# Frailty and Risk Classification for Life Annuity Portfolios

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## Abstract

**:**

## 1. Introduction

## 2. Lifetime and Frailty: The Gompertz-Gamma Model

## 3. Risk Classification Based on a Frailty Model

#### 3.1. Identification of the Risk Classes

#### 3.2. Lifetime and Frailty for the Risk Classes

## 4. Model Calibration

- We first calibrate the mortality model for the general population, based on the life table TG62. As it is common in the literature, we set $\theta =\delta $; then, we need to set three parameters (namely, $\alpha ,\beta ,\delta $), and we assume the following requirements:
- The expected value and variance of the lifetime in the range of ages $[65,95]$ under the Gompertz-Gamma model is the same as in the life table TG62;
- The mean squared error (MSE) between the force of mortality in the range of ages $[65,95]$ under the Gompertz-Gamma model and the life table TG62 is minimum. While the analytical expression for the force of mortality is available for the Gompertz-Gamma model (see (13)), as already noted for the life table TG62, a standard numerical approximation has been used.

- Then, we calibrate the lifetime for standard risks, i.e., individuals in group ${G}_{1}$. We now need to set one parameter (namely, ${z}_{1}$), and we choose it so to minimize the MSE between the force of mortality under the Gompertz-Gamma model for individuals in group ${G}_{1}$ and the table A62I, in the range of ages $[65,95]$. Similarly to the life table TG62, the force of mortality for the life table A62I has been approximated numerically. For the Gompertz-Gamma model, the force of mortality for group ${G}_{1}$ can be obtained through its definition, as ${\overline{\mu}}_{x}\left({G}_{1}\right)=-\frac{{\overline{S}}^{\prime}\left(x\right|{G}_{1})}{\overline{S}\left(x\right|{G}_{1})}$. For simplicity, we omit writing its expression, which cannot be simplified nicely.
- Finally, we calibrate the probability distribution of the lifetime for two additional groups, ${G}_{2}$ and ${G}_{3}$, characterized by higher average frailty levels; we set a (reasonable) benchmark for the reduced expected lifetime in these groups, with respect to the value for standard risks. Such benchmarks are commented on below.

## 5. The Value of the Liabilities of a Life Annuity Portfolio

#### 5.1. The Present Value of Future Benefits

- The expected value $\mathbb{E}\left[P{V}_{t}\right]$,
- The coefficient of variation, $\mathbb{CV}\left[P{V}_{t}\right]$,
- The right tail, measured through the ε-percentiles $P{V}_{t\left[\epsilon \right]}$

#### 5.2. Numerical Investigation

- Portfolio A is the base case; it only consists of standard risks;
- Portfolio E has the largest possible size, including policies in groups ${G}_{2}$ and ${G}_{3}$;
- Portfolio B and C include some policies in group ${G}_{2}$, where C has a larger size;
- Portfolio D has the same size as portfolio C, but with some policies also in group ${G}_{3}$;
- Portfolio F shows adverse-selection: the number of standard risks is lower than in the other cases, but the size is the same as Portfolio A, due to risks in class ${G}_{2}$.

#### 5.3. Sensitivity Analysis

- Assuming the same parameters for the Gompertz-Gamma model referring to the general population as in Table 1, we consider a rating structure arranged on a higher number of risk classes (precisely, four or five);
- We assume a higher degree of heterogeneity in the general population, and we identify three risk classes, assuming the same composition of the general population as in Table 5.

- For Portfolio G, the rating structure is arranged into four classes; details are quoted in Table 14. Note that we keep the same definition for the standard risk class, i.e., class ${G}_{1}$, as for Portfolios A–F, while we arrange preferred risk classes into four groups.
- For Portfolio H, the rating structure is arranged into five classes, with a reduced frailty interval for standard risks. This means that in this case, the definition of what is considered to be a standard risk is more restrictive than in the previous cases. Details are listed in Table 15.
- While Portfolios G and H are based on the same assessment of the heterogeneity of the population as for Portfolios A–F, Portfolio I is designed assuming a stronger degree of heterogeneity of the general population. We identify three risk classes, whose initial relative size in the population is the same as for Portfolios A–F. See Table 16 for details.

- The average survival rate for Portfolios G and H is (almost) the same as for Portfolio E (apart from some rounding in the number of survivors, which must be integer).
- Portfolio G, which has the same class of standard risks, shows a profile of the value of the liabilities that is similar to Portfolio E. Conversely, Portfolio H, which has a reduced class of standard risks, shows a similar dispersion, but a lower expected value of future benefits.
- Portfolio I shows a slightly higher average survival rate, especially at the highest ages (which is due to the lower frailty level for standard risks). From this, it follows a higher expected value of the liabilities, in particular at high ages; conversely, differences in the dispersion are not significant.

## 6. Some Remarks to Conclude

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Expected Value of the Frailty in Group ${\mathit{G}}_{\mathit{j}}$, Age x

## Appendix B. Variance of the Frailty in Group ${\mathit{G}}_{\mathit{j}}$, Age x

## Appendix C. Variance of the Frailty in the General Population, with Respect to the Variance in the Frailty Groups

## Appendix D. Average Survival Function in Group ${\mathit{G}}_{\mathit{j}}$

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**Sample Availability:**Samples of the compounds are available from the authors.

