Dividends to Shareholders

Finally, dividends are given by the difference between the bank’s received cash flows and the payments to debtholders. That is, at time t, the dividends are given by

𝑑_𝑡 =𝛿 𝑡 𝑉_𝑡− 1− 𝜅 𝑐_𝐷𝐷+𝑐_𝑆𝑆+𝑐_𝑇3𝑇3 +𝑐_𝑇2𝑇2 − 𝐶_𝐴,𝑡 (18)

If the bank is not generating enough cash to service its regular debt¹¹, the dividends are negative. In such cases, we assume that new equity will be issued to cover the coupon payments. Thus, negative dividends can be interpreted as an equity issue.

The amount of new equity issued will generate an amount of cash just sufficient to service the bank’s regular debt, which will immediately be paid out to debtholders, such that the issuance has no impact on the bank’s overall capital. The practical interpretation of this assumption is that equity holders, the owners, pay higher-ranked debtholders out of their own pocket.

With this assumption, we need to consider whether, at some point, it would be optimal for the equity owners to stop paying coupons and default on the debt. As shown in Leland’s (1994) model, the owners of equity will at every point in time

assess whether the present value of the bank’s assets is high enough to justify continuing to pay coupons to the debtholders out of their own pocket. If the value of the bank’s assets has reached a level where it is no longer optimal for them to sustain the payouts, they should stop paying and let debtholders take over the bank.

For banks, however, this ability to optimally choose when to pay coupons may not apply. This is because the default boundary is set by regulators. Thus, it is regula-tors who decide whether or not a bank has reached the PONV, and not shareholders as in Leland’s (1994) model. Furthermore, while shareholders might consider ceas-ing to pay coupons, it seems unlikely that any shareholder would voluntarily choose to default, as the PONV is likely to be as high as 4.5% to 7.0% CET1. At least, we have not seen any examples of this in practice. Therefore, we choose to ignore the potential existence of endogenous default boundaries, and thus assume that share-holders will not jump the ship because of choppy waters.

PONV and Costs of Resolution

We assume that the bank reaches PONV when the CET1 ratio falls below the threshold, 𝛺. When this happens, the bank is seized by regulators and will enter resolution. However, at this point, the bank may still have a positive amount of equity¹², although not enough to satisfy the minimum capital requirements. Thus, this is different from Merton’s (1974) classic assumption, in which the firm defaults if the value of assets is lower than the face value of debt at the latter’s maturity.

The bank is assumed to reach PONV at time τ. Hence, we distinguish between the two cases 𝜏 ≤ 𝑇 and 𝜏 >𝑇, with the first corresponding to the case in which the bank goes into resolution in the analyzed period of time. To account for costs of the resolution process¹³, we include the random variable Z, which is uniformly dis-tributed on the interval [0,𝜓]. 𝜓 is the upper bound on the cost distribution, which we will vary in the numerical analysis to study the effects of the resolution costs.

Z is assumed to be independent of the asset value. If the bank goes into resolution at time t, the asset value will drop to 1–𝑍 𝑉_𝑡. Subsequently, losses are allocated according to the waterfall of investors and creditors.

12 This is not always the case. Note that if a large jump in the asset value occurs, equity can be wipe out completely. In this case the bank will not have a positive amount of equity in the event of default. If default is not caused by such jump, however, the bank will be seized by regulators while equity is still positive.

13 See section chapter 2 for a description of the resolution process

In the event of resolution, each liability class will recover a random fraction 𝑅_𝑖∈ 0,1 of its face value, where the subscript i denotes the class of liability. Hence, if the bank ends up in resolution, the promised principal payment on T3 is reduced from 𝑇3 to 𝑅_𝑇3𝑇3. To ensure absolute priority in the hierarchy of creditors, avail-able funds are allocated as a waterfall and a liability class does not experience any losses before the classes ranking junior to it are depleted.

End Value of Payments

To close our model, we need to address what happens if the bank’s asset path ends up at maturity 𝑇, or if PONV 𝜏 is reached. If a bank does not end up in resolution, it will reach the maturity 𝑇. When this happens, we assume that the bank will repay its existing debt in full. The bank might do this by refinancing or by using other options such as selling the bank; this is a choice that we do not consider as it does not affect our analysis. However, a bank does face the possibility at time 𝑇 of ending up in the critical capital zones. At this point, it is still assumed that the bank will make full principle repayments. This might be costly, as new investors might require a greater rate of return, but for the sake of simplicity, we assume that this refinancing risk is a cost absorbed by equity investors.

If the bank has instead reached the PONV at time 𝜏 and enters resolution, we assume that losses will be distributed according to the waterfall, and that the re-spective recovery values for the different debt tranches will be paid. When a bank enters resolution, any of the four tools provided in the BRRD as described in chap-ter 2 might be applied. For example, the bank can be recapitalized and thus con-tinue operations in a post-resolution setting, where existing bondholders of bail-in eligible liabilities can have a fraction or all its bonds converted to equity. For sim-plicity, we assume a recovery payment to bondholders, and do not consider any post-resolution bank and how bondholders in a post-resolution setting might be affected.

Decomposition of Payments

Before we can price the bank’s debt instruments, we need to decompose the cash flows received by investors into earned coupons and principal payments. In this

context, however, it is important to distinguish between the actual and the prom-ised payments. As promprom-ised cash flows are only realized if the bank does not end up in resolution¹⁴, we examine the actual cash flows to investors.

