2.1 On the recent awareness about anomalies in mortality tables

As mentioned in the Introduction, few actuarial work has been focusing on mortality data reliability, but the awareness about such anomalies has recently emerged from the work by Richards (Reference Richards2008), Cairns et al. (Reference Cairns, Blake, Dowd and Kessler2016) and Boumezoued (2016). The last author put into evidence the universality of such anomalies (isolated cohort effects) by highlighting them in a variety of countries from the HMD. By studying the HMD methodology to construct period tables, he proposed to source its fertility counterpart, the HFD, and to apply the philosophy of the work by Cairns et al. (Reference Cairns, Blake, Dowd and Kessler2016) to produce adjusted period mortality tables for a set of countries.

In this paper, we apply and extend the adjustment methodology as described in Boumezoued (2016). This way, we are able to measure the discrepancy between original and adjusted period mortality tables, which are crucial inputs for the measurement and management of longevity risk in the insurance market.

In order to present the up-to-date awareness on such abnormal cohort effects, we summarise below the main conclusions from the work of Boumezoued (2016):

1. 1. While comparing period and cohort mortality tables, anomalies in period tables ones have been highlighted in the form of isolated cohort effects for several countries available in HMD: the period mortality rates for specific generations appear surprisingly low or high compared to the others.

2. 2. The HMD methodology to construct mortality estimates has been identified to embed a strong assumption of uniform distribution of births that is specific to the computation of period mortality tables, which shows that we are facing a universal reliability issue which is shared by most countries.

3. 3. To perform an automatic correction procedure, it is proposed to rely on the HFD, which is considered as the perfect counterpart of the HMD in terms of fertility.

4. 4. The analysis of adjusted mortality tables for several countries led to the conclusion that isolated “cohort effects” in original mortality tables (often for years of birth around 1915, 1920, 1940 and 1945) are in fact universal anomalies that disappear in adjusted tables. Further analysis of adjusted mortality tables shows that, not only the mortality levels for several cohorts are highly over/underestimated, but also that the volatility of mortality improvement rates over the last 30 years was overestimated in original HMD mortality tables, with a large difference for many countries.

In Section 2.2, we detail the correction methodology based on fertility data which is used in the present paper to adjust mortality tables for France and Italy. In Section 2.3, we motivate the need for an extension of the methodology, and we propose a novel approach to produce an adjusted table for West Germany. Finally, Section 2.4 details the main characteristics of the adjusted mortality tables and the key discrepancies with the original data.

As a final remark, let us emphasise that two kinds of mortality tables are provided by the HMD: period and cohort. Anomalies in terms of abnormal cohort effects have been identified in period mortality tables, which are naturally designed to study the dynamics of mortality from one year to the next, and therefore systematically used in practice to calibrate stochastic mortality models. For a review of the definition and construction methods for period and cohort tables, see Boumezoued (2016).

2.2 The correction method based on fertility data

In practice, period mortality tables for 1-year time periods and 1-year age classes are produced based on annual population estimates derived from census, as well as number of deaths combining the information of death reports and population counts. Note that in most countries, census may be performed out of the beginning of the year, or at intervals greater than 1 year, which leads the HMD to perform several adjustments, see Wilmoth et al. (Reference Wilmoth, Andreev, Jdanov and Glei2017) for more details. In our framework, however, we assume that input data provided by the HMD as annual population counts and number of deaths in Lexis triangles are accurate.

