied between 10% and 30%. The five combinations are labeled from #1 to #5. The trigger remains as 6% capital and the size of the CoCo is 5% of the total debt. For a each of the roll-over rate, the firm chooses the conversion ratio and the optimal initial leverage ratio taking into account subsequent recapitalization decisions, in such a way that the initial market value of equity plus debt is maximized. Panel 2-4 show the optimal initial leverage, expected equity issuance rate, and expected default costs for the 5 combinations of conversion ratio and trigger ratios labeled #1-#5. The remaining parameter values are as in the base case reported in Table 1. Expected default costs are measured as the difference of bank's unlevered assets minus total market value of debt and equity as % of the bank unlevered assets.

#1

#2

#3

#4#5

1.06

1.08

1.1

1.12

1.14

1.16

1.18

3.00% 4.00% 5.00% 6.00% 7.00%

Volatility

Panel 1: Optimal combination of conversion rate 

and trigger ratio 

#1

#2#3

#4

#5

88.5%

89.0%

89.5%

90.0%

90.5%

91.0%

91.5%

92.0%

92.5%

0.03 0.05 0.07

Panel 2: Optimal Initial Leverage

1.063.00% 4.00% 5.00% 6.00% 7.00%

Volatility

#5

88.5%

89.0%

0.03 0.05 0.07

Volatility

#1

#2

#3

#4

#5

0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

6.0%

7.0%

3.0% 4.0% 5.0% 6.0% 7.0%

Volatility

Pane 4: Expected Equity Issuance

#1 #2#3

#4#5

0.00%

0.20%

0.40%

0.60%

0.80%

1.00%

1.20%

1.40%

3.00% 5.00% 7.00%

Volatility

Panel 3: Default Costs

Figure 10.1: Optimal CoCo design: Debt Roll-Over rate and Conversion RatiosPanel 1 show the optimal initial leverage ratios as a function of conversion ratio for three different roll over ratios of the senior liabilities of 10%, 20% (base case) and 30%. Panel 2, 3, and 4 show the expected equity issuance, expected default costs, and market value of debt and equity calculated at the optimal initial leverage ratios. These values are measured ex-ante at time zero as percentage of the bank's unlevered assets. Conversion ratio is the market value of equity to which $1 of par valueof CoCo bond will convert when the conversion trigger is breached. The CoCo conversion is triggered when the bank's capital ratio declines below the trigger. The remaining parameter valuesare as in the base case reported in Table 1.

0.88

0.885

0.89

0.895

0.9

0.905

1 1.05 1.1 1.15 1.2 1.25

Optimal Initial Leverage 

Debt Roll‐Over Rate is 10%

D b R ll O R i 20%

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

1 1.05 1.1 1.15 1.2 1.25

Expected Equity Issuance

Debt Roll‐Over Rate is 10%

D b R ll O R i 20%

1 1.05 1.1 1.15 1.2 1.25

Debt Roll‐Over Rate is 10%

Debt Roll‐Over Rate is 20%

Debt Roll‐Over Rate is 30%

1 1.05 1.1 1.15 1.2 1.25

Debt Roll‐Over Rate is 10%

Debt Roll‐Over Rate is 20%

Debt Roll‐Over Rate is 30%

0.0%

0.2%

0.4%

0.6%

0.8%

1.0%

1.2%

1.4%

1.6%

1.8%

1 1.1 1.2

Expected Default Costs

Debt Roll‐Over Rate is 10%

Debt Roll‐Over Rate is 20%

Debt Roll‐Over Rate is 30%

1.052

1.054

1.056

1.058

1.06

1.062

1.064

1 1.05 1.1 1.15 1.2 1.25

Market Value of the Bank's debt and equity divided by Value of Banks Assets

Debt Roll‐Over Rate is 10%Debt Roll‐Over Rate is 20%Debt Roll‐Over Rate is 30%

Figure 10.2: Optimal CoCo design: Debt Roll-Over and Conversion RatiosPanel 1 shows the optimal conversion ratio for 5 different roll-over rates of the straight debt varied between 10% and 30%. The five combinations are labeled from #1 to #5. The trigger remains as 6% capital and the size of the CoCo is 5% of the total debt. For a each of the roll-over rate, the firm chooses the conversion ratio and the optimal initial leverage ratio taking into account subsequent recapitalization decisions, in such a way that the initial market value of equity plus debt is maximized. Panel 2-4 show the optimal initial leverage, expected equity issuance rate, and expected default costs for the 5 combinations of conversion ratio and trigger ratios labeled #1-#5. The remaining parameter values are as in the base case reported in Table 1. Expected default costs are measured as the difference of bank's unlevered assets minus total market value of debt and equity as % of the bank unlevered assets.