The following calculations are identical for all regular debt classes¹⁵ – that is, de-posits, senior, T3, and T2. Consider a bond with face value 𝐵_𝑖, coupon rate 𝑐_𝑖, and recovery rate 𝑅_𝑖, where i denotes the liability class of the bond, such that 𝑖 ∈

𝐷,𝑆,𝑇3,𝑇2 . At time zero, the present value of its cash flows can be decomposed into earned coupons

𝑐_𝑖

𝑚𝑖𝑛 (𝜏,𝑇) 𝑡=1

𝐵_𝑖𝑒^−𝑟𝑡 (19)

and principal payment

𝐵 𝑒^−𝑟𝑇𝟏_𝜏>𝑇 +𝑅_𝑖𝑒^−𝑟𝜏 𝟏_𝜏≤𝑇 (20)

In equation (19), coupons are paid until either maturity, T, or the point when the bank is no longer viable, τ. In equation (20), the 𝟏 _𝜏>𝑇 is an indicator function that is equal to one if the bank does not end up in resolution and zero if it does.

𝟏_𝜏≤𝑇 is similar but corresponds to the opposite situation. Thus, if the bank does not end up in resolution, the bondholders will be repaid the full principal value. If the bank does go into resolution, on the other hand, the principal payment is re-duced by (1–𝑅_𝑖) and received at time 𝜏. If 𝑅_𝑖 = 1, the bond is riskless.

At time T, we assume that the bank is able to refinance its liabilities unless it ends up in resolution. In normal states, this should not be a problem for the bank. How-ever, refinancing the liabilities in cases in which the value of the bank’s assets ends up in a “critical” zone could potentially carry significant refinancing costs for the bank in practice, which will most likely affect the value of equity negatively. How-ever, as our focus is on pricing T3 bonds, this issue does not raise any problems for our exercise.

14 And the bond is held to maturity.

15 The formulas cannot be used for pricing AT1 instruments, since the payoff structure of these bonds is different from regular debt. As noted earlier, we will restrict ourselves from pricing these instruments in this thesis.

Valuation

We assume that the bank’s assets follow the jump-diffusion process in equation (1) under the risk-neutral probability measure. Under the risk-neutral measure, the expected return of all assets is the risk-free rate. Thus, it is essential to ensure that the expected return from assets, considering all components, is indeed the risk-free rate (Hull, 2012). We obtain this by adjusting the drift term of our process. There-fore, we write the drift as

𝑟 − 𝛿 𝑡 − 𝜆𝑘 (21)

By applying Itô’s lemma, we are able to write the process for Vt as 𝑉_𝑡=𝑉₀𝑒𝑥𝑝 ^𝑡𝜇_𝑠 𝑑𝑠+𝜎𝑊_𝑡^ℚ

0

1 +𝜀_𝑖

𝑁_𝑡 𝑖=1

(22) where 𝜇_𝑡 is given by

𝜇_𝑡=𝑟 − 𝛿(𝑡)−1

2𝜎²− 𝜆𝑘 (23)

Thus, the integral in equation (22) takes into account the payout history of the bank up to time t, or whether the MDA has been breached.

Under the risk-neutral probability measure, we are able to price the bank’s liabili-ties as an expected present value of future cash flows, using the risk-free rate as discount rate. Under the risk-neutral measure, the expected discounted value of a bond’s coupon payments in (19) is given by

𝔼 𝑐_𝑖

𝑚𝑖𝑛 𝜏,𝑇 𝑡=1

𝐵_𝑖𝑒^−𝑟𝑡 (24)

In the same way, the expected discounted value of the principal payment in (20) is given by

𝔼 𝐵_𝑖 𝑒^−𝑟𝑇𝟏_𝜏>𝑇 +𝑅_𝑖𝑒^−𝑟𝜏 𝟏_𝜏≤𝑇 (25)

We have now obtained an expression for the value of both the coupon payments

𝐵_𝑖,0=𝔼 ^{𝑚𝑖𝑛 𝜏,𝑇} 𝑐_𝑖

𝑡=1

𝐵_𝑖𝑒^−𝑟𝑡 +𝐵_𝑖 𝑒^−𝑟𝑇𝟏 _𝜏>𝑇 +𝑅_𝑖𝑒^−𝑟𝜏 𝟏 _𝜏≤𝑇 (26) To calculate bond prices using the expressions above, we apply Monte Carlo simu-lation techniques. This is described in chapter 7. For each simusimu-lation, we calculate the term in brackets in equation (26). Then, by averaging over a large number of simulations, we are able to determine the time zero value of the bond.

Before we discuss the simulations, we will address the computation of yields and spreads. So far, we have made the simplifying assumption that all liabilities are issued and sold at par value at time zero. Thus, to obtain consistency in our model, we need to determine the coupon rates such that the market prices of the bonds calculated in equation (26) are indeed equal to the bonds’ par values, i.e. 𝐵_𝑖,0= 𝐵_𝑖. To this end, we assume at time 𝑡= 0 that the bank has issued a certain face value amount of debt, 𝐵_𝑖,0.

Therefore, we find the equilibrium coupon rates by valuing each liability class using equation (26) for a given coupon rate, 𝑐_𝑖. Then, we iterate 𝑐_𝑖 until the market value becomes equal to the face value, 𝐵_𝑖,0=𝐵_𝑖, and 𝑐_𝑖^* is obtained. We then find the equilibrium spread by subtracting the risk-free rate, 𝑠_𝑖^*=𝑐_𝑖^*-𝑟.

Summary

We have now developed a model that can be used to analyze the spreads on the T3 instruments. As we aimed to do at the beginning of this section, we were able to incorporate some of the new bank capital requirements and the consequences of breaching them. Furthermore, using the waterfall approach to consider cash flows, we were able to add the new feature of payment restrictions when a bank enters a critical capital zone. Finally, using a simple yet powerful framework, we are able to calculate the credit spreads of different bank bonds.

In the next chapter, we will apply the developed model numerically for estimated input parameters, and investigate how the spreads on the T3 in our model setting are affected.