# Longevity Risk Management and the Development of a Value-Based Longevity Index

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Value-Based Longevity Index

## 3. Mortality Data Analysis

#### 3.1. Drift of Mortality Intensity

#### 3.2. Volatility of Mortality Intensity

#### 3.3. Cohort Correlations

#### 3.4. Principal Component Analysis

## 4. Mortality Model

#### 4.1. Model Development

#### 4.2. Calibration

#### 4.3. Calibration Results

## 5. Interest Rate Model

#### 5.1. Vasicek Model

#### 5.2. Data and Calibration

#### 5.3. Calibration Results

## 6. Value-Based Longevity Index

#### 6.1. Index Construction

#### 6.2. Hedge Efficiency

## 7. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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1 | The first index-based hedge, q-forward based on J.P. Morgan’s LifeMetrics longevity index, was executed in January 2008 by the U.K. pension insurer Lucida. The first indemnity-based longevity swap was entered in July 2008 by Canada Life with J.P. Morgan as the counterparty. |

2 | We do so in order to keep the model tractable. Calibration results show that the two-factor model fits the observed survival probabilities well. |

3 | We drop the cohort index i for the ease of exposition. |

4 | See Brigo and Mercurio (2006) for the detailed proof. |

5 | Ninety five percent is the lowest correlation level calibrated by the two-factor model of Jevtic et al. (2013). However, this is a rough, but conservative, estimate of the fitting errors of the correlations resulting from their model. |

Initial Age | 1890 | 1895 | 1900 | 1905 |
---|---|---|---|---|

50 | 0.001425 | 0.001639 | 0.001977 | 0.001800 |

65 | 0.007501 | 0.005820 | 0.005335 | 0.005601 |

85 | 0.016177 | 0.017104 | 0.015258 | 0.017705 |

Initial Age | 1890 | 1895 | 1900 | 1905 |
---|---|---|---|---|

50 | 0.001031 | 0.000965 | 0.001748 | 0.001343 |

65 | 0.007080 | 0.005082 | 0.007610 | 0.005494 |

85 | 0.030253 | 0.031574 | 0.033655 | 0.024272 |

Calendar Time 1955 | ||||
---|---|---|---|---|

Cohort | 1890 | 1895 | 1900 | 1905 |

1890 | 1.0000 | |||

1895 | 0.6124 | 1.0000 | ||

1900 | 0.5157 | 0.4758 | 1.0000 | |

1905 | 0.4271 | 0.1857 | 0.5568 | 1.0000 |

Calendar Time 1960 | ||||

1890 | 1.0000 | |||

1895 | 0.4707 | 1.0000 | ||

1900 | 0.1489 | 0.2728 | 1.0000 | |

1905 | 0.2103 | 0.1968 | 0.0949 | 1.0000 |

Calendar Time 1965 | ||||

1890 | 1.0000 | |||

1895 | 0.6583 | 1.0000 | ||

1900 | 0.2881 | 0.4585 | 1.0000 | |

1905 | 0.4376 | 0.4899 | 0.5200 | 1.0000 |

Calendar Time 1970 | ||||

1890 | 1.0000 | |||

1895 | 0.4468 | 1.0000 | ||

1900 | 0.3600 | 0.4117 | 1.0000 | |

1905 | 0.2401 | 0.7432 | 0.6260 | 1.0000 |

Cohort | ${\mathit{\psi}}_{2}$ | ${\mathit{\sigma}}_{2}$ | $\mathit{\rho}$ | ${\mathit{\mu}}_{1}(50)$ | ${\mathit{\mu}}_{1}(65)$ | ${\mathit{\mu}}_{1}(85)$ | ${\mathit{\mu}}_{2}(50)$ | ${\mathit{\mu}}_{2}(65)$ | ${\mathit{\mu}}_{2}(85)$ |
---|---|---|---|---|---|---|---|---|---|

1890 | 0.0032 | 0.0002 | 0.7660 | 0.0091 | 0.0305 | 0.1805 | 0.0091 | 0.0305 | 0.1805 |

1895 | 0.0011 | −0.0001 | 0.9999 | 0.0080 | 0.0326 | 0.1490 | 0.0080 | 0.0326 | 0.1490 |

1900 | 0.0151 | 0.0017 | 0.9999 | 0.0079 | 0.0375 | 0.1442 | 0.0079 | 0.0375 | 0.1442 |

