# Nonparametric Estimation of Extreme Quantiles with an Application to Longevity Risk

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Longevity Risk

## 3. The Conditional Quantile and Longevity Risk

**Proposition**

**1.**

## 4. Estimators of the Unconditional cdf

**Definition**

**1.**

**Theorem**

**1.**

**Proposition**

**2.**

- If $T(\xb7)$ is the cdf of a heavier tailed distribution than ${F}_{X}(\xb7)$ the bias is the sum of two negative terms. The results are greater than the CKE overestimation of the quantile.
- If $T(\xb7)$ is the cdf of a lighter tailed distribution than ${F}_{X}(\xb7)$ the bias is the sum of a negative term and a positive term associated with the positive sign of (22). In this case, we could overestimate or subestimate the quantile; even, the two terms can be compensated.

#### 4.1. Smoothing Parameter of BTKE

**Remark**

**1.**

- If T is heavier tailed than ${F}_{X}$ then $T\left({x}_{p}\right)\le {F}_{X}\left({x}_{p}\right)$, ${M}^{-1}\left[T\left({x}_{p}\right)\right]\le {M}^{-1}\left(p\right)$ and ${b}_{p}\ge {b}_{p}^{*}$.
- If ${F}_{X}$ is heavier tailed than T then $T\left({x}_{p}\right)\ge {F}_{X}\left({x}_{p}\right)$, ${M}^{-1}\left[T\left({x}_{p}\right)\right]\ge {M}^{-1}\left(p\right)$ and ${b}_{p}\le {b}_{p}^{*}$.

## 5. Data Analysis

#### Application to the Annuity Longevity Risk

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proofs

**Proof**

**of**

**Proposition**

**1.**

**Proof**

**of**

**Theorem**

**1.**

## Appendix B. Simulation Study

Distribution | cdf | Parameters |
---|---|---|

We | $1-{e}^{-{\left(\frac{x}{\lambda}\right)}^{\gamma}}$ | $(\lambda ,\gamma )\in \left\{(1,0.75),(1,1.5)\right\}$ |

Ln | ${\int}_{-\infty}^{logx}\frac{1}{\sqrt{2\pi {\sigma}^{2}}}{e}^{-\frac{{(t-\mu )}^{2}}{2{\sigma}^{2}}}dt$ | $(\mu ,\sigma )\in \left\{(0,0.5),(0,1)\right\}$ |

Ln-Pa | $\alpha {\int}_{-\infty}^{logx}\frac{1}{\sqrt{2\pi {\sigma}^{2}}}{e}^{-\frac{{(t-\mu )}^{2}}{2{\sigma}^{2}}}dt$ | $(\alpha ,\mu ,\sigma ,\lambda ,\rho ,c)\in \left\{(0.7,0,1,1,1,-1),(0.7,0,1,1,1.1,-1)\right\}$ |

Ln-Pa | $+(1-\alpha )\left[1-{\left(\frac{x-c}{\lambda}\right)}^{-\rho}\right]$ | $(\alpha ,\mu ,\sigma ,\lambda ,\rho ,c)\in \left\{(0.3,0,1,1,1,-1),(0.3,0,1,1,1.1,-1)\right\}$ |

**Table A2.**Theoretical values of $C{Q}_{{p}_{a}}$ for each distribution in Table A1, given a and ${p}_{a}$.

Distribution | ${\mathit{p}}_{\mathit{a}}=0.950$ | ${\mathit{p}}_{\mathit{a}}=0.990$ | ${\mathit{p}}_{\mathit{a}}=0.995$ | |
---|---|---|---|---|

$a=0.5$ | We$(1,1.5)$ | 2.239 | 2.908 | 3.173 |

Ln$(0,0.5)$ | 3.340 | 4.372 | 4.851 | |

We$(1,0.75)$ | 5.498 | 9.008 | 10.644 | |

Ln$(0,1)$ | 8.662 | 15.465 | 19.281 | |

$a=1$ | We$(1,1.5)$ | 2.518 | 3.155 | 3.411 |

Ln$(0,0.5)$ | 4.599 | 5.772 | 6.317 | |

We$(1,0.75)$ | 6.341 | 9.957 | 11.631 | |

Ln$(0,1)$ | 11.156 | 19.115 | 23.532 | |

Ln-Pa $(0.7,0,1,1,1.1,-1)$ | 11.248 | 41.655 | 78.235 | |

Ln-Pa $(0.7,0,1,1,1,-1)$ | 13.017 | 59.189 | 119.030 | |

Ln-Pa $(0.3,0,1,1,1.1,-1)$ | 20.902 | 92.338 | 174.248 | |

Ln-Pa $(0.3,0,1,1,1,-1)$ | 27.162 | 139.003 | 279.000 | |

$a=2$ | Ln-Pa $(0.7,0,1,1,1.1,-1)$ | 18.094 | 73.965 | 139.512 |

Ln-Pa $(0.7,0,1,1,1,-1)$ | 22.298 | 109.790 | 220.508 | |

Ln-Pa $(0.3,0,1,1,1.1,-1)$ | 33.922 | 149.225 | 281.095 | |

Ln-Pa $(0.3,0,1,1,1,-1)$ | 44.732 | 227.337 | 455.672 |

**Table A3.**Ratios between the MSE (Mean Squared Error) of Beta Transformed Kernel Estimator (BTKE) and Kernel Estimator (KE) of the $C{Q}_{{p}_{a}}$ and MSE of the estimation based on empirical distribution (Ln-Pa distributions with parameters $(\alpha ,\sigma ,\lambda ,\rho )$, given $\mu =0$ and $c=-1$).