Two ingredients are at the core of death rate computation for a given period: the number of deaths in this period, which we assumed to be reliable, divided by the so-called “exposure-to-risk” which represents the “quantity” of individuals at risk of death. In the so-called period mortality tables, the exposure to risk takes the mathematical form of an integral over a 1-year time period and a 1-year age class, see Boumezoued (2016) for more details on the formalism. In this setting, the period mortality rate writes

\begin{equation*}m( {x,t} ) = \frac{{D( {x,t} )}}{{E( {x,t} )}}\end{equation*}

where $D( {x,t} )$ is the number of deaths and $E( {x,t} )$ is the total time lived in the year $\left[ {t,t + 1} \right)$ at age between $x$ and $x + 1.$ The mortality rate is the core object to be modelled by stochastic mortality models, as it is also the case for the death probability $q( {x,t} )$ , more interpretable as the probability for individuals aged $x$ at time $t$ to die in the next year, and linked to the mortality rates through the following equation:

\begin{equation*}q( {x,t} ) = 1 - \exp \left( { - m( {x,t} )} \right)\end{equation*}

In the standard computation practice, the exposure-to-risk component $E( {x,t} )$ is usually approximated based on annual population estimates, as it is done by reference providers as national institutes or the HMD. In other words, the standard practice works with the average between the population at the beginning of the year and that at the end of the year, which is taken as a proxy for the integral over the year in continuous time. This computation amounts to assume that births are uniformly distributed in the year, as well as from one cohort to the next.

Although for several years where population flows are quite stable the assumption may seem reasonable (although not perfect), the approximation error can be huge in situations in which population numbers are fluctuating in the year, that is particularly the case when births are erratic due to demographic shocks (e.g. before and after wars and pandemics). Such specific patterns create a “convexity effect”, and in this context, it is no longer possible to approximate the exposure to risk by a standard average.

To correct for such approximation errors for countries available in the HMD, the approach proposed by Boumezoued (2016) relies on monthly fertility records available from the HFD. Such monthly population estimates are used to construct a “data quality indicator” assessing the reliability of estimates in the (period) mortality table for each generation. The indicator takes the form of a ratio which measures the deviation between an annual and a monthly approximation of the (annual) exposure to risk. For each year of birth, this ratio is then used to adjust the estimates along each diagonal of the mortality table (that is following each cohort), producing an adjusted table which does not present the initial anomalies.

In this latter work, adjusted period tables have been provided for France, Switzerland, Finland, Sweden and Austria, countries for which complete and deep fertility histories were available at time of publication. Meanwhile, the monthly fertility data for Italy has been officially released by the HFD, allowing us to use the same methodology for this country as well. Note also that this is the case of Iceland, which will be added to the set of explanatory countries for West Germany, see the next subsection c). In the following, we present the results in terms of correction indicator for France and Italy; the specific treatment of West Germany, for which the monthly fertility history is not deep enough, is detailed in Section 2.3. The key characteristics of adjusted period tables are then described in Section 2.4.

The number of births by months as well as the correction indicator computed for France and Italy using the method by Boumezoued (2016) is depicted in Figure 2. This figure shows how shocks in birth patterns impact the exposure-to-risk approximation as it is performed in period tables provided by the HMD. The virtue of the correction indicator appears from these graphs: for a given year of birth, a value greater (respectively, lower) than one indicates that the HMD period mortality rates for the generation are overestimated (respectively underestimated). In addition, the ratio measures the magnitude of the discrepancy between the annual exposure-to-risk approximation and its (better) monthly counterpart. It should be noted that although the dynamics of the number of births over the months differs between France and Italy, the 1919–1920 effect shows some universality: the period mortality rates for generation 1919 are underestimated, whereas those for generation 1920 are overestimated. Moreover, in each case the order of magnitude of the error is ±6%, see again Figure 2; this may represent a level of particular attention for practitioners using national mortality data.

Figure 2. Number of births by months and correction indicator for France and Italy.

2.3 An extension of the correction method for West Germany

As mentioned in the previous part, in its present form, the adjustment methodology can only be applied to years of birth for which the number of births by months is provided by the HFD. Unfortunately, this is not the case for West Germany for which the HFD monthly fertility records are provided starting at year 1946, which consequently limits the set of years of birth for the correction ratio (see Figure 3).

Figure 3. Number of births by months and correction indicator for West Germany.

Therefore, there is a need to extend the previous methodology to enable producing an output adjusted table, especially for the typical 1919–1920 phenomenon which, again, lies outside of the historical period.

