# A Managed Volatility Investment Strategy for Pooled Annuity Products

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Pooled Annuities and Investment Risk

## 3. Pooled Annuity Income Modelling Methodology

#### 3.1. Pooled Annuity Product Features

#### 3.2. Mortality Model

#### 3.2.1. Systematic Longevity Risk

#### 3.2.2. Idiosyncratic Longevity Risk

#### 3.3. Economic Scenario Generator (ESG) Models

#### 3.3.1. Economic Scenario Generator

`MATLAB`function

`vgxvarx`, which utilizes the maximum likelihood method. The fitted parameters are given in Appendix B.4 and simulation results from the fitted model are shown in Appendix B.5.

#### 3.3.2. Interest Rate Model

#### 3.4. Equity Volatility Forecast Model

- To generate the series of residuals, subtract the path of realized equity returns by the mean simulated path, then take the square of each difference. Denote the residual at time t as $Re{s}_{t}$, then$$Re{s}_{t}={({\widehat{{r}_{t}}}^{E}-{\widehat{{\mu}_{t}}}^{E})}^{2}$$
- Assume an averaging period of n quarters, calculate the ‘realized variance’ by taking the moving average of residuals for the past n quarters. That is, the first realized variance is the average of the residuals from quarter 1 to quarter n; the second realized variance is the average of the residuals from quarter 2 to quarter $n+1$, and so on. Denote the k-th realized variance as $RVa{r}_{k}$, then$$RVa{r}_{k}=\frac{1}{n}\sum _{t=k}^{k+n-1}Re{s}_{t}$$
- Take the square root of the realized variance to get the realized volatility. Denote the k-th realized volatility as $RVo{l}_{k}$, then$$RVo{l}_{k}=\sqrt{RVa{r}_{k}}.$$
- Fit an AR(1) model to the series of realized volatility and test the significance of autoregression for prediction.

#### 3.5. The Managed-Volatility Framework

#### 3.6. Risk Measures

- Expected retirement income;
- Income variation;
- Access to underlying capital;
- Death benefit and reversionary benefits.

## 4. Investment Strategy Results

#### 4.1. Simulation of Annuity Payments in the Pool

#### 4.2. Balanced Fund with Target Volatility of 1.25 Historical Volatility

#### 4.3. Equity Asset Allocation

- 100% 3-month fixed-income;
- 100% 10-year fixed-income;
- 80% fixed-income, 20% equity, without volatility management.

- 4.
- 80% fixed-income, 20% equity;
- 5.
- 65% fixed-income, 35% equity;
- 6.
- 50% fixed-income, 50% equity.

#### 4.4. Varying the Level of Target Volatility

- Constant target volatility;
- Target volatility that decreases over time.

#### 4.5. Pool Size with Equity Investments

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Mortality Model, Estimated Parameters and Simulation

${\mathit{\delta}}_{1}$ | ${\mathit{\delta}}_{2}$ | ${\mathit{\rho}}_{1}$ | ${\mathit{\rho}}_{2}$ |
---|---|---|---|

−0.1004 | −0.1347 | 1.4285 $\times {10}^{-4}$ | 4.9659 $\times {10}^{-5}$ |

## Appendix B. Economic Series Data, Estimation and Simulations

#### Appendix B.1. Cointegration Test for VAR Model

#### Appendix B.1.1. Stationarity at Level

Aug D-F Test (at Level) | $ln\mathit{CPI}$ | $ln\mathit{XAOA}$ | $ln\mathit{GDP}$ | $\mathit{STY}$ |
---|---|---|---|---|

P-Value | 0.9990 | 0.9990 | 0.9990 | 0.3199 |

Null Hypothesis Result | not rejected | not rejected | not rejected | not rejected |

Stationarity | no | no | no | no |

#### Appendix B.1.2. Johansen Test

r | h | stat | cValue | pValue | eigVal |
---|---|---|---|---|---|

0 | 0 | 46.0198 | 47.8564 | 0.0737 | 0.2351 |

1 | 0 | 24.8498 | 29.7976 | 0.1672 | 0.1396 |

2 | 0 | 12.9739 | 15.4948 | 0.1163 | 0.1142 |

3 | 0 | 3.3944 | 3.8415 | 0.0654 | 0.0421 |

#### Appendix B.2. Stationarity Test at First Difference for VAR Model

Statistic | $\mathbf{\Delta}ln\mathit{CPI}$ | $\mathbf{\Delta}ln\mathit{XAOA}$ | $\mathbf{\Delta}ln\mathit{GDP}$ | $\mathbf{\Delta}\mathit{STY}$ |
---|---|---|---|---|

