# Modeling the Future Value Distribution of a Life Insurance Portfolio

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## Abstract

**:**

## 1. Introduction

- Select a subset of representative policies by means of conditional Latin hypercube sampling;
- Project the risk factors from the evaluation date to the risk horizon by means of outer simulations;
- Compute a rough estimate of each representative policy by means of a very limited (say two) number of inner simulations;
- Create a regression model to approximate the distribution of the value of representative policies;
- Use the regression model to estimate the future value distribution of the entire portfolio.

## 2. The Evaluation Framework

## 3. Problem and Methodology

**Proposition**

**1.**

**Proposition**

**2.**

#### 3.1. The LSMC Method

#### 3.2. The GB2 Model

## 4. Numerical Results

`R`on a computer equipped with an Intel

^{®}Core(TM) i7-1065G7 CPU 1.50 GHz processor with 12 GB of RAM and Windows 10 Home operating system.

#### Full LSMC

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Parameter Values

**Table A1.**Parameters of the reference asset value process, S, and interest rate stochastic process, r.

${\mathit{S}}_{0}$ | ${\mathit{\sigma}}_{\mathit{S}}$ | $\mathit{\lambda}$ | ${\mathit{r}}_{0}$ | $\mathit{\alpha}$ | $\mathit{\theta}$ | ${\mathit{\sigma}}_{\mathit{r}}$ | $\mathit{\gamma}$ | $\mathit{\rho}$ |
---|---|---|---|---|---|---|---|---|

100 | 0.20 | 0.00 | 0.04 | 0.10 | 0.02 | 0.02 | 0.00 | 0.00 |

**Table A2.**Estimated parameters of the stochastic mortality model for Italian male (left) and female (right) aged $x\in \left(\right)open="\{"\; close="\}">55,\cdots ,65$ in 2016.

Age | Male | Female | ||||
---|---|---|---|---|---|---|

$\widehat{\mathit{a}}$ | $\widehat{\mathit{b}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{\mu}}$ | $\widehat{\mathit{a}}$ | $\widehat{\mathit{b}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{\mu}}$ | |

55 | 0.00040 | 0.0881 | 0.00157 | 0.00010 | 0.10017 | 0.00100 |

56 | 0.00700 | 0.0705 | 0.00262 | 0.00001 | 0.11110 | 0.00100 |

57 | 0.00001 | 0.1051 | 0.00100 | 0.00001 | 0.11060 | 0.00100 |

58 | 0.00001 | 0.1045 | 0.00390 | 0.00009 | 0.10740 | 0.00850 |

59 | 0.00040 | 0.0832 | 0.00100 | 0.00001 | 0.11570 | 0.00100 |

60 | 0.00060 | 0.0743 | 0.00100 | 0.00042 | 0.08362 | 0.00669 |

61 | 0.00030 | 0.0907 | 0.00100 | 0.00044 | 0.08505 | 0.00100 |

62 | 0.00010 | 0.1033 | 0.00710 | 0.00001 | 0.11990 | 0.00100 |

63 | 0.00012 | 0.1063 | 0.00750 | 0.00040 | 0.09704 | 0.00182 |

64 | 0.00008 | 0.1112 | 0.00810 | 0.00039 | 0.09860 | 0.00376 |

65 | 0.00020 | 0.1075 | 0.00123 | 0.00049 | 0.09558 | 0.00720 |

## Appendix B. Further Results

**Figure A1.**Boxplots relative to the mean (

**left**) and the 99.5th percentile (

**right**) estimates obtained by running the GB2 and LSMC methods 50 times and varying the number of outer simulations n and that of representative contracts s. The red line refers to the benchmark value based on a nested simulations algorithm with 10,000 × 2500 trajectories applied to the entire portfolio.

**Figure A2.**Boxplots relative to the mean and the 99.5th percentile estimates obtained by running the proposed methodologies 50 times. GB2 stands for the GB2 regression model based on 10,000 outer scenarios and $s=100$ representative policies; LSMC_1 refers to the LSMC method based on 10,000 outer scenarios and $s=100$ representative policies with Hermite polynomials of order 1; LSMC_2 refers to the LSMC method based on 10,000 outer scenarios and $s=100$ representative policies with Hermite polynomials of order 2; LSMC_Full refers to the LSMC method based on 10,000 outer scenarios and constructed on each contract in the insurance portfolio. The red line refers to the benchmark value based on a nested simulations algorithm with 10,000 × 2500 trajectories applied to the entire portfolio.

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**Figure 1.**Q-Q plots relative to the future value distribution of the insurance portfolio. The theoretical distribution is assumed to be the one obtained by nested simulations based on 10,000 × 2500 trajectories. The first row refers to the GB2 regression model based on 10,000 outer scenarios and by varying the number of representative contracts, $s\in \{50,75,100\}$. The second and third rows refer to the LSMC method with Hermite polynomials of orders 1 and 2 based on 10,000 outer scenarios and by varying the number of representative contracts, $s\in \{50,75,100\}$.