2.3.1 Regression with stepwise selection process

The idea now is to try to reconstruct the fertility history for West Germany, by looking at those other countries for which the birth series are available. Although number of births show clear different patterns from one country to another (see Figure 2, as well as Figure 9 in Boumezoued, 2016), the correction indicator, as a quantity without dimension (ratio of population estimates), shows quite similar patterns among the several countries.

Note that the approach developed here is intended to provide a first idea of possible adjustment without fertility data by month; the development of a full methodology to adjust population exposures in the absence of births-by-month data is left for further research. Let us also clarify that the results in terms of SCR will be provided for each country (in fact, each insurance entity within each country), therefore one has to keep in mind that a larger uncertainty remains concerning the calculation based on the Germany West adjusted data.

To infer the correction indicator for West Germany starting from that of other countries, we propose to use a multiple regression approach, while performing an optimal selection procedure. In order to properly detail the regression and prediction methodology, let us introduce some notations. We denote by ${I_C}(t)$ the correction indicator available for country $C$ and year of birth $t$ . We emphasise here that $t$ is annual and take its values in the space of all possible years of birth. The set of countries for which reasonable monthly fertility series are now available is set as

\begin{equation*}{\mathcal S} = \left\{ {{\rm{Finland}},{\rm{\;France}},{\rm{\;Iceland}},{\rm{\;Italy}},{\rm{\;Sweden}},{\rm{\;Switzerland}}} \right\}\end{equation*}

This set is selected in order to satisfy the two following criteria.

1. • Data on births by month are officially released by the HFD (i.e. not stated as “preliminary release”), to avoid any major data issue on birth series.

2. • The monthly fertility series allow constructing correction indicators for years of birth before 1914, which covers the 1919–1920 effect with margin.

The aim now is to find an optimal set of regressors $\;{{\mathcal S}^*} \subset {\mathcal S}$ , and coefficients $\left\{ {{\alpha _C},\;C \in {{\mathcal S}^*}} \right\}$ and $\mu \;$ such that

\begin{equation*}{I_{{\rm{West\;Germany}}}}(t) = \mu + \mathop \sum \limits_{C \in {{\mathcal S}^*}} {\alpha _C}\;{I_C}(t)\end{equation*}

for each $t$ such as the indicator for West Germany, ${I_{{\rm{West\;Germany}}}}(t)$ can be constructed, that is here for each year of birth $t$ between 1947 and 2010 (2010 being chosen as an upper bound for all countries in order to avoid depending on recent revisions of demographic data).

2.3.3 Prediction of the correction indicator

For birth years outside the range of the correction indicator availability for West Germany, that is for $t$ before 1946, the true indicator $I(t)$ can be predicted by its estimate $\hat{I}(t)$ according to the regression formula

\begin{equation*}\hat{I}^{\mathrm{West\;Germany}} (t) = \hat{\mu} + \sum_{C\in S^*} \hat{\alpha}_{C} {I}_C (t)\end{equation*}

The estimated parameters are given in Figure 4 and the reconstructed correction indicator is depicted in Figure 5, see the left panel for the full range of cohorts. The key fact which appears from the observation of the reconstructed correction ratio is the following: although almost no spikes were observed in the available period for regression (plain line), the use of external data allows us to reproduce the generational fertility shocks at the core times in history (including around 1915 and around 1920) – this feature captured here lies in the universality of the abnormal cohort countries in the selected regressors, but also the precise monthly timing of the war which seems to be consistent between the different countries selected.

Figure 4. Results of the linear regression.

Figure 5. Completion by regression of the correction indicator for West Germany.

Note that Figure 5 also provides on the right panel a more detailed analysis of the fitted correction indicator, compared to the original indicator as calculated for cohorts for which births-by-month data are available. This figure shows that the method works for many years, with, however, a very large discrepancy for year 1947, as well as noticeable discrepancies for, for example, years between 1970 and 1975. The correction efficiency of the regression approach presented here will also be evaluated in the next section in which we analyse the main characteristics of the adjusted mortality tables produced.

