# Revisiting Calibration of the Solvency II Standard Formula for Mortality Risk: Does the Standard Stress Scenario Provide an Adequate Approximation of Value-at-Risk?

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## Abstract

**:**

## 1. Introduction

…some risk do not fit naturally into one year VaR framework and it would be excessively dogmatic to insist that longevity trend risk only be measured over a one year horizon.

- EIOPA follows run-off VaR approach and we examine the results using both run-off VaR and one-year VaR approaches.
- EIOPA assumes random walk with drift (RWD) as the default model for forecasting of the time varying parameter, which is driving the projected mortality improvements. We perform formal statistical testing for the purpose of confirming the hypothesis that RWD does not contradict the historic data.
- EIOPA fits the stochastic mortality model using the mortality data for ages 40–120 years (the same as for longevity risk). We fit the stochastic mortality model for ages 25–75 years. We assume, that for most of life assurance protection products the highest exposure to mortality risk is associated with mid-aged assured lives. Therefore, the model fitted using the experience from the target population range can provide the most relevant parametrisation for the forecast.
- EIOPA calibrates VaR using temporary life expectancies, thus implicitly assuming that the benefit payable under life assurance policy decreases with time. Decreasing benefits are common in life assurance products linked to mortgages or other credit instruments. In addition, mortality sum at risk is decreasing with policy duration for some risk and savings products (e.g., traditional endowment insurance) where the total benefit payable on death is fixed and insurer is able to recoup part of the losses by reversing the accumulated savings amount. However, a significant part of life assurance products have fixed sums assured. In this paper we examine the effect on VaR of both benefit formulas: level (fixed) sum assured and sum assured which is decreasing linearly with time.

## 2. Results

## 3. Discussion

## 4. Stochastic Mortality Model

#### 4.1. Overview of the Data

#### 4.2. Lee-Carter Model

- ${m}_{x}\left(t\right)$ is the central mortality rate for age x in year t,
- ${\alpha}_{x}$ is parameter, which represents the general shape of changes in log mortality rates with age,
- $\kappa \left(t\right)$ is the time varying index, which represents the general trend of changes in mortality rates with time,
- ${\beta}_{x}$ is parameter, which determines the impact of the time varying index on age specific log mortality rates,
- ${\epsilon}_{x}\left(t\right)$ are residuals, it is supposed that ${\epsilon}_{x}\left(t\right)$ are independent and identically distributed random variables with zero mean and finite variance.

#### 4.3. Estimation of Parameters

#### 4.4. Modelling of General Mortality Trend Parameter $\kappa \left(t\right)$

## 5. Methodology of VaR Calculations

#### 5.1. Run-off VaR Methodology

#### 5.2. One-Year VaR Methodology

#### 5.3. Modelling Different Mortality Benefit Formulas

#### 5.3.1. Level Benefits

#### 5.3.2. Decreasing Benefits

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Lee-Carter model parameters ${\beta}_{x}$ estimated using SVD and Poisson (before and after smoothing) methods using data of Lithuania (

**a**), Sweden (

**b**), Poland (

**c**) and the Netherlands (

**d**).

**Figure A2.**Lee-Carter model parameters $\kappa \left(t\right)$ estimated using Singular Value Decomposition (SVD) (before and after second step re-estimation) and Poisson (before and after re-estimation using smoothed ${\beta}_{x}$ parameters) methods using data of Lithuania (

**a**), Sweden (

**b**), Poland (

**c**) and the Netherlands (

**d**).

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**Figure 1.**Example of simulations of central mortality rates (for a fixed age) used for the purpose of calculation of run-off VaR (see (

**a**)) and one-year $\mathrm{VaR}$ (see (

**b**)). The calculations are based on the Lithuanian data for a life aged 40 years. In calculations of one-year VaR reserve sensitivity parameter $\delta $ = 10% was used.

**Figure 2.**Calculated run-off $\mathrm{VaR}$ rates at policy inception by country for different ages at policy inception, three different terms to maturity (10 years, 20 years and 30 years) and two benefit formulas (level sum assured and sum assured decreasing with time).

**Figure 3.**Calculated one-year $\mathrm{VaR}$ rates (parameter $\delta =5\%$) at policy inception by country for different ages at policy inception, three different terms to maturity (10 years, 20 years and 30 years) and two benefit formulas (level sum assured and sum assured decreasing with time).

**Figure 4.**Calculated one-year $\mathrm{VaR}$ rates (parameter $\delta =10\%$) at policy inception by country for different ages at policy inception, three different terms to maturity (10 years, 20 years and 30 years) and two benefit formulas (level sum assured and sum assured decreasing with time).

**Figure 5.**Development of periodic life expectancy at birth in Lithuania (

**a**), Sweden (

**b**), Poland (

**c**) and the Netherlands (

**d**) according to data from Human Mortality Database (2018).

**Figure 6.**Development of log mortality rates in Lithuania (

**a**) and Poland (

**b**) according to data from the Human Mortality Database (2018).

**Figure 7.**Confidence intervals for 30-year projection (

**a**) and 1-year projection (

**b**) of mortality rates with and without taking into account the additional volatility from random fluctuation around the modelled mean mortality rates. The calculations were performed for Lithuanian data by assuming Gaussian approximation of Poisson distribution and by allowing for overdisperion.

