1. Introduction
The construction of life expectancy projections has been the subject of much work since the seminal article by Lee and Carter (Lee and Carter [1992]).
With a view to extrapolating trends observed in the past into the future, most of the approaches proposed are based on a “mortality surface”, measuring mortality forces by age and year at the time, which must then be extrapolated over time.
The models inspired by Lee and Carter start by reducing dimension by performing a PCA and then extrapolating one or two time series associated with the projection on the principal axes.
Bongaarts (Bongaarts [2004]) has proposed a different approach, based on parametric adjustments by year of time and extrapolation of the estimated coefficients each year.
In Bongaarts [2004], however, the author uses a rather frustrating parametric representation (Thatcher’s model) which does not allow all ages to be included. Moreover, he limits his extrapolation to 2 out of 3 parameters, treating them independently, which is a questionable approximation.
Models of this type project regular $t\to \mu \left(x,t\right)$ series. However, when we look at annual variations in the mortality rate in France, for example, we see a fairly high degree of volatility:
Figure 1.
Annual variation in mortality rate.
Figure 1.
Annual variation in mortality rate.
The classic models described above cannot account for these shortterm variations. Proposed approaches have been put forward, for example, in Guette [2010] or Currie et al. [2003], but with a slightly different objective, as these works propose to model catastrophes such as wars or epidemics. More recently, an approach using regimeswitching models was proposed in Robben and Antonio [2023].
Our aim here is not to model catastrophes, but to incorporate the above volatility into the modeling to provide a more accurate assessment of residual life expectancies when an unbiased estimate of mortality rates is available.
A specific approach is therefore proposed here, with the aim of accounting for this shortterm volatility and measuring its impact on the anticipation of prospective residual life expectancies.
2. Proposed Stochastic Mortality Model
We draw inspiration here from the fragility models proposed by Vaupel and his coauthors (Vaupel et al. [1979]), by assigning a regular base hazard function of a shock that depends only on time, under a proportional hazards assumption.
The proposed specification is described below, followed by a method for estimating the parameters within the framework of conditional maximum likelihood.
a. Specification
Consider the following specification of the hazard function for the year of time
t:
with the semiparametric form of the basic hazard function
$\mathbf{ln}{\mu}_{0}\left(x,t\right)={\alpha}_{x}+{\beta}_{x}{k}_{t}$. The shocks are assumed to be meancentered, i.e.,
$E\left({Z}_{t}\right)=1$ and the classical identifiability conditions for the basic hazard function are imposed (cf.
Brouhns et al. [2002]):
This involves estimating the parameters of ${Z}_{t}$ and the matrix$\left(\alpha ,\beta ,k\right)$, then extrapolating the time series$t\to k\left(t\right)$.
b. LogLikelihood Determination
For maximum likelihood estimation, we know that everything happens as if the number of deaths observed followed a Poisson distribution,
which leads to the following expression for the conditional likelihood for an observation, noting
$\lambda ={E}_{x,t}\times {\mu}_{0}\left(x,t\right)$:
Likelihood is easy to deduce:
We then choose a Gamma distribution of parameters a and b for the distribution of Z, i.e.,
${f}_{Z}\left(z\right)={z}^{a1}\frac{{b}^{a}{e}^{b\times z}}{\Gamma \left(a\right)}$, which leads to
$P\left(D=d\right)=\frac{{\lambda}^{d}}{d!}\frac{{b}^{a}}{\Gamma \left(a\right)}{\displaystyle \underset{0}{\overset{+\infty}{\int}}{e}^{\left(\lambda +b\right)z}{z}^{d+a1}dz}$. Using the change of variable
$u=\left(\lambda +b\right)z$, we obtain the following expression for the likelihood of an observation
which in turn gives loglikelihood
with $f\left(a,b\right)=\mathbf{ln}\left({b}^{a}\frac{\Gamma \left(d+a\right)}{\Gamma \left(d+1\right)\Gamma \left(a\right)}\right)$ .
