# Minimum Protection in DC Funding Pension Plans and Margrabe Options

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Detailed Design of the Reform Act

## 3. Direct Comparison

#### 3.1. Interest Rate Framework

#### 3.2. Expression of the Liability

**Proposition**

**1.**

#### 3.3. Comparison of the Expectations

## 4. ALM-Based Comparison

**Proposition**

**2.**

#### 4.1. Assets

#### 4.2. Liabilities

#### 4.3. Comparison of Margrabe Option Prices

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Proof of Proposition 1

**Proof.**

## References

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^{1.}“Loi du 18 décembre 2015 visant à garantir la pérennité et le caractère social des pensions complémentaires et visant à renforcer le caractère complémentaire par rapport aux pensions de retraite”, published in the Moniteur Belge on 24 December 2015.

**Figure 3.**Comparison of the expectations of the log-liability (upper graph) and of the liability (lower graph) obtained using the horizontal and the vertical methods.

**Table 1.**Simple example illustrating how the horizontal and vertical methods are applied to the plan contributions.

Scenario 1 | Scenario 2 | |
---|---|---|

2016 guaranteed rate | $2.5\%$ | $2.5\%$ |

2017 guaranteed rate | $3.5\%$ | $2\%$ |

2018 liability (horizontal) | ${1.025}^{2}+1.035=2.086$ | ${1.025}^{2}+1.02=2.071$ |

2018 liability (vertical) | $1.025\xb71.035+1.035=2.096$ | $1.025\xb71.02+1.02=2.066$ |

Highest liability | vertical | horizontal |

k | σ | ${\mathit{r}}_{\mathbf{0}}$ | ${\mathit{r}}_{-\mathbf{1}}$ | ${\mathit{r}}_{-\mathbf{2}}$ |
---|---|---|---|---|

$0.15$ | $0.41\%$ | $1.34\%$ | $1.34\%$ | $1.34\%$ |

θ | K | η | μ |
---|---|---|---|

$1.34\%$ | 10 | $25\%$ | $5\%$ |

**Table 4.**Results of the ALM comparison for some example portfolios. The cheapest liability is the horizontal one when H is displayed, and the vertical one when V is displayed.

Description | Composition | Cheapest Liability | ||||||
---|---|---|---|---|---|---|---|---|

x | y | z | $\mathit{\rho}=-1$ | $\mathit{\rho}=-0.5$ | $\mathit{\rho}=0$ | $\mathit{\rho}=0.5$ | $\mathit{\rho}=1$ | |

Only stocks | 100% | 0% | 0% | H | H | V | V | V |

Only bonds | 0% | 100% | 0% | H | H | H | H | H |

Only cash | 0% | 0% | 100% | V | V | V | V | V |

Stocks and bonds | 50% | 50% | 0% | H | H | V | V | V |

Stocks and cash | 50% | 0% | 50% | H | H | V | V | V |

Bonds and cash | 0% | 50% | 50% | V | V | V | V | V |

Equal repartition | 33% | 33% | 33% | H | H | V | V | V |

Typical insurer portfolio | 10% | 80% | 10% | H | H | V | V | V |

**Table 5.**Prices of the options giving the right to exchange the “Typical insurer” asset portfolio (as defined in Table 4) for the horizontal (resp. vertical) liability.

$\mathit{\rho}=-1$ | $\mathit{\rho}=-0.5$ | $\mathit{\rho}=0$ | $\mathit{\rho}=0.5$ | $\mathit{\rho}=1$ | |
---|---|---|---|---|---|

${p}_{0}^{h}$ | $0.0044$ | $0.0077$ | $0.0112$ | $0.0148$ | $0.0183$ |

${p}_{0}^{v}$ | $0.0108$ | $0.0099$ | $0.0090$ | $0.0082$ | $0.0073$ |

${p}_{0}^{h}-{p}_{0}^{v}$ | $-0.0065$ | $-0.0023$ | $0.0022$ | $0.0066$ | $0.0111$ |

© 2017 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/).

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**MDPI and ACS Style**

Devolder, P.; De Valeriola, S.
Minimum Protection in DC Funding Pension Plans and Margrabe Options. *Risks* **2017**, *5*, 5.
https://doi.org/10.3390/risks5010005

**AMA Style**

Devolder P, De Valeriola S.
Minimum Protection in DC Funding Pension Plans and Margrabe Options. *Risks*. 2017; 5(1):5.
https://doi.org/10.3390/risks5010005

**Chicago/Turabian Style**

Devolder, Pierre, and Sébastien De Valeriola.
2017. "Minimum Protection in DC Funding Pension Plans and Margrabe Options" *Risks* 5, no. 1: 5.
https://doi.org/10.3390/risks5010005