#1

#2

#3

#4

#5

1.09

1.1

1.11

1.12

1.13

1.14

1.15

1.16

1.17

0.1 0.15 0.2 0.25 0.3

Panel 1: Optimal combination of conversion rate 

and trigger ratio 

#1#2

#3#4

#5

89.5%

89.6%

89.7%

89.8%

89.9%

90.0%

90.1%

90.2%

0.1 0.15 0.2 0.25 0.3

Panel 2: Optimal Initial Leverage

1.09

1.1

0.1 0.15 0.2 0.25 0.3

Debt roll‐over rate

89.5%

89.6%

0.1 0.15 0.2 0.25 0.3

Debt Roll‐over Rate

#1

#2

#3#4 #5

3.3%

3.4%

3.4%

3.5%

3.5%

3.6%

0.1 0.15 0.2 0.25 0.3

Debt Roll‐over rate

Panel 4: Expected equity Issuance

#1#2

#3 #4 #5

0.90%

0.92%

0.94%

0.96%

0.98%

1.00%

1.02%

1.04%

1.06%

0.1 0.15 0.2 0.25 0.3

Debt Roll‐over rate

Panel 3: Default Costs

0

0.01

0.02

0.03

0.04

0.05

1 1.05 1.1 1.15 1.2 1.25

Expected Equity Issuance

Transaction Costs=0.3%Transaction Costs=0.5%

0.885

0.89

0.895

0.9

0.905

0.91

1 1.05 1.1 1.15 1.2 1.25

Optimal Initial Leverage

Transaction Costs=0.3%Transaction Costs=0.5%

Figure 11.1: Optimal CoCo design: Transaction Costs and Conversion RatiosPanel 1 show the optimal initial leverage ratios as a function of conversion ratio for three different levels of transaction cost parameter of 0.3%., 0.5% (base case) and 0.7%. Panel 2, 3, and 4 show the expected equity issuance, expected default costs, and market value of debt and equity calculated at the optimal initial leverage ratios. These values are measured ex-ante at time zero as percentage of the bank's unlevered assets. conversion ratio is the market value of equity to which $1 of par value of CoCo bond will convert when the conversion trigger is breached. The CoCo conversion is triggered when the bank's capital ratio declines below the trigger. The remaining parameter values are as in the base case reported in Table 1.

Transaction Costs=0.3%Transaction Costs=0.5%Transaction Costs=0.7%

Transaction Costs=0.3%Transaction Costs=0.5%Transaction Costs=0.7%

1.056

1.057

1.058

1.059

1.06

1.061

1.062

1.063

1 1.05 1.1 1.15 1.2 1.25

Market Value of the Bank's debt and equity divided by Value of Banks 

Assets

Transaction Costs=0.3%

Transaction Costs=0.5%

Transaction Costs=0.7%

0.0075

0.0085

0.0095

0.0105

0.0115

0.0125

0.0135

0.0145

0.0155

0.0165

0.0175

1 1.05 1.1 1.15 1.2 1.25

Expected Default Costs

Transaction Costs=0.3%

Transaction Costs=0.5%

Transaction Costs=0.7%

Figure 11.2: Optimal CoCo design: Transaction Costs and Conversion RatiosPanel 1 shows the optimal conversion ratio for 5 different levels of transaction cost parameter of varied between 0.3% and 0.7%. The five combinations are labeled from #1 to #5. The trigger remains as 6% capital and the size of the CoCo is 5% of the total debt. For a each of the five cases, the firm chooses the conversion ratio and the optimal initial leverage ratio taking into account subsequent recapitalization decisions, in such a way that the initial market value of equity plus debt is maximized. Panel 2-4 show the optimal initial leverage,expected equity issuance rate, and expected default costs for the 5 cases labeled #1-#5. The remaining parameter values are as in the base case reported in Table 1. Expected default costs are measured as the difference of bank's unlevered assets minus total market value of debt and equity as % of the bank unlevered assets.

