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4.2 Results on the empirical data set

4.2.2 Distribution of technical provision

As we concluded that the convergence diagnostics are sufficient, and analyzed the posterior distributions, we now assess the effect of parameter uncertainty on the uncertainty in the technical provision. We have observed from the posteriors distributions of Θpfx corresponding to the Gamma prior that these are very unrealistic, perhaps due to the small size of the data set. For instance, the factors for individuals aged 58 are very low, and the factors for individuals aged 59 are very high. However, the pattern of the distributions of the age-specific factors is smooth when these are based on the model that assumes the lognormal prior. Therefore, we only consider the model that uses the lognormal prior when we obtain a distribution for the technical provision.

The distributions of the technical provisions are compared to the point-estimate of the technical provision that follows from the methodology of Willis Towers Watson. The technical provisions are approached as follows.

Per approach, we use 1,000 samples of the joint posterior distribution to transform 100 scenarios of future paths of κt into mortality rates. Hence, in total, we have simulated 100,000 scenarios to obtain densities for the technical provision. For all scenarios, we obtain mortality rates for ages 45 to 90 up to 120 years into the future. Subsequently, we transform these mortality rates into a table of death probabilities. As also the death probabilities for other ages are required, this table is merged with the mortality table that follows from the methodology of Willis Towers Watson. Recall that they multiply the country-wide mortality table as published

by AG by point-estimates of age-specific factors that are defined in a similar way as in this thesis. The ages in the table of Willis Towers Watson range from age 0 to age 120. The forecasts of the future death probabilities in the table by Willis Towers Watson for ages that are not considered in this thesis, are included in the table for calculating the technical provision. In this way, we ensure that the differences between the provisions solely depend on the mortality rates for ages that are investigated in this paper. Also, we assume that an individual aged 121 dies in that year with probability one. From this table, we calculate the present value of the future pension payments of all individuals. The present value of a pension payment is equal to the accrued pension rights multiplied by the probability that this individual is still alive, multiplied by a discount factor. This discount factor is based on the EIOPA risk-free interest rate term structure. The discount factors are shown in Figure 31. By summing over all present values of future pension payments, we calculate the technical provision.

In this way, we obtain the technical provision for all scenarios.

Figure 31

The discount factors obtained from the EIPOA risk-free interest rate term structure.

In this analysis, we only take males into account that include in the portfolio of the pension fund under consideration. Also, due to privacy reasons, we do not show the value of the technical provision. Instead, all technical provisions are divided by the value obtained when the methodology of Willis Towers Watson is applied. The distribution of the technical provision is estimated by means of four approaches:

• The first approach is the technical provision that follows from the model of Willis Towers Watson. The provisions that result from the other approaches will be divided by this value. Thus, this value serves as a benchmark.

• The second approach bases the technical provision on the mortality rates that follow from the Bayesian

• The third approach uses the mortality rates that follow from the Bayesian approach with a fixed βx to calculate the technical provision. In this approach βxis equal to the estimate from the Lee-Carter model fitted on the country-wide population. The uncertainties that follow from the other parameters, as well as κt, drive the volatility in the technical provision.

• In the fourth approach, we use the frequentist estimates of the parameters with fixed βx estimated from the Dutch population. Hence, we do not take parameter uncertainty into account. The resulting volatility is caused by the projections of the time series κt.19

Figure 32

The distribution of the technical provision based on all four approaches relative to the technical provision that follows from the approach of Willis Towers Watson. Thus, the vertical dashed line at one corresponds to the approach of Willis Towers Watson. The green plane depicts the distribution of the technical provision based on the Bayesian model with varying βx. The orange plane corresponds to the Bayesian model with a fixed βx. In these two models, we also take parameter uncertainty into account. The blue plane corresponds to the frequentist approach to the model that uses a fixed βx. This approach only takes uncertainty that follows from the time series κtinto account.