$\mathit{n}=500$ | $\mathit{n}=5000$ | |||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{p}}_{\mathit{a}}=\mathbf{0}.\mathbf{950}$ | ${\mathit{p}}_{\mathit{a}}=\mathbf{0}.\mathbf{990}$ | ${\mathit{p}}_{\mathit{a}}=\mathbf{0}.\mathbf{950}$ | ${\mathit{p}}_{\mathit{a}}=\mathbf{0}.\mathbf{990}$ | ${\mathit{p}}_{\mathit{a}}=\mathbf{0}.\mathbf{995}$ | ||||||||||||||||

CKE | BTKE | CKE | BTKE | CKE | BTKE | CKE | BTKE | CKE | BTKE | |||||||||||

$\mathit{a}$ | Distribution | Ch | Ln | Ch | Ln | Ch | Ln | Ch | Ln | Ch | Ln | |||||||||

$0.5$ | We$(1,1.5)$ | 0.87 | 0.99 | 1.52 | 0.85 | 1.21 | 1.48 | 0.94 | 0.99 | 1.15 | 0.91 | 1.03 | 1.16 | 0.91 | 1.02 | 1.08 | ||||

We$(1,0.75)$ | 0.89 | 1.27 | 2.45 | 0.90 | 1.43 | 1.76 | 0.97 | 1.03 | 2.71 | 0.97 | 1.11 | 1.73 | 0.96 | 1.13 | 1.49 | |||||

Ln$(0,0.5)$ | 0.95 | 0.92 | 0.94 | 0.96 | 0.76 | 0.86 | 0.99 | 0.98 | 0.97 | 0.99 | 0.95 | 0.88 | 1.00 | 0.92 | 0.83 | |||||

Ln$(0,1)$ | 1.00 | 0.84 | 0.87 | 1.05 | 0.63 | 0.77 | 1.00 | 0.97 | 0.97 | 1.01 | 0.88 | 0.93 | 1.02 | 0.83 | 0.90 | |||||

1 | We$(1,1.5)$ | 0.82 | 1.06 | 1.47 | 0.73 | 1.10 | 1.16 | 0.92 | 0.97 | 1.15 | 0.91 | 1.01 | 1.07 | 0.91 | 1.04 | 1.04 | ||||

We$(1,0.75)$ | 0.91 | 1.16 | 1.43 | 0.92 | 1.22 | 1.13 | 0.97 | 1.04 | 1.18 | 0.98 | 1.10 | 0.17 | 0.97 | 1.14 | 1.09 | |||||

Ln$(0,0.5)$ | 0.99 | 0.93 | 0.96 | 0.99 | 0.79 | 0.87 | 1.00 | 0.99 | 0.99 | 1.00 | 0.95 | 0.97 | 1.00 | 0.93 | 0.96 | |||||

Ln$(0,1)$ | 1.02 | 0.84 | 0.89 | 1.00 | 0.57 | 0.71 | 1.01 | 0.96 | 0.98 | 1.01 | 0.88 | 0.93 | 1.00 | 0.83 | 0.89 | |||||

Ln-Pa $(0.7,1,1,1.1)$ | 0.90 | 1.01 | 0.91 | 1.00 | 0.74 | 0.53 | 0.97 | 0.94 | 0.92 | 0.97 | 0.89 | 0.85 | 0.99 | 0.95 | 0.79 | |||||

Ln-Pa $(0.7,1,1,1)$ | 0.86 | 0.93 | 0.84 | 1.00 | 0.75 | 0.52 | 0.97 | 0.95 | 0.92 | 0.98 | 0.92 | 0.84 | 0.99 | 0.91 | 0.66 | |||||

Ln-Pa $(0.3,1,1,1.1)$ | 0.90 | 0.89 | 0.82 | 1.00 | 0.82 | 0.54 | 0.98 | 0.94 | 0.90 | 0.98 | 0.91 | 0.85 | 0.96 | 0.94 | 0.81 | |||||

Ln-Pa $(0.3,1,1,1)$ | 0.86 | 0.92 | 0.84 | 1.00 | 1.05 | 0.72 | 0.97 | 0.91 | 0.89 | 0.97 | 0.95 | 0.86 | 1.00 | 1.05 | 0.85 | |||||

2 | Ln-Pa $(0.7,1,1,1.1)$ | 0.95 | 0.78 | 0.66 | 0.99 | 2.70 | 59.93 | 0.98 | 0.92 | 0.89 | 0.97 | 0.86 | 0.72 | 1.00 | 0.88 | 0.67 | ||||

Ln-Pa $(0.7,1,1,1)$ | 0.94 | 0.75 | 0.63 | 1.00 | 0.95 | 49.04 | 0.98 | 0.90 | 0.87 | 0.99 | 0.93 | 0.69 | 1.00 | 0.82 | 0.74 | |||||