Mean | 0.0066 | 0.0223 | 0.0079 | −0.0005 |

Std Dev | 0.0057 | 0.0693 | 0.0055 | 0.0053 |

Skewness | 1.9863 | −0.8634 | 0.5771 | −1.0974 |

Kurtosis | 12.6015 | 4.6751 | 5.2732 | 14.2357 |

First Quantile | 0.0030 | −0.0108 | 0.0047 | −0.0021 |

Median | 0.0063 | 0.0273 | 0.0077 | 0.0001 |

Third Quantile | 0.0091 | 0.0663 | 0.0112 | 0.0021 |

Min | −0.0045 | −0.2256 | −0.0068 | −0.0286 |

Max | 0.0377 | 0.1952 | 0.0296 | 0.0218 |

Aug D-F Test (First Difference) | $\mathbf{\Delta}ln\mathit{CPI}$ | $\mathbf{\Delta}ln\mathit{XAOA}$ | $\mathbf{\Delta}ln\mathit{GDP}$ | $\mathbf{\Delta}\mathit{STY}$ |
---|---|---|---|---|

p-value | 0.0010 | 0.0010 | 0.0010 | 0.0010 |

Null Hypothesis result | rejected | rejected | rejected | rejected |

stationarity | yes | yes | yes | yes |

#### Appendix B.3. Optimal Number of Legs for VAR Model

VAR (1) | VAR (2) | VAR (3) | VAR (4) | |
---|---|---|---|---|

AIC | −2.1280 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{3}$ | −2.1154 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{3}$ | −2.0959 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{3}$ | −2.0854 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{3}$ |

#### Appendix B.4. Estimated Parameters for VAR Model

#### Appendix B.5. Simulation Versus Actual Results

Simulation | Actual | |||||||
---|---|---|---|---|---|---|---|---|

Statistic | $\mathbf{\Delta}ln\mathit{CPI}$ | $\mathbf{\Delta}ln\mathit{XAOA}$ | $\mathbf{\Delta}ln\mathit{GDP}$ | $\mathbf{\Delta}\mathit{STY}$ | $\mathbf{\Delta}ln\mathit{CPI}$ | $\mathbf{\Delta}ln\mathit{XAOA}$ | $\mathbf{\Delta}ln\mathit{GDP}$ | $\mathbf{\Delta}\mathit{STY}$ |

Mean | 0.0067 | 0.0223 | 0.0080 | −0.0005 | 0.0066 | 0.0223 | 0.0079 | −0.0005 |

Std Dev | 0.0001 | 0.0007 | 0.0001 | 0.0001 | 0.0057 | 0.0693 | 0.0055 | 0.0053 |

Skewness | −0.2898 | −0.4418 | −0.3923 | −0.0616 | 1.9863 | −0.8634 | 0.5771 | −1.0974 |

Kurtosis | 2.4385 | 3.5483 | 2.9874 | 4.0682 | 12.6015 | 4.6751 | 5.2732 | 14.2357 |

First Quantile | 0.0066 | 0.0220 | 0.0080 | −0.0005 | 0.0030 | −0.0108 | 0.0047 | −0.0021 |

Median | 0.0067 | 0.0224 | 0.0080 | −0.0005 | 0.0063 | 0.0273 | 0.0077 | 0.0001 |

Third Quantile | 0.0067 | 0.0228 | 0.0080 | −0.0004 | 0.0091 | 0.0663 | 0.0112 | 0.0021 |

Min | 0.0065 | 0.0203 | 0.0079 | −0.0006 | −0.0045 | −0.2256 | −0.0068 | −0.0286 |

Max | 0.0068 | 0.0242 | 0.0081 | −0.0003 | 0.0377 | 0.1952 | 0.0296 | 0.0218 |

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**Figure 1.**The $95\%$ confidence interval of annuity payments. Managed-Volatility vs. Fixed Allocation (65%/35%)—Nominal.

**Figure 2.**The $95\%$ confidence interval of annuity payments. Managed-Volatility vs. Fixed Allocation (65%/35%)—Real.

**Figure 3.**The $50\%$ confidence interval of annuity payments. Managed-Volatility vs. Fixed Allocation (65%/35%)—Nominal.

