# Analyzing How the Social Security Reserve Fund in Spain Affects the Sustainability of the Pension System

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## Abstract

**:**

## 1. Introduction

## 2. The Spanish Public Pensions System and the Social Security Reserve Fund

## 3. Literature Review

## 4. An Alternative Statistical Model for Evaluating the Sustainability of the Pension System

#### 4.1. A Statistical Model for the Total Payments Made from the Pensions System

#### 4.2. The Basic Model of Sustainability

#### 4.3. Computing Probabilities of Unsustainability

#### 4.4. Beyond Unsustainability

#### 4.5. Theoretical Extension of the Combination of Private and Public Pension Systems

## 5. Assessing the Performance of the Proposed Model

#### 5.1. Preliminary Analysis

#### 5.2. Probability of Unsustainability

#### 5.2.1. Probability of Unsustainability with Homogeneous or Heterogeneous Pensioners

#### With Homogeneous Pensioners

#### With Heterogeneous Pensioners

#### 5.2.2. Probability of Unsustainability and Deficit

#### 5.3. Expected Time to Unsustainability

#### 5.4. Size of the Reserve Fund Required for a Given Probability of Unsustainability

## 6. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Volterra Integral Equation

## Appendix B. Bessel Function

## Notes

1 | We are grateful to the Ministry of Inclusion, Social Security and Migrations for providing the raw data. |

2 | Note that $D\left(t\right)$ is a non-random variable; for the sake of simplicity, this potential risk to the pensions system is omitted. Nor do we explicitly consider the returns generated by the Reserve Fund (which are defined by the internal rate of return of each asset at the time of purchase) or the adjustments made to reflect the amortisation or disposal of assets. Therefore, the only random component of our model is that of the pensions payments made, $X\left(t\right)$. Nevertheless, in the empirical part of this study, since the variable $R\left(t\right)$ is the balance of the Reserve Fund, which incorporates information on endowments, provisions and net generated returns, the latter are implicitly taken into account. |

3 | Note that, although $R\left(t\right)$ is expressed in continuous time, it could be taken as an approximate representation of (yearly) social security flows. |

4 | Note that, as we use the equation $D\left(t\right)=(1+\tau )E\left[N\right(t\left)\right]E\left[Y\right]$ to calculate the value of $\tau $, we are implicitly assuming that p varies with time, although it is not considered random, as opposed to the number of pensioners and the aggregate annual payment they receive. Extending this model to incorporate p as a random variable is a question that remains to be addressed in future research. |

5 | See Feller (1971, chp. 2) for details about this special function, which can be implemented in most statistical software packages, such as R, Matlab, Mathematica and WinRATS. |

6 | These empirical results should be taken with caution due to the small sample considered. The reason for this limitation is that the Reserve Fund only began operating in 2000. |

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**Figure 1.**Evolution of the total payments in contributory pensions, the balance of the Social Security Reserve Fund and revenues from Social Security contributions in Spain during the period 2001–2019. Source: Devised by the authors, based on data offered in Table 1.

**Figure 2.**Empirical (thick) and fitted (thin) distribution of the total annual payments. Source: Devised by the authors.

**Figure 4.**Probabilities of unsustainability for the Non-mixture model and for the mixture model. Source: Devised by the authors.

**Figure 5.**Probabilities ${\tilde{\psi}}_{z}\left(r\right)$ in 2016, 2017, 2018 and 2019. Source: Devised by the authors.

**Figure 6.**(

**left**); Expected time to unsustainability; (

**right**) The thick line represents the exact value and the thin one, the estimated value and the corresponding 95% confidence interval. The thick line represents the upper bound of the interval and the thin line, the lower bound. Source: Devised by the authors.

**Figure 7.**Required balance of the Social Security Reserve Fund for a given probability of unsustainability. Devised by the authors.

Year (t) | Number of Contributors $\left(\mathit{N}\right(\mathit{t}\left)\right)$ | Contributions $\left(\mathit{D}\right(\mathit{t}\left)\right)$ | Reserve Fund $\left(\mathit{R}\right(\mathit{t}\left)\right)$ | Total Annual Payment $\left(\mathit{X}\right(\mathit{t}\left)\right)$ |
---|---|---|---|---|

2000 | 15,063 | 60,539.03 | 604 | 49,829 |

2001 | 15,650 | 63,466.69 | 2433 | 51,060 |

2002 | 16,126 | 65,132.58 | 6169 | 51,635 |

2003 | 16,614 | 67,624.07 | 12,025 | 53,077 |

2004 | 17,082 | 69,744.38 | 19,330 | 54,356 |

2005 | 17,835 | 72,177.67 | 27,185 | 55,930 |

2006 | 18,596 | 75,334.90 | 35,879 | 57,940 |

2007 | 19,152 | 78,846.55 | 45,716 | 59,585 |

2008 | 19,006 | 80,723.89 | 57,223 | 63,129 |

2009 | 17,917 | 79,447.23 | 60,022 | 67,158 |

2010 | 17,582 | 78,538.38 | 64,375 | 70,449 |

2011 | 17,326 | 78,420.62 | 66,815 | 73,891 |

2012 | 16,739 | 75,341.94 | 63,008 | 77,254 |

2013 | 16,228 | 72,926.41 | 53,744 | 80,710 |

2014 | 16,492 | 73,823.12 | 41,634 | 83,498 |

2015 | 17,017 | 74,438.01 | 32,481 | 85,503 |

2016 | 17,518 | 76,461.03 | 15,020 | 87,811 |

2017 | 18,127 | 79,485.10 | 8095 | 89,242 |

2018 | 18,701 | 82,836.57 | 5043 | 91,615 |

2019 | 19,189 | 88,057.37 | 2153 | 95,719 |

Year | $\widehat{\mathit{\tau}}$ |
---|---|

2000–2004 | ≈0 |

2005 | 0.031 |

2006 | 0.070 |

2007 | 0.126 |

2008 | 0.123 |

2009 | 0.135 |

2010 | 0.122 |

2011 | 0.121 |

2012 | 0.077 |

2013 | 0.042 |

2014 | 0.055 |

2015 | 0.064 |

2016 | 0.093 |

2017 | 0.136 |

2018 | 0.184 |

2019 | 0.258 |

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Gómez-Déniz, E.; Pérez-Rodríguez, J.V.; Sosvilla-Rivero, S.
Analyzing How the Social Security Reserve Fund in Spain Affects the Sustainability of the Pension System. *Risks* **2022**, *10*, 120.
https://doi.org/10.3390/risks10060120

**AMA Style**

Gómez-Déniz E, Pérez-Rodríguez JV, Sosvilla-Rivero S.
Analyzing How the Social Security Reserve Fund in Spain Affects the Sustainability of the Pension System. *Risks*. 2022; 10(6):120.
https://doi.org/10.3390/risks10060120

**Chicago/Turabian Style**

Gómez-Déniz, Emilio, Jorge V. Pérez-Rodríguez, and Simón Sosvilla-Rivero.
2022. "Analyzing How the Social Security Reserve Fund in Spain Affects the Sustainability of the Pension System" *Risks* 10, no. 6: 120.
https://doi.org/10.3390/risks10060120