**Figure 1.**Comparison between the life tables TG62 and A62I (force of mortality, probability density function of ${T}_{65}$, survival function). (

**a**) Force of mortality ${\mu}_{x}$; (

**b**) Probability density function ${f}_{65}\left(t\right)$; (

**c**) Survival function $S\left(x\right)$.

**Figure 2.**Comparison between the life table TG62 and the Gompertz-Gamma model for the general population (force of mortality, probability density function of ${T}_{65}$, survival function). (

**a**) Force of mortality ${\mu}_{x}$; (

**b**) Probability density function ${f}_{65}\left(t\right)$; (

**c**) Survival function $S\left(x\right)$.

**Figure 3.**Comparison between the life table A62I and the Gompertz-Gamma model for group ${G}_{1}$ (force of mortality, probability density function of ${T}_{65}$, survival function). (

**a**) Force of mortality ${\mu}_{x}$; (

**b**) Prob. density function ${f}_{65}\left(t\right)$; (

**c**) survival function $S\left(x\right)$.

Parameters | |
---|---|

α | $4.88661\times {10}^{-06}$ |

β | 0.111902 |

δ | 18.408049 |

θ | 18.408049 |

${\alpha}^{\prime}$ | $4.886619\times {10}^{-06}$ |

${\delta}^{\prime}$ | $2.372256\times {10}^{-06}$ |

**Table 2.**Summary statistics of the probability distribution of the lifetime ${T}_{65}$. General population.

TG62 | Gompertz-Gamma (Population) | |
---|---|---|

$\mathbb{E}\left[{T}_{65}\right]$ | 21.70 | 21.67 |

$\mathbb{CV}\left[{T}_{65}\right]$ | 41.93% | 41.73% |

$\mathrm{Mo}\left[{T}_{65}\right]$ | 25 | 24.71 |

${T}_{65[0.25]}$ | 15.57 | 15.43 |

${T}_{65[0.75]}$ | 28.12 | 28.38 |

$\mathrm{IQR}={T}_{65[0.75]}-{T}_{65[0.25]}$ | 12.55 | 12.95 |

${T}_{65[0.95]}$ | 35.92 | 35.45 |

${T}_{65[0.99]}$ | 40.25 | 39.64 |

**Table 3.**Gompertz-Gamma model: expected value and coefficient of variation of the frailty at age x. General population.

$\mathbb{E}\left[{\mathit{Z}}_{\mathit{x}}\right]$ | |
---|---|

65 | 0.996594 |

70 | 0.994053 |

75 | 0.989638 |

80 | 0.982007 |

85 | 0.968933 |

90 | 0.946874 |

95 | 0.910599 |

100 | 0.853391 |

105 | 0.768868 |

110 | 0.655299 |

115 | 0.520714 |

$\mathbb{CV}\left[{Z}_{x}\right]$ | 23.308% |

**Table 4.**Summary statistics of the probability distribution of the lifetime ${T}_{65}$. Risk group ${G}_{1}$.

A62I | Gompertz-Gamma (Group ${\mathit{G}}_{1}$) | |
---|---|---|

$\mathbb{E}\left[{T}_{65}\right]$ | 24.08 | 22.81 |

$\mathbb{CV}\left[{T}_{65}\right]$ | 59.92% | 40.13% |

$\mathrm{Mo}\left[{T}_{65}\right]$ | 25 | 26 |

${T}_{65[0.25]}$ | 18.85 | 16.61 |

${T}_{65[0.75]}$ | 30.05 | 29.62 |

$\mathrm{IQR}={T}_{65[0.75]}-{T}_{65[0.25]}$ | 11.19 | 13.01 |

${T}_{65[0.95]}$ | 37.34 | 36.55 |

${T}_{65[0.99]}$ | 41.32 | 40.60 |

Group | Frailty Interval $({\mathit{z}}_{\mathit{j}-1},{\mathit{z}}_{\mathit{j}}]$ | Relative Size at Age 65 of Group ${\mathit{G}}_{\mathit{j}}$ in the General Population ${\mathit{\rho}}_{\mathit{j};65}$ | Expected Value of the Frailty $\mathbb{E}\left[{\mathit{Z}}_{65}\right|{\mathit{G}}_{\mathit{j}}]$ | Coefficient of Variation $\mathbb{CV}\left[{\mathit{Z}}_{65}\right|{\mathit{G}}_{\mathit{j}}]$ | Expected Lifetime $\mathbb{E}\left[{\mathit{T}}_{65}\right|{\mathit{G}}_{\mathit{j}}]$ |
---|---|---|---|---|---|

${G}_{1}$ | $(\phantom{0.00000}0,1.038741]$ | 60.121% | 0.845593 | 15.243% | 22.81 |

${G}_{2}$ | $(1.038741,1.307144]$ | 30.111% | 1.152338 | 6.479% | 20.36 |

${G}_{3}$ | $(1.307144,\infty \phantom{.00000})$ | 9.769% | 1.445866 | 8.736% | 18.71 |

Population | $(\phantom{0.00000}0,\infty \phantom{.00000})$ | 100% | 0.996594 | 23.308% | 21.67 |