**Figure 4.**The $50\%$ confidence interval of annuity payments. Managed-Volatility vs. Fixed Allocation (65%/35%)—Real.

$\mathit{\theta}$ | $\mathit{\kappa}$ | $\mathit{\sigma}$ | $\mathit{\lambda}$ | ${\mathit{\sigma}}_{1}$ | ${\mathit{\sigma}}_{2}$ | ${\mathit{\sigma}}_{3}$ | ${\mathit{\sigma}}_{4}$ |
---|---|---|---|---|---|---|---|

0.0345 | 0.0532 | 0.0542 | −0.0580 | 0.0088 | 0.0041 | 0.0000 | 0.0025 |

Parameter | Value | Std Error | t-Statistic |
---|---|---|---|

a | 0.0028 | 0.0030 | 0.9436 |

b | 0.9627 | 0.0390 | 24.6907 |

${\sigma}^{2}$ | $2.77\times {10}^{-5}$ | $2.97\times {10}^{-6}$ | 9.3195 |

Annuity Payment | Mean | 2.5% | 25% | 50% | 75% | 97.5% |
---|---|---|---|---|---|---|

Age 80 | ||||||

Managed-Volatility | 17,076 | 1979 | 6833 | 11,946 | 21,308 | 64,504 |

Fixed Allocation | 12,741 | 1284 | 4472 | 8537 | 15,775 | 50,797 |

Age 90 | ||||||

Managed-Volatility | 21,732 | 790 | 4472 | 10,247 | 24,330 | 113,346 |

Fixed Allocation | 15,574 | 364 | 2416 | 6149 | 15,849 | 85,344 |

PV Annuity Payments | Mean | 2.5% | 25% | 50% | 75% | 97.5% |
---|---|---|---|---|---|---|

Nominal | ||||||

Managed-Volatility | 362,034 | 122,504 | 204,108 | 278,783 | 411,766 | 1,118,248 |

Fixed Allocation | 295,151 | 111,769 | 170,489 | 229,722 | 324,410 | 889,271 |

Real | ||||||

Managed-Volatility | 213,224 | 93,966 | 141,926 | 181,039 | 243,751 | 515,499 |

Fixed Allocation | 180,308 | 89,340 | 124,175 | 155,512 | 199,716 | 426,634 |

Nominal | ||||||
---|---|---|---|---|---|---|

Break Even Year | Mean | 2.5% | 25% | 50% | 75% | 97.5% |

Managed-Volatility | 15 | NA | 19 | 16 | 14 | 11 |

Fixed Allocation | 17 | NA | 21 | 17 | 15 | 12 |

t | 1 | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 |
---|---|---|---|---|---|---|---|---|---|---|---|

Managed-Volatility | 0.053 | 0.147 | 0.262 | 0.400 | 0.580 | 0.812 | 1.108 | 1.506 | 2.098 | 3.013 | 5.406 |

Fixed Allocation | 0.077 | 0.198 | 0.307 | 0.432 | 0.587 | 0.784 | 1.005 | 1.289 | 1.704 | 2.349 | 4.160 |

t | 1 | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 |
---|---|---|---|---|---|---|---|---|---|---|---|

Managed-Volatility | 0.036 | 0.096 | 0.168 | 0.241 | 0.319 | 0.397 | 0.474 | 0.549 | 0.622 | 0.693 | 0.775 |

Fixed Allocation | 0.052 | 0.128 | 0.194 | 0.256 | 0.322 | 0.387 | 0.450 | 0.516 | 0.582 | 0.650 | 0.736 |

Age 80 | Age 90 | |||||
---|---|---|---|---|---|---|

Annuity Payment | Mean | 2.5% | 97.5% | Mean | 2.5% | 97.5% |

All 3-mth FI | 4649 | 164 | 24,175 | 5571 | 15 | 34,202 |

All 10-year FI | 5882 | 239 | 29,299 | 7340 | 26 | 44,811 |

80/20 10-year FI/Equity | 9045 | 638 | 37,899 | 10,885 | 121 | 62,853 |

Annuity Payment | Age 80 | Age 90 | |||||
---|---|---|---|---|---|---|---|

FI/Equity | Asset Allocation | Mean | 2.5% | 97.5% | Mean | 2.5% | 97.5% |

80%/20% | Managed-Volatility | 10,596 | 894 | 42,687 | 12,806 | 199 | 71,993 |

Fixed Allocation | 9045 | 638 | 37,899 | 10,885 | 121 | 62,853 | |

65%/35% | Managed-Volatility | 17,076 | 1979 | 64,504 | 21,732 | 790 | 113,346 |

Fixed Allocation | 12,741 | 1284 | 50,797 | 15,574 | 364 | 85,344 | |

50%/50% | Managed-Volatility | 28,266 | 3822 | 100,717 | 40,403 | 2692 | 176,100 |

