# A Bridge between Local GAAP and Solvency II Frameworks to Quantify Capital Requirement for Demographic Risk

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## Abstract

**:**

## 1. Introduction

## 2. Technical Profit and Gain/Loss Decomposition in a Solvency II Framework

## 3. The Demographic Profit and Its Factorisation

## 4. Model Algebra and Underlying Recursive Formula

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

## 5. The Profit Formation

## 6. The Application of the Model to Non-Participating Life Policies

- For each policyholder, 10 million simulations have been made. Therefore, the results are particularly consistent, especially in terms of volatility;12
- The total amount of the sums insured at the inception of the policy ($t=0$) is equal to approximately 1.5 billion euros. It is therefore noted that any Capital Requirement, in terms of magnitude, must be compared with the value just mentioned, although at first glance it may seem particularly high.

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Homans’ Revised Decomposition

## Notes

1 | From now on, each random variable will be indicated with the tilde. |

2 | “Complete technical provisions” $V{B}_{t}$ indicates the sum of the pure mathematical reserve (expected present value of the benefits net of the expected present value of the premiums) and the expense reserve. Both are calculated on locked and prudential (demographic and financial) bases: hence both demographic and financial bases used are the same as applied in the pricing phase. |

3 | By “actual financial return rate” we mean the yield obtained by the company, different from the technical rate ${j}^{*}$ used in the pricing phase. |

4 | The rates are calculated on unitary insured sums, so as to be able to distinguish the trends of the main quantities from the monetary amounts of the insured sums. |

5 | In this context ${\tilde{x}}_{t+1}={\tilde{z}}_{t}+1$ for term Insurance and Endowment Policies; ${\tilde{x}}_{t+1}=0$ for Pure Endowment policies. |

6 | The technical bases of the first order are those used in the pricing phase, they are therefore the prudential ones that lead to an expected profit. The second order bases, on the other hand, are the realistic assumptions of the Undertaking: therefore q indicates the best estimate of the probability of death of the individual policyholder while j indicates the best estimate of the return deriving from the investment of premiums and reserves. |

7 | As is well-known, a Pure Endowment is a policy that, in the face of a single premium or regular (constant) Premiums, entitles to receive a certain insured sum upon maturity, only if the insured is alive on that date. Hence, the best estimate in t is equal to the first part of Equation (17), hence $b{e}_{t}{=}_{n-t}{p}_{x+t}\xb7{\left[{\prod}_{h=0}^{n-t-1}(1+{i}_{t}(0,h,h+1))\right]}^{-1}-\pi \xb7{\ddot{a}}_{(x+t):(n-t)}$. |

8 | A Term Insurance is a policy that pays the beneficiary a certain insured sum on a generic policy anniversary t if the policyholder dies in the time span $[t-1,t)$: the best estimate at time t is therefore ${\sum}_{k=0}^{n-t-1}{}_{k/1}{q}_{x+t}{\left[{\prod}_{h=0}^{k}(1+{i}_{t}(0,h,h+1))\right]}^{-1}-\pi \xb7{\ddot{a}}_{(x+t):(n-t)}$. |

9 | CV stands for Coefficient of Variation of initial sums insured. |

10 | ${\lambda}_{t}$ is calculated as ${\lambda}_{t}={\displaystyle \frac{({q}_{t}^{*}-{q}_{t})}{{q}_{t}}}$ and while in the Pure Endowment ${\lambda}_{t}$ was set equal to 15%, using ISTAT2014 entails that when $t=0$, ${\lambda}_{0}=24.64\%$, then it increases until $t=19$, where ${\lambda}_{19}=32.34\%$. |

11 | $\mathbb{E}\left[{}_{1}{\tilde{y}}_{1}^{MCV}\right]=\pi \xb7{w}_{0}\xb7(1+{j}^{*})-\mathbb{E}\left[{\tilde{be}}_{1}^{Rf\left(1\right),q}\right]\xb7{w}_{0}{\xb7}_{1}{p}_{40}$ This formulation is easily obtainable by computing the expected value on Equation (3), when t = 0. |