Remark: It is worth mentioning that although this extrapolation over another time period may induce some instability issues in several regression contexts due to non-stationarity effects, we rely here on the fact that the selected regressors are “mirror” countries which show similar error patterns (due to external and internal shocks), which are thus likely to reproduce the 1919–1920 effect (in between the Spanish flu and the end of the First World War). The illustrative results detailed in the next part indeed show the efficiency of the method to adjust the 1919–1920 effects, while providing an underlying interpretation of the correction indicator dynamics based on “explanatory” countries. At this stage, we argue that significant theoretical work remains to be done in this direction, which is out of scope of the present paper.

2.4 Analysis of adjusted mortality tables

Let us denote by $I( b )$ the correction indicator for year of birth $b$ . Then the adjusted period mortality rates are constructed as

\begin{equation*}\tilde{m}(x, t )= \frac{{m}\left(x, t \right)}{I(t-x)}\end{equation*}

Then matrices of crude and adjusted mortality improvement rates $r( {x,t} )$ or $\tilde{r}( {x,t} )$ can be analysed and compared; mortality improvement rates are here defined as

\begin{equation*}{r}\left(x, t \right)=\frac{m\left(x, t+1 \right)}{{m}\left(x, t \right)}-1\ \text{and}\ \tilde{r}\left(x, t \right)= \frac{\tilde{m}\left(x, t+1 \right)}{\tilde{m}\left(x,t\right)}-1.\end{equation*}

The data used to specify the crude death rates $m( {x,t} )$ by single year and 1-year age bands were downloaded from the HMD on 1 September 2015 for France (civil population), Italy (civil population) and West Germany (total population).

2.4.1 Adjusted mortality tables for France and Italy

The matrices of crude and adjusted mortality improvement rates by age and time are depicted in Figure 6. The matrices are centred on the 1919–1920 effects of interest, and the colour scale is fixed. The correction of these effects for France and Italy, and also the correction for France of the additional anomalies around years of birth 1915 and 1940 can clearly be observed. Note that for all the three countries, we use both male and female adjusted tables, although we represent that related to the total population as a purpose of illustration.

Figure 6. Crude and adjusted mortality improvements for France and Italy.

2.4.2 Adjusted mortality tables for West Germany

As stated in Section 2.3, the efficiency of the regression approach to reconstruct the correction indicator for West Germany is to be assessed by comparing the crude and adjusted period mortality table. The results of the method are depicted on improvement rate matrices in Figure 7 for the two criteria selected (BIC or R-squared); from left to right: crude data, updated data with BIC selection, updated data with adjusted R-squared selection.

Figure 7. From left to right: crude mortality improvements for West Germany, and the adjusted versions according to the BIC and R-squared criteria.

The correction process appears to be efficient concerning the 1919–1920 effects, which are the generations of interest in this paper. It should be noted that the 1915 effect is also clearly removed, but additionally that the correction specific to year of birth 1945 correction is not fully performed. Although in our context, this is a second-order issue as our main interest lies in the proper correction of the huge 1915 and 1920 anomalies, developing an improved correction method for such generations is left for further research.

Recall that the retained criterion is R-squared for the reasons detailed in Section 2.3, including in particular the fact that the adjusted table shows even less isolated cohort trends.

2.4.3 Adjustement diagnosis plots

In this part, we complete the analysis of the correction efficiency by studying the regularity of the mortality surface after the adjustment process. To do so, we rely on a similar quantity as the so-called empirical concavity function as introduced and studied in Cairns et al. (Reference Cairns, Blake, Dowd and Kessler2016); our formulation is homogeneous to the discrete version of the second-order derivative in time of the mortality rate and writes

\begin{equation*}C( {x,t} ) = \left| {\frac{{m\left( {x,t + 1} \right) - 2m( {x,t} ) + m\left( {x,t - 1} \right)}}{2}} \right|\end{equation*}