Fixed Allocation | 18,244 | 2272 | 66,516 | 23,501 | 1044 | 120,537 |

PV Annuity Payments | Nominal | Real | |||||
---|---|---|---|---|---|---|---|

FI/Equity | Asset Allocation | Mean | 2.5% | 97.5% | Mean | 2.5% | 97.5% |

80%/20% | Managed-volatility | 264,253 | 104,840 | 774,528 | 165,073 | 85,286 | 376,239 |

Fixed Allocation | 239,543 | 100,366 | 682,581 | 152,110 | 82,250 | 336,479 | |

65%/35% | Managed-volatility | 362,034 | 122,504 | 1,118,248 | 213,224 | 93,966 | 515,499 |

Fixed Allocation | 295,151 | 111,769 | 889,271 | 180,308 | 89,340 | 426,634 | |

50%/50% | Managed-volatility | 535,537 | 149,816 | 1,647,287 | 292,319 | 105,250 | 748,161 |

Fixed Allocation | 377,269 | 128,044 | 1,161,115 | 219,743 | 96,991 | 535,033 |

Break Even Year | Nominal | |||
---|---|---|---|---|

FI/Equity | Asset Allocation | Mean | 2.5% | 97.5% |

80%/20% | Managed-Volatility | 18 | NA | 13 |

Fixed Allocation | 19 | NA | 14 | |

65%/35% | Managed-Volatility | 15 | NA | 11 |

Fixed Allocation | 17 | NA | 12 | |

50%/50% | Managed-Volatility | 14 | 31 | 9 |

Fixed Allocation | 15 | 41 | 11 |

Age 80 | Age 90 | |||||
---|---|---|---|---|---|---|

Mean | 2.5% | 97.5% | Mean | 2.5% | 97.5% | |

Fixed Allocation | 12,741 | 1284 | 50,797 | 15,574 | 364 | 85,344 |

1 historical vol | 13,633 | 1415 | 53,270 | 16,798 | 422 | 91,590 |

1.25 historical vol | 17,076 | 1979 | 64,504 | 21,732 | 790 | 113,346 |

1.5 historical vol | 21,520 | 2733 | 77,124 | 28,697 | 1441 | 138,148 |

Nominal | Real | |||||
---|---|---|---|---|---|---|

Mean | 2.5% | 97.5% | Mean | 2.5% | 97.5% | |

Fixed Allocation | 295,151 | 111,769 | 889,271 | 180,308 | 89,340 | 426,634 |

1 historical vol | 310,310 | 113,091 | 945,729 | 188,229 | 89,956 | 447,627 |

1.25 historical vol | 362,034 | 122,504 | 1,118,248 | 213,224 | 93,966 | 515,499 |

1.5 historical vol | 429,562 | 133,505 | 1,319,962 | 244,708 | 98,498 | 604,932 |

Nominal | |||
---|---|---|---|

Mean | 2.5% | 97.5% | |

Fixed Allocation | 17 | NA | 12 |

1 historical vol | 16 | NA | 12 |

1.25 historical vol | 15 | NA | 11 |

1.5 historical vol | 15 | 36 | 10 |

Nominal | Real | |||||
---|---|---|---|---|---|---|

Mean | 2.5% | 97.5% | Mean | 2.5% | 97.5% | |

Fixed Target Vol | 362,034 | 122,504 | 1,118,248 | 213,224 | 93,966 | 515,499 |

Trend Down Vol | 358,390 | 122,142 | 1,101,347 | 212,132 | 93,835 | 511,014 |

Step Down Vol | 355,997 | 121,905 | 1,090,013 | 211,391 | 93,731 | 507,902 |

Nominal | |||
---|---|---|---|

Mean | 2.5%-Tile | 97.5%-Tile | |

Fixed Target Vol | 15 | NA | 11 |

Trend Down Vol | 15 | NA | 11 |

Step Down Vol | 15 | NA | 11 |

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## Share and Cite

**MDPI and ACS Style**

Li, S.; Labit Hardy, H.; Sherris, M.; Villegas, A.M.
A Managed Volatility Investment Strategy for Pooled Annuity Products. *Risks* **2022**, *10*, 121.
https://doi.org/10.3390/risks10060121

**AMA Style**

Li S, Labit Hardy H, Sherris M, Villegas AM.
A Managed Volatility Investment Strategy for Pooled Annuity Products. *Risks*. 2022; 10(6):121.
https://doi.org/10.3390/risks10060121

**Chicago/Turabian Style**

Li, Shuanglan, Héloïse Labit Hardy, Michael Sherris, and Andrés M. Villegas.
2022. "A Managed Volatility Investment Strategy for Pooled Annuity Products" *Risks* 10, no. 6: 121.
https://doi.org/10.3390/risks10060121