1905 | 0.0031 | −0.0041 | 0.8377 | 0.0074 | 0.0344 | 0.1465 | 0.0074 | 0.0344 | 0.1465 |

a | b | c | d |
---|---|---|---|

0.2280 | −0.0037 | −10.3270 | 0.0343 |

Cohort | ${\mathit{\psi}}_{2}$ | ${\mathit{\sigma}}_{2}$ | $\mathit{\rho}$ | ${\mathit{\mu}}_{1}(50)$ | ${\mathit{\mu}}_{1}(65)$ | ${\mathit{\mu}}_{1}(85)$ | ${\mathit{\mu}}_{2}(50)$ | ${\mathit{\mu}}_{2}(65)$ | ${\mathit{\mu}}_{2}(85)$ |
---|---|---|---|---|---|---|---|---|---|

1890 | 0.0721 | −0.0001 | 0.7306 | −0.0068 | −0.0145 | 0.0227 | 0.0157 | 0.0460 | 0.1528 |

1895 | 0.0632 | −0.0073 | 0.8710 | −0.0368 | −0.0292 | −0.0343 | 0.0444 | 0.0607 | 0.1844 |

1900 | 0.0598 | 0.0000 | 0.9767 | −0.0241 | −0.0045 | −0.0255 | 0.0321 | 0.0449 | 0.1815 |

1905 | 0.0817 | −0.0000 | 0.8482 | −0.0072 | 0.0106 | −0.0011 | 0.0146 | 0.0256 | 0.1409 |

Calendar Time 1955 | ||||
---|---|---|---|---|

Cohort | 1890 | 1895 | 1900 | 1905 |

1890 | 1.0000 | |||

1895 | 0.4710 | 1.0000 | ||

1900 | 0.3427 | 0.4019 | 1.0000 | |

1905 | 0.5471 | 0.6195 | 0.5193 | 1.0000 |

Calendar Time 1960 | ||||

1890 | 1.0000 | |||

1895 | −0.1862 | 1.0000 | ||

1900 | 0.3584 | −0.1156 | 1.0000 | |

1905 | 0.6125 | −0.2554 | 0.5283 | 1.0000 |

Calendar Time 1965 | ||||

1890 | 1.0000 | |||

1895 | 0.2348 | 1.0000 | ||

1900 | 0.6964 | 0.2709 | 1.0000 | |

1905 | 0.8403 | 0.2587 | 0.7707 | 1.0000 |

Calendar Time 1970 | ||||

1890 | 1.0000 | |||

1895 | 0.4367 | 1.0000 | ||

1900 | 0.7740 | 0.4174 | 1.0000 | |

1905 | 0.9324 | 0.4593 | 0.8174 | 1.0000 |

Calendar Time 1955 | |||
---|---|---|---|

Cohort | 1890 | 1895 | B |

1895 | 0 | ||

1900 | 0 | 0 | |

1905 | 0 | 0 | 0 |

Calendar Time 1960 | |||

1895 | 0.0484 | ||

1900 | 0 | 0 | |

1905 | 0 | 0 | 0 |

Calendar Time 1965 | |||

1895 | 0.0026 | ||

1900 | 0 | 0 | |

1905 | 0 | 0 | 0 |

Calendar Time 1970 | |||

1895 | 0 | ||

1900 | 0 | 0.0391 | |

1905 | 0 | 0 | 0 |

Inputs | k | $\mathit{\theta}$ | $\mathit{\sigma}$ |
---|---|---|---|

Initial Value | 0.1386 | 0.0542 | 0.0009 |

Upper Bound | 2.7726 | 0.0660 | 0.0043 |

Lower Bound | 0.0693 | 0.0375 | 0.0002 |

Portfolio Size | 200 | 1000 | 100,00 |

Index Swap | 12.60% | 63.23% | 95.73% |

s-Forward | 11.45% | 52.31% | 68.61% |

© 2018 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**

Chang, Y.; Sherris, M.
Longevity Risk Management and the Development of a Value-Based Longevity Index. *Risks* **2018**, *6*, 10.
https://doi.org/10.3390/risks6010010

**AMA Style**

Chang Y, Sherris M.
Longevity Risk Management and the Development of a Value-Based Longevity Index. *Risks*. 2018; 6(1):10.
https://doi.org/10.3390/risks6010010

**Chicago/Turabian Style**

Chang, Yang, and Michael Sherris.
2018. "Longevity Risk Management and the Development of a Value-Based Longevity Index" *Risks* 6, no. 1: 10.
https://doi.org/10.3390/risks6010010