As a function of the parameters
$\left(\alpha ,\beta ,k\right)$ and conditional on
$\left(a,b\right)$, the loglikelihood for an observation is of the form
$l\left(\alpha ,\beta ,k\right)=\mathbf{ln}P\left(D=d\right)$ with
$\lambda ={E}_{x,t}\times {\mu}_{0}\left(x,t\right)={E}_{x,t}\times {e}^{{\alpha}_{x}+{\beta}_{x}{k}_{t}}$. Conditional on
$\left(a,b\right)$, the loglikelihood has the following form (with one additive constant):
It can then be maximized by $\left(\alpha ,\beta ,k\right)$ under the constraint that $\sum _{x={x}_{m}}^{{x}_{M}}{\beta}_{x}=1$ and $\sum _{t={t}_{m}}^{{t}_{M}}{k}_{t}=0$.
c. Parameter Estimation
Parameter estimation can be carried out in two stages: in the first stage, the fragility parameter is estimated, then, in the second stage, the above loglikelihood is maximized at $\left(\alpha ,\beta ,k\right)$.
The condition
$E\left({Z}_{t}\right)=1$ implies
$a=b$. We also have
$V\left({Z}_{t}\right)=\frac{a}{{b}^{2}}=\frac{1}{a}$, so the disturbance control parameter
${Z}_{t}$ is the inverse of the variance
$a={\sigma}_{Z}^{2}$. A direct estimate of this parameter can be made as follows, observing that the mean annual output intensities are of the form
from which we derive
$E\left(\overline{\mu}\left(t\right)\right)={\overline{\mu}}_{0}\left(t\right)$,
$V\left(\overline{\mu}\left(t\right)\right)=V\left({Z}_{t}\right){\overline{\mu}}_{0}^{2}\left(t\right)$ then
$V\left({Z}_{t}\right)=\frac{V\left(\overline{\mu}\left(t\right)\right)}{E{\left(\overline{\mu}\left(t\right)\right)}^{2}}$. The variance of
${Z}_{t}$ is therefore equal to the square of the coefficient of variation of
$\overline{\mu}\left(t\right)$:
$\sigma \left({Z}_{t}\right)=cv\left(\overline{\mu}\left(t\right)\right)$. It is then straightforward to propose as estimator
With $\widehat{\mu}\left(t\right)=\frac{{\displaystyle \sum _{x={x}_{m}}^{{x}_{M}}{E}_{x,t}\widehat{\mu}\left(x,t\right)}}{{\displaystyle \sum _{x={x}_{m}}^{{x}_{M}}{E}_{x,t}}}$ and $\widehat{\mu}\left(x,t\right)=\frac{{D}_{x,t}}{{E}_{x,t}}$ as estimators, the Hoem estimator of the hazard function.
Once the fragility parameter has been estimated, the aim is to maximize the previously expressed loglikelihood. The partial derivatives of the loglikelihood for an observation are as follows:
with
p one of the parameters
$\left(\alpha ,\beta ,k\right)$. We also have
$\frac{\partial \lambda}{\partial \alpha}=\lambda $,
$\frac{\partial \lambda}{\partial \beta}=k\lambda $ and
$\frac{\partial \lambda}{\partial k}=\beta \lambda $.
Finally, it remains to estimate
$\left(\alpha ,\beta ,k\right)$, which is a solution of the firstorder conditions:
This system is nonlinear.
d. Calculating Prospective Residual Life Expectancies
In the proposed model, the calculation of prospective life expectancy
takes the form
because the Laplace transform of a Gamma distribution is
$E\left({e}^{x{Z}_{t}}\right)={\left(\frac{b}{b+x}\right)}^{a}$. As
$E\left({e}^{\mu \left(x,t\right)}\right)=E\left({e}^{{Z}_{t}\times {\mu}_{0}\left(x,t\right)}\right)$, we deduce that:
Note that when $b=a\to +\infty $, $E\left({e}^{\mu \left(x,t\right)}\right)\to {e}^{{\mu}_{0}\left(x,t\right)}$ and then we find the classic formula $e\left(x,t\right)={\displaystyle \sum _{i\ge 0}{\displaystyle \prod _{j=0}^{i}\mathbf{exp}\left({\mu}_{0}\left(x+j,t+j\right)\right)}}$.
3. Numerical Application
We use data for metropolitan France for the period 20002020, for ages 0 to 105 inclusive, from the Agalva and Blanpain study [2021]. All calibration was performed in R.
Prospective analyses are then carried out over the entire age range and for the years from 2021 to 2060.
a. Model Adjustment
All these steps are discussed in turn in the following subsections. Throughout the study, all the results obtained are compared with those given by a LeeCarter model calibrated on the same data.
Estimation of Gamma Distribution Parameters
The estimation of the pair of parameters $\left(a,b\right)$ has been made with the raw data and we find ${\sigma}_{Z}^{2}=4,3\text{\hspace{0.33em}}\%$ , i.e., $a=b=550$.
Estimation of Model Parameters
As shown in the following figure, this leads to coefficients $\left(\alpha ,\beta ,k\right)$ very close to those of the reference LeeCarter model:
Figure 2.
Comparison of different coefficients with those of a LeeCarter model.
Figure 2.
Comparison of different coefficients with those of a LeeCarter model.
It can be seen that the time parameter shows a slower rate of decline from the 18^{ème} year of the observation period.
b. Projected Mortality Forces
Here, we compare projected mortality forces with and without shocks integrated into the model, as a function of age and year.
First, we look at the evolution of the mortality force as a function of age, for a few fixed years:
Figure 3.
Mortality forces by age.
Figure 3.
Mortality forces by age.
The mortality forces of the two models are very close, except at the highest ages, where the model with shocks tends to predict higher mortality forces. A closer look at the age range [90; 105] reveals the following:
Figure 4.
Mortality rates by age from 90 to 105 years.
Figure 4.
Mortality rates by age from 90 to 105 years.
The maximum relative deviation over the entire prospective range is 3.6% at age 80 for the year 2060.
We now compare the mortality forces of the two models over the entire prospective analysis period, for a few selected ages.
Figure 5.
Mortality rates by year for selected ages.
Figure 5.
Mortality rates by year for selected ages.
Overall, the mortality model studied tends to predict higher mortality forces than the reference LeeCarter model at older ages, and the gap increases over time.
Before calculating prospective residual life expectancies, it’s worth looking at the overall impact on mortality forces. To this end, we calculate the following ratio:
with
${\mu}_{LC}\left(x,t\right)$ the mortality force derived from the reference LeeCarter model and
$\mu \left(x,t\right)$ model. This ratio is shown in the following figure:
Figure 6.
Mortality force ratio over the entire age range and prospective analysis period (20212060).
Figure 6.
Mortality force ratio over the entire age range and prospective analysis period (20212060).
The average value of this ratio for all ages and years combined is 100%, which means that the model studied is equivalent to the LeeCarter model.
The impact of the introduction of shocks on adjustment is therefore negligible when assessed on a very global basis. However, the difference increases over time, leading the two models to diverge in the medium term and at older ages.
If we restrict ourselves to ages over 65, we obtain a weighted average equal to 99.8%.
In addition, the average mortality rate of the population, calculated on the basis of exposure to risk in 2020, evolves as follows:
Figure 7.
Average mortality rate per year, all ages combined, from 2021 to 2060.
Figure 7.
Average mortality rate per year, all ages combined, from 2021 to 2060.
It can be seen that the model studied is on average more pessimistic than the LeeCarter model on mortality improvement in future mortality.
c. Estimating Prospective Residual Life Expectancies
It is then possible to look at the consequences of the mortality model studied on prospective residual life expectancies, first by variable age for a few fixed years, then by variable year for a few fixed ages.
As shown in the following figure, it appears that the mortality model studied does not greatly change prospective residual life expectancies compared to the reference mortality model, except possibly at high ages:
Figure 8.
Trends in prospective residual life expectancy by age.
Figure 8.
Trends in prospective residual life expectancy by age.
An enlargement of these graphs at older ages is shown below, with the algebraic difference between the two models for each year:
Figure 9.
Change in prospective residual life expectancy from age 90 to 105.
Figure 9.
Change in prospective residual life expectancy from age 90 to 105.
The maximum absolute difference between the two models is found at age 96 for the year 2060, and is worth 0.18, or around 65 days.
In this analysis, we return to the observation made in the previous section: it is at older ages that the difference with the reference LeeCarter model is most marked.
Figure 10.
Prospective residual life expectancy by year.
Figure 10.
Prospective residual life expectancy by year.