#1

#2

#3

#4

#5

1.06

1.08

1.1

1.12

1.14

1.16

1.18

1.2

1.22

0 30% 0 40% 0 50% 0 60% 0 70%

Panel 1: Optimal combination of conversion rate 

and trigger ratio #1

#2

#3

#4

#5

89.2%

89.4%

89.6%

89.8%

90.0%

90.2%

90.4%

90.6%

90.8%

91.0%

Panel 2: Optimal Initial Leverage

#1

1.06

1.08

1.1

0.30% 0.40% 0.50% 0.60% 0.70%

Transaction Costs

#5

89.2%

89.4%

89.6%

0.30% 0.50% 0.70%

Transaction Costs

#1#2

#3#4

#5

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

4.0%

4.5%

0.30% 0.40% 0.50% 0.60% 0.70%

Transaction Costs

Panel 4: Expected equity Issuance

#1 #2#3 #4

#5

0.90%

0.92%0.94%0.96%

0.98%1.00%

1.02%1.04%1.06%

1.08%

0.30% 0.50% 0.70%

Transaction Costs

Panel 3: Default Costs

0

0.01

0.02

0.03

0.04

0.05

1 1.05 1.1 1.15 1.2 1.25

Expected Equity Issuance

C P b bili 50%/

0.8840.8860.8880.890.8920.8940.8960.8980.9

0.902

1 1.05 1.1 1.15 1.2 1.25

Optimal Initial Leverage

Figure 12.1: Optimal CoCo design: Delayed Conversion/trigger and Conversion RatiosPanel 1 shows the optimal initial leverage ratios for different degrees to which the trigger is delayed as a function of conversion ratio. With delayed conversion, once the bank capital declines below the trigger, the actual conversion is not instantaneous, but rather probabilistic. For example, 50% annual probability means that when below the trigger, the conversion will occur with 50% probability (not instantly) within a year period if the bank ratio remains under the trigger. The base case (instantaneous conversion) is also presented for comparison reasons. Panel 2, 3, and 4 show the expected equity issuance, expected default costs, and market value of debt and equity calculated at the optimal initial leverage ratios. These values are measured ex-ante at time zero as percentage of the bank's unlevered assets. conversion ratio is the market value of equity to which $1 of par value of CoCo bond willconvert when the conversion trigger is breached. The CoCo conversion is triggered when the bank's capital ratio declines below the trigger. The remaining parameter values are as in the base case reported in Table 1.

0.0075

0.0095

0.0115

0.0135

0.0155

0.0175

1 1.05 1.1 1.15 1.2 1.25

Expected Default Costs

Conv. Probability=50%/yearConv. Probability=80%/yearBase Case (no Delay)

01 1.05 1.1 1.15 1.2 1.25

Conv. Probability=50%/yearConv. Probability=80%/yearBase Case (no Delay)

0.8841 1.05 1.1 1.15 1.2 1.25

Conv. Probability=50%/yearConv. Probability=80%/yearBase Case (no Delay)

1.0565

1.057

1.0575

1.058

1.0585

1.059

1.0595

1.06

1.0605

1 1.05 1.1 1.15 1.2 1.25

Market Value of the Bank's debt and equity divided by Value of 

Banks Assets

Conv. Probability=50%/yearConv. Probability=80%/yearBase Case (no Delay)

Figure 12.2: Optimal CoCo design: Delayed Conversion/Trigger and Conversion RatiosPanel 1 shows the optimal conversion ratio for 5 different delayed conversion conditions varied between 50% and 86% annual probability of trigger. With a delayed conversion, once the bank capital declines below the trigger, the actual conversion is not instantaneous, but rather probabilistic. For example, 50% annual probability means that when the assets ratio is below the trigger, the conversion will occur with 50% probability (not instantly) within a year period if the bank ratio remains under the trigger. The five cases are labeled from #1 to #5. The contractual trigger remains as 6% capital and the size of the CoCo is 5% of the total debt. For a each of the 5 cases, the firm chooses the conversion ratio and the optimal initial leverage ratio taking into account subsequent recapitalization decisions, in such a way that the initial market value of equity plus debt is maximized. Panel 2-4 show the optimal initial leverage, expected equity issuance rate, and expected default costs for the 5 combinations of conversion ratio and trigger ratios labeled #1-#5. The remaining parameter values are as in the base case reported in Table 1. Expected default costs are measured as the difference of bank's unlevered assets minus total market value of debt and equity as % of the bank unlevered assets.