Groups | Portfolio | |||||
---|---|---|---|---|---|---|

A | B | C | D | E | F | |

${G}_{1}$ | 1 000 | 1 000 | 1 000 | 1 000 | 1 000 | 500 |

${G}_{2}$ | 0 | 200 | 250 | 200 | 501 | 500 |

${G}_{3}$ | 0 | 0 | 0 | 50 | 162 | 0 |

All | 1 000 | 1 200 | 1 250 | 1 250 | 1 663 | 1 000 |

Group ${\mathit{G}}_{1}$ | Group ${\mathit{G}}_{2}$ | Group ${\mathit{G}}_{3}$ | |
---|---|---|---|

Benefit amount ${b}_{j}$ | 4.483 | 5.034 | 5.492 |

$\frac{{b}_{j}}{{b}_{1}}-1$ | 0% | 12.302% | 22.515% |

**Table 8.**Average benefit amount for the portfolio: additional amount with respect to the base case, $\frac{{\overline{b}}_{t}}{{b}_{1}}-1$.

Time t | Portfolio A | Portfolio B | Portfolio C | Portfolio D | Portfolio E | Portfolio F |
---|---|---|---|---|---|---|

0 | 0% | 2.050% | 2.460% | 2.869% | 5.899% | 6.151% |

5 | 0% | 2.022% | 2.434% | 2.826% | 5.820% | 6.112% |

10 | 0% | 1.981% | 2.381% | 2.740% | 5.694% | 6.032% |

15 | 0% | 1.913% | 2.296% | 2.633% | 5.481% | 5.893% |

20 | 0% | 1.786% | 2.150% | 2.402% | 5.116% | 5.654% |

25 | 0% | 1.592% | 1.918% | 2.073% | 4.503% | 5.227% |

30 | 0% | 1.268% | 1.555% | 1.587% | 3.574% | 4.522% |

35 | 0% | 0.868% | 1.001% | 1.120% | 2.287% | 3.189% |

40 | 0% | 0.000% | 0.000% | 0.000% | 0.879% | 1.757% |

45 | 0% | 0.000% | 0.000% | 0.000% | 0.000% |

Time t | Portfolio A | Portfolio B | ||||||||

Total Size | Relative Group Size | Total Size | Relative Group Size | |||||||

${\mathit{n}}_{\mathit{t}}$ | $\frac{{\mathit{n}}_{\mathit{t}}}{{\mathit{n}}_{0}}$ | Group ${\mathit{G}}_{1}$ | Group ${\mathit{G}}_{2}$ | Group ${\mathit{G}}_{3}$ | ${\mathit{n}}_{\mathit{t}}$ | $\frac{{\mathit{n}}_{\mathit{t}}}{{\mathit{n}}_{0}}$ | Group ${\mathit{G}}_{1}$ | Group ${\mathit{G}}_{2}$ | Group ${\mathit{G}}_{3}$ | |

0 | 1 000 | 100.00% | 100% | 0% | 0% | 1 200 | 100.00% | 83.333% | 16.667% | 0.000% |

5 | 961 | 96.10% | 100% | 0% | 0% | 1 150 | 95.83% | 83.565% | 16.435% | 0.000% |

10 | 896 | 89.60% | 100% | 0% | 0% | 1 068 | 89.00% | 83.895% | 16.105% | 0.000% |

15 | 793 | 79.30% | 100% | 0% | 0% | 939 | 78.25% | 84.452% | 15.548% | 0.000% |

20 | 642 | 64.20% | 100% | 0% | 0% | 751 | 62.58% | 85.486% | 14.514% | 0.000% |

25 | 444 | 44.40% | 100% | 0% | 0% | 510 | 42.50% | 87.059% | 12.941% | 0.000% |

30 | 235 | 23.50% | 100% | 0% | 0% | 262 | 21.83% | 89.695% | 10.305% | 0.000% |

35 | 79 | 7.90% | 100% | 0% | 0% | 85 | 7.08% | 92.941% | 7.059% | 0.000% |

40 | 13 | 1.30% | 100% | 0% | 0% | 13 | 1.08% | 100.000% | 0.000% | 0.000% |

45 | 1 | 0.10% | 100% | 0% | 0% | 1 | 0.08% | 100.000% | 0.000% | 0.000% |

50 | 0 | 0.00% | 0 | 0.00% | ||||||

Time t | Portfolio C | Portfolio D | ||||||||

Total Size | Relative Group Size | Total Size | Relative Group Size | |||||||

${\mathit{n}}_{\mathit{t}}$ | $\frac{{\mathit{n}}_{\mathit{t}}}{{\mathit{n}}_{0}}$ | Group ${\mathit{G}}_{1}$ | Group ${\mathit{G}}_{2}$ | Group ${\mathit{G}}_{3}$ | ${\mathit{n}}_{\mathit{t}}$ | $\frac{{\mathit{n}}_{\mathit{t}}}{{\mathit{n}}_{0}}$ | Group ${\mathit{G}}_{1}$ | Group ${\mathit{G}}_{2}$ | Group ${\mathit{G}}_{3}$ | |

0 | 1 250 | 100.00% | 80.000% | 20.000% | 0.000% | 1 250 | 100.00% | 80.000% | 16.000% | 4.000% |

5 | 1 198 | 95.84% | 80.217% | 19.783% | 0.000% | 1 197 | 95.76% | 80.284% | 15.789% | 3.926% |

