# Special-Rate Life Annuities: Analysis of Portfolio Risk Profiles

^{*}

## Abstract

**:**

## 1. Introduction and Motivation

## 2. Literature Review

^{®}process is described and commented on1.

## 3. The Products

- Given the single premium amount, a lifestyle annuity pays out benefits higher than a standard life annuity because of risk factors (e.g., smoking and drinking habits, marital status, occupation, height and weight, blood pressure and cholesterol levels), which might result, to some extent, in a shorter life expectancy. Specific lifestyle annuities are the following ones.
- (a)
- Smoker life annuities: if the applicant has smoked at least a given number of cigarettes for a certain number of years, then they are eligible for a smoker annuity.
- (b)
- Mortality differences between married and unmarried individuals underpin the use of special rates in pricing the unmarried lives annuities. The observed higher mortality rates of unmarried individuals justify a higher annuity rate.

- The enhanced life annuity pays out an income to a person with a reduced life expectancy, in particular because of a personal history of medical conditions. Of course, the “enhancement” in the annuity benefit (compared to a standard-rate life annuity, same premium) comes from the use of a higher mortality assumption.
- The impaired life annuity pays out a higher income than an enhanced life annuity, as a result of medical conditions which significantly shorten the life expectancy of the annuitant (e.g., diabetes, chronic asthma, cancer, etc.).
- Finally, care annuities are aimed at individuals with very serious impairments or individuals who are already in a senescent-disability (or long-term care) state. These annuities are frequently placed in the context of long-term care insurance products, and labeled as providing benefits “in point of need” (see, for example, Pitacco 2014).

## 4. The Mortality Model

- Heterogeneity in mortality inside the (total) life annuity portfolio (see Section 4.1 and Section 4.2);
- Possible future mortality trends (see Section 4.1);
- The individual age-pattern of mortality (see Section 4.2).

#### 4.1. General Aspects

- Some risk factors are unobservable;
- Diverse pathologies entitle one to the same annuity rate.

- An assumed future mortality trend can be taken into account by adopting projected life tables or projected mortality laws;
- Representing uncertainty in future mortality trend, which implies systematic risk (that is, the aggregate longevity risk), calls for the use of stochastic mortality models.

#### 4.2. Age Pattern of Mortality

## 5. The Actuarial Model

#### 5.1. Portfolio Structures

- Sub-portfolio $\mathrm{SP}1$ initially consisting of ${n}_{1}$ standard life annuities;
- Sub-portfolio $\mathrm{SP}2$ initially consisting of ${n}_{2}$ enhanced life annuities;
- Sub-portfolio $\mathrm{SP}3$ initially consisting of ${n}_{3}$ impaired life annuities.

- The lifetime distribution for annuitants in the sub-portfolio $\mathrm{SP}k$ follows the Gompertz law with parameters ${M}_{k}$ and ${D}_{k}$, $k=1,2,3$ (see Section 4.2);
- All the annuitants are age x at policy issue;
- The individual lifetimes in each sub-portfolio and in the portfolio $\mathrm{P}$ are independent random variables;
- The same benefit b is paid by all the life annuity policies;
- Each sub-portfolio is closed to new entries (and hence consists of a generation of policies).

#### 5.2. Actuarial Values

- $\omega $ denotes the maximum attainable age;
- ${a}_{h\rceil}=\frac{1-{(1+i)}^{-h}}{i}$ is the present value of an annuity-certain, with i denoting the interest rate used for discounting;
- ${}_{h|1}{q}_{x}^{\left(k\right)}=\frac{{\ell}_{x+h}^{\left(k\right)}-{\ell}_{x+h+1}^{\left(k\right)}}{{\ell}_{x}^{\left(k\right)}}$ is the probability of a person age x dying between age $x+h$ and $x+h+1$, according to the biometric model with parameters ${M}_{k}$ and ${D}_{k}$.

#### 5.3. The Risk Index

#### 5.4. Cash Flows

## 6. Portfolio Risk Profiles: Deterministic Approach

#### 6.1. Impact of the Portfolio Structure

#### 6.1.1. Cases 1.1

#### 6.1.2. Cases 1.2

#### 6.1.3. Cases 1.3

#### 6.1.4. Cases 1.4

#### 6.1.5. Some Comments

#### 6.2. Impact of Lifetime Distributions

#### 6.2.1. Cases 2.1

#### 6.2.2. Cases 2.2

#### 6.2.3. Cases 2.3

#### 6.2.4. Some Comments

## 7. Portfolio Risk Profiles: Stochastic Approach

#### 7.1. Impact of the Portfolio Structure

#### 7.1.1. Cases 1.1

#### 7.1.2. Cases 1.4

#### 7.1.3. Some Comments

#### 7.2. Impact of the Lifetime Distribution

#### 7.2.1. Cases 2.1

#### 7.2.2. Some Comments

#### 7.3. Meeting the Annual Payouts

#### 7.3.1. The Percentile Principle

#### 7.3.2. Numerical Results

#### 7.3.3. Some Comments

## 8. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Note

1 | Stage-Gate ^{®} is a registered trademark of Stage-Gate Inc. (www.stage-gate.com (accessed on 10 January 2022)). |

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**Figure 3.**Impact of the number of enhanced annuities: empirical distributions of the annual benefit payout at time $t=5$.

**Figure 4.**Impact of the number of enhanced annuities: empirical distributions of the annual benefit payout at time $t=10$.

