# Immunization and Hedging of Post Retirement Income Annuity Products

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Stochastic Models and Calibration

#### 2.1. Mortality Model

#### 2.2. Interest Rate Model

## 3. Life Annuity Immunization

#### 3.1. Duration, Convexity, Delta and Gamma

#### 3.2. Whole-Life Annuities

#### 3.3. Fixed-Income Securities: Coupon Bonds and Annuity Bonds

#### 3.4. Longevity-Linked Securities: Longevity Bonds

## 4. Bond Markets—Coupon, Annuity and Longevity Bonds

- Life Annuity
- List of Government Coupon Bonds
- List of Coupon Bonds Based on Securities Offered on FIIG
- List of Waratah Annuity Bonds Offered by the NSW Government
- List of Hypothetical Annuity Bonds Based on Securities Offered on FIIG
- List of Assumed Longevity Bonds

## 5. Duration-Convexity Immunization

#### Immunization Portfolio Results

## 6. Delta-Gamma Hedging

#### Hedge Portfolio Results

## 7. Stochastic Assessment of Liability Hedging

#### 7.1. Surplus Analysis

#### 7.2. Portfolio Hedge Effectiveness

## 8. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Historical (1960–2009) and Projected (2010–2050) Mortality Rates for Australian Population Males Aged 50–100.

**Figure 3.**Asset and Liability Cash Flows—Immunization. (

**a**) Only Coupon Bonds; (

**b**) Only Annuity Bonds; (

**c**) Only Longevity Bonds; (

**d**) Coupon Bonds and Longevity Bonds; (

**e**) Annuity Bonds and Longevity Bonds.

**Figure 4.**Asset and Liability Cash Flows—Hedging. (

**a**) Only Coupon Bonds; (

**b**) Only Annuity Bonds; (

**c**) Only Longevity Bonds; (

**d**) Coupon Bonds and Longevity Bonds; (

**e**) Coupon Bonds and Longevity Bonds.

**Figure 6.**Surplus Distribution—Immunization. (

**a**) Time horizon—1 year; (

**b**) Time horizon—10 years; (

**c**) Time horizon—50 years.

**Figure 7.**Surplus Distribution—Delta-Gamma hedging. (

**a**) Time horizon—1 year; (

**b**) Time horizon—10 years; (

**c**) Time horizon—50 years.

**Table 1.**Parameters of the Calibrated Mortality Model—Australian Population Males Aged 50 to 100 for years 1960 to 2009.

Parameter | Estimate | Standard Error |
---|---|---|

${a}_{1}$ | 0.00621 | 1.48e-04 |

${a}_{2}$ | 0.000742 | 1.93e-05 |

${\sigma}_{1}$ | 0.000204 | 5.92e-07 |

${\sigma}_{2}$ | 0.0000148 | 7.79e-09 |

c | 1.092 | 6.19e-06 |

**Table 2.**Parameters of the Calibrated Interest Rate Model—Australian Interest Rate Data 4 January 1993 to 31 July 2014.

Parameter ^{1} | Estimate | Standard Error |
---|---|---|

${\overline{\kappa}}_{r}$ | 0.445 | 0.0022 |

${\overline{\theta}}_{r}$ | 0.0523 | 0.0012 |

${\sigma}_{r}$ | 0.0414 | 0.0013 |

${\lambda}_{r}$ | −0.111 | 0.0022 |

$\tilde{\mathcal{D}}=\frac{\mathcal{D}}{P}$ | ${\tilde{\mathsf{\Delta}}}_{{Y}_{1}\left(t\right)}=\frac{{\mathsf{\Delta}}_{{Y}_{1}\left(t\right)}}{P}$ | ${\tilde{\mathsf{\Delta}}}_{{Y}_{2}\left(t\right)}=\frac{{\mathsf{\Delta}}_{{Y}_{2}\left(t\right)}}{P}$ | ${\tilde{\mathsf{\Delta}}}_{r\left(t\right)}=\frac{{\mathsf{\Delta}}_{r\left(t\right)}}{P}$ |

$\tilde{\mathcal{C}}=\frac{\mathcal{C}}{P}$ | ${\tilde{\mathsf{\Gamma}}}_{{Y}_{1}\left(t\right)}=\frac{{\mathsf{\Gamma}}_{{Y}_{1}\left(t\right)}}{P}$ | ${\tilde{\mathsf{\Gamma}}}_{{Y}_{2}\left(t\right)}=\frac{{\mathsf{\Gamma}}_{{Y}_{2}\left(t\right)}}{P}$ | ${\tilde{\mathsf{\Gamma}}}_{r\left(t\right)}=\frac{{\mathsf{\Gamma}}_{r\left(t\right)}}{P}$ |

Fixed-Income Securities (k) | Longevity-Linked Securities (j) | Liabilities |
---|---|---|

${a}_{k}={\sum}_{t\ge 1}^{}{A}_{k,t}\xb7B(0,t)$ | ${a}_{j}={\sum}_{t\ge 1}^{}{A}_{j,t}\xb7{S}_{x}(0,t)B(0,t)$ | $l={\sum}_{t\ge 1}^{}{L}_{t}\xb7{S}_{x}(0,t)B(0,t)$ |

$\mathcal{D}\left[{a}_{k}\right]={\sum}_{t\ge 1}^{}{A}_{k,t}\xb7t\xb7B(0,t)$ | $\mathcal{D}\left[{a}_{j}\right]={\sum}_{t\ge 1}^{}{A}_{j,t}\xb7t\xb7{S}_{x}(0,t)B(0,t)$ | $\mathcal{D}\left[l\right]={\sum}_{t\ge 1}^{}{L}_{t}\xb7t\xb7{S}_{x}(0,t)B(0,t)$ |