10 | 1 111 | 88.88% | 80.648% | 19.352% | 0.000% | 1 109 | 88.72% | 80.794% | 15.509% | 3.697% |

15 | 975 | 78.00% | 81.333% | 18.667% | 0.000% | 973 | 77.84% | 81.501% | 15.005% | 3.494% |

20 | 778 | 62.24% | 82.519% | 17.481% | 0.000% | 774 | 61.92% | 82.946% | 14.083% | 2.972% |

25 | 526 | 42.08% | 84.411% | 15.589% | 0.000% | 522 | 41.76% | 85.057% | 12.644% | 2.299% |

30 | 269 | 21.52% | 87.361% | 12.639% | 0.000% | 266 | 21.28% | 88.346% | 10.150% | 1.504% |

35 | 86 | 6.88% | 91.860% | 8.140% | 0.000% | 86 | 6.88% | 91.860% | 6.977% | 1.163% |

40 | 13 | 1.04% | 100.000% | 0.000% | 0.000% | 13 | 1.04% | 100.000% | 0.000% | 0.000% |

45 | 1 | 0.08% | 100.000% | 0.000% | 0.000% | 1 | 0.08% | 100.000% | 0.000% | 0.000% |

50 | 0 | 0.00% | 0 | 0.00% | ||||||

Time t | Portfolio E | Portfolio F | ||||||||

Total Size | Relative Group Size | Total Size | Relative Group Size | |||||||

${\mathit{n}}_{\mathit{t}}$ | $\frac{{\mathit{n}}_{\mathit{t}}}{{\mathit{n}}_{0}}$ | Group ${\mathit{G}}_{1}$ | Group ${\mathit{G}}_{2}$ | Group ${\mathit{G}}_{3}$ | ${\mathit{n}}_{\mathit{t}}$ | $\frac{{\mathit{n}}_{\mathit{t}}}{{\mathit{n}}_{0}}$ | Group ${\mathit{G}}_{1}$ | Group ${\mathit{G}}_{2}$ | Group ${\mathit{G}}_{3}$ | |

0 | 1 663 | 100.00% | 60.132% | 30.126% | 9.741% | 1 000 | 100.00% | 50.00% | 50.00% | 0.00% |

5 | 1 586 | 95.37% | 60.593% | 29.887% | 9.521% | 954 | 95.40% | 50.31% | 49.69% | 0.00% |

10 | 1 461 | 87.85% | 61.328% | 29.500% | 9.172% | 879 | 87.90% | 50.97% | 49.03% | 0.00% |

15 | 1 267 | 76.19% | 62.589% | 28.808% | 8.603% | 762 | 76.20% | 52.10% | 47.90% | 0.00% |

20 | 991 | 59.59% | 64.783% | 27.548% | 7.669% | 594 | 59.40% | 54.04% | 45.96% | 0.00% |

25 | 648 | 38.97% | 68.519% | 25.309% | 6.173% | 386 | 38.60% | 57.51% | 42.49% | 0.00% |

30 | 316 | 19.00% | 74.367% | 21.519% | 4.114% | 185 | 18.50% | 63.24% | 36.76% | 0.00% |

35 | 95 | 5.71% | 83.158% | 14.737% | 2.105% | 54 | 5.40% | 74.07% | 25.93% | 0.00% |

40 | 14 | 0.84% | 92.857% | 7.143% | 0.000% | 7 | 0.70% | 85.71% | 14.29% | 0.00% |

45 | 1 | 0.06% | 100.000% | 0.000% | 0.000% | 0 | 0.06% | |||

50 | 0 | 0.00% | 0.00% |

**Table 10.**Expected present value of future benefits, per policy in-force: $\frac{\mathbb{E}\left[P{V}_{t}\right]}{{n}_{t}}$.

Time t | Portfolio A | Portfolio B | Portfolio C | Portfolio D | Portfolio E | Portfolio F |
---|---|---|---|---|---|---|

Abs.Value | % of the Value Obtained for Portfolio A | |||||

0 | 100.00 | 100.00% | 100.00% | 100.00% | 100.01% | 100.01% |

5 | 81.26 | 99.71% | 99.65% | 99.60% | 99.18% | 99.13% |

10 | 64.00 | 99.37% | 99.24% | 99.15% | 98.24% | 98.10% |

15 | 48.62 | 99.00% | 98.80% | 98.66% | 97.24% | 96.94% |

20 | 35.44 | 98.63% | 98.35% | 98.22% | 96.25% | 95.67% |

25 | 24.66 | 98.32% | 97.98% | 97.89% | 95.45% | 94.47% |

30 | 16.35 | 98.18% | 97.77% | 97.82% | 95.13% | 93.55% |

35 | 10.34 | 98.31% | 98.04% | 97.94% | 95.72% | 93.71% |

40 | 6.32 | 100.00% | 100.00% | 100.00% | 97.63% | 95.54% |

45 | 3.93 | 100.00% | 100.00% | 100.00% | 100.00% |

**Table 11.**Coefficient of variation of the present value of future benefits: $\mathbb{CV}\left[P{V}_{t}\right]$.

Time t | Portfolio A | Portfolio B | Portfolio C | Portfolio D | Portfolio E | Portfolio F |
---|---|---|---|---|---|---|

