Convertible Debt

A part of the bank’s debt is issued in the form of AT1 instruments. These are CoCos that are either written down or convert to equity if the CET1 ratio falls below a pre-defined contractual trigger point and/or at the PONV. As the objective of the two types of instruments is the same, we assume that all CoCos are issued as the type that converts to equity, and thus ignore instruments with a contractual principal write-down mechanism. A further characteristic of AT1 capital is that coupon payments are discretionary and may be cancelled at any time, for any rea-son. Furthermore, mandatory payout restrictions will be imposed on the bank if the CET1 ratio falls below certain contractually defined trigger points. This will be discussed below.

In our model as well as in practice, the trigger points are based on the bank’s CET1 ratio, i.e. book values and not market values. Flannery (2002, 2016) and Pennacchi et al. (2010) argue in favor of basing CoCo trigger points on market values instead of book values because market values are forward looking, constantly updated, and less prone to manipulation. However, the findings of Sundaresan and Wang (2010) suggest that the use of market-based trigger points can be problematic because they can lead to either no equilibrium or multiple equilibria, questioning the applicability of contracts based on market-based triggers. This is an interesting field for future research, but for now we limit ourselves to applying trigger levels based on book values.

According to market practice, there are both high- and low-trigger CoCos, referring to the point at which the instrument is converted to equity (BIS CoCo Primer, 2013). We assume that the bank issues both types of instruments, and denote the high- and low-trigger points as H and L, respectively. In practice, most AT1 CoCos are issued with CET1 capital triggers of either 7.0% (high) or 5.125% (low). We assume that a fraction 𝛾, 0 <𝛾 < 1, of the total amount of AT1 capital is issued as high-trigger instruments, while the remaining (1–𝛾) is issued as low-trigger in-struments. If the CET1 ratio falls below H, an amount 𝛾𝐴 will be converted to

equity, and the remaining principal of AT1 will decrease to (1–𝛾)𝐴. Furthermore, we assume that the bank does not issue any new AT1 capital between time zero and time T.

At time t, we denote the aggregate amount of AT1 capital that has been converted up to time t by 𝛬_𝑡. Thus, at every point in time, the remaining amount of AT1 capital will be

𝐴_𝑡 = (1− 𝛬_𝑡)𝐴 (8)

We assume that the conversion ratio of AT1 to CET1 is equal to one. The conver-sion ratio is the book value of equity received by CoCo investors for each Euro of face value converted. While, due to the issue of equity dilution, it would be im-portant to keep track of the allocation of post-conversion equity between original and converted shareholders when pricing AT1 capital and equity, we can ignore this issue as this is not our goal.

Taking into account the possibility of AT1 conversion, the level of equity is given by

𝐶𝐸𝑇1_𝑡 =𝑉_𝑡− 𝐷_𝑡− 𝑆_𝑡− 𝑇3_𝑡− 𝑇2_𝑡−(1− 𝛬_𝑡)𝐴 (9)

Before we continue, let us consider an example of the AT1 conversion process. Part (A) of Figure (11) below shows the initial balance sheet of a bank. The asset value is 100, and assets are funded by four types of liabilities: 50 by deposits, 40 by senior debt, 5 by AT1 capital, and 5 by equity. The bank is assumed to have issued 50.0%

of high-trigger (i.e. 7.0%) and 50.0% of low-trigger (i.e. 5.125%) AT1 capital, and to reach the PONV when the CET1 ratio drops to 6.0%. The level of RWA is assumed to be 40% of total assets, such that the bank has 40 in RWAs, which means that the initial CET1 ratio is 12.5%. In (B), the asset value drops to 97.5, resulting in a decline in equity to 2.5. Correspondingly, the RWA drops to 39, leading to a CET1 ratio of 6.4%. Because this is below the 7.0% trigger, high-trigger CoCos will convert to equity, leaving 2.5 in AT1 capital and increasing equity to 5. This leads to the final balance sheet in (C) with a CET1 ratio of 12.8%.

Figure 10: Example of CoCo Conversion

(A) Initial balance sheet; (B) after drop in asset value;(C) after conversion of AT1 capital to equity. Source:

Authors’ own creation.

Cash Flows

Every period, the bank’s assets are assumed to generate a stream of cash flows at a state-dependent rate, 𝛿 𝑡 , 0 <𝛿(𝑡) < 1. Thus, at time t, the available cash flow is 𝛿(𝑡)𝑉_𝑡. We assume that the bank uses the periodical cash flows to service its debt and to pay out dividends to its shareholders. The interpretation of this assumption is that the bank is financing its payments by asset sales. This is a common assump-tion in the capital structure literature, e.g. Leland (1994), Leland and Toft (1996), and Glasserman and Nouri (2012).