$\mathcal{C}\left[{a}_{k}\right]={\sum}_{t\ge 1}^{}{A}_{k,t}\xb7{t}^{2}\xb7B(0,t)$ | $\mathcal{C}\left[{a}_{j}\right]={\sum}_{t\ge 1}^{}{A}_{j,t}\xb7{t}^{2}\xb7{S}_{x}(0,t)B(0,t)$ | $\mathcal{C}\left[l\right]={\sum}_{t\ge 1}^{}{L}_{t}\xb7{t}^{2}\xb7{S}_{x}(0,t)B(0,t)$ |

Fixed-Income Securities (k) | Longevity-Linked Securities (j) | Liabilities |
---|---|---|

${a}_{k}={\sum}_{t\ge 1}^{}{A}_{k,t}\xb7B(0,t)$ | ${a}_{j}={\sum}_{t\ge 1}^{}{A}_{j,t}\xb7{S}_{x}(0,t)B(0,t)$ | $l={\sum}_{t\ge 1}^{}{L}_{t}\xb7{S}_{x}(0,t)B(0,t)$ |

${\mathsf{\Delta}}_{r\left(0\right)}\left[{a}_{k}\right]=\frac{\partial [{\sum}_{t\ge 1}^{}{A}_{k,t}\xb7B(0,t)]}{\partial r\left(0\right)}$ | ${\mathsf{\Delta}}_{r\left(0\right)}\left[{a}_{j}\right]=\frac{\partial [{\sum}_{t\ge 1}^{}{A}_{j,t}\xb7{S}_{x}(0,t)B(0,t)]}{\partial r\left(t\right)}$ | ${\mathsf{\Delta}}_{r\left(0\right)}\left[l\right]=\frac{\partial [{\sum}_{t\ge 1}^{}{L}_{t}\xb7{S}_{x}(0,t)B(0,t)]}{\partial r\left(0\right)}$ |

${\mathsf{\Gamma}}_{r\left(0\right)}\left[{a}_{k}\right]=\frac{{\partial}^{2}[{\sum}_{t\ge 1}^{}{A}_{k,t}\xb7B(0,t)]}{\partial {\left(r\left(0\right)\right)}^{2}}$ | ${\mathsf{\Gamma}}_{r\left(0\right)}\left[{a}_{j}\right]=\frac{{\partial}^{2}[{\sum}_{t\ge 1}^{}{A}_{j,t}\xb7{S}_{x}(0,t)B(0,t)]}{\partial {\left(r\left(0\right)\right)}^{2}}$ | ${\mathsf{\Gamma}}_{r\left(0\right)}\left[l\right]=\frac{{\partial}^{2}[{\sum}_{t\ge 1}^{}{L}_{t}\xb7{S}_{x}(0,t)B(0,t)]}{\partial {\left(r\left(0\right)\right)}^{2}}$ |

- | ${\mathsf{\Delta}}_{{Y}_{1}\left(0\right)}\left[{a}_{j}\right]=\frac{\partial [{\sum}_{t\ge 1}^{}{A}_{j,t}\xb7{S}_{x}(0,t)B(0,t)]}{\partial {Y}_{1}\left(0\right)}$ | ${\mathsf{\Delta}}_{{Y}_{1}\left(0\right)}\left[l\right]=\frac{\partial [{\sum}_{t\ge 1}^{}{L}_{t}\xb7{S}_{x}(0,t)B(0,t)]}{\partial {Y}_{1}\left(0\right)}$ |

- | ${\mathsf{\Gamma}}_{{Y}_{1}\left(0\right)}\left[{a}_{j}\right]=\frac{{\partial}^{2}[{\sum}_{t\ge 1}^{}{A}_{j,t}\xb7{S}_{x}(0,t)B(0,t)]}{\partial {\left({Y}_{1}\left(0\right)\right)}^{2}}$ | ${\mathsf{\Gamma}}_{{Y}_{1}\left(0\right)}\left[l\right]=\frac{{\partial}^{2}[{\sum}_{t\ge 1}^{}{L}_{t}\xb7{S}_{x}(0,t)B(0,t)]}{\partial {\left({Y}_{1}\left(0\right)\right)}^{2}}$ |

- | ${\mathsf{\Delta}}_{{Y}_{2}\left(0\right)}\left[{a}_{j}\right]=\frac{\partial [{\sum}_{t\ge 1}^{}{A}_{j,t}\xb7{S}_{x}(0,t)B(0,t)]}{\partial {Y}_{2}\left(0\right)}$ | ${\mathsf{\Delta}}_{{Y}_{2}\left(0\right)}\left[l\right]=\frac{\partial [{\sum}_{t\ge 1}^{}{L}_{t}\xb7{S}_{x}(0,t)B(0,t)]}{\partial {Y}_{2}\left(0\right)}$ |

- | ${\mathsf{\Gamma}}_{{Y}_{2}\left(0\right)}\left[{a}_{j}\right]=\frac{{\partial}^{2}[{\sum}_{t\ge 1}^{}{A}_{j,t}\xb7{S}_{x}(0,t)B(0,t)]}{\partial {\left({Y}_{2}\left(0\right)\right)}^{2}}$ | ${\mathsf{\Gamma}}_{{Y}_{2}\left(0\right)}\left[l\right]=\frac{{\partial}^{2}[{\sum}_{t\ge 1}^{}{L}_{t}\xb7{S}_{x}(0,t)B(0,t)]}{\partial {\left({Y}_{2}\left(0\right)\right)}^{2}}$ |

**Table 6.**These are details of the life annuity with monthly payments. The deltas with respect to the mortality risk factors are negative. Increases in these factors produce lower survival probabilities used for the discount factors and hence lower annuity values. The interest rate delta is also negative. Increases in the short rate produce lower zero coupon bond prices and hence lower annuity values. For a 65 year old, the Fisher-Weil duration is 8.12 years. Interest rate sensitivity for the stochastic interest rate model is lower than the Fisher-Weil duration. The delta for the mortality risk factor ${Y}_{1}\left(t\right)$ is of a similar magnitude as the duration, with opposite sign. ${Y}_{1}\left(t\right)$ reflects the level of mortality, whereas ${Y}_{2}\left(t\right)$ captures the impact of age.