0 | 1.30% | 1.20% | 1.17% | 1.18% | 1.04% | 1.87% |

5 | 1.48% | 1.37% | 1.34% | 1.35% | 1.19% | 1.55% |

10 | 1.75% | 1.62% | 1.60% | 1.60% | 1.39% | 1.80% |

15 | 2.10% | 1.96% | 1.91% | 1.93% | 1.70% | 2.19% |

20 | 2.64% | 2.45% | 2.41% | 2.43% | 2.17% | 2.80% |

25 | 3.55% | 3.34% | 3.31% | 3.31% | 3.04% | 3.97% |

30 | 5.62% | 5.38% | 5.32% | 5.35% | 4.96% | 6.54% |

35 | 11.10% | 10.78% | 10.78% | 10.73% | 10.28% | 13.82% |

40 | 32.19% | 32.19% | 32.19% | 32.19% | 31.40% | 44.42% |

45 | 136.25% | 136.25% | 136.25% | 136.25% | 136.25% |

Time t | Portfolio A | Portfolio B | Portfolio C | Portfolio D | Portfolio E | Portfolio F |
---|---|---|---|---|---|---|

0 | 102.11% | 101.96% | 101.90% | 101.94% | 101.72% | 103.06% |

5 | 102.43% | 102.25% | 102.21% | 102.22% | 101.94% | 102.52% |

10 | 102.86% | 102.69% | 102.63% | 102.64% | 102.30% | 102.99% |

15 | 103.43% | 103.24% | 103.13% | 103.23% | 102.80% | 103.61% |

20 | 104.43% | 104.04% | 103.98% | 104.00% | 103.57% | 104.67% |

25 | 105.86% | 105.53% | 105.54% | 105.43% | 105.08% | 106.55% |

30 | 109.34% | 109.04% | 108.82% | 108.93% | 108.27% | 110.73% |

35 | 118.51% | 118.05% | 117.97% | 117.83% | 116.92% | 123.53% |

40 | 158.21% | 158.21% | 158.21% | 158.21% | 155.67% | 180.28% |

45 | 342.00% | 342.00% | 342.00% | 342.00% | 342.00% |

Time t | Portfolio A | Portfolio B | Portfolio C | Portfolio D | Portfolio E | Portfolio F |
---|---|---|---|---|---|---|

0 | 103.07% | 102.81% | 102.70% | 102.76% | 102.44% | 104.35% |

5 | 103.46% | 103.18% | 103.06% | 103.13% | 102.73% | 103.63% |

10 | 104.12% | 103.70% | 103.77% | 103.73% | 103.22% | 104.17% |

15 | 104.98% | 104.58% | 104.48% | 104.58% | 104.00% | 105.08% |

20 | 106.36% | 105.90% | 105.73% | 105.79% | 105.15% | 106.59% |

25 | 108.28% | 107.88% | 107.78% | 107.83% | 106.98% | 109.49% |

30 | 113.66% | 112.77% | 112.77% | 112.75% | 111.91% | 115.31% |

35 | 126.74% | 125.91% | 125.69% | 125.67% | 124.79% | 133.88% |

40 | 185.49% | 185.49% | 185.49% | 185.49% | 181.62% | 222.70% |

45 | 570.00% | 570.00% | 570.00% | 570.00% | 570.00% |

Group ${\mathit{G}}_{1}$ | Group ${\mathit{G}}_{2}$ | Group ${\mathit{G}}_{3}$ | Group ${\mathit{G}}_{4}$ | Population | |
---|---|---|---|---|---|

Frailty interval $({\mathit{z}}_{\mathit{j}\mathbf{-}\mathbf{1}}\mathbf{,}{\mathit{z}}_{\mathit{j}}]$ | $(0,1.038741]$ | $(1.038741,1.186127]$ | $(1.186127,1.410339]$ | $(1.410339,\infty )$ | $(0,\infty )$ |

Relative size in the population ${\mathit{\rho}}_{\mathit{j};\mathbf{65}}$ | 60.121% | 20.000% | 15.000% | 4.879% | 100% |

Expected value of the frailty $\mathbb{E}\left[{\mathit{Z}}_{\mathbf{65}}\right|{\mathit{G}}_{\mathit{j}}]$ | 0.845593 | 1.107415 | 1.277892 | 1.538161 | 0.996594 |

Coefficient of variation $\mathbb{CV}\left[{\mathit{Z}}_{\mathbf{65}}\right|{\mathit{G}}_{\mathit{j}}]$ | 15.243% | 3.806% | 4.871% | 7.706% | 23.308% |

Expected lifetime $\mathbb{E}\left[{\mathit{T}}_{\mathbf{65}}\right|{\mathit{G}}_{\mathit{j}}]$ | 22.81 | 20.65 | 19.59 | 18.26 | 21.67 |

Benefit amount ${\mathit{b}}_{\mathit{j}}$ | 4.483 | 4.963 | 5.238 | 5.632 | |

as a % of the standard benefit, $\frac{{\mathit{b}}_{\mathit{j}}}{{\mathit{b}}_{\mathbf{1}}}\mathbf{-}\mathbf{1}$ | 0% | 10.708% | 16.855% | 25.6347% |