12 | With an Intel i7 8700K processor (working in parallel—6 Cores, 12 Threads) it requires about 18 min. |

13 | In all tables, T stands for Theoretical values, i.e., the exact characteristics of the random variables computed using closed formulas. |

14 | Results are strongly influenced by the risk-free curve used. For instance, using the EIOPA curve for August 2020 we obtain an expected profit in $t=19$ in Table 4, because of a spot rate equal to 0.69%. |

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Individual age at policy issue | 40 |

Gender | Male |

Policy duration | 20 |

Expenses loadings | 0% |

Risk-free rates | 1% |

Number of policyholders | 15,000 |

Initial sums insured | 1,510,653,999 |

CV9 | 1.99 |

I order demographic basis | 0.85·ISTAT2016 |

II order demographic basis | ISTAT2016 |

Technical rate | 1% |

Individual age at policy issue | 40 |

Policy duration | 20 |

Premium type | Annual premiums (20) |

Acquisition loading | 50% |

Collection loading | 2.5% |

Management loading | 0.15% |

Number of policyholders $\left({l}_{0}\right)$ | 15,000 |

Expected value of the single insured sum | 100,000 |

CV of the sums insured | 1.99% |

Pure Endowment | Endowment | Term Insurance | |
---|---|---|---|

First order—q* | ${\alpha}_{t}\xb7$ISTAT2016 | ISTAT2014 | ISTAT2014 |

Second order—q | ISTAT2016 | ISTAT2016 | ISTAT2016 |

Pure Endowment | t = 0 | t = 10 | t = 19 |
---|---|---|---|

$\mathbb{E}\left[{}_{1}{\tilde{y}}_{t+1}^{MCV}\right]\left(T\right)$ | 150,339,562 | −8,882,965 | −19,893,883 |

$\mathbb{E}\left[{}_{1}{\tilde{y}}_{t+1}^{MCV}\right]$ | 150,349,827 | −8,760,096 | −19,894,122 |

$\mathbb{E}\left[{}_{1}{\tilde{y}}_{t+1}^{MCV}\right]$ on ${w}_{t}$ | 9.95% | −0.61% | −1.38% |

$\sigma {(}_{1}{\tilde{y}}_{t+1}^{MCV})$ | 4,345,500 | 5,937,483 | 1,821,044 |

$\gamma {(}_{1}{\tilde{y}}_{t+1}^{MCV})$ | −0.04 | −0.02 | 0.97 |

SCR | −138,942,272 | 24,208,313 | 23,333,458 |

SCR on ${w}_{t}$ | −9.20% | 1.62% | 1.60% |

Pure Endowment | t = 0 | t = 10 | t = 19 |
---|---|---|---|

$\mathbb{E}\left[{}_{1}{\tilde{y}}_{t+1}^{LG}\right]\left(T\right)$ | 2528 | 230,554 | 1,040,705 |

$\sigma {(}_{1}{\tilde{y}}_{t+1}^{LG})\left(T\right)$ | 16,455 | 615,730 | 1,799,850 |

$\gamma {(}_{1}{\tilde{y}}_{t+1}^{LG})\left(T\right)$ | 2.41 | 1.54 | 1.09 |

SCR | 19,463 | 792,091 | 2,397,872 |

SCR on ${w}_{t}$ | 0.01% | 0.05% | 0.16% |

Endowment | t = 0 | t = 10 | t = 19 |
---|---|---|---|

$\mathbb{E}\left[{}_{1}{\tilde{y}}_{t+1}^{MCV}\right]\left(T\right)$ | 153,901,015 | −9,096,636 | −20,084,746 |

$\mathbb{E}\left[{}_{1}{\tilde{y}}_{t+1}^{MCV}\right]$ | 153,905,829 | −9,138,027 | −20,084,746 |