Code | Maturity | TTM | Freq | Price | ${\tilde{\mathsf{\Delta}}}_{{\mathit{Y}}_{\mathbf{1}}\mathbf{\left(}\mathit{t}\mathbf{\right)}}$ | ${\tilde{\mathsf{\Delta}}}_{{\mathit{Y}}_{\mathbf{2}}\mathbf{\left(}\mathit{t}\mathbf{\right)}}$ | ${\tilde{\mathsf{\Delta}}}_{\mathit{r}\mathbf{\left(}\mathit{t}\mathbf{\right)}}$ | ${\tilde{\mathsf{\Gamma}}}_{{\mathit{Y}}_{\mathbf{1}}\mathbf{\left(}\mathit{t}\mathbf{\right)}}$ | ${\tilde{\mathsf{\Gamma}}}_{{\mathit{Y}}_{\mathbf{2}}\mathbf{\left(}\mathit{t}\mathbf{\right)}}$ | ${\tilde{\mathsf{\Gamma}}}_{\mathit{r}\mathbf{\left(}\mathit{t}\mathbf{\right)}}$ | $\tilde{\mathcal{D}}$ | $\tilde{\mathcal{C}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|

IA-WL | ∞ | ∞ | 12 | 127.67 | −7.79 | −5.08E+03 | −2.27 | 98.20 | 7.85E+07 | 5.78 | 8.12 | 109.23 |

**Table 7.**These are semi-annual coupon paying bonds available in the bond market. Codes used are those for the ASX. Maturities range up to 18.8 years and Fisher-Weil durations range up to 11.82 years with the longest duration exceeding that of the life annuity. The interest rate deltas range up to 2.62 and are all similar for bonds maturing longer than 4 years. Fisher-Weil convexity varies much more than interest rate gamma across the maturity range of the bonds.

Code | Sector | Coupon | Maturity | TTM | FV | Freq | Price | ${\tilde{\mathsf{\Delta}}}_{\mathit{r}\mathbf{\left(}\mathit{t}\mathbf{\right)}}$ | ${\tilde{\mathsf{\Gamma}}}_{\mathit{r}\mathbf{\left(}\mathit{t}\mathbf{\right)}}$ | $\tilde{\mathcal{D}}$ | $\tilde{\mathcal{C}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|

GSBS-CB-14 | Government | 4.50 % | 21/10/2014 | 0.31 | 100 | 2 | 101.39 | −0.29 | 0.09 | 0.31 | 0.10 |

GSBS-CB-15 | Government | 4.75 % | 21/10/2015 | 1.31 | 100 | 2 | 102.65 | −1.03 | 1.08 | 1.28 | 1.65 |

GSBM-CB-17 | Government | 4.25 % | 21/07/2017 | 3.06 | 100 | 2 | 102.05 | −1.79 | 3.35 | 2.85 | 8.53 |

GSBA-CB-18 | Government | 5.50 % | 21/01/2018 | 3.56 | 100 | 2 | 106.13 | −1.90 | 3.83 | 3.21 | 11.10 |

GSBS-CB-18 | Government | 3.25 % | 21/10/2018 | 4.31 | 100 | 2 | 95.33 | −2.15 | 4.77 | 4.02 | 16.93 |

GSBG-CB-23 | Government | 5.50 % | 21/04/2023 | 8.81 | 100 | 2 | 101.44 | −2.47 | 6.49 | 6.99 | 56.56 |

GSBG-CB-24 | Government | 2.75 % | 21/04/2024 | 9.82 | 100 | 2 | 79.06 | −2.62 | 7.16 | 8.39 | 77.99 |

GSBG-CB-25 | Government | 3.25 % | 21/04/2025 | 10.82 | 100 | 2 | 80.99 | −2.60 | 7.13 | 8.85 | 89.19 |

GSBG-CB-26 | Government | 4.25 % | 21/04/2026 | 11.82 | 100 | 2 | 88.01 | −2.56 | 6.97 | 9.04 | 96.85 |

GSBG-CB-27 | Government | 4.75 % | 21/04/2027 | 12.82 | 100 | 2 | 91.41 | −2.54 | 6.91 | 9.35 | 106.67 |

GSBG-CB-29 | Government | 3.25 % | 21/04/2029 | 14.82 | 100 | 2 | 74.11 | −2.61 | 7.22 | 11.04 | 147.40 |

GSBG-CB-33 | Government | 4.50 % | 21/04/2033 | 18.82 | 100 | 2 | 83.37 | −2.55 | 6.94 | 11.82 | 186.06 |

GSBG-CB-15 | Government | 6.25 % | 15/04/2015 | 0.79 | 100 | 2 | 103.77 | −0.68 | 0.47 | 0.78 | 0.61 |

GSBK-CB-16 | Government | 4.75 % | 15/06/2016 | 1.96 | 100 | 2 | 102.13 | −1.39 | 1.97 | 1.89 | 3.66 |

GSBC-CB-17 | Government | 6.00 % | 15/02/2017 | 2.63 | 100 | 2 | 107.13 | −1.63 | 2.77 | 2.43 | 6.23 |

GSBE-CB-19 | Government | 5.25 % | 15/03/2019 | 4.71 | 100 | 2 | 103.81 | −2.15 | 4.88 | 4.17 | 18.85 |

GSBG-CB-20 | Government | 4.50 % | 15/04/2020 | 5.80 | 100 | 2 | 98.56 | −2.33 | 5.68 | 5.10 | 28.25 |

GSBI-CB-21 | Government | 5.75 % | 15/05/2021 | 6.88 | 100 | 2 | 104.08 | −2.38 | 6.01 | 5.74 | 36.91 |

GSBM-CB-22 | Government | 5.75 % | 15/07/2022 | 8.05 | 100 | 2 | 105.28 | −2.40 | 6.22 | 6.38 | 47.39 |

**Table 8.**These are hypothetical coupon paying bonds with coupons and maturities corresponding to index linked bonds available on the FIIG web site. We do not include inflation in the analysis so we have used these as hypothetical coupon paying bonds with quarterly frequency. These hypothetical bonds have longer duration compared to the Government Coupon bonds. They also have quarterly coupon cash flows.