The concavity function for both crude and adjusted tables are depicted in Figure 8. Recall that classical mortality surfaces for medium-sized countries show relatively small concavity function values, as it is observed for generations before 1915 in both tables. The graphs show that the concavity function is a key tool to detect anomalies in crude tables, as already highlighted by Cairns et al. (Reference Cairns, Blake, Dowd and Kessler2016). They also show that the adjusted table encompasses concavity function magnitudes for cohorts 1915–1920 in the same order as the previous generations, although still high values can be observed for the highest ages in these cohorts. This unsmoothness could be expected due to the apparent period effect for year 2003 which corresponds to the European heat wave. Finally, this figure shows some room for improvement of the method, especially regarding the 1939–1945 cohorts for which the adjustment is not well performed (as already discussed for Figure 7).

Figure 8. Convexity matrix for the raw and adjusted West Germany mortality table.

Figure 9. Historical period life expectancy at age 30, truncated at age 95, before and after HFD correction.

2.4.6 Death curves

In mirror to the force of mortality, the next two graphs in Figure 11 present the death curves of France female population over the same four periods between years 1964 and 2009. It is noticeable how adjusted tables increased the regularity of the death curves. Results are similar for all countries and gender in our study.

In summary to the past section, analytical data show that the correction added to HMD data effectively removes artificial cohort effects around generations 1918–1923, while keeping main mortality characteristics.

Figure 10. Comparative period force of mortality before and after HFD adjustment, France – Women population.

Figure 11. Period-based death functions, before (left) and after (right) HFD correction.

3.1 Internal model specification

The simplified prototype of internal model used for this analysis works as follows: a set of mortality models is fitted based on national population mortality data, and the “best” model is selected based on both statistical and qualitative criteria. The selected mortality model is then used to draw simulated forecasts, and a prospective mortality table corresponding to the 99.5th percentile in terms of improvements is built. This allows to provide the longevity shock at the core of the Solvency II economic capital calculation, here based on an annuity product portfolio. Longevity risk can be divided into two different risk sources: a level component and a trend component, on which this article focuses on.

The internal model process workflow is illustrated in Figure 12: the usual workflow is represented, with the additional step of mortality tables adjustment. Indeed, this correction process is directly performed on national population mortality tables and is then used in the model calibration and selection processes.

Figure 12. The mortality table adjustment process within the internal model workflow.

The simplified prototype of internal model, employed here to assess longevity risk, relies on several classical stochastic mortality models, which we briefly describe in the following.

The famous model by Lee and Carter (Reference Lee and Carter1992), also referred to as M1, decomposes mortality as a static age-structure $( {\beta _x^{( 1 )}} )$ , a general level driven by a stochastic process in time $( {\kappa _t^{( 2 )}\;} )$ and an age-specific sensitivity $( {\beta _x^{( 2 )}} )$ to this general level as follows:

\begin{equation*}\log ( {\;m( {x,t} )\;} ) = \;\beta _x^{(1)} + \beta _x^{(2)}\kappa _t^{(2)}\end{equation*}

In order to account for possible cohort effects, while relying on a reasonable parametrisation, one can also consider the age–period–cohort model (M3) as proposed in Currie (Reference Currie, Durban and Eilers2004), which is a special case of Renshaw & Haberman (Reference Renshaw and Haberman2006), where the age, period and cohort components influence mortality independently in the following form:

\begin{equation*}\log ( {m( {x,t} )}) = \beta _x^{(1)} + \kappa _t^{(2)} + \gamma _{t - x}^{(3)}\;\end{equation*}

with ${\gamma _c}\;$ the additional cohort-related factor which allows to adjust mortality rates for a “generation” (or diagonal) which originates from year $c = t - x{\rm{\;}}$ .

