# Testing the Least-Squares Monte Carlo Method for the Evaluation of Capital Requirements in Life Insurance

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## Abstract

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## 1. Introduction

## 2. The Evaluation Framework

## 3. Numerical Results

## 4. Conclusions

- In the case of policy with long maturity, VaR estimates are biased even when a relevant number of simulation runs and several basis functions are used.
- The choice of the number and the type of basis functions seems not to be an issue in a low-dimensional framework as standard polynomials of degree 3 give the highest explanatory power and increasing the degree of the polynomials or using different types of basis functions do not improve the accuracy of VaR estimates.
- Further research is needed to understand if also in a high-dimensional setting polynomials of relatively low degree are able to fit well the insurer’s loss distribution. If not, basis functions obtained according to a certain optimal criterion will, arguably, be of great importance.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

- $\tilde{\mathcal{P}}\left(A\right)=\mathcal{P}\left(A\right),A\in {\mathcal{F}}_{t},\phantom{\rule{1.em}{0ex}}0\le t\le \tau ;$ and
- ${E}_{\tilde{\mathcal{P}}}\left[X\right|{\mathcal{F}}_{t}]={E}_{\mathcal{Q}}\left[X\right|{\mathcal{F}}_{t}],\phantom{\rule{1.em}{0ex}}\forall X\in {\mathcal{F}}_{t}$.

## References

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1. | In Appendix A we give a sketch of how such optimal basis functions are derived. |

**Figure 1.**This figure presents the box plots relative to VaR estimates for the equity-linked policy with maturity $T=5$ years. Each box plot was generated by considering 100 estimates. Each estimate in the first four plots was computed with $N=5\times {10}^{4}$ simulations and a number of basis functions equal to $6,10,15$ and 21. The second four plots were generated similarly except that each VaR were computed with $N=5\times {10}^{5}$ simulations. In the last four plots, each VaR was obtained with $N={10}^{6}$ simulations. The segment labeled B represents the benchmark.

**Figure 2.**This figure presents the box plots relative to VaR estimates for the equity-linked policy with maturity $T=10$ years. Each box plot was generated by considering 100 estimates. Each estimate in the first four plots was computed with $N=5\times {10}^{4}$ simulations and a number of basis functions equal to $6,10,15$, and 21. The second four plots were computed similarly except that each VaR has been computed with $N=5\times {10}^{5}$ simulations. In the last four plots, each VaR was obtained with $N={10}^{6}$ simulations. The segment labeled B represents the benchmark.

**Figure 3.**This figure presents the box plots relative to VaR estimates for the equity-linked policy with maturity $T=20$ years. Each box plot was generated by considering 100 estimates. Each estimate in the first four plots was computed with $N=5\times {10}^{4}$ simulations and a number of basis functions equal to $6,10,15$, and 21. The second four plots were computed similarly except that each VaR has been computed with $N=5\times {10}^{5}$ simulations. In the last four plots, each VaR was obtained with $N={10}^{6}$ simulations. The segment labeled B represents the benchmark.

**Figure 4.**This graph presents the box plots of VaR estimates for the policy with maturity $T=5$ years, computed considering different number of optimal basis functions (ox with $x=6,10,15,21$). Beside each plot obtained through optimal basis functions a plot with the VaRs obtained with monomials in the proxy function is inserted ($mx$ with $x=6,10,15,21$). The eight boxes on the left were obtained by considering $N=5\times {10}^{4}$ simulation runs, the eight boxes in the middle were generated by considering $N=5\times {10}^{5}$ simulations, and the eight boxes on the were been generated with $N={10}^{6}$ simulations. The last segment on the right, B, represents the benchmark.

**Figure 5.**This graph presents the box plots of VaR estimates for the policy with maturity $T=10$ years, computed considering different number of optimal basis functions (ox with $x=6,10,15,21$). Beside each plot obtained through optimal basis functions a plot with the VaRs obtained with monomials in the proxy function is inserted ($mx$ with $x=6,10,15,21$). The eight boxes on the left were obtained by considering $N=5\times {10}^{4}$ simulation runs, the eight boxes in the middle were generated by considering $N=5\times {10}^{5}$ simulations, and the eight boxes on the right were generated with $N={10}^{6}$ simulations. The last segment on the right, B, represents the benchmark.

**Figure 6.**This graph presents the box plots of VaR estimates for the policy with maturity $T=20$ years, computed considering different number of optimal basis functions (ox with $x=6,10,15,21$). Beside each plot obtained through optimal basis functions a plot with the VaRs obtained with monomials in the proxy function is inserted ($mx$ with $x=6,10,15,21$). The eight boxes on the left were obtained by considering $N=5\times {10}^{4}$ simulation runs, the eight boxes in the middle were generated by considering $N=5\times {10}^{5}$ simulations, and the eight boxes on the right were generated with $N={10}^{6}$ simulations. The last segment on the right, B, represents the benchmark.

**Table 1.**This table illustrates the MAPE of VaRs computed with the LSMC method for the equity-linked policy with maturity $T=5$ years, $T=15$ years, and $T=20$ years. Each value was computed by considering a sample of 100 estimated VaRs. Four possible choices for the number of basis functions ($M=6,10,15,21$) and three possible choices of simulations runs ($N=5\times {10}^{4}$, $N=5\times {10}^{5}$, and $N={10}^{6}$) were considered.