Code | Sector | Coupon | Maturity | TTM | FV | Freq | Price | ${\tilde{\mathsf{\Delta}}}_{\mathit{r}\mathbf{\left(}\mathit{t}\mathbf{\right)}}$ | ${\tilde{\mathsf{\Gamma}}}_{\mathit{r}\mathbf{\left(}\mathit{t}\mathbf{\right)}}$ | $\tilde{\mathcal{D}}$ | $\tilde{\mathcal{C}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|

ACG-CB-15 | Government | 4.00 % | 20/08/2015 | 1.14 | 100 | 4 | 101.26 | −0.93 | 0.87 | 1.12 | 1.26 |

ACG-CB-20 | Government | 4.00 % | 20/08/2020 | 6.15 | 100 | 4 | 95.06 | −2.38 | 5.91 | 5.42 | 31.87 |

ACG-CB-22 | Government | 1.25 % | 21/02/2022 | 7.65 | 100 | 4 | 74.54 | −2.63 | 7.08 | 7.21 | 54.15 |

ACG-CB-25 | Government | 3.00 % | 20/09/2025 | 11.23 | 100 | 4 | 77.73 | −2.63 | 7.23 | 9.22 | 96.65 |

ACG-CB-30 | Government | 2.50 % | 20/09/2030 | 16.24 | 100 | 4 | 63.77 | −2.65 | 7.40 | 12.31 | 181.46 |

SAFA-CB-15 | Semi-govern | 4.00 % | 20/08/2015 | 1.14 | 100 | 4 | 101.26 | −0.93 | 0.87 | 1.12 | 1.26 |

TCV-CB-20 | Semi-govern | 4.00 % | 15/08/2020 | 6.13 | 100 | 4 | 95.13 | −2.37 | 5.90 | 5.40 | 31.72 |

ACT-CB-30 | Semi-govern | 3.50 % | 17/06/2030 | 15.98 | 100 | 4 | 74.83 | −2.60 | 7.16 | 11.41 | 161.28 |

QTC-CB-30 | Semi-govern | 2.75 % | 20/08/2030 | 16.15 | 100 | 4 | 66.80 | −2.63 | 7.31 | 12.01 | 174.92 |

NSWTC-CB-20 | Semi-govern | 3.75 % | 20/11/2020 | 6.40 | 100 | 4 | 93.20 | −2.41 | 6.07 | 5.65 | 34.59 |

NSWTC-CB-25 | Semi-govern | 2.75 % | 20/11/2025 | 11.40 | 100 | 4 | 75.47 | −2.63 | 7.28 | 9.41 | 100.52 |

NSWTC-CB-35 | Semi-govern | 2.50 % | 20/11/2035 | 21.41 | 100 | 4 | 56.85 | −2.61 | 7.25 | 14.30 | 265.62 |

ELECTRANET-CB-15 | Infrastructure | 5.21 % | 20/08/2015 | 1.14 | 100 | 4 | 102.74 | −0.92 | 0.86 | 1.11 | 1.25 |

LANECOVE-CB-20 | Infrastructure | 4.50 % | 9/09/2020 | 6.20 | 100 | 4 | 97.46 | −2.37 | 5.88 | 5.40 | 31.89 |

SYDAIR-CB-20 | Infrastructure | 3.76 % | 20/11/2020 | 6.40 | 100 | 4 | 93.25 | −2.41 | 6.07 | 5.64 | 34.58 |

SYDAIR-CB-30 | Infrastructure | 3.12 % | 20/11/2030 | 16.40 | 100 | 4 | 70.42 | −2.61 | 7.21 | 11.82 | 172.67 |

RABO-CB-20 | ADI-IB | 1.51 % | 28/08/2020 | 6.17 | 100 | 4 | 81.37 | −2.50 | 6.36 | 5.85 | 35.42 |

CBA-CB-20 | ADI-Major Bank | 3.60 % | 20/11/2020 | 6.40 | 100 | 4 | 92.35 | −2.41 | 6.09 | 5.67 | 34.79 |

ALE-CB-23 | Other Financials | 3.40 % | 20/11/2023 | 9.40 | 100 | 4 | 84.89 | −2.57 | 6.94 | 7.86 | 69.45 |

ENVESTRA-CB-25 | Energy | 3.04 % | 20/08/2025 | 11.15 | 100 | 4 | 78.49 | −2.61 | 7.19 | 9.12 | 94.87 |

**Table 9.**These are annuity bonds with monthly payments. Terms to maturity are relatively short compared to the coupon paying bonds with a maximum of around 9 years. Fisher-Weil durations are between 3 and 5 years. Interest rate deltas do not vary much. Similar comments apply to interest rate gamma and Fisher-Weil convexity. Since the life annuity is assumed to have monthly payments these annuity bonds have the potential to better match the cash flows for the liability but suffer from having short maturities.

Code | Sector | Annuity Payment | Maturity | TTM | Freq | No. of Payment | Price | ${\tilde{\mathsf{\Delta}}}_{\mathit{r}\mathbf{\left(}\mathit{t}\mathbf{\right)}}$ | ${\tilde{\mathsf{\Gamma}}}_{\mathit{r}\mathbf{\left(}\mathit{t}\mathbf{\right)}}$ | $\tilde{\mathcal{D}}$ | $\tilde{\mathcal{C}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|

NSWWAB1-AB-21 | Semi-govern | 1.00 | 15/10/2021 | 7.30 | 12 | 111 | 74.60 | −1.79 | 3.77 | 3.43 | 16.17 |

NSWWAB2-AB-21 | Semi-govern | 1.00 | 15/10/2021 | 7.30 | 12 | 108 | 74.60 | −1.79 | 3.77 | 3.43 | 16.17 |

NSWWAB3-AB-22 | Semi-govern | 1.00 | 15/01/2022 | 7.55 | 12 | 108 | 76.62 | −1.81 | 3.86 | 3.54 | 17.22 |