Below is an enlargement for age 96 with the algebraic difference between the two models:
Figure 11.
Trend in prospective residual life expectancy by year at age 96.
Figure 11.
Trend in prospective residual life expectancy by year at age 96.
d. Sensitivity to Fragility Parameter
With this level of volatility, we note that
$P\left({Z}_{t}\ge 1,09\right)\approx 5\text{\hspace{0.33em}}\%$; 9% being the excess mortality rate (
https://actudactuaires.typepad.com/laboratoire/2021/03/mortalit%C3%A9enfranceen2020suite.html) for the year 2020, we can deduce that the probability of observing excess mortality at this level is of the order of 5%. Furthermore,
$Va{R}_{99,5\text{\hspace{0.17em}}\%}\left({Z}_{t}\right)\approx 1,15$, which corresponds to the mortality shock for the “mortality” risk module of the delegated regulation.
On the basis of the central table
${\mu}_{0}\left(x,t\right)$ adjusted above, the prospective residual life expectancies associated with a volatility coefficient of 5.5% are recalculated using
which leads to the following results:
Figure 12.
Evolution of prospective residual life expectancies from age 65 to 105 for selected years, with a new volatility coefficient.
Figure 12.
Evolution of prospective residual life expectancies from age 65 to 105 for selected years, with a new volatility coefficient.
There is no significant difference between the two models with this new volatility setting.
e. Consequences for the Capital Requirement of an Annuity Plan
The presence of the frailty factor therefore has no material impact on central tendency indicators (mortality forces, residual life expectancies, etc.).
However, the random nature of the mortality distribution in a given year has consequences for the assessment of the capital required to protect against adverse deviations in mortality. In the specific context of a life annuity plan, we are therefore led, following a logic analogous to that of the Solvency 2 standard, to consider the 99.5% quantile of the distribution of residual life expectancies induced by frailty. For each age from 60 to 100, we obtain the following results for the ratio between this quantile and the expectation:
Figure 13.
Ratio between 99.5% quantile and expectation (SCR).
Figure 13.
Ratio between 99.5% quantile and expectation (SCR).
Weighted by the age structure of the French population, the average ratio is around 101.3%.
For its part, the delegated regulation (EU Delegated Regulation n°2015/35:
https://eurlex.europa.eu/legalcontent/FR/TXT/?uri=CELEX:32015R0035) imposes a 20% discount on death rates when calculating the SCR associated with longevity risk (cf. art. 138 of delegated regulation EU n°2015/35), which leads to a capital requirement equal to 10% of the expectation.
This means that the volatility observed in annual death rates explains around 12% of the longevity SCR.
4. Conclusions and Discussion
The use of a Gamma frailty model enables us to correctly account for the annual variations in mortality levels observed throughout France.
Incorporating these variations into the fitting of a forwardlooking logPoisson model poses no major difficulty, and a twostage parameter estimation process enables us to use conventional likelihood maximization algorithms.
The results obtained show that the impact of this additional volatility is negligible on the central tendency indicators.
On the other hand, there is a material impact on the capital requirement associated with longevity risk, with just under 15% of this requirement being induced by the presence of this volatility. The remainder is associated with uncertainty about the trend in death rates.
Thus, while the main hazard associated with the construction of a prospective mortality table remains the uncertainty attached to the determination of the trend (see Juillard et al. [2008] and Juillard and Planchet [2006] for detailed analyses on this point), taking into account these shortterm fluctuations in mortality levels provides a better understanding of the determinants of longevity risk.
Appendix A
Table A1.
Calibrated model coefficients.
Table A1.
Calibrated model coefficients.
Alpha 
Age 
Model studied 
LC reference model 