Ln-Pa $(0.3,1,1,1.1)$ | 0.96 | 0.72 | 0.72 | 1.00 | 0.46 | 0.46 | 0.98 | 0.89 | 0.86 | 0.98 | 0.97 | 0.84 | 0.99 | 1.01 | 0.79 | |||||

Ln-Pa $(0.3,1,1,1)$ | 0.94 | 0.39 | 0.78 | 1.00 | 0.70 | 0.70 | 0.98 | 0.90 | 0.87 | 0.99 | 1.00 | 0.84 | 1.00 | 0.90 | 0.66 |

## Appendix C. Estimated Conditional Significant Levels

**Table A4.**Quotients between empirical and theoretical $(1-{p}_{a})$ significance levels of estimated $C{Q}_{{p}_{a}}$ for the age-at-death of the population aged over 65 in Spain, separately for men and women, from 2011 to 2017, using a Beta Transformed Kernel Estimator (BTKE) or a Generalized Pareto Distribution (GPD).

BTKE for Each a | GPD for Each a | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{p}}_{\mathit{a}}$ | Year | 65 | 75 | 85 | 95 | 100 | 65 | 75 | 85 | 95 | 100 | ||

Men | 0.990 | 2011 | 1.00 | 1.01 | 0.98 | 0.90 | 1.03 | 0.97 | 0.97 | 0.98 | 1.96 | 14.52 | |

2012 | 1.00 | 0.99 | 0.99 | 0.98 | 0.93 | 1.00 | 0.99 | 0.99 | 1.96 | 14.19 | |||

2013 | 0.98 | 1.00 | 0.98 | 0.98 | 0.94 | 0.98 | 0.97 | 0.98 | 2.22 | 15.33 | |||

2014 | 1.00 | 1.00 | 0.96 | 0.97 | 0.82 | 0.96 | 1.00 | 0.96 | 2.33 | 16.43 | |||

2015 | 1.01 | 0.99 | 1.00 | 1.01 | 0.92 | 0.97 | 0.99 | 1.00 | 1.43 | 10.42 | |||

2016 | 0.99 | 0.98 | 0.97 | 0.91 | 0.98 | 0.99 | 0.98 | 1.00 | 0.91 | 3.04 | |||

2017 | 1.01 | 0.99 | 0.97 | 0.97 | 0.69 | 0.98 | 0.99 | 0.97 | 0.97 | 1.88 | |||

0.995 | 2011 | 0.99 | 1.01 | 0.97 | 1.04 | 1.03 | 0.99 | 0.97 | 0.97 | 3.92 | 29.05 | ||

2012 | 0.99 | 1.00 | 0.95 | 1.03 | 0.93 | 0.99 | 0.96 | 0.95 | 3.92 | 28.37 | |||

2013 | 0.97 | 0.99 | 0.94 | 0.99 | 0.71 | 0.97 | 0.99 | 0.94 | 4.45 | 30.66 | |||

2014 | 0.98 | 0.98 | 0.96 | 0.92 | 0.93 | 0.98 | 0.98 | 1.00 | 4.66 | 32.87 | |||

2015 | 1.01 | 1.01 | 0.96 | 0.95 | 1.02 | 0.96 | 0.97 | 0.99 | 2.85 | 20.84 | |||

2016 | 1.00 | 1.00 | 0.97 | 1.04 | 1.08 | 0.96 | 0.96 | 0.97 | 0.92 | 6.07 | |||

2017 | 0.98 | 0.97 | 1.01 | 1.04 | 0.99 | 0.98 | 0.97 | 0.96 | 0.96 | 3.76 | |||

0.999 | 2011 | 1.00 | 0.99 | 0.92 | 0.87 | 1.29 | 0.94 | 0.99 | 2.07 | 19.59 | 145.24 | ||

2012 | 0.97 | 0.96 | 0.97 | 0.96 | 0.00 | 0.97 | 0.96 | 2.05 | 19.58 | 141.86 | |||

2013 | 0.98 | 1.00 | 0.97 | 1.20 | 0.00 | 0.98 | 1.04 | 2.21 | 22.25 | 153.30 | |||

2014 | 1.00 | 1.01 | 0.99 | 0.99 | 0.00 | 1.00 | 1.12 | 2.29 | 23.28 | 164.34 | |||

2015 | 0.93 | 0.95 | 1.00 | 0.84 | 1.02 | 0.93 | 0.95 | 1.48 | 14.27 | 104.19 | |||

2016 | 1.00 | 0.96 | 0.91 | 0.98 | 1.08 | 0.95 | 0.96 | 0.95 | 3.92 | 30.37 | |||

2017 | 0.95 | 1.00 | 0.91 | 0.60 | 0.99 | 0.98 | 0.93 | 0.99 | 2.29 | 18.81 | |||

Women | 0.990 | 2011 | 1.00 | 0.99 | 0.96 | 0.92 | 1.02 | 0.97 | 0.99 | 0.96 | 1.95 | 11.38 | |