#1

#2

#3

#4

#5

1.19

1.2

1.21

1.22

1.23

1.24

1.25

1.26

40% 50% 60% 70% 80% 90%

Panel 1: Optimal combination of conversion rate 

and trigger ratio 

#1 #2 #3#4#5

0.0%

10.0%

20.0%

30.0%

40.0%

50.0%

60.0%

70.0%

80.0%

90.0%

100.0%

Panel 2: Optimal Initial Leverage

1.19

1.2

40% 50% 60% 70% 80% 90%

Annual Probability of Conversion when capital fall below the trigger

0.0%

10.0%

20.0%

30.0%

40% 60% 80% 100%

Annual Probability of Conversion when capital fall below the trigger

#1

#2 #3#4#5

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

4.0%

40.0% 50.0% 60.0% 70.0% 80.0% 90.0% 100.0%

Annual Probability of Conversion when capital fall below the trigger

Pane 4: Expected Equity Issuance

#1#2 #3#4#5

0.00%

0.20%

0.40%

0.60%

0.80%

1.00%

1.20%

1.40%

40.00% 60.00% 80.00% 100.00%

Annual Probability of Conversion when capital fall below the trigger

Panel 3: Default Costs

0.885

0.89

0.895

0.9

0.905

1 1.05 1.1 1.15 1.2 1.25

Optimal Initial Leverage

0%

20%

40%

60%

80%

100%

120%

0 0.1 0.2 0.3 0.4 0.5

(Premature) Trigger Probability per Year as a function of Bank Asset Ratio

Figure 13.1: Optimal CoCo design: premature Conversion/Trigger and Conversion RatiosPanel 1 shows the probability of (premature) triggering as a function of bank capital ratio ratiо, where the probability is described by the following function: xp( 6% - cap.ratio ) ϕ. The parameter ϕ is takes values of 20 and 60. With premature conversion/trigger, the CoCo conversion can be triggered prematurely with some probability even if the bank capital ratio still remains above the contractual trigger of 6%. Panel 2 depicts the optimal initial leverage ratios for two cases of premature triggering (more premature and less premature) as a function of conversion ratio The base case (instantaneous conversion at trigger of 6%) is also presented for comparison reasons. Panel 3, and 4 show the expected equity issuance, expected default costs calculated at the optimal initial leverage ratios. These values are measured ex-ante at time zero as percentage of the bank's unlevered assets. Conversion ratio is the market value of equity to which $1 of par value of CoCo bond will convert when the conversion trigger is breached. The CoCo conversion is triggered when the bank's capital ratio declines below the trigger. The remaining parameter values are as in the base case reported in Table 1.

0.00750.00850.00950.01050.01150.01250.01350.01450.01550.01650.0175

1 1.05 1.1 1.15 1.2 1.25

Expected Default Costs

More Premature ConversionLess Premature ConversionBase Case (no Premature Trigger)

0.8851 1.05 1.1 1.15 1.2 1.25

More Premature ConversionLess Premature ConversionBase Case (no Premature Trigger)

0%0 0.1 0.2 0.3 0.4 0.5

More Premature ConversionLess Premature ConversionBase Case (No Premature Conversion)

00.0050.010.0150.020.0250.030.0350.040.0450.05

1 1.05 1.1 1.15 1.2 1.25

Expected Equity Issuance

More Premature ConversionLess Premature ConversionBase Case (no Premature Trigger)

Figure 13.2: Optimal CoCo design: Premature Conversion/Trigger and Conversion RatiosPanel 1 shows the optimal conversion ratio for 5 different delayed conversion conditions varied between 50% and 86% annual probability of trigger. With a delayed conversion, once the bank capital declines below the trigger, the actual conversion is not instantaneous, but rather probabilistic. For example, 50% annual probability means that when the assets ratio is below the trigger, the conversion will occur with 50% probability (not instantly) within a year period if the bank ratio remains under the trigger. The five cases are labeled from #1 to #5. The contractual trigger remains as 6% capital and the size of the CoCo is 5% of the total debt. For a each of the 5 cases, the firm chooses the conversion ratio and the optimal initial leverage ratio taking into account subsequent recapitalization decisions, in such a way that the initial market value of equity plus debt is maximized. Panel 2-4show the optimal initial leverage, expected equity issuance rate, and expected default costs for the 5 combinations of conversion ratio and trigger ratios labeled #1-#5. The remaining parameter values are as in the base casereported in Table 1. Expected default costs are measured as the difference of bank's unlevered assets minus total market value of debt and equity as % of the bank unlevered assets.