$\mathit{T}=5$ Years | $\mathit{T}=10$ Years | $\mathit{T}=20$ Years | |||||||
---|---|---|---|---|---|---|---|---|---|

$\mathbf{M}$\$\mathbf{N}$ | $\mathbf{5}\times {\mathbf{10}}^{\mathbf{4}}$ | $\mathbf{5}\times {\mathbf{10}}^{\mathbf{5}}$ | ${\mathbf{10}}^{\mathbf{6}}$ | $\mathbf{5}\times {\mathbf{10}}^{\mathbf{4}}$ | $\mathbf{5}\times {\mathbf{10}}^{\mathbf{5}}$ | ${\mathbf{10}}^{\mathbf{6}}$ | $\mathbf{5}\times {\mathbf{10}}^{\mathbf{4}}$ | $\mathbf{5}\times {\mathbf{10}}^{\mathbf{5}}$ | ${\mathbf{10}}^{\mathbf{6}}$ |

6 | 2.32% | 2.37% | 2.42% | 2.74% | 2.24% | 2.58% | 3.22% | 2.35% | 2.64% |

10 | 1.21% | 0.46% | 0.36% | 2.24% | 0.70% | 0.42% | 4.19% | 1.16% | 0.93% |

15 | 1.20% | 0.48% | 0.34% | 2.43% | 0.75% | 0.47% | 4.72% | 1.32% | 1.01% |

21 | 1.25% | 0.47% | 0.35% | 2.36% | 0.74% | 0.44% | 4.22% | 1.25% | 0.98% |

**Table 2.**This table reports the running times (in seconds) of the policy illustrated in Table 1 with maturity $T=20$ years. The MAPE of VaRs computed with nested simulations and the running time needed to obtain each VaR are also reported.

$\mathit{T}=20$ Years | |||
---|---|---|---|

$\mathbf{M}$\$\mathbf{N}$ | $\mathbf{5}\times {\mathbf{10}}^{\mathbf{4}}$ | $\mathbf{5}\times {\mathbf{10}}^{\mathbf{5}}$ | ${\mathbf{10}}^{\mathbf{6}}$ |

6 | 3.54 s | 34.45 s | 63.76 s |

10 | 3.83 s | 34.21 s | 66.17 s |

15 | 3.68 s | 33.25 s | 71.84 s |

21 | 3.60 s | 31.38 s | 76.89 s |

**Table 3.**This table illustrates the MAPE of VaRs computed by the LSMC method with optimal basis functions in the sense of Bauer and Ha (2013) (in brackets are reported the corresponding values obtained with monomials as basis functions). Four possible choices for the number of basis functions ($M=6,10,15,21$) and three possible choices of simulations runs ($N=5\times {10}^{4},5\times {10}^{5}$, and $N={10}^{6}$) were considered. Three different maturities, $T=5,10,$ and 20, years were considered. Each value was computed by considering a sample of 100 estimated VaRs.

$\mathit{T}=5$ Years | $\mathit{T}=10$ Years | $\mathit{T}=20$ Years | |||||||
---|---|---|---|---|---|---|---|---|---|

$\mathbf{M}$\$\mathbf{N}$ | $\mathbf{5}\times {\mathbf{10}}^{\mathbf{4}}$ | $\mathbf{5}\times {\mathbf{10}}^{\mathbf{5}}$ | ${\mathbf{10}}^{\mathbf{6}}$ | $\mathbf{5}\times {\mathbf{10}}^{\mathbf{4}}$ | $\mathbf{5}\times {\mathbf{10}}^{\mathbf{5}}$ | ${\mathbf{10}}^{\mathbf{6}}$ | $\mathbf{5}\times {\mathbf{10}}^{\mathbf{4}}$ | $\mathbf{5}\times {\mathbf{10}}^{\mathbf{5}}$ | ${\mathbf{10}}^{\mathbf{6}}$ |

6 | 3.21% | 2.41% | 2.43% | 4.30% | 3.36% | 3.35% | 7.14% | 6.69% | 6.88% |

(3.21%) | (2.41%) | (2.43%) | (3.58%) | (2.64%) | (2.58%) | (5.92%) | (2.92%) | (2.67%) | |

10 | 2.93% | 0.89% | 0.69% | 4.54% | 1.38% | 1.19% | 6.88% | 2.43% | 1.65% |

(2.81%) | (0.85%) | (0.67%) | (4.53%) | (1.32%) | (1.14%) | (4.56%) | (2.62%) | (1.67%) | |

15 | 3.02% | 0.88% | 0.70% | 4.77% | 1.41% | 1.20% | 7.87% | 2.68% | 1.75% |

(3.13%) | (0.91%) | (0.70%) | (4.98%) | (1.48%) | (1.23%) | (8.42%) | (2.83%) | (1.77%) | |

21 | 3.42% | 0.87% | 0.78% | 4.83% | 1.49% | 1.18% | 8.96% | 2.84% | 1.93% |

(3.08%) | (0.90%) | (0.70%) | (4.77%) | (1.38%) | (1.18%) | (8.51%) | (2.74%) | (1.88%) |

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**MDPI and ACS Style**

Costabile, M.; Viviano, F.
Testing the Least-Squares Monte Carlo Method for the Evaluation of Capital Requirements in Life Insurance. *Risks* **2020**, *8*, 48.
https://doi.org/10.3390/risks8020048

**AMA Style**

Costabile M, Viviano F.
Testing the Least-Squares Monte Carlo Method for the Evaluation of Capital Requirements in Life Insurance. *Risks*. 2020; 8(2):48.
https://doi.org/10.3390/risks8020048

**Chicago/Turabian Style**

Costabile, Massimo, and Fabio Viviano.
2020. "Testing the Least-Squares Monte Carlo Method for the Evaluation of Capital Requirements in Life Insurance" *Risks* 8, no. 2: 48.
https://doi.org/10.3390/risks8020048