Age 
Model studied 
LC reference model 

Age 
Model studied 
LC reference model 
0 
5,6542 
5,6553 

36 
7,0173 
7,0186 

72 
4,0422 
4,0456 
1 
7,3447 
7,3457 

37 
6,9390 
6,9396 

73 
3,9592 
3,9625 
2 
8,2790 
8,2804 

38 
6,8474 
6,8480 

74 
3,8668 
3,8713 
3 
8,6679 
8,6695 

39 
6,7630 
6,7633 

75 
3,7867 
3,7882 
4 
8,9672 
8,9721 

40 
6,6729 
6,6733 

76 
3,6851 
3,6862 
5 
9,0765 
9,0803 

41 
6,5862 
6,5867 

77 
3,5790 
3,5803 
6 
9,1958 
9,2028 

42 
6,4878 
6,4880 

78 
3,4774 
3,4791 
7 
9,2892 
9,2924 

43 
6,3872 
6,3877 

79 
3,3674 
3,3679 
8 
9,4087 
9,4124 

44 
6,2849 
6,2854 

80 
3,2163 
3,2216 
9 
9,3935 
9,4009 

45 
6,1800 
6,1803 

81 
3,0838 
3,0881 
10 
9,4223 
9,4329 

46 
6,0917 
6,0922 

82 
2,9510 
2,9556 
11 
9,3577 
9,3662 

47 
5,9813 
5,9817 

83 
2,8178 
2,8236 
12 
9,2768 
9,2809 

48 
5,8840 
5,8845 

84 
2,6937 
2,7014 
13 
9,1674 
9,1742 

49 
5,7947 
5,7952 

85 
2,5725 
2,5850 
14 
8,9411 
8,9443 

50 
5,7068 
5,7077 

86 
2,4381 
2,4498 
15 
8,6907 
8,6949 

51 
5,6157 
5,6166 

87 
2,3015 
2,3126 
16 
8,4690 
8,4721 

52 
5,5399 
5,5411 

88 
2,1619 
2,1733 
17 
8,2022 
8,2047 

53 
5,4596 
5,4605 

89 
2,0266 
2,0390 
18 
7,9805 
7,9814 

54 
5,3712 
5,3722 

90 
1,8871 
1,9001 
19 
7,7424 
7,7441 

55 
5,2853 
5,2866 

91 
1,7522 
1,7651 
20 
7,6642 
7,6654 

56 
5,2062 
5,2078 

92 
1,6216 
1,6351 
21 
7,6042 
7,6051 

57 
5,1270 
5,1289 

93 
1,4936 
1,5072 
22 
7,5935 
7,5952 

58 
5,0598 
5,0611 

94 
1,3657 
1,3777 
23 
7,5824 
7,5837 

59 
4,9902 
4,9925 

95 
1,2453 
1,2594 
24 
7,5520 
7,5540 

60 
4,9156 
4,9178 

96 
1,1239 
1,1367 
25 
7,5513 
7,5531 

61 
4,8551 
4,8570 

97 
1,0154 
1,0262 
26 
7,5195 
7,5211 

62 
4,7853 
4,7877 

98 
0,9056 
0,9177 
27 
7,4982 
7,4995 

63 
4,7165 
4,7182 

99 
0,8059 
0,8166 
28 
7,4795 
7,4816 

64 
4,6552 
4,6571 

100 
0,7093 
0,7208 
29 
7,4393 
7,4409 

65 
4,5875 
4,5894 

101 
0,6286 
0,6345 
30 
7,3880 
7,3899 

66 
4,5141 
4,5163 

102 
0,5389 
0,5481 
31 
7,3594 
7,3607 

67 
4,4558 
4,4580 

103 
0,4581 
0,4679 
32 
7,3008 
7,3012 

68 
4,3774 
4,3802 

104 
0,4095 
0,4185 
33 
7,2536 
7,2544 

69 
4,2982 
4,3005 

105 
0,4625 
0,4652 
34 
7,1764 
7,1771 

70 
4,2170 
4,2209 




35 
7,1106 
7,1110 

71 
4,1381 
4,1415 




Beta 
Age 
Model studied

LC reference model


Age 
Model studied

LC reference model


Age 
Model studied

LC reference model

0 
0,0033 
0,0033 

36 
0,0120 
0,0120 

72 
0,0070 
0,0071 
1 
0,0115 
0,0115 

37 
0,0115 
0,0115 

73 
0,0070 
0,0071 
2 
0,0114 
0,0114 

38 
0,0123 
0,0123 

74 
0,0075 
0,0077 
3 
0,0118 
0,0117 

39 
0,0119 
0,0118 

75 
0,0090 
0,0091 
4 
0,0130 
0,0131 

40 
0,0129 
0,0129 

76 
0,0094 
0,0094 
5 
0,0121 
0,0118 

41 
0,0134 
0,0135 

77 
0,0097 
0,0097 
6 
0,0097 
0,0102 

42 
0,0140 
0,0140 

78 
0,0096 
0,0097 
7 
0,0145 
0,0146 

43 
0,0137 
0,0137 

79 
0,0099 
0,0098 
8 
0,0119 
0,0121 

44 
0,0135 
0,0136 

80 
0,0136 
0,0129 
9 
0,0126 
0,0127 

45 
0,0132 
0,0133 

81 
0,0139 
0,0134 
10 
0,0144 
0,0138 

46 
0,0136 
0,0137 

82 
0,0141 
0,0137 
11 
0,0114 
0,0118 

47 
0,0128 
0,0127 

83 
0,0142 