NSWWAB4-AB-22 | Semi-govern | 1.00 | 15/04/2022 | 7.80 | 12 | 108 | 78.60 | −1.83 | 3.95 | 3.64 | 18.28 |

NSWWAB5-AB-22 | Semi-govern | 1.00 | 15/07/2022 | 8.05 | 12 | 108 | 80.56 | −1.86 | 4.04 | 3.75 | 19.38 |

NSWWAB6-AB-22 | Semi-govern | 1.00 | 15/10/2022 | 8.30 | 12 | 108 | 82.48 | −1.88 | 4.13 | 3.85 | 20.50 |

NSWWAB7-AB-23 | Semi-govern | 1.00 | 15/01/2023 | 8.55 | 12 | 108 | 84.36 | −1.90 | 4.21 | 3.95 | 21.64 |

NSWWAB8-AB-23 | Semi-govern | 1.00 | 15/04/2023 | 8.80 | 12 | 108 | 86.22 | −1.92 | 4.29 | 4.06 | 22.81 |

NSWWAB9-AB-23 | Semi-govern | 1.00 | 15/07/2023 | 9.05 | 12 | 108 | 87.05 | −1.96 | 4.41 | 4.20 | 24.28 |

NSWWAB10-AB-23 | Semi-govern | 1.00 | 15/07/2023 | 9.05 | 12 | 105 | 84.06 | −2.02 | 4.57 | 4.35 | 25.14 |

**Table 10.**These are hypothetical annuity bonds with maturities corresponding to index linked bonds available on FIIG. We do not include inflation in the analysis so we have used these as hypothetical annuity bonds with quarterly frequency. Terms to maturity are longer than for the Waratah annuity bonds. We do not adjust pricing for credit risk.

Code | Sector | Annuity Payment | Maturity | TTM | Freq | No. of Payment | Price | ${\tilde{\mathsf{\Delta}}}_{\mathit{r}\mathbf{\left(}\mathit{t}\mathbf{\right)}}$ | ${\tilde{\mathsf{\Gamma}}}_{\mathit{r}\mathbf{\left(}\mathit{t}\mathbf{\right)}}$ | $\tilde{\mathcal{D}}$ | $\tilde{\mathcal{C}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|

MPC-AB-25 | Infrastructure | 1.00 | 31/12/2025 | 11.51 | 4 | 46 | 34.49 | −2.11 | 5.05 | 5.23 | 37.95 |

MPC-AB-33 | Infrastructure | 1.00 | 31/12/2033 | 19.52 | 4 | 78 | 46.99 | −2.33 | 5.99 | 7.90 | 91.30 |

CIVICNEXUS-AB-32 | Infrastructure | 1.00 | 15/09/2032 | 18.22 | 4 | 73 | 45.57 | −2.30 | 5.87 | 7.48 | 81.61 |

PHF-AB-29 | Other Financials | 1.00 | 15/09/2029 | 15.22 | 4 | 61 | 41.31 | −2.23 | 5.58 | 6.52 | 60.83 |

PJS-AB-30 | Other Financials | 1.00 | 15/06/2030 | 15.97 | 4 | 64 | 42.46 | −2.25 | 5.67 | 6.77 | 65.88 |

Novacare-AB-33 | Other Financials | 1.00 | 15/04/2033 | 18.81 | 4 | 76 | 46.97 | −2.28 | 5.83 | 7.55 | 84.60 |

Praeco-AB-20 | Other Corporate | 1.00 | 15/08/2020 | 6.13 | 4 | 25 | 21.82 | −1.66 | 3.31 | 2.96 | 11.98 |

Boral-AB-20 | Other Corporate | 1.00 | 16/11/2020 | 6.39 | 4 | 26 | 22.54 | −1.70 | 3.43 | 3.07 | 12.92 |

WYUNA-AB-22 | Other Corporate | 1.00 | 30/03/2022 | 7.75 | 4 | 31 | 12.58 | −1.84 | 3.95 | 3.61 | 17.69 |

JEM(CCV)-AB-22 | Other Corporate | 1.00 | 15/06/2022 | 7.96 | 4 | 32 | 26.52 | −1.88 | 4.09 | 3.79 | 19.62 |

JEM-AB-35 | Other Corporate | 1.00 | 28/06/2035 | 21.01 | 4 | 84 | 48.67 | −2.35 | 6.08 | 8.32 | 102.30 |

JEM(NSWSch)-AB-31 | Other Corporate | 1.00 | 28/02/2031 | 16.68 | 4 | 67 | 43.66 | −2.26 | 5.72 | 6.98 | 70.48 |

JEM(NSWSch)-AB-35 | Other Corporate | 1.00 | 28/11/2035 | 21.43 | 4 | 86 | 49.44 | −2.34 | 6.06 | 8.38 | 104.73 |

ANU-AB-29 | Other Corporate | 1.00 | 7/10/2029 | 15.28 | 4 | 62 | 42.16 | −2.20 | 5.49 | 6.43 | 60.14 |

**Table 11.**These longevity bonds are hypothetical bonds with maturities at 5 year intervals up to a maximum of 50 years. They are based on a cohort aged 65 at issue. Fisher-Weil durations at the longer maturities do not vary much with a maximum of 8.48 years. The interest rate deltas also show very little variation with maturity. The deltas for ${Y}_{1}\left(t\right)$ in the mortality model are of a similar magnitude to the Fisher-Weil durations. The gammas for ${Y}_{1}\left(t\right)$ are of a similar magnitude to the convexity. The deltas for the ${Y}_{2}\left(t\right)$ are larger and reflect the impact of age.