As an alternative, the model by Plat (Reference Plat2011) combines the Cairns-Blake-Dowd model from Cairns et al. (Reference Cairns, Blake and Dowd2006), with some features of the Lee–Carter model to produce a model that is suitable for full age ranges and captures year-of-birth effects, here simplified into

\begin{equation*}\log ( {m( {x,t} )}) = \;\beta _x^{(1)} + \;\kappa _t^{(1)} + \;\kappa _t^{(2)}( {x - \overline x } ) + \;\gamma _{ct - x}^{(1)}\end{equation*}

where $\overline x $ denotes the mean age over the range of ages used in the calibration. This formulation can be seen as an extension of the age–period–cohort (M3) model where the joint age and period effect

\begin{equation*}\kappa _t^{(2)}( {x - \overline x } )\end{equation*}

have been added.

To select the most adequate modelling to one set of historical mortality data, each model is calibrated independently in a first phase. Calibration results are compared in a second phase, putting emphasis on a set of necessary or wishful properties, for example

1. • quality of fit (BIC criterion) of historical mortality tables;

2. • randomness of standardised residuals;

3. • stability of model parameters to the age band and historical length of mortality data used for calibration;

4. • back test of projected mortality improvements against historical improvements.

For further discussion regarding interesting properties and tests for model selection, the reader is referred to, for example, Cairns et al. (Reference Cairns, Blake, Dowd, Coughlan, Epstein, Ong and Balevich2009). Calibration results will be discussed in the next section.

Let us recall that the data used to specify the crude death rates $m( {x,t} )$ by single year and one-year age bands were downloaded from the HMD on 1 September 2015 for France (civil population), Italy (civil population) and West Germany (total population). As for calibration purposes, the datasets are restricted to the most recent decades of historic, retaining an age band characteristic for a standard insured population. Mortality at higher ages than the maximum of the age band is commonly assessed based on a table closure technique. References and discussions on table closure methodologies can be found in Quashie and Denuit (Reference Quashie and Denuit2005).

3.2 Model fitting

For the purpose of illustration, we restrict our graphical analysis to France female population only, as results are very similar for Italy and West Germany for both genders. Any specific outcome for Italy or Germany will be mentioned. We are considering the population aged from 30 to 95 and years from 1984 to 2009 (Italy), 2011 (Germany) or 2013 (France), which corresponds to HMD availability at the time of the study.

3.2.3 Plat model

Fitting results for the Plat model are depicted in Figure 15, for which similar conclusions as for the M3 model hold for the common parameters, the main changes being observed for the cohort component. As for the additional $\kappa _t^{(2)}$ parameter, this shows a relative stability from crude to adjusted data.

Figure 13. Lee–Carter model (M1) components fitting.

3.3 Impact on future life expectancy

In this section, we propose to highlight potential changes on life expectancy through two complementary approaches. On the one hand, projected period mortality tables are compared with and without exposure retreatment on the basis of residual period life expectancy at age 30 and truncated at age 95. On the other hand, cohort mortality tables are compared with and without exposure retreatment for all active generations being of age 30 to 95 years in 2015. The figures depicted in this section present the outcome of the models stochastic projection and differences between life expectancies for the median (baseline), 0.5th percentile (mortality shock) and 99.5th percentile (longevity shock) scenarios.

3.3.1 M1 model

As it was expected based on the reduced impact on the fit of M1 model component, life expectancy analysis shows very limited change between both sets of population’s exposure data, as depicted in Figure 16. For both genders and for each country in our study, period-based life expectancy between age 30 and 95 as well as live generation life expectancy does not vary more than half a month.

Figure 14. Age–period–cohort model (M3) components fitting.

Figure 15. Plat model components fitting.

Figure 16. Period and cohort life expectancy as of 2015 for French females, M1 model.

Let us recall that the LEs for non-observed cohorts are obtained by using the projections of each stochastic mortality models; in particular, models including a cohort component rely on a time series forecast of this cohort component.

