# Contingent Convertible Debt: The Impact on Equity Holders

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Model

**Proof.**

#### 2.1. The Debt Ratio $\mathbb{P}$-Dynamics

**Assumption**

**1.**

**Assumption**

**2.**

#### 2.2. Conversion

#### 2.3. Default

## 3. Stochastic Optimum Control Problem

**Assumption**

**3.**

- 1.
- The asset returns are no longer uncertain, that is, ${R}_{t}={m}_{T}$, ${\sigma}_{t}^{2}=0;$
- 2.
- There is no more possibility of conversion, that is, the CoCo debt becomes a standard debt;
- 3.
- The dividends are the remaining part of the returns once the interest rate payment on the debts is deducted:$${\delta}_{t}{A}_{t}=max\left(\right)open="("\; close=")">{m}_{T}{A}_{t}-{\mu}_{t}{D}_{t-1},0$$
- 4.
- If the dividend payment ${m}_{T}{A}_{t}-{\mu}_{t}{D}_{t-1}$ is positive, then there is no variation of the debt value, that is, ${D}_{t}={D}_{t-1}$ or, equivalently, ${\eta}_{t}=0;$
- 5.
- The risk-free rate ${r}_{t}$ is constant and equal to $r.$

**Lemma**

**1.**

## 4. Numerical Results

#### 4.1. Data

#### 4.2. Empirical Results

#### 4.3. Sensitivity Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. The Floating Rates

#### Appendix A.1. The $\mathbb{Q}$-Dynamics of the Debt Ratio

#### Appendix A.2. Credit Sensitive Debt

**Lemma**

**A1.**

#### Appendix A.3. Convertible Contingent Debt

**Lemma**

**A2.**

**Remark**

**A1.**

#### Appendix A.4. Conditional Probabilities

**Lemma A3**(No conversion risk-neutral probability)

**.**

**Lemma A4**(Conversion and survival risk-neutral probability)

**.**

**Lemma A5**(Survival risk-neutral probabilities)

**.**

#### Appendix A.5. Approximations

#### Appendix A.5.1. Approximation of b(x,0)

#### Appendix A.5.2. Approximation of b(x,y) and c(x)

## Appendix B. Proofs

#### Appendix B.1. Standard Bond Floating Coupon Rate

**Proof of**

**Lemma A1.**

#### Appendix B.2. Convertible Bond Floating Coupon Rate

**Proof of**

**Lemma A2.**

#### Appendix B.3. The Equity Value Variation

#### Appendix B.4. The Value of Expected Discounted Dividends at T

**Proof of**

**Lemma 1.**

#### Appendix B.5. Proofs of Lemmas A3–A5

**Proof of**

**Lemma A3.**

**Proof of**

**Lemma A4.**

**Proof of**

**Lemma A5.**

## Appendix C. Calibration of Default and Conversion Probabilities

## Appendix D. Finding the Optimal Dividend Rate Sequence

#### Appendix D.1. Post-Conversion Optimal Dividend Rates

#### Appendix D.2. Pre-Conversion Optimal Dividend Rates

#### Appendix D.3. Proofs

#### Appendix D.3.1. Proof of Equation (A7)

#### Appendix D.3.2. Proof of Equation (A12)

#### Appendix D.3.3. Proof of Equation (A14)

## References

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**Figure 2.**One-year conversion and default probabilities. All curves are obtained from a Monte Carlo simulation based on $2\times {10}^{6}$ paths. The parameters are described in Table 2. Each column corresponds to a specific bank and each line corresponds to a specific year. The dark grey dotted line represents the conversion probability. The black line represents the one-year default probability in the presence of CoCos in the debt structure. The light gray circle dash line represents the one-year default probability without CoCos in the debt structure. The vertical dashed-dotted line corresponds to the trigger level ($\alpha $) of 1–5.125% of risk-weighted assets. The dark grey dotted line corresponds to the debt ratio observed for the specified bank at the specified year.

**Figure 3.**CoCos and standard debts coupons. All curves are obtained from a Monte Carlo simulation based on $2\times {10}^{6}$ paths. The parameters are described in Table 2. Each column corresponds to a specific bank and each line corresponds to a specific year. The dark grey dotted line represents the coupon on the CoCo debt. The black line represents the coupon on the standard debt in the presence of CoCos in the debt structure. The light gray circle dash line represents the coupon on the standard debt without CoCos in the debt structure. The vertical dashed-dotted line corresponds to the trigger level ($\alpha $) of 1–5.125% of risk-weighted assets. The dark grey dotted line corresponds to the debt ratio observed for the specified bank at the specified year.

**Figure 4.**Cost of the debt. All curves are obtained from a Monte Carlo simulation based on $2\times {10}^{6}$ paths. The parameters are described in Table 2. Each column corresponds to a specific bank and each line corresponds to a specific year. The black line represents the cost of debt when there is CoCos in the debt structure. The light gray dashed line represents the cost of debt without CoCos in the debt structure. The vertical black dashed-dotted line corresponds to the trigger level ($\alpha $) of 1–5.125% of risk-weighted assets. The dark grey dotted line corresponds to the debt ratio observed for the specified bank at the specified year.