Group ${\mathit{G}}_{1}$ | Group ${\mathit{G}}_{2}$ | Group ${\mathit{G}}_{3}$ | Group ${\mathit{G}}_{4}$ | Group ${\mathit{G}}_{5}$ | Population | |
---|---|---|---|---|---|---|

Frailty interval $({\mathit{z}}_{\mathit{j}\mathbf{-}\mathbf{1}}\mathbf{,}{\mathit{z}}_{\mathit{j}}]$ | $(0,0.921533]$ | $(0.921533,1.038742]$ | $(1.038742,1.186128]$ | $(1.186128,1.307152]$ | $(1.307152,\infty )$ | $(0,\infty )$ |

Relative size in the population ${\mathit{\rho}}_{\mathit{j};\mathbf{65}}$ | 40.000% | 20.121% | 20.000% | 10.111% | 9.768% | 100% |

Expected value of the frailty $\mathbb{E}\left[{\mathit{Z}}_{\mathbf{65}}\right|{\mathit{G}}_{\mathit{j}}]$ | 0.778312 | 979346 | 1.107417 | 1.241204 | 1.445874 | 0.996594 |

Coefficient of variation $\mathbb{CV}\left[{\mathit{Z}}_{\mathbf{65}}\right|{\mathit{G}}_{\mathit{j}}]$ | 13.398% | 3.440% | 3.806% | 2.787% | 8.736% | 23.308% |

Expected lifetime $\mathbb{E}\left[{\mathit{T}}_{\mathbf{65}}\right|{\mathit{G}}_{\mathit{j}}]$ | 23.43 | 21.58 | 20.65 | 19.80 | 18.71 | 21.67 |

Benefit amount ${\mathit{b}}_{\mathit{j}}$ | 4.362 | 4.744 | 4.963 | 5.182 | 5.492 | |

as a % of the standard benefit, $\frac{{\mathit{b}}_{\mathit{j}}}{{\mathit{b}}_{\mathbf{1}}}\mathbf{-}\mathbf{1}$ | 0% | 8.776% | 13.781% | 18.802% | 25.915% | |

as a % of the standard benefit in Portfolio A, $\frac{{\mathit{b}}_{\mathit{j}}}{{\mathit{b}}_{\mathbf{1}}[\mathbf{Ptf}\mathbf{.}\mathbf{A}]}\mathbf{-}\mathbf{1}$ | $-2.700\%$ | 5.839% | 10.708% | 15.594% | 22.515% |

Group ${\mathit{G}}_{1}$ | Group ${\mathit{G}}_{2}$ | Group ${\mathit{G}}_{3}$ | Population | |
---|---|---|---|---|

Frailty interval $({\mathit{z}}_{\mathit{j}-\mathbf{1}},{\mathit{z}}_{\mathit{j}}]$ | $(0,1.04219]$ | $(1.04219,1.41721]$ | $(1.41721,\infty )$ | $(0,\infty )$ |

Relative size in the population ${\mathit{\rho}}_{\mathit{j};\mathbf{65}}$ | 60.121% | 30.111% | 9.768% | 100% |

Expected value of the frailty $\mathbb{E}\left[{\mathit{Z}}_{\mathbf{65}}\right|{\mathit{G}}_{\mathit{j}}]$ | 0.789420 | 1.199340 | 1.619084 | 0.993896 |

Coefficient of variation $\mathbb{CV}\left[{\mathit{Z}}_{\mathbf{65}}\right|{\mathit{G}}_{\mathit{j}}]$ | 20.728% | 8.686% | 11.507% | 31.623% |

Expected lifetime $\mathbb{E}\left[{\mathit{T}}_{\mathbf{65}}\right|{\mathit{G}}_{\mathit{j}}]$ | 23.25 | 19.93 | 17.79 | 21.71 |

Benefit amount ${\mathit{b}}_{\mathit{j}}$ | 4.397 | 5.147 | 5.783 | |

as a % of the standard benefit, $\frac{{\mathit{b}}_{\mathit{j}}}{{\mathit{b}}_{\mathbf{1}}}-\mathbf{1}$ | 0% | 17.071% | 31.531% | |

as a % of the standard benefit in Portfolio A, $\frac{{\mathit{b}}_{\mathit{j}}}{{\mathit{b}}_{\mathbf{1}}[\mathbf{Ptf}\mathbf{.}\mathbf{A}]}\mathbf{-}\mathbf{1}$ | $-1.920\%$ | 14.824% | 29.006% |

Time t | Portfolio G | |||||
---|---|---|---|---|---|---|

Total Size | Relative Group Size | |||||

${\mathit{n}}_{\mathit{t}}$ | $\frac{{\mathit{n}}_{\mathit{t}}}{{\mathit{n}}_{0}}$ | Group ${\mathit{G}}_{1}$ | Group ${\mathit{G}}_{2}$ | Group ${\mathit{G}}_{3}$ | Group ${\mathit{G}}_{4}$ | |