The results are presented inFigure 32. Surprisingly, all distributions of the provision are below the provision that follows from the methodology of Willis Towers Watson. Note that the approach of Willis Towers Watson is based on the characteristics of the portfolio in terms of income and working sector occupancy, and the model in this paper is based on historical data. As the privacy of the pension fund under consideration must be guaranteed, we are not allowed to speculate based on the characteristics of the portfolio of the pension fund.

The difference might simply be caused by the fact that individuals in the portfolio have experienced higher mortality rates in their past, relative to other subpopulations with the same level of income and working sector occupancy. However, as the approaches are very different, it is difficult to analyze the sources of the deviation.

We observe that the distribution of the technical provision that is based on the Bayesian model that uses

19As the volatility of βx is very high in this model, the frequentist estimate of this value is very arbitrary. Since the technical provision depends a lot on this value, the results might provide deceitful insights, this approach is left out for clarity.

a varying βx is often much lower than the distributions that follow from the other approaches. This directly follows from the volatility in the samples of βx. As there is a lot of uncertainty in the values of βxfor almost all x, in all samples there is always at least one βx that is below one. For these ages, the death probabilities converge rapidly towards one. This affects the value of the technical provision severely as this leads to very low probabilities that individuals survive up to certain ages, which in turn leads to very low present values of future pension payments and low technical provisions. Also, the distribution is very wide. Mostly when the future pattern of κt is slightly decreasing, the future death probabilities for ages with low values of βx do not converge to one very fast, which leads to higher values of the technical provision. We conclude that without fixing βx, the confidence intervals are too wide when the data set is small. This approach leads to unrealistic wide intervals for the technical provisions.

The distributions for the technical provision that are based on the model with the fixed βx is more con-centrated and closer to the approach of Willis Towers Watson. However, we emphasize that the volatility is underestimated as we disregard the volatility in βx. Fixing the value of βx leads to more satisfactory distri-butions for the technical provision. Although, it is still surprising that the distribution is largely below the provision calculated from the approach of Willis Towers Watson. We observe that the distribution is centred lower when we base the provision on the frequentist approach. This is caused by the unrealistic values of Θpfx. For some ages, the value of Θpfx is very high. As substantiated before, this affects the technical provision severely through the low probabilities for individuals to reach certain ages. Note that very low values of Θpfx have less effect on the provision as the death probabilities are in fact low for all considered ages. Thus, reducing the death probabilities to even lower values hardly increases the probability of reaching certain ages. This explains why the effects of high and low values of Θpfx do not cancel each other out. Therefore, the calculated technical provision is lower when it is based on the frequentist estimates of Θpfx. The widths of the distributions are in line with earlier results in this paper. Firstly, the high volatility of βx leads to high volatility in future death counts, and therefore also to unrealistic uncertainty in the technical provisions. Secondly, the effect of parameter uncertainty, besides that of βx, on future death counts is very small. When we calculate the distribution of the technical provision, including this uncertainty only leads to a slide widening of the confidence interval.

Besides disregarding the uncertainty in the technical provision that follows from the uncertainty in βx, we also disregard the uncertainty that follows from individual mortality risk. InSubsubsection 4.1.3 we modeled this uncertainty by drawing death counts from a binomial distribution. We found that for small pension funds this source of uncertainty is relevant, mostly on age group level. For larger pension funds this is no longer relevant. As we consider a pension fund that is comparable to the smallest simulated data set, we believe that considering this certainty increases the volatility in the technical provision. However, the effect of individual mortality risk on the distribution of technical provision, should be investigated to a further extent in order to draw conclusions.

5 Conclusion

The technical provision of a pension fund is an important component for assessing the financial health of a pension fund. This is calculated as the present value of the future pension payments to the individuals in the portfolio. The value depends severely on the mortality assumptions, which have been studied widely. For years, the technical provision is calculated as a point-estimate, and the underlying uncertainty has remained unconsidered.