**Figure 5.**Cost of the capital. All curves are obtained from a Monte Carlo simulation based on $2\times {10}^{6}$ paths. The parameters are described in Table 2. Each column corresponds to a specific bank and each line corresponds to a specific year. The black line represents the cost of capital when there is CoCos in the debt structure. The light gray dashed line represents the cost of capital without CoCos in the debt structure. The vertical dashed-dotted line corresponds to the trigger level ($\alpha $) of 1–5.125% of risk-weighted assets. The dark grey dotted line corresponds to the debt ratio observed for the specified bank at the specified year.

**Figure 6.**Optimal dividend at time $T=0$. All curves are obtained from a Monte Carlo simulation based on $2\times {10}^{6}$ paths. The parameters are described in Table 2. Each column corresponds to a specific bank and each line corresponds to a specific year. The black line represents the optimal dividend rate when there is CoCos in the debt structure. The light gray dashed line represents the optimal dividend rate without CoCos in the debt structure. The vertical dashed-dotted line corresponds to the trigger level ($\alpha $) of 1–5.125% of risk-weighted assets. The dark grey dotted line corresponds to the debt ratio observed for the specified bank at the specified year.

**Figure 7.**(Standardized) value of discounted cumulated dividends at time $T=0$. All curves are obtained from a Monte Carlo simulation based on $2\times {10}^{6}$ paths. The parameters are described in Table 2. Each column corresponds to a specific bank and each line corresponds to a specific year. The black line represents the case when there is CoCos in the debt structure. The light gray dashed line represents the case without CoCos in the debt structure. The vertical dashed-dotted line corresponds to the trigger level ($\alpha $) of 1–5.125% of risk-weighted assets. The dark grey dotted line corresponds to the debt ratio observed for the specified bank at the specified year.

**Figure 9.**Optimal dividend and discounted cumulated dividends at time $T=0$, with high Return on Equity.

**Figure 10.**Optimal dividend and discounted cumulated dividends at time $T=0$, with higher default probabilities.

Société Générale | Royal Bank of Canada | Bank of America | |
---|---|---|---|

Equity/Asset | 5.41% | 6.43% | 10.62% |

Deposit/Asset | 40.60% | 77.37% | 58.86% |

Bonds (including CoCo)/Asset | 53.99% | 16.20% | 32.52% |

Société Générale | Royal Bank of Canada | Bank of America | |||||||
---|---|---|---|---|---|---|---|---|---|

2006 | 2008 | 2015 | 2006 | 2008 | 2015 | 2006 | 2008 | 2015 | |

Returns | |||||||||

${r}_{t}$ | $3.76\%$ | $1.99\%$ | $-0.2\%$ | $4.07\%$ | $0.89\%$ | $0.51\%$ | $4.94\%$ | $0.28\%$ | $0.73\%$ |

${r}_{t}^{E}$ | $20.04\%$ | $7.02\%$ | $6.23\%$ | $23.21\%$ | $17.64\%$ | $18.42\%$ | $18.07\%$ | $1.81\%$ | $6.27\%$ |

m | $3.07\%$ | $3.43\%$ | $1.26\%$ | $3.5\%$ | $4.35\%$ | $2.25\%$ | $3.84\%$ | $4.66\%$ | $0.36\%$ |

${\sigma}^{2}$ | $2\%$ | $2\%$ | $2\%$ | ||||||

Initial debt structure | |||||||||

${F}_{0}$ | $31.41\%$ | $29.71\%$ | $44.35\%$ | $79.58\%$ | $86.39\%$ | $83.45\%$ | $54.78\%$ | $56.13\%$ | $70.29\%$ |

${y}_{0}$ | $1\%$ | $1\%$ | $1\%$ | ||||||

Conversion risk | |||||||||

$\alpha $ | $98.47\%$ | $98.43\%$ | $98.63\%$ | $97.86\%$ | $98.03\%$ | $98.03\%$ | $96.30\%$ | $96.28\%$ | $96.65\%$ |

${\beta}_{C}$ | $74.48$ | $66.55$ | $39.85$ | ||||||

${\theta}_{C}$ | $97.42\%$ | $96.82\%$ | $94.69\%$ | ||||||

${\rho}_{C}$ | $90\%$ | $90\%$ | $90\%$ | ||||||

Default risk | |||||||||

${\theta}_{D}$ | $107.7\%$ | $104.11\%$ | $107.29\%$ | ||||||

${\beta}_{D}$ | $47.2$ | $69.9$ | $40.46$ | ||||||

${\lambda}_{D}$ | 0 | 0 | 0 | ||||||

${\rho}_{D}$ | $0.4$ | $0.4$ | $0.4$ | ||||||

Numerical scheme | |||||||||

T | 30 | 30 | 30 | ||||||

${\Delta}_{x}$ | $0.002$ | $0.002$ | $0.002$ | ||||||

${\Delta}_{t}$ | 1 | 1 | 1 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Boursicot, D.; Gauthier, G.; Pourkalbassi, F.
Contingent Convertible Debt: The Impact on Equity Holders. *Risks* **2019**, *7*, 47.
https://doi.org/10.3390/risks7020047

**AMA Style**

Boursicot D, Gauthier G, Pourkalbassi F.
Contingent Convertible Debt: The Impact on Equity Holders. *Risks*. 2019; 7(2):47.
https://doi.org/10.3390/risks7020047

**Chicago/Turabian Style**

Boursicot, Delphine, Geneviève Gauthier, and Farhad Pourkalbassi.
2019. "Contingent Convertible Debt: The Impact on Equity Holders" *Risks* 7, no. 2: 47.
https://doi.org/10.3390/risks7020047