0 | 1 663 | 100.000% | 60.132% | 20.024% | 14.973% | 4.871% |

5 | 1 586 | 95.370% | 60.593% | 19.924% | 14.754% | 4.729% |

10 | 1 461 | 87.853% | 61.328% | 19.713% | 14.442% | 4.517% |

15 | 1 267 | 76.188% | 62.589% | 19.416% | 13.812% | 4.183% |

20 | 991 | 59.591% | 64.783% | 18.769% | 12.815% | 3.633% |

25 | 648 | 38.966% | 68.519% | 17.593% | 11.111% | 2.778% |

30 | 316 | 19.002% | 74.367% | 15.190% | 8.544% | 1.899% |

35 | 96 | 5.773% | 82.292% | 11.458% | 5.208% | 1.042% |

40 | 14 | 0.842% | 92.857% | 7.143% | 0.000% | 0.000% |

45 | 1 | 0.060% | 100.000% | 0.000% | 0.000% | 0.000% |

50 | 0 | 0.000% |

**Table 18.**Portfolio G: portfolio average benefit amount (additional amount with respect to the standard benefit for Portfolio A, $\frac{{\overline{b}}_{t}}{{b}_{1}[\mathrm{Ptf}.\mathrm{A}]}-1$) and summary statistics of the probability distribution of insurer’s liabilities.

Time t | $\frac{{\overline{\mathit{b}}}_{\mathit{t}}}{{\mathit{b}}_{1}[\mathbf{Ptf}.\mathbf{A}]}-1$ | $\frac{\mathbb{E}\left[{\mathit{PV}}_{\mathit{t}}\right]}{{\mathit{n}}_{\mathit{t}}}$ | $\mathbb{CV}\left[{\mathit{PV}}_{\mathit{t}}\right]$ | 95th perc.of ${\mathit{PV}}_{\mathit{t}}$ | 99th perc. of ${\mathit{PV}}_{\mathit{t}}$ |
---|---|---|---|---|---|

as a % of the Value | as a % of $\mathbb{E}\left[{\mathit{PV}}_{\mathit{t}}\right]$ | as a % of $\mathbb{E}\left[{\mathit{PV}}_{\mathit{t}}\right]$ | |||

for Ptf.A | |||||

0 | 5.917% | 100.01% | 1.04% | 101.68% | 102.48% |

5 | 5.833% | 99.17% | 1.18% | 101.94% | 102.71% |

10 | 5.703% | 98.23% | 1.41% | 102.32% | 103.31% |

15 | 5.480% | 97.22% | 1.69% | 102.77% | 103.81% |

20 | 5.101% | 96.22% | 2.19% | 103.63% | 105.19% |

25 | 4.469% | 95.42% | 3.05% | 105.01% | 107.18% |

30 | 3.553% | 95.05% | 4.96% | 108.27% | 111.75% |

35 | 2.372% | 95.42% | 10.28% | 117.02% | 124.71% |

40 | 0.765% | 97.69% | 31.38% | 155.57% | 182.05% |

45 | 0.000% | 100.00% | 136.25% | 342.00% | 570.00% |

Time t | Portfolio H | ||||||
---|---|---|---|---|---|---|---|

Total Size | Relative Group Size | ||||||

${\mathit{n}}_{\mathit{t}}$ | $\frac{{\mathit{n}}_{\mathit{t}}}{{\mathit{n}}_{0}}$ | Group ${\mathit{G}}_{1}$ | Group ${\mathit{G}}_{2}$ | Group ${\mathit{G}}_{3}$ | Group ${\mathit{G}}_{4}$ | Group ${\mathit{G}}_{5}$ | |

0 | 1 663 | 100.000% | 39.99% | 20.14% | 20.02% | 10.10% | 9.74% |

5 | 1 586 | 95.370% | 40.42% | 20.18% | 19.92% | 9.96% | 9.52% |

10 | 1 461 | 87.853% | 41.14% | 20.19% | 19.71% | 9.79% | 9.17% |

15 | 1 268 | 76.248% | 42.35% | 20.19% | 19.40% | 9.46% | 8.60% |

20 | 991 | 59.591% | 44.60% | 20.18% | 18.77% | 8.78% | 7.67% |

25 | 648 | 38.966% | 48.46% | 20.06% | 17.59% | 7.72% | 6.17% |

30 | 315 | 18.942% | 55.24% | 19.37% | 15.24% | 6.03% | 4.13% |

35 | 96 | 5.773% | 65.63% | 16.67% | 11.46% | 4.17% | 2.08% |

40 | 14 | 0.842% | 78.57% | 14.29% | 7.14% | 0.00% | 0.00% |

45 | 1 | 0.060% | 100.00% | 0.00% | 0.00% | 0.00% | 0.00% |

50 | 0 | 0.000% |

**Table 20.**Portfolio H: portfolio average benefit amount (additional amount with respect to the standard benefit for Portfolio A, $\frac{{\overline{b}}_{t}}{{b}_{1}[\mathrm{Ptf}.\mathrm{A}]}-1$) and summary statistics of the probability distribution of insurer’s liabilities.