At Willis Towers Watson, the fund-specific mortality is estimated on the basis of the characteristics of the portfolio. However, as of the Solvency II agreement, pension funds are obliged to valuate their liabilities in a stochastic fashion. In order to find stochastic mortality rates, historical data is needed. Although, since pension funds are often small such that there is little data available, it is difficult to obtain credible confidence intervals. Multiple population mortality modelling often includes age-specific factors that can be assumed to be constant over time, and forms a possible solution. This is a model that could approach the mortality of a small fund by simultaneously estimating the mortality of a large population. Further, many studies have shown that there is an inverse link between income and mortality rates. However, due to privacy regulations, there is no income-related information available for the country-wide population.

In this paper, we propose a Bayesian Lee-Carter based multiple population mortality model that incorporates income. The model is designed to be applied to aggregated data of multiple pension funds in which information on income is included. The numbers of historical years are allowed to be inconsistent for the pension funds.

Also, as there are often at most ten years of historical data available, the general time trend is obtained from estimating the Lee-Carter model on the Dutch population and used as input for the model. We investigated two types Bayesian models. The first model assumes the age-specific factors to be independent of each other, and the second model assumes dependence among ages and uses a lognormal smoothing prior.

The model is first applied to various simulated data sets of different sizes, both in terms of the number of historical years and exposure. After this, the model is applied to an empirical data set that is obtained from thirteen pension funds. From applying the model to the largest simulated data set, we found that the resulting posterior distributions were centred closely around the parameters that are used for the data generating process.

This confirms the correctness of the model. Although, when the model is applied to the smallest data set, which in terms of size corresponds to the available data set, the volatility of the posterior distribution of the age-specific parameter that relates to the extent to which each age group is exposed to the general time trend in mortality was very high. Interestingly, increasing the data set in terms of historical years leads to more credible estimates for this age-specific time trend parameter, whereas it hardly affects the confidence intervals of the other parameters. We also observed that when we use the model that assumes independence among ages, mostly for the smaller data sets, the confidence intervals of the age-specific factors are very wide and noisy, and thus not interpretable. When the model that uses the smoothing prior is applied, the intervals are more concentrated and smoother. Even for the smallest data set, the factors that should follow from the data generating process are closely followed by these confidence intervals.

We also investigated three different sources of uncertainty that result in uncertainty in future death counts.

The first uncertainty follows from the projections of the future general time trend, which is input for our model, the second is the individual mortality risk, and the third is the volatility in the parameters, which is obtained from the Bayesian model in this paper. We have found that individual risk is negligible when the number of individuals is high due to the law of large numbers. However, for the small data sets, this effect overrules the effect of including parameter uncertainty. We found that the effect of the inclusion of parameter uncertainty is small. However, when the number of historical years is small, the time trend in mortality is very volatile due to the volatility in the age-specific time trend parameter. Therefore, when the time horizon of the predictions increases, the effect of parameter uncertainty leads to higher standard deviations in future deaths.

When we apply the model to the empirical data set, we observe similar estimation results as for the smallest simulated data set. As the confidence intervals for the age-specific parameter that relates to the time trend were unrealistic, we calibrated an adjusted model that uses the age-specific time trend parameter as input besides the general time trend, as well. The value of this parameter is also obtained from estimating the conventional Lee-Carter model on the Dutch population. From the results that follow from both approaches, we obtained the distribution of technical provision. We compared these with the distribution that follows from the frequentist estimate of the model to investigate the effect of parameter uncertainty on the distribution of the technical provision. The distribution that follows from the model with a variable age-specific time trend parameter is very wide and low compared to the other distributions. This is a direct consequence of the very high volatility in the future mortality rates for different ages. For the model with a fixed time trend, we found that the distribution of the technical provision is hardly affected by parameter uncertainty. However, the effect of uncertainty in the age-specific time trend parameter is disregarded in this analysis. For the available data set, it is impossible to obtain a reliable confidence interval for this parameter. Another source of uncertainty that is not taken into account in this analysis is individual mortality risk. Although we found that it is not relevant for large portfolios, taking this uncertainty into account perhaps leads to more volatility in the technical provision for the pension fund that we considered in this paper. Hence, the volatility that we found for the technical provision could be considered a lower bound for the true volatility.