3.4 Model selection process

As far as the model selection process is concerned, in all cases of our study, the improved BIC criterion as depicted in Figure 18 shows that the retreated exposure allows the Lee–Carter (M1) model to better capture mortality dynamics embedded in HMD table. BIC criterion does not show a significant change for the M3 model nor the Plat model. This seems to be explained by the fact that the cohort parameter captures the irregularities of the original mortality tables, leading to relatively unchanged residuals between the originals and the adjusted mortality tables.

Figure 17. Period and cohort life expectancy as of 2015 for French females, M3 model.

The selected model based on a BIC criterion is still the (simplified) Plat model in each configuration. Finally, for several populations as for Italian female table, retreated exposures allow for M1 to achieve a higher BIC level than the M3 model.

As highlighted in the former section, the retreatment of the exposure reduces fluctuations of the component related to the cohort effect. Therefore, models that do not take “cohort effect” into account as the Lee–Carter model improves their efficiency to fit historical mortality data; models that capture “cohort effect” also benefit from an improved stability (less fluctuation) in the cohort parameter estimate.

3.5 Solvency capital requirement

The trend component of the longevity SCR (called “SCR” in the following paragraphs) is here given as a change of future mortality improvements. The model described previously gives a full distribution of mortality improvement rate by gender, age and year. Note that the approach taken here does not consider a run-off distribution in order to remain compliant with the Solvency 2 framework, nor corresponds to taking the percentile each year, which woud be indeed too conservative. Also the methodology is based on a key indicatorFootnote ³ that allows to select the 0.5th percentile of the distribution of longevity improvements, denoted $IR_{x,{t_0} \to t}^{SCR}$ . To obtain the final mortality table for SCR calculations, the improvements of the best estimate assumptions

\begin{equation*}IR_{x,{t_0} \to t}^{BE} = \frac{{q_{x,t}^{BE} - q_{x,{t_0}}^{BE}}}{{q_{x,{t_0}}^{BE}}},\;for\;t > {t_0}\end{equation*}

are replaced by the ones obtained from the model mentioned above for all years from the valuation year t ₀ onwards. The initial mortality rates of the trend SCR component for year ${t_0}$ do not change from the best estimate mortality rates for this same year. For future years $t > {t_0}$ , the mortality rates are defined as follows for the best estimate ( $q_{x,t}^{BE}$ ) and the SCR ( $q_{x,t}^{SCR}$ ), respectively:

\begin{equation*}q_{x,t}^{BE} = q_{x,{t_0}}^{BE} \cdot \left( {1 + IR_{x,{t_0} \to t}^{BE}} \right)\end{equation*}

\begin{equation*}q_{x,t}^{SCR} = q_{x,{t_0}}^{BE} \cdot \left( {1 + IR_{x,{t_0} \to t}^{SCR}} \right)\end{equation*}

We depict in Figure 19 the impact on mortality rate scenarios of the change of data source (HMD data with HFD adjustment). As the improvements distribution is changed with the data adjustment, we observe in Figure 19 a slight impact on mortality rates due to lower cohort effects, as stated previously during the calibration of mortality models. Note that both the baseline scenario (“Base”) and the SCR scenario are changed in this figure.

Figure 18. BIC criteria for model calibration.

Figure 19. Baseline and longevity trend SCR future mortality for French females born in 1955.

As stated previously, there is less volatility in mortality assumptions when using the HFD adjustment. The level of mortality rates is, however, not impacted. The best estimate used are already without this excess of volatility; therefore, in the following analysis, we will only focus on the impact of SCR from changing the HMD data with HFD-adjusted data while keeping the same best estimate assumptions.