Code | Maturity | TTM | Freq | Price | ${\tilde{\mathsf{\Delta}}}_{{\mathit{Y}}_{1}\mathbf{\left(}\mathit{t}\mathbf{\right)}}$ | ${\tilde{\mathsf{\Delta}}}_{{\mathit{Y}}_{2}\mathbf{\left(}\mathit{t}\mathbf{\right)}}$ | ${\tilde{\mathsf{\Delta}}}_{\mathit{r}\mathbf{\left(}\mathit{t}\mathbf{\right)}}$ | ${\tilde{\mathsf{\Gamma}}}_{{\mathit{Y}}_{1}\mathbf{\left(}\mathit{t}\mathbf{\right)}}$ | ${\tilde{\mathsf{\Gamma}}}_{{\mathit{Y}}_{2}\mathbf{\left(}\mathit{t}\mathbf{\right)}}$ | ${\tilde{\mathsf{\Gamma}}}_{\mathit{r}\mathbf{\left(}\mathit{t}\mathbf{\right)}}$ | $\tilde{\mathcal{D}}$ | $\tilde{\mathcal{C}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|

LB65-19 | 30/06/2019 | 5 | 1 | 420.94 | −2.83 | −1.00E+03 | −1.71 | 9.92 | 1.31E+06 | 3.23 | 2.86 | 10.17 |

LB65-24 | 30/06/2024 | 10 | 1 | 699.86 | −4.74 | −1.96E+03 | −2.11 | 29.81 | 5.66E+06 | 4.89 | 4.83 | 31.23 |

LB65-29 | 30/06/2029 | 15 | 1 | 866.81 | −6.19 | −2.94E+03 | −2.26 | 53.41 | 1.45E+07 | 5.58 | 6.36 | 57.03 |

LB65-34 | 30/06/2034 | 20 | 1 | 956.71 | −7.18 | −3.86E+03 | −2.33 | 74.98 | 2.88E+07 | 5.87 | 7.43 | 81.38 |

LB65-39 | 30/06/2039 | 25 | 1 | 998.30 | −7.77 | −4.59E+03 | −2.35 | 90.50 | 4.70E+07 | 5.98 | 8.07 | 99.45 |

LB65-44 | 30/06/2044 | 30 | 1 | 1,013.60 | −8.03 | −5.05E+03 | −2.36 | 98.84 | 6.47E+07 | 6.03 | 8.36 | 109.46 |

LB65-49 | 30/06/2049 | 35 | 1 | 1,017.70 | −8.12 | −5.24E+03 | −2.36 | 101.87 | 7.64E+07 | 6.04 | 8.46 | 113.21 |

LB65-54 | 30/06/2054 | 40 | 1 | 1,018.40 | −8.13 | −5.30E+03 | −2.36 | 102.52 | 8.10E+07 | 6.04 | 8.48 | 114.03 |

LB65-59 | 30/06/2059 | 45 | 1 | 1,018.40 | −8.13 | −5.30E+03 | −2.36 | 102.58 | 8.19E+07 | 6.04 | 8.48 | 114.11 |

LB65-64 | 30/06/2064 | 50 | 1 | 1,018.40 | −8.13 | −5.30E+03 | −2.36 | 102.59 | 8.20E+07 | 6.04 | 8.48 | 114.12 |

Bond | Weight | Bond | Weight | |
---|---|---|---|---|

Only Coupon Bonds | ||||

GSBS-CB-14 | 0.03 | GSBK-CB-16 | 0.01 | |

GSBS-CB-15 | 0.04 | GSBC-CB-17 | 0.08 | |

GSBS-CB-18 | 0.09 | GSBE-CB-19 | 0.01 | |

GSBG-CB-23 | 0.06 | GSBG-CB-20 | 0.05 | |

GSBG-CB-24 | 0.04 | GSBI-CB-21 | 0.03 | |

GSBG-CB-25 | 0.05 | ACG-CB-22 | 0.05 | |

GSBG-CB-26 | 0.01 | ACT-CB-30 | 0.01 | |

GSBG-CB-27 | 0.09 | NSWTC-CB-25 | 0.01 | |

GSBG-CB-29 | 0.07 | NSWTC-CB-35 | 0.24 | |

GSBG-CB-15 | 0.02 | SYDAIR-CB-20 | 0.02 | |

Only Annuity Bonds | ||||

Praeco-AB-20 | 0.05 | JEM(NSWSch)-AB-35 | 0.94 | |

JEM-AB-35 | 0.01 | - | - | |

Only Longevity Bonds | ||||

LB65-19 | 0.06 | LB65-64 | 0.08 | |

LB65-59 | 0.86 | - | - | |

Coupon Bonds and Longevity Bonds | ||||

GSBS-CB-14 | 0.04 | LB65-64 | 0.08 | |

LB65-59 | 0.87 | - | - | |

Annuity Bonds and Longevity Bonds | ||||

LB65-19 | 0.06 | LB65-64 | 0.08 | |

LB65-59 | 0.86 | - | - |

Bond | Weight | Bond | Weight |
---|---|---|---|

Only Coupon Bonds | |||

GSBS-CB-18 | 0.65 | RABO-CB-20 | 0.35 |

Only Annuity Bonds | |||

NSWWAB10-AB-23 | 0.24 | JEM-AB-35 | 0.76 |

Only Longevity Bonds | |||

LB65-19 | 0.22 | LB65-34 | −1.34 |

LB65-24 | −0.12 | LB65-39 | 2.23 |

Coupon Bonds and Longevity Bonds | |||

GSBG-CB-15 | 0.02 | LB65-34 | −1.44 |

LB65-19 | 0.13 | LB65-39 | 2.28 |

Annuity Bonds and Longevity Bonds | |||

LB65-19 | 0.22 | LB65-34 | −1.34 |

LB65-24 | −0.12 | LB65-39 | 2.23 |

Measure | CB | AB | LB | LB & CB | LB & AB |
---|---|---|---|---|---|

Time Horizon 1 year | |||||

$Mean$ | −0.0183% | −0.0187% | −0.0008% | −0.0002% | −0.0008% |

$SE\left(Mean\right)$ | 0.0090% | 0.0090% | 0.0002% | 0.0000% | 0.0002% |

$SD$ | 1.28% | 1.28% | 0.03% | 0.01% | 0.03% |

$SE\left(SD\right)$ | 0.01% | 0.01% | 0.00% | 0.00% | 0.00% |