Furthermore, the results regarding the effect of income on mortality rates were unexpected. We found that the mortality rates are the highest for individuals with the highest incomes. This is due to the high observed death rates in the two highest income classes. However, we attribute this to the estimation of the unobserved incomes of the retirees. In order to obtain valuable insights into the dependence between mortality rates, more useful data is required.

We conclude that for the determination of the stochastic technical provision, taking uncertainty from pro-jecting the general time trend into account is of importance. The volatility that follows from the uncertainty in the parameters except the age-specific time trend parameter is less relevant. Further investigation is needed in order to analyze the effect of including the uncertainty in this parameter on the distribution of the technical provision. The data that was available for this paper consists of too few historical years and too low exposures

found to lead to unrealistic distributions. However, we found that when the amount of data increases, which could be realized by finding more pension funds that are willing to provide data, or when there are more his-torical years, which results from simply keeping track of the data, this model would serve as a valuable tool to obtain stochastic fund-specific mortality rates for small pension funds.

6 Discussion

In this section, some possible pitfalls of this research are discussed as well as some recommendations for further research. The results from the model calibration on simulated results were as expected. Namely, the posterior distributions have unrealistic wide intervals when the simulated data set is small, and when the size of the data set increases, the distributions are more centred and located around the parameters that are expected from the data generating process.

However, the results from applying the model to the empirical data set showed some unexpected results.

Whereas many studies have shown that mortality rates negatively depend on income, we found that the expected mortality rates are the highest for the two highest income classes. This should be attributed to the fact that the observed death rates in this income class were spuriously high. We believe that this follows from the method that we used to estimate the incomes, which relied on many assumptions. For instance, the available data consists of many separate files that we combined. As the exact compositions of the data sets are not of importance for Willis Towers Watson, the validity of the data is not assessed. As a result, we experienced many difficulties when we prepared the data. To illustrate the inaccuracy of the files, mortality data from one year of one of the pension funds consists of many duplicating individuals so that this year had to be disregarded in this research.

The most severe pitfall concerning the estimation of the incomes is that we had to make two assumptions that are very questionable. On the one hand, the unobserved incomes of the actives are estimated based on the progress of their accrued pension rights. Besides the fact that these are obtained by combining separate files with mortality rates which leads to inaccuracies, the amount of accrued rights could alternate for many reasons. For instance, when an individual postpones his pension payments, the accrued rights increase as he is expected to receive fewer pension payments. In the method, this unfairly leads to higher accrual percentages and thus higher incomes. On the other hand, the incomes of retirees are solely based on their constant amount of accrued rights. We assume that retirees with more accrued pension rights have higher incomes. However, the same accrued rights could follow from working for a long time with a low salary, and from working for a short time with a high salary. The results regarding the dependence between mortality and income would improve considerably when the incomes of all individuals are known. However, as this information is usually not available in the pension funds, more information such as the last earned salary or the date of the start of employment would lead to more accurate results.

Further, we found that the distribution of the technical provision was estimated to be below the point-estimate of the technical provision calculated using the methodology of Willis Towers Watson. At Willis Towers Watson, the fund-specific mortality is based on the characteristics of the portfolio of the pension fund in terms of income and occupied working sector. Hence, the difference could have been caused by the fact that in the past the death rates were higher than expected from the characteristics of the pension funds. However, it is also possible that this difference is due to the way the exposures are obtained in this paper. As the mortality data was based on snapshots, we only have information on which years an individual was in the portfolio. In that case, we assumed that the individual entered on July 1st. This method could lead to an underestimation