**Figure 5.**Impact of cannibalization effect: empirical distributions of the annual benefit payout at time $t=5$.

**Figure 6.**Impact of cannibalization effect: empirical distributions of the annual benefit payout at time $t=10$.

**Figure 7.**Impact of different dispersion parameters: empirical distributions of the total payout at time $t=5$.

**Figure 8.**Impact of different dispersion parameters: empirical distributions of the total payout at time $t=10$.

Rating Class | k | ${\mathit{M}}_{\mathit{k}}$ | ${\mathit{D}}_{\mathit{k}}$ |
---|---|---|---|

Standard | 1 | 90 | 5 |

Enhanced | 2 | 80 | 8 |

Impaired | 3 | 70 | 13 |

Portfolio | ${\mathit{n}}_{2}$ | $\mathit{\rho}(\mathrm{10,000},{\mathit{n}}_{2},0)$ |
---|---|---|

P01 | 100 | $0.002378268$ |

P02 | 200 | $0.002381505$ |

P03 | 300 | $0.002384564$ |

P04 | 400 | $0.002387452$ |

P05 | 500 | $0.002390176$ |

P06 | 600 | $0.002392740$ |

P07 | 700 | $0.002395152$ |

P08 | 800 | $0.002397415$ |

P09 | 900 | $0.002399535$ |

P10 | 1000 | $0.002401517$ |

Portfolio | ${\mathit{n}}_{3}$ | $\mathit{\rho}(\mathrm{10,000},0,{\mathit{n}}_{3})$ |
---|---|---|

P01 | 100 | $0.002382799$ |

P02 | 200 | $0.002390524$ |

P03 | 300 | $0.002398028$ |

P04 | 400 | $0.002405318$ |

P05 | 500 | $0.002412401$ |

P06 | 600 | $0.002419283$ |

P07 | 700 | $0.002425969$ |

P08 | 800 | $0.002432465$ |

P09 | 900 | $0.002438776$ |

P10 | 1000 | $0.002444908$ |

Portfolio | ${\mathit{n}}_{2}$ | ${\mathit{n}}_{3}$ | $\mathit{\rho}(\mathrm{10,000},{\mathit{n}}_{2},{\mathit{n}}_{3})$ |
---|---|---|---|

P01 | 500 | 250 | $0.002407197$ |

P02 | 600 | 300 | $0.002412496$ |

P03 | 700 | 350 | $0.002417448$ |

P04 | 800 | 400 | $0.002422070$ |

P05 | 900 | 450 | $0.002426375$ |

P06 | 1000 | 500 | $0.002430381$ |

Portfolio | ${\mathit{n}}_{1}$ | ${\mathit{n}}_{2}$ | ${\mathit{n}}_{3}$ | $\mathit{\rho}({\mathit{n}}_{1},{\mathit{n}}_{2},{\mathit{n}}_{3})$ |
---|---|---|---|---|

P01 | 9750 | 500 | 250 | $0.002340041$ |

P02 | 9700 | 600 | 300 | $0.002352541$ |

P03 | 9650 | 700 | 350 | $0.002364783$ |

P04 | 9600 | 800 | 400 | $0.002376774$ |

P05 | 9550 | 900 | 450 | $0.002388521$ |

P06 | 9500 | 1000 | 500 | $0.002400030$ |

Portfolio | ${\mathit{D}}_{2}$ | $\mathit{\rho}(\mathrm{10,000},1000,0)$ |
---|---|---|

P01 | 4 | $0.002315649$ |

P02 | 5 | $0.002341998$ |

P03 | 6 | $0.002364379$ |

P04 | 7 | $0.002383918$ |

P05 | 8 | $0.002401517$ |

P06 | 9 | $0.002417793$ |

P07 | 10 | $0.002433146$ |

P08 | 11 | $0.002447834$ |

P09 | 12 | $0.002462022$ |

P10 | 13 | $0.002475816$ |

Portfolio | ${\mathit{D}}_{3}$ | $\mathit{\rho}(\mathrm{10,000},0,1000)$ |
---|---|---|

P01 | 11 | $0.002422885$ |

P02 | 12 | $0.002433728$ |

P03 | 13 | $0.002444908$ |

P04 | 14 | $0.002456358$ |

P05 | 15 | $0.002468019$ |

Portfolio | ${\mathit{D}}_{2}={\mathit{D}}_{3}$ | $\mathit{\rho}(\mathrm{10,000},1000,500)$ |
---|---|---|

P01 | 4 | $0.002360437$ |

P02 | 5 | $0.002366549$ |

P03 | 6 | $0.002373800$ |

P04 | 7 | $0.002382362$ |

P05 | 8 | $0.002392205$ |

P06 | 9 | $0.002403223$ |

P07 | 10 | $0.002415280$ |

P08 | 11 | $0.002428234$ |

P09 | 12 | $0.002441949$ |

P10 | 13 | $0.002456293$ |

Portfolio | ${\mathit{n}}_{1}$ | ${\mathit{n}}_{2}$ | ${\mathit{n}}_{3}$ |
---|---|---|---|

P01 | 10,000 | 0 | 0 |

P02 | 10,000 | 1000 | 0 |

P03 | 10,000 | 0 | 500 |

P04 | 10,000 | 1000 | 500 |

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**MDPI and ACS Style**

Pitacco, E.; Tabakova, D.Y.
Special-Rate Life Annuities: Analysis of Portfolio Risk Profiles. *Risks* **2022**, *10*, 65.
https://doi.org/10.3390/risks10030065

**AMA Style**

Pitacco E, Tabakova DY.
Special-Rate Life Annuities: Analysis of Portfolio Risk Profiles. *Risks*. 2022; 10(3):65.
https://doi.org/10.3390/risks10030065

**Chicago/Turabian Style**

Pitacco, Ermanno, and Daniela Y. Tabakova.
2022. "Special-Rate Life Annuities: Analysis of Portfolio Risk Profiles" *Risks* 10, no. 3: 65.
https://doi.org/10.3390/risks10030065