$\mathbb{E}\left[{}_{1}{\tilde{y}}_{t+1}^{MCV}\right]$ on ${w}_{t}$ | 10.19% | −0.61% | −1.38% |

$\sigma {(}_{1}{\tilde{y}}_{t+1}^{MCV})$ | 4,588,717 | 6,020,325 | 0 |

$\gamma {(}_{1}{\tilde{y}}_{t+1}^{MCV})$ | −0.05 | −0.02 | 0 |

SCR | −141,802,756 | 24,821,250 | 20,084,746 |

SCR on ${w}_{t}$ | −9.39% | 1.66% | 1.38% |

Endowment | t = 0 | t = 10 | t = 19 |
---|---|---|---|

$\mathbb{E}\left[{}_{1}{\tilde{y}}_{t+1}^{LG}\right]\left(T\right)$ | 260,745 | 395,352 | 0 |

$\sigma {(}_{1}{\tilde{y}}_{t+1}^{LG})\left(T\right)$ | 750,933 | 595,986 | 0 |

$\gamma {(}_{1}{\tilde{y}}_{t+1}^{LG})\left(T\right)$ | −2.41 | −1.54 | 0 |

SCR | 3,289,428 | 2,091,067 | 0 |

SCR on ${w}_{t}$ | 0.22% | 0.14% | 0% |

Term Insurance | t = 0 | t = 10 | t = 19 |
---|---|---|---|

$\mathbb{E}\left[{}_{1}{\tilde{y}}_{t+1}^{MCV}\right]\left(T\right)$ | 20,898,387 | −271,353 | −269,009 |

$\mathbb{E}\left[{}_{1}{\tilde{y}}_{t+1}^{MCV}\right]$ | 20,891,192 | −269,927 | −269,285 |

$\mathbb{E}\left[{}_{1}{\tilde{y}}_{t+1}^{MCV}\right]$ on ${w}_{t}$ | 1.38% | −0.02% | −0.02% |

$\sigma {(}_{1}{\tilde{y}}_{t+1}^{MCV})$ | 777,012 | 1,224,059 | 1,820,823 |

$\gamma {(}_{1}{\tilde{y}}_{t+1}^{MCV})$ | −2.42 | −1.48 | −0.97 |

SCR | −17,219,206 | 24,821,250 | 6,769,046 |

SCR on ${w}_{t}$ | −1.14% | 0.36% | 0.47% |

Term Insurance | t = 0 | t = 10 | t = 19 |
---|---|---|---|

$\mathbb{E}\left[{}_{1}{\tilde{y}}_{t+1}^{LG}\right]\left(T\right)$ | 266,582 | 800,159 | 2,160,710 |

$\sigma {(}_{1}{\tilde{y}}_{t+1}^{LG})\left(T\right)$ | 767,745 | 1,206,226 | 1,799,850 |

$\gamma {(}_{1}{\tilde{y}}_{t+1}^{LG})\left(T\right)$ | −2.41 | −1.54 | −1.09 |

SCR | 3,335,518 | 4,265,074 | 4,320,054 |

SCR on ${w}_{t}$ | 0.22% | 0.29% | 0.29% |

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**MDPI and ACS Style**

Clemente, G.P.; Della Corte, F.; Savelli, N.
A Bridge between Local GAAP and Solvency II Frameworks to Quantify Capital Requirement for Demographic Risk. *Risks* **2021**, *9*, 175.
https://doi.org/10.3390/risks9100175

**AMA Style**

Clemente GP, Della Corte F, Savelli N.
A Bridge between Local GAAP and Solvency II Frameworks to Quantify Capital Requirement for Demographic Risk. *Risks*. 2021; 9(10):175.
https://doi.org/10.3390/risks9100175

**Chicago/Turabian Style**

Clemente, Gian Paolo, Francesco Della Corte, and Nino Savelli.
2021. "A Bridge between Local GAAP and Solvency II Frameworks to Quantify Capital Requirement for Demographic Risk" *Risks* 9, no. 10: 175.
https://doi.org/10.3390/risks9100175