Time t | $\frac{{\overline{\mathit{b}}}_{\mathit{t}}}{{\mathit{b}}_{1}[\mathbf{Ptf}.\mathbf{A}]}-1$ | $\frac{\mathbb{E}\left[{\mathit{PV}}_{\mathit{t}}\right]}{{\mathit{n}}_{\mathit{t}}}$ | $\mathbb{CV}\left[{\mathit{PV}}_{\mathit{t}}\right]$ | 95th perc. of ${\mathit{PV}}_{\mathit{t}}$ | 99th perc. of ${\mathit{PV}}_{\mathit{t}}$ |
---|---|---|---|---|---|

as a % of the Value | as a % of $\mathbb{E}\left[{\mathit{PV}}_{\mathit{t}}\right]$ | as a % of $\mathbb{E}\left[{\mathit{PV}}_{\mathit{t}}\right]$ | |||

for Ptf. A | |||||

0 | 6.009% | 100.01% | 1.03% | 101.70% | 102.38% |

5 | 5.918% | 99.16% | 1.20% | 101.98% | 102.82% |

10 | 5.771% | 98.19% | 1.39% | 102.29% | 103.20% |

15 | 5.524% | 97.10% | 1.70% | 102.81% | 103.94% |

20 | 5.080% | 95.99% | 2.18% | 103.58% | 105.22% |

25 | 4.340% | 95.04% | 3.05% | 105.02% | 107.21% |

30 | 3.141% | 94.48% | 4.94% | 108.26% | 111.58% |

35 | 1.547% | 94.36% | 10.06% | 116.76% | 124.25% |

40 | −0.523% | 95.00% | 31.01% | 152.56% | 179.67% |

45 | −2.700% | 99.35% | 134.82% | 334.93% | 558.22% |

Time t | Portfolio I | ||||
---|---|---|---|---|---|

Total Size | Relative Group Size | ||||

${\mathit{n}}_{\mathit{t}}$ | $\frac{{\mathit{n}}_{\mathit{t}}}{{\mathit{n}}_{0}}$ | Group ${\mathit{G}}_{1}$ | Group ${\mathit{G}}_{2}$ | Group ${\mathit{G}}_{3}$ | |

0 | 1 663 | 100.000% | 60.132% | 30.126% | 9.741% |

5 | 1 587 | 95.430% | 60.744% | 29.805% | 9.452% |

10 | 1 462 | 87.913% | 61.696% | 29.343% | 8.960% |

15 | 1 267 | 76.188% | 63.457% | 28.335% | 8.208% |

20 | 989 | 59.471% | 66.431% | 26.694% | 6.876% |

25 | 648 | 38.966% | 71.296% | 23.611% | 5.093% |

30 | 320 | 19.242% | 78.750% | 18.438% | 2.813% |

35 | 103 | 6.194% | 88.350% | 10.680% | 0.971% |

40 | 18 | 1.082% | 94.444% | 5.556% | 0.000% |

45 | 1 | 0.060% | 100.000% | 0.000% | 0.000% |

50 | 0 | 0.000% |

**Table 22.**Portfolio I: portfolio average benefit amount (additional amount with respect to the standard benefit for Portfolio A, $\frac{{\overline{b}}_{t}}{{b}_{1}[\mathrm{Ptf}.\mathrm{A}]}-1$) and summary statistics of the probability distribution of insurer’s liabilities.

Time t | $\frac{{\overline{\mathit{b}}}_{\mathit{t}}}{{\mathit{b}}_{1}[\mathbf{Ptf}.\mathbf{A}]}-1$ | $\frac{\mathbb{E}\left[{\mathit{PV}}_{\mathit{t}}\right]}{{\mathit{n}}_{\mathit{t}}}$ | $\mathbb{CV}\left[{\mathit{PV}}_{\mathit{t}}\right]$ | 95th perc. of ${\mathit{PV}}_{\mathit{t}}$ | 99th perc. of ${\mathit{PV}}_{\mathit{t}}$ |
---|---|---|---|---|---|

as a % of the Value | as a % of $\mathbb{E}\left[{\mathit{PV}}_{\mathit{t}}\right]$ | as a % of $\mathbb{E}\left[{\mathit{PV}}_{\mathit{t}}\right]$ | |||

for Ptf. A | |||||

0 | 6.472% | 100.01% | 1.03% | 101.71% | 102.44% |

5 | 6.304% | 99.09% | 1.19% | 101.97% | 102.73% |

10 | 6.013% | 98.07% | 1.40% | 102.29% | 103.16% |

15 | 5.500% | 97.07% | 1.69% | 102.78% | 103.86% |

20 | 4.673% | 96.32% | 2.18% | 103.57% | 105.13% |

25 | 3.422% | 96.31% | 3.04% | 104.99% | 107.14% |

30 | 1.641% | 98.01% | 4.89% | 108.16% | 111.53% |

35 | −0.129% | 102.99% | 9.85% | 116.45% | 124.06% |

40 | −1.920% | 112.69% | 26.89% | 147.46% | 168.03% |

45 | −1.920% | 131.86% | 127.36% | 339.18% | 508.78% |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Olivieri, A.; Pitacco, E.
Frailty and Risk Classification for Life Annuity Portfolios. *Risks* **2016**, *4*, 39.
https://doi.org/10.3390/risks4040039

**AMA Style**

Olivieri A, Pitacco E.
Frailty and Risk Classification for Life Annuity Portfolios. *Risks*. 2016; 4(4):39.
https://doi.org/10.3390/risks4040039

**Chicago/Turabian Style**

Olivieri, Annamaria, and Ermanno Pitacco.
2016. "Frailty and Risk Classification for Life Annuity Portfolios" *Risks* 4, no. 4: 39.
https://doi.org/10.3390/risks4040039