2012 | 1.02 | 0.99 | 1.00 | 0.96 | 0.97 | 0.98 | 0.99 | 0.96 | 0.96 | 2.13 | |||

2013 | 1.00 | 0.98 | 0.98 | 0.98 | 0.90 | 1.00 | 0.98 | 0.98 | 0.98 | 1.91 | |||

2014 | 1.00 | 1.00 | 0.97 | 0.95 | 1.00 | 0.97 | 1.00 | 0.97 | 0.95 | 2.77 | |||

2015 | 0.99 | 1.01 | 0.98 | 0.98 | 0.90 | 0.99 | 0.96 | 0.98 | 0.98 | 2.47 | |||

2016 | 0.98 | 1.00 | 0.99 | 0.98 | 0.93 | 0.98 | 1.00 | 0.99 | 0.98 | 0.96 | |||

2017 | 0.98 | 1.00 | 0.98 | 0.98 | 0.98 | 0.98 | 1.00 | 0.98 | 0.98 | 2.42 | |||

0.995 | 2011 | 0.97 | 1.00 | 1.00 | 0.99 | 0.85 | 0.97 | 1.00 | 0.95 | 3.89 | 22.75 | ||

2012 | 1.00 | 1.00 | 1.00 | 0.95 | 0.94 | 1.00 | 0.95 | 0.95 | 0.95 | 4.25 | |||

2013 | 1.00 | 0.98 | 0.98 | 1.01 | 0.90 | 1.00 | 0.98 | 0.98 | 0.89 | 3.82 | |||

2014 | 0.97 | 0.99 | 0.98 | 1.02 | 0.86 | 0.97 | 0.99 | 0.98 | 1.02 | 5.53 | |||

2015 | 0.99 | 0.99 | 0.99 | 0.95 | 0.95 | 0.99 | 0.99 | 0.99 | 0.95 | 4.94 | |||

2016 | 0.99 | 1.00 | 1.01 | 0.96 | 1.01 | 0.99 | 1.00 | 0.95 | 0.96 | 1.16 | |||

2017 | 0.98 | 0.99 | 0.98 | 0.93 | 0.91 | 0.98 | 0.99 | 0.98 | 0.99 | 4.83 | |||

0.999 | 2011 | 0.97 | 0.94 | 1.01 | 0.84 | 0.66 | 2.06 | 2.29 | 3.63 | 19.46 | 113.77 | ||

2012 | 1.00 | 0.97 | 0.97 | 1.00 | 0.89 | 0.90 | 0.99 | 0.97 | 3.80 | 21.26 | |||

2013 | 0.99 | 0.96 | 0.94 | 0.81 | 0.90 | 0.99 | 0.96 | 0.94 | 3.47 | 19.11 | |||

2014 | 0.98 | 0.98 | 1.00 | 0.95 | 1.14 | 0.98 | 0.98 | 0.98 | 5.11 | 27.65 | |||

2015 | 0.98 | 0.96 | 0.97 | 1.05 | 0.95 | 0.98 | 0.96 | 0.97 | 4.74 | 24.71 | |||

2016 | 0.99 | 0.94 | 0.99 | 1.06 | 0.75 | 0.99 | 0.94 | 0.99 | 1.06 | 5.78 | |||

2017 | 0.99 | 1.00 | 0.93 | 0.90 | 0.91 | 0.99 | 0.96 | 1.00 | 4.35 | 24.16 |

**Table A5.**Quotients between empirical and theoretical $(1-{p}_{a})$ significance levels of estimated $C{Q}_{{p}_{a}}$ for the age-at-death of the population aged over 65 in Spain separately for men and women, from 2011 to 2017, using Gompertz and Gompertz-G distributions.

Gompertz for Each a | Gompertz-G for Each a | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{p}}_{\mathit{a}}$ | Year | 65 | 75 | 85 | 95 | 100 | 65 | 75 | 85 | 95 | 100 | ||

Men | 0.990 | 2011 | 1.29 | 1.33 | 1.57 | 2.74 | 3.47 | 0.32 | 0.27 | 0.14 | 0.14 | 0.51 | |

2012 | 1.25 | 1.27 | 1.50 | 2.57 | 4.77 | 0.66 | 0.53 | 0.34 | 0.42 | 0.93 | |||

2013 | 1.17 | 1.26 | 1.49 | 2.84 | 3.18 | 0.40 | 0.35 | 0.25 | 0.14 | 0.24 | |||

2014 | 1.07 | 1.15 | 1.29 | 2.81 | 3.61 | 0.44 | 0.38 | 0.28 | 0.23 | 0.23 | |||

2015 | 1.08 | 1.15 | 1.27 | 2.59 | 3.47 | 0.36 | 0.36 | 0.33 | 0.41 | 0.51 | |||

2016 | 0.95 | 1.01 | 1.09 | 2.38 | 4.01 | 0.30 | 0.28 | 0.20 | 0.21 | 0.33 | |||

2017 | 0.94 | 0.95 | 1.01 | 1.85 | 4.26 | 0.24 | 0.20 | 0.12 | 0.06 | 0.00 | |||

0.995 | 2011 | 1.45 | 1.52 | 1.83 | 2.70 | 4.63 | 0.22 | 0.18 | 0.11 | 0.21 | 0.26 | ||