$Va{R}_{0.5\%}$ | −3.32% | −3.31% | −0.08% | −0.02% | −0.08% |

$SE\left(Va{R}_{0.5\%}\right)$ | 0.03% | 0.04% | 0.00% | 0.00% | 0.00% |

$E{S}_{0.5\%}$ | −3.73% | −3.74% | −0.09% | −0.02% | −0.09% |

$SE\left(E{S}_{0.5\%}\right)$ | 0.08% | 0.08% | 0.01% | 0.00% | 0.01% |

Time Horizon 10 year | |||||

$Mean$ | −0.0591% | −0.0597% | 0.0016% | −0.0017% | 0.0016% |

$SE\left(Mean\right)$ | 0.0218% | 0.0218% | 0.0013% | 0.0010% | 0.0013% |

$SD$ | 3.08% | 3.08% | 0.19% | 0.14% | 0.19% |

$SE\left(SD\right)$ | 0.02% | 0.02% | 0.00% | 0.00% | 0.00% |

$Va{R}_{0.5\%}$ | −8.90% | −8.93% | −0.45% | −0.35% | −0.45% |

$SE\left(Va{R}_{0.5\%}\right)$ | 0.11% | 0.13% | 0.00% | 0.00% | 0.00% |

$E{S}_{0.5\%}$ | −10.13% | −10.12% | −0.49% | −0.40% | −0.49% |

$SE\left(E{S}_{0.5\%}\right)$ | 0.17% | 0.16% | 0.02% | 0.02% | 0.02% |

Time Horizon 50 year | |||||

$Mean$ | −0.1377% | −0.1280% | −0.0079% | −0.0083% | −0.0079% |

$SE\left(Mean\right)$ | 0.0242% | 0.0241% | 0.0017% | 0.0014% | 0.0017% |

$SD$ | 3.42% | 3.41% | 0.25% | 0.20% | 0.25% |

$SE\left(SD\right)$ | 0.02% | 0.02% | 0.00% | 0.00% | 0.00% |

$Va{R}_{0.5\%}$ | −10.55% | −10.44% | −0.60% | −0.51% | −0.60% |

$SE\left(Va{R}_{0.5\%}\right)$ | 0.20% | 0.19% | 0.01% | 0.01% | 0.01% |

$E{S}_{0.5\%}$ | −12.63% | −12.61% | −0.90% | −0.78% | −0.90% |

$SE\left(E{S}_{0.5\%}\right)$ | 0.27% | 0.27% | 0.10% | 0.10% | 0.10% |

Measure | CB | AB | LB | LB & CB | LB & AB |
---|---|---|---|---|---|

Time Horizon 1 year | |||||

$Mean$ | −0.0179% | −0.0181% | −0.0038% | −0.0038% | −0.0038% |

$SE\left(Mean\right)$ | 0.0090% | 0.0090% | 0.0002% | 0.0002% | 0.0002% |

$SD$ | 1.28% | 1.28% | 0.02% | 0.02% | 0.02% |

$SE\left(SD\right)$ | 0.01% | 0.01% | 0.00% | 0.00% | 0.00% |

$Va{R}_{0.5\%}$ | −3.33% | −3.32% | −0.06% | −0.06% | −0.06% |

$SE\left(Va{R}_{0.5\%}\right)$ | 0.04% | 0.03% | 0.00% | 0.00% | 0.00% |

$E{S}_{0.5\%}$ | −3.74% | −3.74% | −0.07% | −0.07% | −0.07% |

$SE\left(E{S}_{0.5\%}\right)$ | 0.08% | 0.08% | 0.01% | 0.01% | 0.01% |

Time Horizon 10 year | |||||

$Mean$ | 0.0225% | −0.0473% | −0.0449% | −0.0439% | −0.0449% |

$SE\left(Mean\right)$ | 0.0251% | 0.0218% | 0.0019% | 0.0015% | 0.0019% |

$SD$ | 3.55% | 3.08% | 0.26% | 0.22% | 0.26% |

$SE\left(SD\right)$ | 0.03% | 0.02% | 0.00% | 0.00% | 0.00% |

$Va{R}_{0.5\%}$ | −9.65% | −8.90% | −0.73% | −0.62% | −0.73% |

$SE\left(Va{R}_{0.5\%}\right)$ | 0.12% | 0.12% | 0.01% | 0.01% | 0.01% |

$E{S}_{0.5\%}$ | −11.07% | −10.15% | −0.87% | −0.78% | −0.87% |

$SE\left(E{S}_{0.5\%}\right)$ | 0.18% | 0.17% | 0.04% | 0.05% | 0.04% |

Time Horizon 50 year | |||||

$Mean$ | −0.1397% | −0.1324% | −0.1148% | −0.1134% | −0.1148% |

$SE\left(Mean\right)$ | 0.0275% | 0.0241% | 0.0037% | 0.0035% | 0.0037% |

$SD$ | 3.89% | 3.41% | 0.52% | 0.49% | 0.52% |

$SE\left(SD\right)$ | 0.03% | 0.02% | 0.00% | 0.00% | 0.00% |

$Va{R}_{0.5\%}$ | −11.39% | −10.43% | −1.97% | −1.96% | −1.97% |

$SE\left(Va{R}_{0.5\%}\right)$ | 0.20% | 0.16% | 0.05% | 0.06% | 0.05% |

$E{S}_{0.5\%}$ | −13.43% | −12.65% | −3.34% | −3.32% | −3.34% |

$SE\left(E{S}_{0.5\%}\right)$ | 0.27% | 0.28% | 0.23% | 0.23% | 0.23% |

Volatility | Value-at-Risk | Expected Shortfall |
---|---|---|

$HE\left(\sigma \right)=\frac{\sigma \left({S}_{t}\right)-\sigma \left({S}_{t}^{*}\right)}{\sigma \left({S}_{t}\right)}$ | $HE\left(Va{R}_{\alpha}\right)=\frac{Va{R}_{\alpha}\left({S}_{t}\right)-Va{R}_{\alpha}\left({S}_{t}^{*}\right)}{Va{R}_{\alpha}\left({S}_{t}\right)}$ | $HE\left(E{S}_{\alpha}\right)=\frac{E{S}_{\alpha}\left({S}_{t}\right)-E{S}_{\alpha}\left({S}_{t}^{*}\right)}{E{S}_{\alpha}\left({S}_{t}\right)}$ |

