Replacement rate

In order to illustrate the difference in benefit payments of the different mortality gains allocation methods, this section elaborates on the different replacement rates per age for a pension fund where the mortality gains are distributed according to capital and the risk premium method, the fair mortality gains method is left out of account since the risk premium method is very similar to the fair mortality gains method and due to the computational complexity of the fair mortality gains method. In order to demonstrate the difference in replacement rate, a deterministic economic scenario is used to reduce variation induced by stochastic economic scenarios.

Warm-up period

The warm-up period of the simulation is determined as described in section 3.4.1. Since the variable evaluated is the replacement rate, this variable is visualized in order to determine the warm-up period.

Figure 4.3: Welch Graphical Procedure to determine model warm-up time, where the weights ware as explained in section 3.4.1.

Reading example:the observed variable in the graph is the average replacement rate, av- eraged over all retirees. Considering year 200, the blue line (w₌1) represents the average replacement rate averaged over year 199, 200 and 201 and the orange line (w₌5) the average over 195 to 205.

4.1. ALLOCATION OF MORTALITY GAINS

Figure 4.3 is a visualization of the Welch Graphical Procedure using ten simulations of a length of 1000 years, where there is no distinction between salary classes and adjusted mortality factors based on salary classes with a pension fund population of 10000 pension fund participants. This figure illustrates the need of such a warm-up period because in the first 100-200 years of the simulation, there is variation induced by starting the simulation model. After 250 years the weighted average of the observed variable seem to have converged to a stable replacement rate. So taking a warm-up period of longer than 250 years should reduce the variation induced by starting the simulation.

Comparison replacement rate per age

To visualize the replacement rate per age given a certain mortality gain allocation, a simulation was performed with a duration of 500 years using a warm-up period of 300 years, not distinguish- ing between salary classes and corresponding differentiations in death probability, keeping the economic scenario constant, with a pension fund pool size of 10000 pension fund participants, whilst varying in the method of allocating mortality gains, using the distribution proportional to capital method and the risk premium method.

Figure 4.4 demonstrates that using this method of allocating mortality gains, favors the younger retirees in terms of replacement rate, whilst older retirees receive a lower replacement rate, that converges to a benefit payment of 0. One can argue if it is a desirable situation that when retirees reach a certain age, they receive close to zero benefit payments. In figure 4.4, the black bars show the average plus and minus the standard deviation of the replacement rate for that age, in the simulation run there has only been one pension fund participant of age 111, which is why the standard deviation is missing in that instance.

When using the risk premium method to allocate the mortality gains, the older pension fund participants receive a greater benefit payment than at a lower age, figure 4.5 demonstrates this. This is caused by the fact that older pension fund participants benefit more from the mortality gains when the risk premium method is used for allocating mortality gains. Using the risk premium method, the benefit payments do not drastically decrease as pension fund participants get older. It would seem as if figure 4.5 implies that the majority of the benefit payments are given to the higher aged pension fund participants, however when looking at the age distribution of the pension fund participants, one can see that this is not the case as shown in figure 4.6. Figure 4.6 demonstrates that as pension fund participants age, the number of pension fund participants of a certain age decrease at an increasing rate, this is what one would expect since death probabilities increase as pension fund participants age.

Individual case

Information about the average replacement rate and its standard deviation per age group can give insights on the development of the received benefit payments as pension fund participants

Figure 4.4: Average replacement rate per age with 10000 pension fund participants allocating mortality gains based on capital.

Reading example:The graph shows the replacement rate per age averaged over a simula- tion of 500 years and its standard deviation. Considering age 80, the replacement rates of all pension fund participants of age 80 in every simulation year are registered, at the end of the simulation these values are used to compute the average and standard deviation of the replacement rate for age 80.

age. However it can also be interesting to see what these figures mean for an individual pension fund participant. This can be done by tracking the replacement rate of the individual pension fund participant from age of retirement till death. In this section the replacement rates of three example pension fund participants will be shown, these pension fund participants will join the pension fund at an age of 25, 45 and 65, this directly demonstrates the influence of the way of allocating mortality gains on these individual pension fund participants their replacement rates. Figure 4.7 demonstrates the difference in benefit payments for the three individual pension fund participants. It should be noted that for this simulation a deterministic economic scenario is used with a yearly return on equity and bonds of 6% and 2% respectively and that no inflation is taken into account, also the lifespan of the tracked individual pension fund participants was forcefully extended to an age of 110 years, to better demonstrate the development of the annuity payments over time. The last earned salary for all pension fund participants in this case is 54288. In this simulation the pension fund participant joining at the age of 25 benefits most because

4.1. ALLOCATION OF MORTALITY GAINS

Figure 4.5: Average replacement rate per age with 10000 pension fund participants allocating mortality gains based on the risk premium method.

Reading example:The graph shows the replacement rate per age averaged over a simula- tion of 500 years and its standard deviation. Considering age 80, the replacement rates of all pension fund participants of age 80 in every simulation year are registered, at the end of the simulation these values are used to compute the average and standard deviation of the replacement rate for age 80.

of the steady positive equity and bond returns. All cases show that using the mortality gains allocating method of distributing proportional to capital yields the highest benefit payments for pension fund participants in the years directly after retirement, after that the received benefit payment declines rapidly as older pension fund participants do not benefit more from mortality gains using this allocation method, resulting that when an extreme age of for example 110 is reached, the benefit payment is close to zero. Note that on an age over 90, the annuity payment is still a fraction of the initial benefit payment. When allocating mortality gains according to the risk premium method, the benefit payment is lowest around retirement age and increases as one ages. As stated earlier, using different economic scenarios might change the shape of the lines in the graph. One of the strong points of the risk premium method is that the annuity payments do not converge to zero over time, preventing a period in which a pension fund participant would receive no or or little income. Since the risk premium method does not converge to zero as pension fund participants age, does not favor young pension fund participants over old pension fund participants and is less computational complex than the fair mortality gains method which is

Figure 4.6: Age distribution with 10000 pension fund participants.

Reading example:this graph shows what fraction of the retired population has which age. Considering age 90, the graph shows that less than 2% of the population was of age 90, where the fractions are computed by counting the pension fund participants of year 90 over multiple years of simulation divided by total count of retired pension fund participants.

infeasible to implement using the Ethereum blockchain because of its computational complexity, the risk premium method is the preferred way of allocating mortality gains. From here on forward the risk premium method will be used for the subsequent analyses.