2012 | 1.43 | 1.43 | 1.70 | 3.18 | 7.44 | 0.49 | 0.42 | 0.28 | 0.45 | 1.40 | |||

2013 | 1.41 | 1.48 | 1.67 | 3.42 | 3.54 | 0.32 | 0.29 | 0.15 | 0.07 | 0.00 | |||

2014 | 1.25 | 1.30 | 1.56 | 3.17 | 4.66 | 0.35 | 0.32 | 0.23 | 0.23 | 0.00 | |||

2015 | 1.22 | 1.29 | 1.48 | 3.30 | 4.70 | 0.33 | 0.33 | 0.32 | 0.42 | 0.20 | |||

2016 | 1.09 | 1.13 | 1.35 | 2.86 | 4.77 | 0.25 | 0.23 | 0.15 | 0.20 | 0.43 | |||

2017 | 1.07 | 1.09 | 1.19 | 2.50 | 5.74 | 0.17 | 0.15 | 0.10 | 0.05 | 0.00 | |||

0.999 | 2011 | 2.10 | 2.16 | 2.90 | 5.20 | 10.28 | 0.10 | 0.08 | 0.11 | 0.17 | 0.00 | ||

2012 | 1.96 | 2.09 | 2.53 | 6.90 | 16.28 | 0.35 | 0.33 | 0.30 | 0.96 | 2.33 | |||

2013 | 2.04 | 2.18 | 2.83 | 4.96 | 7.08 | 0.13 | 0.11 | 0.03 | 0.00 | 0.00 | |||

2014 | 1.82 | 2.06 | 2.50 | 5.12 | 8.16 | 0.20 | 0.18 | 0.11 | 0.00 | 0.00 | |||

2015 | 1.83 | 2.01 | 2.34 | 5.60 | 6.13 | 0.28 | 0.29 | 0.22 | 0.14 | 0.00 | |||

2016 | 1.68 | 1.74 | 2.15 | 5.18 | 9.76 | 0.15 | 0.14 | 0.10 | 0.28 | 2.17 | |||

2017 | 1.48 | 1.59 | 1.86 | 5.18 | 4.95 | 0.08 | 0.04 | 0.03 | 0.00 | 0.00 | |||

Women | 0.990 | 2011 | 1.33 | 1.37 | 1.50 | 2.69 | 4.26 | 1.13 | 1.16 | 1.06 | 1.52 | 2.43 | |

2012 | 1.40 | 1.42 | 1.60 | 2.88 | 4.72 | 0.50 | 0.47 | 0.39 | 0.48 | 0.59 | |||

2013 | 1.39 | 1.42 | 1.56 | 2.84 | 4.12 | 1.04 | 1.06 | 1.05 | 1.48 | 2.30 | |||

2014 | 1.37 | 1.42 | 1.56 | 2.98 | 4.87 | 0.86 | 0.89 | 0.81 | 1.26 | 2.39 | |||

2015 | 1.44 | 1.48 | 1.66 | 3.29 | 5.30 | 1.44 | 1.48 | 1.66 | 3.39 | 5.94 | |||

2016 | 1.37 | 1.42 | 1.57 | 3.07 | 5.33 | 0.59 | 0.58 | 0.51 | 0.69 | 1.08 | |||

0.995 | 2017 | 1.37 | 1.41 | 1.53 | 2.95 | 5.22 | 1.23 | 1.27 | 1.36 | 2.22 | 3.53 | ||

2011 | 1.55 | 1.60 | 1.84 | 3.58 | 6.30 | 1.20 | 1.21 | 1.21 | 1.53 | 3.15 | |||

2012 | 1.70 | 1.73 | 2.00 | 3.76 | 6.32 | 0.46 | 0.44 | 0.37 | 0.48 | 0.83 | |||

2013 | 1.70 | 1.76 | 2.04 | 3.62 | 6.27 | 1.16 | 1.23 | 1.16 | 1.84 | 2.57 | |||

2014 | 1.66 | 1.78 | 1.95 | 3.83 | 7.24 | 0.93 | 0.95 | 0.89 | 1.53 | 2.96 | |||

2015 | 1.83 | 1.84 | 2.06 | 4.58 | 6.89 | 1.83 | 1.84 | 2.06 | 4.77 | 7.84 | |||

2016 | 1.71 | 1.74 | 1.98 | 3.79 | 6.99 | 0.57 | 0.57 | 0.52 | 0.78 | 1.31 | |||

2017 | 1.74 | 1.77 | 1.95 | 3.78 | 7.06 | 1.46 | 1.50 | 1.61 | 2.69 | 4.01 | |||

0.999 | 2011 | 2.71 | 2.72 | 3.34 | 6.11 | 14.10 | 1.71 | 1.70 | 1.49 | 3.14 | 4.59 | ||

2012 | 2.93 | 2.96 | 3.57 | 7.55 | 13.29 | 0.42 | 0.40 | 0.36 | 0.69 | 0.89 | |||

2013 | 2.88 | 2.84 | 3.31 | 7.05 | 9.56 | 1.51 | 1.59 | 1.59 | 2.71 | 4.48 | |||

2014 | 2.92 | 2.95 | 3.47 | 7.64 | 15.96 | 1.20 | 1.28 | 1.35 | 2.74 | 6.27 | |||

