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
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 , then
- 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 , and so on. Denote the k-th realized variance as , then
- Take the square root of the realized variance to get the realized volatility. Denote the k-th realized volatility as , then
- 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
−0.1004 | −0.1347 | 1.4285 | 4.9659 |
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) | ||||
---|---|---|---|---|
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 | ||||
---|---|---|---|---|
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) | ||||
---|---|---|---|---|
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 | −2.1154 | −2.0959 | −2.0854 |
Appendix B.4. Estimated Parameters for VAR Model
Appendix B.5. Simulation Versus Actual Results
Simulation | Actual | |||||||
---|---|---|---|---|---|---|---|---|
Statistic | ||||||||
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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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 |
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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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
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 StyleLi, 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