# A Discussion of a Risk-Sharing Pension Plan

## Abstract

**:**

## 1. Introduction

## 2. Risk-Sharing Plan

#### 2.1. Investment Strategy

#### 2.2. Accumulation of Contributions

#### 2.3. Funding Level

## 3. Illustrations

#### 3.1. Model Parameterization

#### 3.2. Comparison Plans

#### 3.3. Preliminary Investigation

#### 3.3.1. Simple Membership Profile

#### 3.3.2. Plans Which Vary Only the Investment Strategy

#### 3.3.3. Plans Which Vary Only the AAFs

#### 3.3.4. Devastation

#### 3.3.5. Preliminary Findings

#### 3.4. Further Investigation

#### 3.4.1. Selected Risk-Sharing Pension Plans

#### 3.4.2. Membership Profile

#### 3.4.3. Quantiles of the Average AAF

#### 3.4.4. Analyzing Benefit Stability

#### 3.4.5. Disappointment

## 4. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

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^{1.}Based on the average annual 10-year US Treasury bond returns obtained from https://fred.stlouisfed.org/series/ IRLTLT01USA156N over the years 1956–1999.

**Figure 1.**The average AAF for each generation in the model, as the investment risk adjustment (a) is varied and the AAF adjustment (β) is set equal to zero. The target funding level is $\overline{F}=100$%. In this simple membership, each of the 40 members starts with $1 at time 0, and makes no further contribution. (

**a**) 75% quantile, $\overline{F}=100$%; (

**b**) Median, $\overline{F}=100$%; (

**c**) 25% quantile, $\overline{F}=100$%.

**Figure 2.**The average AAF for each generation in the model, as the AAF adjustment (β) is varied and the investment risk adjustment (a) is set equal to zero. The target funding level is $\overline{F}=100$%. In this simple membership, each of the 40 members starts with $1 at time 0, and makes no further contribution. (

**a**) 75% quantile, $\overline{F}=100$%; (

**b**) Median, $\overline{F}=100$%; (

**c**) 25% quantile, $\overline{F}=100$%.

**Figure 3.**Devastation. The number of consecutive years in which there is no retirement benefit paid to a retiring generation, plotted against the probability. The investment risk adjustment (a) is varied and the AAF adjustment (β) is set equal to zero, and the target funding level $\overline{F}=100$%.

**Figure 4.**Devastation. The number of consecutive years in which there is no retirement benefit paid to a retiring generation, plotted against the probability. The AAF adjustment (β) is varied and the investment risk adjustment (a) is set equal to zero, and the target funding level $\overline{F}=100$%.

**Figure 5.**The annualized average accumulation factor for each generation in the model, for the more realistic membership, as both the investment risk adjustment (a) and the AAF adjustment (β) are varied. The target funding level is $\overline{F}=100$%. In this membership profile, the kth generation contributes $$(40-k+1)$ at time 0, and contributes a further $1 per annum at the end of each year that they are not retired. (

**a**) 75% quantile, $\overline{F}=100$%; (

**b**) Median, $\overline{F}=100$%; (

**c**) 25% quantile, $\overline{F}=100$%.

**Figure 6.**The annualized average accumulation factor for each generation in the model, for the more realistic membership, as both the investment risk adjustment (a) and the AAF adjustment (β) are varied. The target funding level is $\overline{F}=120$%. In this membership profile, the kth generation contributes $$(40-k+1)$ at time 0, and contributes a further $1 per annum at the end of each year that they are not retired. (

**a**) 75% quantile, $\overline{F}=120$%; (

**b**) Median, $\overline{F}=120$%; (

**c**) 25% quantile, $\overline{F}=120$%.

**Figure 7.**Disappointment for the more realistic membership when the target funding level $\overline{F}=100$%. The probability of a declining average AAF plotted against the number of consecutive years that the decline occurs over. Both the investment risk adjustment (a) and the AAF adjustment (β) are varied.

**Figure 8.**Disappointment for the more realistic membership when the target funding level $\overline{F}=120$%. The probability of a declining average AAF plotted against the number of consecutive years that the decline occurs over. Both the investment risk adjustment (a) and the AAF adjustment (β) are varied.

**Figure 9.**Disappointment for the more realistic membership when the target funding level $\overline{F}=100$%. The probability of a declining AAF plotted against the number of consecutive years that the decline occurs over. Both the investment risk adjustment (a) and the AAF adjustment (β) are varied.

**Figure 10.**Disappointment for the more realistic membership when the target funding level $\overline{F}=120$%. The probability of a declining AAF plotted against the number of consecutive years that the decline occurs over. Both the investment risk adjustment (a) and the AAF adjustment (β) are varied.

Plan Type | Investment Risk Adjustment a | AAF Adjustment β |
---|---|---|

Benchmark plan | 0 | 0 |

Risk-sharing | $0.2$ | $0.2$ |

Risk-sharing | $0.2$ | $0.4$ |

Risk-sharing | $0.4$ | $0.2$ |

DC plan | N/A | N/A |

Plan Type | a | β | Target Funding Level $\overline{\mathit{F}}=100$% | ||

IQR Instability | Qu. Inequity | Median Inequity | |||

Risk-sharing | $0.2$ | $0.2$ | $9.4\%$ | $10.3\%$ | $3.8\%$ |

Risk-sharing | $0.2$ | $0.4$ | $15.7\%$ | $15.7\%$ | $3.7\%$ |

Risk-sharing | $0.4$ | $0.2$ | $9.7\%$ | $10.5\%$ | $3.8\%$ |

DC | N/A | N/A | $40.7\%$ | $40.7\%$ | $3.3\%$ |

Plan Type | $\mathit{a}$ | $\mathit{\beta}$ | Target Funding Level $\overline{\mathit{F}}=\mathbf{120}$% | ||

IQR Instability | Qu. Inequity | Median Inequity | |||

Risk-sharing | $0.2$ | $0.2$ | $9.5\%$ | $12.1\%$ | $4.4\%$ |

Risk-sharing | $0.2$ | $0.4$ | $14.9\%$ | $19.4\%$ | $8.9\%$ |

Risk-sharing | $0.4$ | $0.2$ | $9.5\%$ | $12.2\%$ | $4.6\%$ |

DC | N/A | N/A | $40.7\%$ | $40.7\%$ | $3.3\%$ |

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Donnelly, C.
A Discussion of a Risk-Sharing Pension Plan. *Risks* **2017**, *5*, 12.
https://doi.org/10.3390/risks5010012

**AMA Style**

Donnelly C.
A Discussion of a Risk-Sharing Pension Plan. *Risks*. 2017; 5(1):12.
https://doi.org/10.3390/risks5010012

**Chicago/Turabian Style**

Donnelly, Catherine.
2017. "A Discussion of a Risk-Sharing Pension Plan" *Risks* 5, no. 1: 12.
https://doi.org/10.3390/risks5010012