2015 | 3.20 | 3.37 | 4.12 | 8.53 | 15.21 | 3.33 | 3.53 | 4.28 | 9.80 | 18.06 | |||

2016 | 2.97 | 3.15 | 3.44 | 7.75 | 14.07 | 0.62 | 0.60 | 0.58 | 1.06 | 2.01 | |||

2017 | 2.97 | 3.16 | 3.46 | 7.75 | 15.27 | 2.33 | 2.31 | 2.56 | 4.35 | 5.70 |

## Appendix D. Split Lifetime Annuities Premiums

**Table A6.**Temporary annuity premium assuming $w=C{Q}_{{p}_{a}}$ (upper table) and “tail premium”, for $r=0.2$ and year 2017. The probabilities ${}_{t}{P}_{a}$ were estimated with the Beta Transformed Kernel Estimator (BTKE).

Temporary Annuity Premium | |||||
---|---|---|---|---|---|

${\mathit{p}}_{\mathit{a}}$ | $\mathit{a}=\mathbf{75}$ | $\mathit{a}=\mathbf{85}$ | $\mathit{a}=\mathbf{95}$ | $\mathit{a}=\mathbf{100}$ | |

0.990 | Men | 4.8503 | 4.5314 | 3.7946 | 3.3886 |

Women | 4.8959 | 4.6740 | 3.9961 | 3.5580 | |

0.995 | Men | 4.8590 | 4.5736 | 3.9088 | 3.4867 |

Women | 4.9007 | 4.7025 | 4.0777 | 3.6583 | |

0.999 | Men | 4.8708 | 4.6315 | 4.0247 | 3.6779 |

Women | 4.9074 | 4.7396 | 4.1900 | 3.9485 | |

Tail Premium | |||||

${\mathit{p}}_{\mathit{a}}$ | $\mathit{a}=\mathbf{75}$ | $\mathit{a}=\mathbf{85}$ | $\mathit{a}=\mathbf{95}$ | $\mathit{a}=\mathbf{100}$ | |

0.990 | Men | 0.0351 | 0.1750 | 0.4674 | 0.7252 |

Women | 0.0209 | 0.1198 | 0.4014 | 0.6602 | |

0.995 | Men | 0.0264 | 0.1328 | 0.3532 | 0.6272 |

Women | 0.0161 | 0.0914 | 0.3198 | 0.5600 | |

0.999 | Men | 0.0146 | 0.0749 | 0.2373 | 0.4360 |

Women | 0.0093 | 0.0542 | 0.2075 | 0.2697 |

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**Figure 1.**$C{Q}_{{p}_{a}}$ for the year 2015 of the age-at-death variable for the population in Spain estimated from the Beta Transformed Kernel Estimator (BTKE), kernel estimator (KE) and empirical distribution, for men (

**a**) and women (

**b**).

**Figure 2.**$C{Q}_{{p}_{a}}$ of year 2011 (thin lines) and 2017 (thick lines) of the age-at-death variable estimated from Beta Transformed Kernel Estimator (BTKE), (

**a**) ${p}_{a}=0.99$, (

**b**) ${p}_{a}=0.995$ and (

**c**) ${p}_{a}=0.999$.

**Figure 3.**${}_{t}{P}_{a}$ for the year 2017 estimated from Beta Transformed Kernel Estimator (BTKE), for men (solid line) and women (dashed line), with $a=75,85,95,100$.

**Table 1.**Descriptive statistics of age-at-death (in years) for the population aged 65 or over in Spain (2011).

Gender | Year | Population | Deaths | Rate | Min. | Mean | Max. | Variance |
---|---|---|---|---|---|---|---|---|

2011 | 3,768,467 | 157,757 | 0.042 | 65 | 81.34 | 111.17 | 62.22 | |

2012 | 3,717,697 | 164,281 | 0.044 | 65 | 81.62 | 112.17 | 62.89 | |

2013 | 3,676,272 | 159,705 | 0.043 | 65 | 81.64 | 109.08 | 63.93 | |

Men | 2014 | 3,613,455 | 162,066 | 0.045 | 65 | 81.80 | 109.33 | 65.47 |

2015 | 3,565,325 | 173,472 | 0.049 | 65 | 82.09 | 109.75 | 65.92 | |

2016 | 3,510,560 | 169,414 | 0.048 | 65 | 82.13 | 114.75 | 67.21 | |

2017 | 3,449,614 | 175,086 | 0.051 | 65 | 82.38 | 109.50 | 68.20 | |

2011 | 4,995,737 | 168,547 | 0.034 | 65 | 85.58 | 111.33 | 60.62 | |

2012 | 4,940,008 | 177,790 | 0.036 | 65 | 85.84 | 111.42 | 59.59 | |

2013 | 4,897,713 | 171,411 | 0.035 | 65 | 85.87 | 111.08 | 60.78 | |

Women | 2014 | 4,828,972 | 175,202 | 0.036 | 65 | 86.03 | 112.83 | 60.96 |

2015 | 4,770,536 | 189,889 | 0.040 | 65 | 86.36 | 111.17 | 60.35 | |

2016 | 4,711,636 | 182,392 | 0.039 | 65 | 86.39 | 114.50 | 61.78 | |

2017 | 4,995,737 | 190,600 | 0.038 | 65 | 86.64 | 116.17 | 62.06 |

**Table 2.**Results of estimated $C{Q}_{{p}_{a}}$ and $90\%$ confidence interval based on Beta Transformed Kernel Estimator (BTKE) for the age-at-death of the population aged over 65 in Spain separately for men and women and conditional ages of the deceased population ($a=65,75$), from 2011 to 2017.