Feature | Value |
---|---|

Policyholder age | {55, …, 65} |

Sex | {Male, Female} |

Maturity | {10, 15, 20, 25, 30} |

Product type | {Unit-linked, Term Insurance, Life Annuity} |

**Table 2.**This table reports the MAPE of the estimates obtained by running 50 times the GB2 and LSMC methods with n = 10,000 and $s=50$. The benchmark values are based on a nested simulations algorithm with 10,000 $\times 2500$ trajectories applied to the entire portfolio.

5th Perc. | 10th Perc. | Median | Mean | 90th Perc. | 95th Perc. | 99th Perc. | 99.5th Perc. | |
---|---|---|---|---|---|---|---|---|

GB2 | 2.812% | 2.180% | 1.798% | 2.594% | 3.832% | 4.016% | 6.154% | 4.375% |

LSMC_1 | 3.238% | 3.000% | 2.399% | 2.557% | 2.398% | 2.174% | 2.436% | 2.722% |

LSMC_2 | 2.762% | 2.754% | 2.567% | 2.557% | 2.436% | 2.114% | 2.356% | 2.841% |

**Table 3.**This table reports the MAPE of the estimates obtained by running 50 times the GB2 and LSMC methods with n = 10,000 and $s=75$. The benchmark values are based on a nested simulations algorithm with 10,000 × 2500 trajectories applied to the entire portfolio.

5th Perc. | 10th Perc. | Median | Mean | 90th Perc. | 95th Perc. | 99th Perc. | 99.5th Perc. | |
---|---|---|---|---|---|---|---|---|

GB2 | 1.971% | 1.782% | 0.806% | 0.542% | 3.605% | 3.949% | 6.094% | 3.867% |

LSMC_1 | 2.500% | 1.338% | 1.530% | 1.392% | 1.251% | 1.657% | 0.941% | 1.678% |

LSMC_2 | 1.828% | 1.047% | 1.756% | 1.392% | 1.307% | 1.485% | 1.842% | 2.142% |

**Table 4.**This table reports the MAPE of the estimates obtained by running 50 times the GB2 and LSMC methods with n = 10,000 and $s=100$. The benchmark values are based on a nested simulations algorithm with 10,000 × 2500 trajectories applied to the entire portfolio.

5th Perc. | 10th Perc. | Median | Mean | 90th Perc. | 95th Perc. | 99th Perc. | 99.5th Perc. | |
---|---|---|---|---|---|---|---|---|

GB2 | 1.986% | 1.745% | 0.519% | 0.347% | 1.129% | 1.313% | 2.856% | 1.944% |

LSMC_1 | 1.629% | 1.504% | 0.440% | 0.627% | 0.764% | 0.824% | 0.958% | 2.561% |

LSMC_2 | 1.148% | 1.145% | 0.578% | 0.627% | 0.762% | 0.986% | 2.101% | 2.334% |

**Table 5.**This table reports the MPE and MAPE of the mean estimates obtained by running 50 times the GB2 and LSMC methods and varying the number of outer simulations (Outer) and that of representative contracts s. The benchmark value is based on a nested simulations algorithm with 10,000 × 2500 trajectories applied to the entire portfolio.

s = 50 | s = 75 | s = 100 | |||||
---|---|---|---|---|---|---|---|

Outer | Method | MPE | MAPE | MPE | MAPE | MPE | MAPE |

GB2 | 3.612% | 3.612% | 0.163% | 0.983% | −0.240% | 0.923% | |

1000 | LSMC_1 | −3.475% | 3.475% | −2.104% | 2.221% | −1.017% | 1.364% |

LSMC_2 | −3.475% | 3.475% | −2.104% | 2.221% | −1.017% | 1.364% | |

GB2 | 2.981% | 2.981% | 0.715% | 0.747% | −0.301% | 0.474% | |

5000 | LSMC_1 | −2.840% | 2.840% | −1.533% | 1.533% | −1.029% | 1.092% |

LSMC_2 | −2.840% | 2.840% | −1.533% | 1.533% | −1.029% | 1.092% | |

GB2 | 2.594% | 2.594% | 0.491% | 0.542% | 0.179% | 0.347% | |

10,000 | LSMC_1 | −2.557% | 2.557% | −1.392% | 1.392% | −0.490% | 0.627% |

LSMC_2 | −2.557% | 2.557% | −1.392% | 1.392% | −0.490% | 0.627% |

**Table 6.**This table reports the MPE and MAPE of the 99.5th percentile estimates obtained by running the GB2 and LSMC methods 50 times and varying the number of outer simulations (Outer) and that of representative contracts s. The benchmark value is based on a nested simulations algorithm with 10,000 × 2500 trajectories applied to the entire portfolio.