**Figure 8.**Lee-Carter model parameters ${\alpha}_{x}$ (panel (

**a**)), ${\beta}_{x}$ (panel (

**b**)) and $\kappa \left(t\right)$ (panel (

**c**)) estimated using Poisson regression (with re-fitting). Panel (

**d**) shows distribution of Pearson residuals for ages 45–65 in Poland.

**Figure 9.**Fitted and forecasted $\kappa \left(t\right)$ parameters modelled as RWD using data of Lithuania (

**a**), Sweden (

**b**), Poland (

**c**) and the Netherlands (

**d**). The shaded error represents $95\%$ confidence interval of the projections assuming normal distribution of errors.

**Figure 10.**Calculation of the Best Estimate (orange line) and run-off VaR (green line) for term life assurance with level sum insured for the policy term of 30 years. The red line represents $0.5\%$ confidence interval of the survival probability, which is equivalent to $99.5\%$ confidence interval of the cumulative death probability.

**Figure 11.**Calculation of Best Estimate (orange area) and run-off VaR (green area) for term life assurance with linearly decreasing sum insured for the policy term of 30 years. The red line represents $0.5\%$ confidence interval of survival probability, which is equivalent to $99.5\%$ confidence interval of the cumulative death probability.

**Table 1.**Number of Solvency II submissions (solo undertakings) in 2017 by the method used for calculation of the Solvency Capital Requirement (SCR). Data source: EIOPA.

Type of Insurer | Internal Model | Partial Internal Model | Standard Formula | Total |
---|---|---|---|---|

Life insurance companies | 21 | 29 | 545 | 595 |

Non-life insurance companies | 37 | 42 | 1519 | 1598 |

Reinsurance companies | 15 | 4 | 253 | 272 |

Composite insurance companies | 8 | 30 | 368 | 406 |

Total | 81 | 105 | 2685 | 2871 |

**Table 2.**Proportion of variance explained by the first principal component and estimated overdispersion parameter for fitted Lee-Carter model. Overdispersion was estimated as the sum of squared Pearson residuals divided by the residual degrees of freedom.

Statistics | Lithuania | Sweden | Poland | The Netherlands |
---|---|---|---|---|

SVD estimation: proportion of | ||||

variance explained by the first | 0.78 | 0.92 | 0.96 | 0.95 |

principal component | ||||

Poisson regression: estimated | ||||

overdispersion parameter ${\sigma}^{2}$ | 1.63 | 1.45 | 3.95 | 1.34 |

**Table 3.**Statistical testing for unit-roots for the time series parameter of $\kappa \left(t\right)$.

Test Statistics | Lithuania | Sweden | Poland | The Netherlands |
---|---|---|---|---|

Philips-Perron test | ||||

$Z\left(\rho \right)$ | −6.5 [−17.9] | −2.7 [−19.8] | −19.1 [−17.9] | −4.1 [−19.8] |

$Z\left(t\right)$ | −1.7 [−3.6] | −1.6 [−3.5] | −3.6 [−3.6] | −2.2 [−3.5] |

Augmented Dickey-Fuller test | ||||

${Z}_{DF}\left(\rho \right)$ | −5.8 [−17.9] | −3.5 [−19.8] | −17.8 [−17.9] | −5.4 [−19.8] |

${Z}_{DF}\left(t\right)$ | −1.1 [−3.6] | −2.7 [−3.5] | −2.3 [−3.6] | −1.9 [−3.5] |

**Table 4.**Estimated Random Walk with Drift (RWD) parameters of time series $\kappa \left(t\right)$ derived from the re-fitted Poisson regression.

Parameter | Lithuania | Sweden | Poland | The Netherlands |
---|---|---|---|---|

$\mu $ | −1.25 | −0.79 | −1.05 | −0.63 |

${\sigma}^{2}$ | 6.62 | 0.65 | 1.14 | 0.78 |

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**MDPI and ACS Style**

Gylys, R.; Šiaulys, J.
Revisiting Calibration of the Solvency II Standard Formula for Mortality Risk: Does the Standard Stress Scenario Provide an Adequate Approximation of Value-at-Risk? *Risks* **2019**, *7*, 58.
https://doi.org/10.3390/risks7020058

**AMA Style**

Gylys R, Šiaulys J.
Revisiting Calibration of the Solvency II Standard Formula for Mortality Risk: Does the Standard Stress Scenario Provide an Adequate Approximation of Value-at-Risk? *Risks*. 2019; 7(2):58.
https://doi.org/10.3390/risks7020058

**Chicago/Turabian Style**

Gylys, Rokas, and Jonas Šiaulys.
2019. "Revisiting Calibration of the Solvency II Standard Formula for Mortality Risk: Does the Standard Stress Scenario Provide an Adequate Approximation of Value-at-Risk?" *Risks* 7, no. 2: 58.
https://doi.org/10.3390/risks7020058