- | $Va{R}_{\alpha}=sup\{x:Pr({S}_{t}<x)\le \alpha \}$ | $E{S}_{\alpha}=\frac{1}{\alpha}{\int}_{0}^{\alpha}Va{R}_{u}\left({S}_{t}\right)du$ |

Measure | t = 1 | t = 10 | t = 50 |
---|---|---|---|

Delta-Gamma Hedging: AB compared to CB | |||

$HE\left(\sigma \right)$ | 0 % | −13 % | −13 % |

$HE\left(Va{R}_{0.5\%}\right)$ | −0 % | −8 % | −8 % |

$HE\left(E{S}_{0.5\%}\right)$ | −0 % | −8 % | −6 % |

Immunization: AB compared to CB | |||

$HE\left(\sigma \right)$ | 2 % | 1 % | −7 % |

$HE\left(Va{R}_{0.5\%}\right)$ | 0 % | 0 % | −0 % |

$HE\left(E{S}_{0.5\%}\right)$ | −0 % | 0 % | −1 % |

Delta-Gamma Hedging: LB compared to CB | |||

$HE\left(\sigma \right)$ | −98 % | −93 % | −87 % |

$HE\left(Va{R}_{0.5\%}\right)$ | −98 % | −92 % | −83 % |

$HE\left(E{S}_{0.5\%}\right)$ | −98 % | −92 % | −75 % |

Immunization: LB compared to CB | |||

$HE\left(\sigma \right)$ | −98 % | −94 % | −93 % |

$HE\left(Va{R}_{0.5\%}\right)$ | −98 % | −95 % | −94 % |

$HE\left(E{S}_{0.5\%}\right)$ | −98 % | −95 % | −93 % |

Delta-Gamma Hedging: CB and LB compared to only CB | |||

$HE\left(\sigma \right)$ | −98 % | −94 % | −87 % |

$HE\left(Va{R}_{0.5\%}\right)$ | −98 % | −94 % | −83 % |

$HE\left(E{S}_{0.5\%}\right)$ | −98 % | −93 % | −75 % |

Immunization: CB and LB compared to only CB | |||

$HE\left(\sigma \right)$ | −100 % | −96 % | −94 % |

$HE\left(Va{R}_{0.5\%}\right)$ | −100 % | −96 % | −95 % |

$HE\left(E{S}_{0.5\%}\right)$ | −100 % | −96 % | −94 % |

Delta-Gamma Hedging: AB and LB compared to only CB | |||

$HE\left(\sigma \right)$ | −98 % | −93 % | −87 % |

$HE\left(Va{R}_{0.5\%}\right)$ | −98 % | −92 % | −83 % |

$HE\left(E{S}_{0.5\%}\right)$ | −98 % | −92 % | −75 % |

Immunization: AB and LB compared to only CB | |||

$HE\left(\sigma \right)$ | −98 % | −94 % | −93 % |

$HE\left(Va{R}_{0.5\%}\right)$ | −98 % | −95 % | −94 % |

$HE\left(E{S}_{0.5\%}\right)$ | −98 % | −95 % | −93 % |

Measure | t = 1 | t = 10 | t = 50 |
---|---|---|---|

Using CB: Delta-Gamma Hedging compared to Immunization | |||

$HE\left(\sigma \right)$ | − 0 % | 15 % | 14 % |

$HE\left(Va{R}_{0.5\%}\right)$ | 0 % | 8 % | 8 % |

$HE\left(E{S}_{0.5\%}\right)$ | 0 % | 9 % | 6 % |

Using AB: Delta-Gamma Hedging compared to Immunization | |||

$HE\left(\sigma \right)$ | −0 % | 0 % | 0 % |

$HE\left(Va{R}_{0.5\%}\right)$ | 0 % | −0 % | −0 % |

$HE\left(E{S}_{0.5\%}\right)$ | 0 % | 0 % | 0 % |

Using LB: Delta-Gamma Hedging compared to Immunization | |||

$HE\left(\sigma \right)$ | −13 % | 41 % | 113 % |

$HE\left(Va{R}_{0.5\%}\right)$ | −20 % | 62 % | 228 % |

$HE\left(E{S}_{0.5\%}\right)$ | −20 % | 77 % | 272 % |

Using CB and LB: Delta-Gamma Hedging compared to Immunization | |||

$HE\left(\sigma \right)$ | 292 % | 57 % | 146 % |

$HE\left(Va{R}_{0.5\%}\right)$ | 289 % | 77 % | 288 % |

$HE\left(E{S}_{0.5\%}\right)$ | 304 % | 95 % | 327 % |

Using AB and LB: Delta-Gamma Hedging compared to Immunization | |||

$HE\left(\sigma \right)$ | −13 % | 41 % | 113 % |

$HE\left(Va{R}_{0.5\%}\right)$ | −20 % | 62 % | 228 % |

$HE\left(E{S}_{0.5\%}\right)$ | −20 % | 77 % | 272 % |

© 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/).

## Share and Cite

**MDPI and ACS Style**

Liu, C.; Sherris, M.
Immunization and Hedging of Post Retirement Income Annuity Products. *Risks* **2017**, *5*, 19.
https://doi.org/10.3390/risks5010019

**AMA Style**

Liu C, Sherris M.
Immunization and Hedging of Post Retirement Income Annuity Products. *Risks*. 2017; 5(1):19.
https://doi.org/10.3390/risks5010019

**Chicago/Turabian Style**

Liu, Changyu, and Michael Sherris.
2017. "Immunization and Hedging of Post Retirement Income Annuity Products" *Risks* 5, no. 1: 19.
https://doi.org/10.3390/risks5010019