s = 50 | s = 75 | s = 100 | |||||
---|---|---|---|---|---|---|---|

Outer | Method | MPE | MAPE | MPE | MAPE | MPE | MAPE |

GB2 | 3.936% | 6.570% | −1.512% | 5.453% | 1.410% | 4.494% | |

1000 | LSMC_1 | −2.664% | 3.715% | −6.308% | 6.478% | −2.961% | 4.253% |

LSMC_2 | −0.252% | 6.487% | −4.211% | 7.150% | −1.438% | 5.517% | |

GB2 | 4.110% | 4.723% | 3.813% | 4.018% | 0.081% | 2.653% | |

5000 | LSMC | −2.908% | 3.001% | −4.708% | 4.722% | −1.659% | 2.006% |

LSMC_2 | −1.787% | 3.484% | −3.118% | 4.017% | −0.462% | 3.110% | |

GB2 | 4.157% | 4.375% | 3.737% | 3.867% | 0.421% | 1.944% | |

10,000 | LSMC_1 | −2.643% | 2.722% | −1.560% | 1.678% | −2.522% | 2.561% |

LSMC_2 | −2.259% | 2.841% | −0.131% | 2.142% | −1.007% | 2.334% |

**Table 7.**Percentage of the runtime required by the GB2 and LSMC methods with respect to the nested simulations approach. Note that the computational demand to construct the benchmark with a nested simulations approach based on 10,000 × 2500 scenarios applied to the entire portfolio is about 187,200 s.

Method | $\mathit{n}=1000$ | $\mathit{n}=5000$ | n = 10,000 | ||||||
---|---|---|---|---|---|---|---|---|---|

$\mathit{s}=50$ | $\mathit{s}=75$ | $\mathit{s}=100$ | $\mathit{s}=50$ | $\mathit{s}=75$ | $\mathit{s}=100$ | $\mathit{s}=50$ | $\mathit{s}=75$ | $\mathit{s}=100$ | |

GB2 | 0.069% | 0.078% | 0.098% | 0.337% | 0.380% | 0.501% | 0.660% | 0.832% | 1.021% |

LSMC_1 | 0.005% | 0.006% | 0.007% | 0.012% | 0.018% | 0.019% | 0.036% | 0.045% | 0.047% |

LSMC_2 | 0.005% | 0.006% | 0.007% | 0.013% | 0.019% | 0.020% | 0.037% | 0.046% | 0.047% |

**Table 8.**This table reports the MPE and MAPE relative to the 5th percentile, the mean, and the 99.5th percentile estimates obtained by applying different methodologies. GB2 stands for the GB2 regression model based on n = 10,000 outer scenarios and $s=100$ representative policies; LSMC_1 refers to the LSMC method based on n = 10,000 outer scenarios and $s=100$ representative policies with Hermite polynomials of order 1; LSMC_2 refers to the LSMC method based on n = 10,000 outer scenarios and $s=100$ representative policies with Hermite polynomials of order 2; LSMC_Full refers to the LSMC method based on n = 10,000 outer scenarios and constructed on each contract in the insurance portfolio. The results are compared with the corresponding benchmark value based on nested simulations with $10000\times 2500$ trajectories applied to the entire portfolio.

Method | 5th Perc. | Mean | 99.5th Perc. | |||
---|---|---|---|---|---|---|

MPE | MAPE | MPE | MAPE | MPE | MAPE | |

GB2 | −1.986% | 1.986% | 0.179% | 0.347% | 0.421% | 1.944% |

LSMC_1 | −1.472% | 1.629% | −0.490% | 0.627% | −2.522% | 2.561% |

LSMC_2 | −0.742% | 1.148% | −0.490% | 0.627% | −1.007% | 2.334% |

LSMC_Full | −0.501% | 1.032% | −0.084% | 0.461% | −0.420% | 1.070% |

**Table 9.**Runtime, in seconds, of GB2 model and LSMC methods based on 10,000 × 2 simulations and $s=100$ representative contracts (GB2, LSMC_1, LSMC_2). LSMC_Full refers to the LSMC method applied to each contract in the insurance portfolio.

Method | Time |
---|---|

GB2 | 1911.445 |

LSMC_1 | 87.824 |

LSMC_2 | 88.290 |

LSMC_Full | 7847.960 |

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## Share and Cite

**MDPI and ACS Style**

Costabile, M.; Viviano, F.
Modeling the Future Value Distribution of a Life Insurance Portfolio. *Risks* **2021**, *9*, 177.
https://doi.org/10.3390/risks9100177

**AMA Style**

Costabile M, Viviano F.
Modeling the Future Value Distribution of a Life Insurance Portfolio. *Risks*. 2021; 9(10):177.
https://doi.org/10.3390/risks9100177

**Chicago/Turabian Style**

Costabile, Massimo, and Fabio Viviano.
2021. "Modeling the Future Value Distribution of a Life Insurance Portfolio" *Risks* 9, no. 10: 177.
https://doi.org/10.3390/risks9100177