If the CET1 level of a bank falls below a regulatory threshold, i.e. the MDA trigger, restrictions will be imposed on the bank’s payment of coupons on AT1 instruments and dividends to shareholders. To capture the possibility of mandatory payout restrictions, we set the payout rate to be state dependent. Thus, in states where the CET1 ratio is below the MDA trigger, 𝛼, we assume that the payout ratio is reduced by a fixed amount, 𝜑, representing the restrictions on payments. The 𝜑 is set to be a fixed amount because it limits the computational challenges for the model. By introducing the MDA restrictions to the cash flow modeling, the model incorporates an important goal of the bank capital regulation that focuses on lim-iting the payout in difficult times for the bank.

To complicate matters further, breaching the MREL requirement can also trigger payout restrictions. As described in chapter 3, one of the immediate consequences of breaching the MREL is restriction on payments⁹. We assume that the bank takes

9 As described in chapter 3, there can be multiple consequences of breaching the MREL requirement, however, most of these are difficult to incorporate into a model. Therefore, we assume in this setting

V 100,00 D 50,00 V 97,50 D 50,00 V 97,50 D 50,00

S 40,00 S 40,00 S 40,00

A 5,00 A 5,00 A 2,50

CET1 5,00 CET1 2,50 CET1 5,00

RWA 40,00 RWA 39,00 RWA 39,00

CET1 ratio 12,5% CET1 ratio 6,4% CET1 ratio 12,8%

(A) (B) (C)

Assets Liabilities Assets Liabilities Assets Liabilities

the stacking approach to the CBR, which means that CET1 cannot count towards the MREL and the CBR at the same time – that is, the CET1 cannot be counted twice. Therefore, if the bank breaches the MREL, the amount of CET1 used to satisfy the CBR will instead be used to comply with the MREL. This can in turn cause the bank to breach the MDA, which means that payout restrictions will be imposed on the bank as if the MDA were breached in a “normal” way.

To satisfy the MREL, the bank must hold an MREL ratio above the minimum requirement, which is assumed to be given by 𝜉. The MREL ratio at time t is given by

MREL ratio_𝑡 =𝐶𝐸𝑇1_𝑡+ 1− 𝛬_𝑡 𝐴+𝑇2 +𝑇3

𝑅𝑊𝐴_𝑡 (10)

If this requirement is not met, it will be considered as breaching the MDA.

To capture the possibility of mandatory payout restrictions, triggered by a breach of either the MREL or the MDA, we set the payout rate to be state dependent.

Hence, at time t, the payout is given as

𝛿 𝑡 =𝛿 − 𝜑𝟏 _{𝑀𝐷𝐴(𝑡)} (11)

where 𝟏 _{𝑀𝐷𝐴(𝑡)} is an indicator function that is equal to one if the CET1 ratio is below the MDA threshold of 𝛼, or if the MREL ratio is below the requirement of 𝜉, and zero if the CET1 ratio is greater than 𝛼 and the MREL ratio is greater than 𝜉. That is, we see the two cases as equivalent, and we have that

𝟏 _{𝑀𝐷𝐴(𝑡)} = 1 if 𝐶𝐸𝑇1 ratio_𝑡 <𝛼 or 𝑀𝑅𝐸𝐿 ratio_𝑡<𝜉

0 if 𝐶𝐸𝑇1 ratio_𝑡>𝛼 and 𝑀𝑅𝐸𝐿 ratio_𝑡 >𝜉 (12)

Thus, in states where payout is restricted, assets will decrease by a smaller amount than in normal states, reflecting the fact that less cash is paid out.

Coupon Payments and Dividends

Every period, the available cash flow, 𝛿(𝑡)𝑉_𝑡, is allocated between debtholders and shareholders according to a waterfall hierarchy. The highest priority is given to holders of regular debt, i.e. deposits, senior debt, T3, and T2. If cash is available,

that the only consequence of breaching the MREL is the MDA restriction, and do not consider any

coupons are paid on AT1 instruments, and finally, any remaining cash is paid out to shareholders as dividends. If the cash flow is insufficient to cover coupon pay-ments on regular debt, additional equity will have to be raised in a new issue, as shown by Leland and Toft (1996) and Glasserman and Nouri (2012).