$\mathit{a}=65$ | $\mathit{a}=75$ | ||||
---|---|---|---|---|---|

${\mathit{p}}_{\mathit{a}}$ | Year | Men | Women | Men | Women |

0.990 | 2011 | 98.49 | 101.23 | 99.08 | 101.45 |

(98.40;98.59) | (101.15;101.31) | (98.97;99.18) | (101.37;101.53) | ||

2012 | 98.57 | 101.33 | 99.11 | 101.53 | |

(98.47;98.66) | (101.25;101.41) | (99.01;99.21) | (101.45;101.61) | ||

2013 | 98.54 | 101.40 | 99.14 | 101.61 | |

(98.44;98.63) | (101.31;101.48) | (99.04;99.25) | (101.53;101.69) | ||

2014 | 98.56 | 101.50 | 99.14 | 101.72 | |

(98.47;98.65) | (101.42;101.57) | (99.04;99.24) | (101.64;101.79) | ||

2015 | 98.74 | 101.78 | 99.29 | 101.98 | |

(98.64;98.83) | (101.70;101.85) | (99.19;99.38) | (101.90;102.05) | ||

2016 | 98.62 | 101.76 | 99.19 | 101.97 | |

(98.53;98.71) | (101.68;101.84) | (99.09;99.29) | (101.89;102.05) | ||

2017 | 98.75 | 101.92 | 99.31 | 102.14 | |

(98.66;98.84) | (101.84;101.99) | (99.21;99.40) | (102.06;102.21) | ||

0.995 | 2011 | 99.99 | 102.59 | 100.49 | 102.79 |

(99.87;100.10) | (102.48;102.69) | (100.37;100.61) | (102.69;102.90) | ||

2012 | 100.04 | 102.70 | 100.49 | 102.89 | |

(99.93;100.15) | (102.60;102.79) | (100.37;100.61) | (102.79;102.99) | ||

2013 | 100.09 | 102.81 | 100.56 | 103.00 | |

(99.98;100.20) | (102.71;102.91) | (100.45;100.68) | (102.90;103.11) | ||

2014 | 100.10 | 102.85 | 100.62 | 103.04 | |

(99.99;100.21) | (102.75;102.94) | (100.50;100.74) | (102.95;103.14) | ||

2015 | 100.24 | 103.11 | 100.75 | 103.31 | |

(100.14;100.35) | (103.02;103.21) | (100.63;100.86) | (103.21;103.40) | ||

2016 | 100.14 | 103.13 | 100.66 | 103.33 | |

(100.03;100.26) | (103.04;103.23) | (100.54;100.78) | (103.23;103.43) | ||

2017 | 100.26 | 103.28 | 100.76 | 103.47 | |

CI | (100.15;100.37) | (103.19;103.38) | (100.64;100.88) | (103.38;103.57) | |

0.999 | 2011 | 102.89 | 105.31 | 103.27 | 105.47 |

(102.69;103.09) | (105.13;105.50) | (103.05;103.49) | (105.28;105.66) | ||

2012 | 102.90 | 105.44 | 103.30 | 105.59 | |

(102.70;103.10) | (105.26;105.61) | (103.09;103.52) | (105.41;105.77) | ||

2013 | 103.06 | 105.40 | 103.46 | 105.56 | |

(102.86;103.26) | (105.21;105.58) | (103.24;103.68) | (105.37;105.75) | ||

2014 | 103.16 | 105.63 | 103.57 | 105.82 | |

(102.96;103.35) | (105.47;105.80) | (103.36;103.78) | (105.65;106.00) | ||

2015 | 103.26 | 105.89 | 103.68 | 106.03 | |

(103.06;103.45) | (105.72;106.05) | (103.47;103.89) | (105.86;106.20) | ||

2016 | 103.24 | 105.89 | 103.67 | 106.03 | |

(103.04;103.43) | (105.72;106.05) | (103.46;103.88) | (105.86;106.20) | ||

2017 | 103.19 | 106.01 | 103.63 | 106.16 | |

(102.99;103.38) | (105.84;106.18) | (103.42;103.85) | (105.98;106.33) |

**Table 3.**Results of estimated $C{Q}_{{p}_{a}}$ and $90\%$ confidence interval based on Beta Transformed Kernel Estimator (BTKE) for the age-at-death of the population aged over 65 in Spain separately for men and women and conditional ages of the deceased population ($a=85,95$), from 2011 to 2017.