#1

#2

#3 #4 #5

1.131.141.151.161.171.181.191.21.211.22

0 20 40 60 80 100

Panel 1: Optimal combination of conversion rate 

and trigger ratio 

#1

#2 #3 #4 #5

89.3%89.4%89.4%89.5%89.5%89.6%89.6%89.7%89.7%89.8%

0 50 100

Panel 2: Optimal Initial Leverage

#3 #4 #5

1.131.141.15

0 20 40 60 80 100

more premature ‐‐‐‐> less premature

#1

89.3%89.4%

%

0 50 100

more premature ‐‐‐‐>less premature

#1

#2#3

#4 #5

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

0 20 40 60 80 100

Coefficient for the probability of premature trigger

more premature   ‐‐‐‐>     less premature

Pane 4: Expected Equity Issuance

#1#2

#3 #4 #5

0.00%

0.20%

0.40%

0.60%

0.80%

1.00%

1.20%

1.40%

0 50 100

Coefficient for the probability of premature trigger

more premature ‐‐‐‐>    less premature

Panel 3: Default Costs

0.883

0.8835

0.884

0.8845

0.885

0.8855

0.886

0.8865

0 0.2 0.4 0.6 0.8 1

Conversion ratio

Panel 1: Optimal Initial Leverage

‐0.04%

‐0.04%

‐0.03%

‐0.03%

‐0.02%

‐0.02%

‐0.01%

‐0.01%

0.00%0 0.2 0.4 0.6 0.8 1

Conversion ratio

Panel 2: Expected Equity Issuance/Money Burning

Figure 14: "Money Burning" Effect of CoCo with Accretive Conversion TermsPanel 1 shows the optimal initial leverage ratios for accretive conversion terms. Conversion ratios vary from 0 (total loss for CoCo holders) to 1 (а par for par conversion). Panel 2 show the expected money burning, (negative of equity issuance). Panel 3 depicts the spread of the CoCo as a function of conversion ratio. Panel 3 shows total deadweight costs calculated as expected default costs plus expected money burning. All these measures are calculated at the optimal initial leverage ratios. These values are measured ex-ante at time zero as percentage of the bank's unlevered assets. The remainingparameter values are as in the base case reported in Table 1.

1.58%

1.60%

1.62%

1.64%

1.66%

1.68%

1.70%

1.72%

0 0.2 0.4 0.6 0.8 1

Panel 4: Expected Default Costs + Money Burning costs

0

50

100

150

200

250

300

350

400

450

0 0.2 0.4 0.6 0.8 1

Conversion Ratio

Panel 3: Spread of the CoCo, in b.p..

0.8830 0.2 0.4 0.6 0.8 1

Conversion ratio

‐0.04%

0.04%

Conversion ratio

Panel 4: Expected Default Costs andPanel 3 Trigger Probabilit

0200400600800100012001400160018002000

0.06 0.08 0.1 0.12 0.14

Bank Capital

Panel 2: Spread of the CoCo bond

Remaining Maturity=1 dayRemaining Maturity= 0.5 yearRemaining Maturity= 1 year

‐5.00%‐4.00%‐3.00%‐2.00%‐1.00%0.00%1.00%

0.06 0.08 0.1 0.12 0.14

Bank Capital 

Panel 1: Expected Equity Issuance/Money Burning

Remaining Maturity=1 dayRemaining Maturity= 0.5 yearRemaining Maturity= 1 year

Figure 15: Maturity Effect of CoCo with Accretive Conversion TermsPanel 1 depicts expected money burning (negative of equity issuance) for the case of total write down conversion term (total loss for CoCo holders) as a function of bank's capital for three cases of remaining maturity of 1 day, 0.5 year and 1 year. Expected money burning is calculated as percentage of bank unlevered assets. Panel 2 depicts the spread of the CoCo. Panel 4 depicts total deadweight costs comprised of default costs and "money burnt". All these measures are calculated at the optimal initial leverage ratios. The trigger is set at 6% (i.e., leverage of 94%) The remaining

0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

0.06 0.08 0.1 0.12 0.14

Panel 4: Expected Default Costs and Money Burning Costs 

Remaining Maturity=1 dayRemaining Maturity= 0.5 yearRemaining Maturity= 1 year

0

0.2

0.4

0.6

0.8

1

1.2

0.06 0.08 0.1 0.12 0.14

Bank Capital

Panel 3: Trigger Probability

Remaining Maturity=1 dayRemaining Maturity= 0.5 yearRemaining Maturity= 1 year

Remaining Maturity= 0.5 yearRemaining Maturity= 1 year

Remaining Maturity= 0.5 yearRemaining Maturity= 1 year