0,0137 
12 
0,0165 
0,0162 

48 
0,0122 
0,0122 

84 
0,0115 
0,0107 
13 
0,0143 
0,0142 

49 
0,0108 
0,0108 

85 
0,0083 
0,0073 
14 
0,0151 
0,0151 

50 
0,0106 
0,0106 

86 
0,0065 
0,0057 
15 
0,0155 
0,0151 

51 
0,0099 
0,0099 

87 
0,0059 
0,0054 
16 
0,0180 
0,0182 

52 
0,0093 
0,0093 

88 
0,0056 
0,0052 
17 
0,0171 
0,0171 

53 
0,0093 
0,0093 

89 
0,0057 
0,0054 
18 
0,0185 
0,0187 

54 
0,0095 
0,0094 

90 
0,0051 
0,0049 
19 
0,0183 
0,0180 

55 
0,0093 
0,0092 

91 
0,0049 
0,0048 
20 
0,0170 
0,0169 

56 
0,0082 
0,0080 

92 
0,0044 
0,0046 
21 
0,0158 
0,0158 

57 
0,0082 
0,0081 

93 
0,0039 
0,0041 
22 
0,0146 
0,0146 

58 
0,0067 
0,0066 

94 
0,0026 
0,0030 
23 
0,0156 
0,0156 

59 
0,0061 
0,0059 

95 
0,0023 
0,0027 
24 
0,0132 
0,0134 

60 
0,0050 
0,0049 

96 
0,0016 
0,0024 
25 
0,0126 
0,0127 

61 
0,0042 
0,0041 

97 
0,0008 
0,0014 
26 
0,0113 
0,0115 

62 
0,0039 
0,0038 

98 
0,0005 
0,0004 
27 
0,0111 
0,0112 

63 
0,0040 
0,0039 

99 
0,0013 
0,0006 
28 
0,0107 
0,0108 

64 
0,0042 
0,0042 

100 
0,0002 
0,0001 
29 
0,0088 
0,0088 

65 
0,0038 
0,0038 

101 
0,0018 
0,0022 
30 
0,0103 
0,0102 

66 
0,0046 
0,0046 

102 
0,0044 
0,0047 
31 
0,0111 
0,0110 

67 
0,0050 
0,0051 

103 
0,0046 
0,0048 
32 
0,0107 
0,0106 

68 
0,0047 
0,0047 

104 
0,0046 
0,0053 
33 
0,0105 
0,0104 

69 
0,0059 
0,0059 

105 
0,0046 
0,0051 
34 
0,0106 
0,0105 

70 
0,0055 
0,0055 




35 
0,0114 
0,0113 

71 
0,0063 
0,0064 




Kappa 

Age 
Model studied

LC reference model


Age 
Model studied

LC reference model


2000 
24,4565 
24,4761 

2030 
43,8008 
43,8043 

2001 
24,4627 
24,5265 

2031 
45,9908 
45,9945 

2002 
21,2073 
21,2070 

2032 
48,1809 
48,1847 

2003 
18,4710 
18,4083 

2033 
50,3709 
50,3750 

2004 
10,9651 
10,9513 

2034 
52,5610 
52,5652 

2005 
9,4399 
9,4231 

2035 
54,7510 
54,7554 

2006 
6,0386 
6,0366 

2036 
56,9410 
56,9456 

2007 
3,1599 
3,1578 

2037 
59,1311 
59,1358 

2008 
1,4649 
1,4631 

2038 
61,3211 
61,3260 

2009 
1,6981 
1,6984 

2039 
63,5112 
63,5163 

2010 
1,1863 
1,1865 

2040 
65,7012 
65,7065 

2011 
4,7123 
4,7140 

2041 
67,8912 
67,8967 

2012 
6,6593 
6,6691 

2042 
70,0813 
70,0869 

2013 
8,5651 
8,5639 

2043 
72,2713 
72,2771 

2014 
12,6304 
12,6243 

2044 
74,4614 
74,4673 

2015 
9,8526 
9,8338 

2045 
76,6514 
76,6575 

2016 
13,5803 
13,5743 

2046 
78,8414 
78,8478 

2017 
15,8335 
15,8386 

2047 
81,0315 
81,0380 

2018 
15,6887 
15,6650 

2048 
83,2215 
83,2282 

2019 
16,4993 
16,4493 

2049 
85,4116 
85,4184 

2020 
16,1564 
16,2296 

2050 
87,6016 
87,6086 

2021 
24,0904 
24,0924 

2051 
89,7916 
89,7988 

2022 
26,2805 
26,2826 

2052 
91,9817 
91,9891 

2023 
28,4705 
28,4728 

2053 
94,1717 
94,1793 

2024 
30,6606 
30,6630 

2054 
96,3618 
96,3695 

2025 
32,8506 
32,8532 

2055 
98,5518 
98,5597 

2026 
35,0406 
35,0435 

2056 
100,7418 
100,7499 

2027 
37,2307 
37,2337 

2057 
102,9319 
102,9401 

2028 
39,4207 
39,4239 

2058 
105,1219 
105,1304 

2029 
41,6108 
41,6141 

2059 
107,3120 
107,3206 





2060 
109,5020 
109,5108 

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