$\mathit{a}=85$ | $\mathit{a}=95$ | ||||
---|---|---|---|---|---|

${\mathit{p}}_{\mathit{a}}$ | Year | Men | Women | Men | Women |

0.990 | 2011 | 100.72 | 102.35 | 104.27 | 105.25 |

(100.60;100.84) | (102.26;102.45) | (103.99;104.55) | (105.08;105.42) | ||

2012 | 100.64 | 102.39 | 104.57 | 105.36 | |

(100.52;100.76) | (102.30;102.49) | (104.29;104.85) | (105.20;105.53) | ||

2013 | 100.68 | 102.51 | 104.47 | 105.30 | |

(100.56;100.80) | (102.41;102.60) | (104.19;104.75) | (105.13;105.48) | ||

2014 | 100.68 | 102.52 | 104.67 | 105.51 | |

(100.56;100.79) | (102.43;102.61) | (104.40;104.93) | (105.35;105.66) | ||

2015 | 100.73 | 102.72 | 104.73 | 105.70 | |

(100.62;100.85) | (102.63;102.81) | (104.47;105.00) | (105.54;105.85) | ||

2016 | 100.59 | 102.71 | 104.61 | 105.65 | |

(100.48;100.71) | (102.61;102.80) | (104.34;104.89) | (105.49;105.81) | ||

2017 | 100.62 | 102.84 | 104.52 | 105.65 | |

(100.50;100.74) | (102.75;102.92) | (104.25;104.78) | (105.49;105.81) | ||

0.995 | 2011 | 101.98 | 103.66 | 105.33 | 106.33 |

(101.82;102.15) | (103.53;103.78) | (104.96;105.71) | (106.11;106.55) | ||

2012 | 101.86 | 103.66 | 105.95 | 106.37 | |

(101.71;102.02) | (103.55;103.78) | (105.57;106.32) | (106.15;106.58) | ||

2013 | 101.95 | 103.70 | 105.32 | 106.31 | |

(101.79;102.11) | (103.58;103.83) | (104.93;105.70) | (106.09;106.54) | ||

2014 | 102.01 | 103.78 | 105.58 | 106.71 | |

(101.85;102.17) | (103.67;103.89) | (105.21;105.94) | (106.51;106.92) | ||

2015 | 102.04 | 104.07 | 105.58 | 106.68 | |

(101.88;102.20) | (103.95;104.18) | (105.22;105.95) | (106.48;106.88) | ||

2016 | 101.96 | 104.00 | 105.66 | 106.63 | |

(101.80;102.12) | (103.88;104.11) | (105.28;106.04) | (106.43;106.84) | ||

2017 | 101.91 | 104.10 | 105.71 | 106.62 | |

(101.76;102.07) | (103.99;104.22) | (105.35;106.08) | (106.42;106.83) | ||

0.999 | 2011 | 104.34 | 106.19 | 108.47 | 108.61 |

(104.06;104.63) | (105.98;106.41) | (107.61;109.32) | (108.17;109.04) | ||

2012 | 104.66 | 106.22 | 109.40 | 108.51 | |

(104.37;104.94) | (106.01;106.44) | (108.56;110.25) | (108.07;108.94) | ||

2013 | 104.46 | 106.18 | 107.16 | 108.09 | |

(104.17;104.75) | (105.96;106.40) | (106.29;108.03) | (107.65;108.52) | ||

2014 | 104.65 | 106.55 | 107.58 | 109.18 | |

(104.37;104.92) | (106.35;106.75) | (106.75;108.42) | (108.76;109.60) | ||

2015 | 104.78 | 106.57 | 107.37 | 108.62 | |

(104.50;105.05) | (106.37;106.77) | (106.53;108.20) | (108.21;109.04) | ||

2016 | 104.68 | 106.56 | 107.80 | 108.89 | |

(104.40;104.96) | (106.36;106.76) | (107.00;108.61) | (108.47;109.30) | ||

2017 | 104.69 | 106.62 | 107.43 | 108.51 | |

(104.42;104.96) | (106.41;106.82) | (106.62;108.23) | (108.10;108.92) |

**Table 4.**Estimates of a single annuity premium (${\rho}_{a})$ for $r=0.2$ and year 2017, at $a=75,85,95,100$. The probabilities ${}_{t}{P}_{a}$ are estimated with the Beta Transformed Kernel Estimator (BTKE).

$\mathit{a}=75$ | $\mathit{a}=85$ | $\mathit{a}=95$ | $\mathit{a}=100$ | |
---|---|---|---|---|

Men | 4.8854 | 4.7064 | 4.2620 | 4.1138 |

Women | 4.9168 | 4.7938 | 4.3975 | 4.2183 |

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Bolancé, C.; Guillen, M.
Nonparametric Estimation of Extreme Quantiles with an Application to Longevity Risk. *Risks* **2021**, *9*, 77.
https://doi.org/10.3390/risks9040077

**AMA Style**

Bolancé C, Guillen M.
Nonparametric Estimation of Extreme Quantiles with an Application to Longevity Risk. *Risks*. 2021; 9(4):77.
https://doi.org/10.3390/risks9040077

**Chicago/Turabian Style**

Bolancé, Catalina, and Montserrat Guillen.
2021. "Nonparametric Estimation of Extreme Quantiles with an Application to Longevity Risk" *Risks* 9, no. 4: 77.
https://doi.org/10.3390/risks9040077