This decrease of captured volatility in the SCR scenario can be seen for various different cohorts. This can be illustrated with the impact of longevity trend SCR on the evolution of the cohort life expectancy. The cohort life expectancy including future mortality improvements is defined as follows for $A \in \left\{ {BE,\,SCR} \right\}$ :

\begin{equation*}e_{x,t}^{{\rm{Cohort}}}\left( A \right) = \mathop \sum \limits_{k = 1}^\infty \mathop \prod \limits_{i = 0}^{k - 1} \left( {1 - q_{x + i,t + i}^A} \right)\end{equation*}

The impact on cohort life expectancy from replacing the best estimate improvements by the SCR improvements is defined as follows:

\begin{equation*}I{E_{x,t}} = \frac{{e_{x,t}^{{\rm{Cohort}}}\left( {SCR} \right) - e_{x,t}^{{\rm{Cohort}}}\left( {BE} \right)}}{{e_{x,t}^{{\rm{Cohort}}}\left( {BE} \right)}}\end{equation*}

The impact is illustrated in Figure 20 for French females in 2015 born between 1925 and 1975, when having on one hand the best estimate assumptions with the improvement remaining the same, and on the other hand mortality rates using longevity SCR improvements. For the latter, we perform the calculations with the data from HMD only, and with the HMD data adjusted with the HFD data.

Figure 20. Longevity SCR impact on cohort life expectancy in 2015 for French females born between 1925 and 1975.

Figure 21. Impact on longevity trend SCR.

With the change of data, we observe a lower volatility on the indicator $I{E_{x,t}}$ for some cohorts, notably females being born in the early 1940s during World War II, that is, being between 70 and 75 in 2015. There is a small difference for older people (those being born between the World Wars) and almost no difference for those being born after 1945.

In order to compute the final impact depicted in Figure 21, the portfolios used for the study are originated from the three countries France, West Germany and Italy, including various products. For France, the portfolio is made of a savings and retirement business made of individual and group products, with on the in-force an average age around 65/70 for the annuity phase. For Germany, these are individual investment and savings products, with an average age of policyholders around 47. For Italy, the products are investment and savings products (62%) and long-term care products (38%) exposed to longevity risk. The savings and retirement businesses for these three countries are a mix of general account, with various guaranteed rates, and unit-linked products for the accumulation phase. Long-term care are a pure general account products.

Using the full internal model methodology and the aggregation with other risks, with on the one hand the HMD data and on the other hand the HMD data adjusted with HFD, we observe that the adjustment leads to a small reduction of longevity SCR (Figure 21).

This negative impact is consistent with the results found previously on indicator $I{E_{x,t}}$ : the removal of non-necessary volatility on several cohorts leads to a decrease of the longevity SCR. This is even clearer for the French portfolio, when compared to the German and Italian portfolios, as the French portfolio contains more annuitants close to ages 70–75 at which the reduction of non-necessary volatility is the strongest.

Looking now at the total life SCR of each of these entities with the other risks and the diversification between these risks calculated with an internal model methodology, the impacts on total life SCR are negative but almost non-significant (Figure 22) even as longevity SCR roughly represents 20% life SCR pre-diversification.

Figure 22. Impact on life SCR.

Even if the quantitative impacts are very moderate, the treatment of HMD data with the HFD has a positive effect on the data quality and therefore on the longevity risk assessment due to a lower volatility in future mortality assumptions.

3.6 Stability of the longevity risk assessment

The stability from a year to another of the longevity risk assessment can be roughly estimated by looking at the evolution of the impact of longevity trend SCR mortality improvements on the cohort life expectancy of the portfolio, that is, the difference for a policyholder at age $x$ in year $t$ :

\begin{equation*}e_{x,t}^{{\rm{Cohort}}}\left( {{\rm{SCR}}} \right) - e_{x,t}^{{\rm{Cohort}}}\left( {{\rm{BE}}} \right)\end{equation*}

and its evolution between t and t + 1. This indicator is calculated on the whole portfolio mix. There again, the best estimate does not change but SCR mortality improvements change as we consider the improvements calibrated with HMD data, and also the improvements calibrated with HMD retreated with HFD data.

As shown in Figure 23, the retreatment brings a more stable longevity risk assessment between 2015 and 2016 for France and Italy but not for Germany, notably due to women. Overall, the volatility of longevity SCR shock over time decreases.

Figure 23. Evolution of cohort life expectancy gap between baseline and longevity SCR trend between 2015 and 2016.