The Inflation Risk Model
3.4 The Results
3.4.5 Application of CEC and REW Measures
65 65, 65, 0, 64;k 65, 65, 0, 64;k
V W Y I ≈E D W Y I . This is the result that we use in Section 3.3.7.
Once we have a random variable D W
(
65,Y65, 0,I64;k)
in a numeric form, we can use it to calculate the approximate values of VaRα and CVaRα which will give us an idea of the pensioner’s left tail risk of discounted utility from future consumption and bequest. We define this measure more precisely in Section 3.4.7 and investigate its use.3.4.5 Application of CEC and REW Measures
In this section, we investigate the relations amongst the results from CEC and REW measures for different cases and for different parameter setup. The parameters which we change here are the RRA coefficient γ and a bequest motive coefficient bt. In order to focus our investigation on the differences in the results in different cases, we assume in this section that inflation is constant and equal to 4%. Further investigation of the effects of the stochastic inflation will be presented in Section 3.4.6.
Before presenting the results using CEC and REW measures, we show two examples of mean consumption and mean wealth paths. In Figures 3.12 and 3.13, we show mean pension wealth and mean consumption development during the retirement for the pensioner who optimally consumes, allocates assets and annuitises in Cases 3.1–3.6. The mean is calculated from the sample of 2,000 simulated pension wealth and consumption paths. We assume γ = − and 9 bt = , pension wealth at age 65 is 0 200,000, inflation is constant and equal to 4%, and other parameters are as stated in Section 3.4.1.
Figure 3.12 Mean pension wealth development in the retirement period in Cases 3.1–
3.6, for γ = − and 9 bt = 0
Figure 3.13 Mean optimal consumption development in the retirement in Cases 3.1–
3.6, for γ = − and 9 bt = 0
Figures 3.12 and 3.13 give an idea of very different mean wealth and mean consumption paths in Cases 3.1–3.6. Then the CEC and REW are measures which summarise into a single number the complexity of these future developments. We can see that both mean pension wealth and mean consumption paths are very different from case to case.
Pension wealth in Cases 3.1 to 3.6, gamma=-9, b=0
0 50.000 100.000 150.000 200.000
65 70 75 80 85 90 95 Age
Pension wealth
Case 3.1 Case 3.2 Case 3.3 Case 3.4
Case 3.5 Case 3.6
Consumption in Cases 3.1 to 3.6, gamma=-9, b=0
20.000 25.000 30.000 35.000 40.000
65 70 75 80 85 90 95 Age
Pension wealth
Case 3.1 Case 3.2 Case 3.3 Case 3.4
Case 3.5 Case 3.6
In Case 3.2, where optimal nominal annuitisation is allowed at age 65 only and no real annuities are allowed, it is optimal to annuitise almost all pension wealth (more than 97% of the available pension wealth). However, some of the pension wealth is saved again afterwards, which means that the pensioner does not spend all his income from social security and nominal annuities during retirement, but saves one part in the early pension years and then consumes it afterwards. The pensioner in this example uses his only opportunity to purchase annuity, or in the other words he uses the only opportunity to benefit from the survival credits. Apart from the survival credits the pensioner protects himself from the decreasing income in real terms during retirement.. As he has access to annuities at age 65 only, he converts almost all his pension wealth into nominal annuities at that age. This is interesting result because the pensioner optimally purchases more annuities than he needs for the consumption, and actually uses annuity as a risk free investment for saving and not only as an instrument for providing income in retirement. Also, income from nominal annuity is beneficial as risk free investment in early years of retirement because income from nominal annuities is increased by inflation rate in the early years compared to the risk free investment. We emphasise here that increase in pension wealth is not in a contradiction with the assumption, stated in Section 3.1.2, that the pensioner never annuitises any part of his income. The pension wealth can increase if it is optimal for the pensioner not to consume all his income in earlier years, but annuities are bought from the available pension wealth at the beginning of the year only. An increase in the pension wealth can also happen if the return on investment is significantly better than expected. However, in all our examples where purchasing annuities is allowed wherever during retirement, the mean consumption will always be higher than the mean income. Increase in the mean pension wealth happens only in the cases where no annuities are allowed after retirement such as in the Case 3.2 in Figure 3.13.
The mean wealth and optimal consumption in Cases 3.4, 3.5 and 3.6 are almost the same. It means that whichever option the pensioner chooses amongst these three, he would have very similar mean pension wealth and mean optimal consumption paths.
We also see that the Case 3.1 is the worst one in terms of consumption. It provides a lower consumption than any other case at all ages.
Figures 3.14 and 3.15 show the same results as Figures 3.12 and 3.13 but for γ = − 1 and bt = . 1
Figure 3.14. Mean pension wealth development in the retirement in Cases 3.1–3.6, for γ = − and 1 bt = 1
Figure 3.15 Mean optimal consumption development in the retirement in Cases 3.1–
3.6, for γ = − and 1 bt = 1
Comparing Figures 3.12 and 3.13 and Figures 3.14 and 3.15, we observe less differences of the mean pension wealth and mean optimal consumption paths in the latter two. Regarding mean pension wealth in Figure 3.14, we observe two groups of the patterns of mean pension wealth developments. In the first group are the mean pension wealth paths for Cases 3.4, 3.5 and 3.6, and in the second one are the mean
Consumption in Cases 3.1 to 3.6, gamma=-1, b=1
20.000 25.000 30.000 35.000 40.000
65 70 75 80 85 90 95 Age
Pension wealth
Case 3.1 Case 3.2 Case 3.3 Case 3.4
Case 3.5 Case 3.6
Pension wealth in Cases 3.1 to 3.6, gamma=-1, b=1
0 50.000 100.000 150.000 200.000
65 70 75 80 85 90 95 Age
Pension wealth
Case 3.1 Case 3.2 Case 3.3 Case 3.4
Case 3.5 Case 3.6
pension wealth paths for Cases 3.1, 3.2 and 3.3. We observe the similar patterns in Figure 3.15, apart from mean optimal consumption path in the later years of retirement when income decreases due to the influence of the inflation.
Table 3.7 shows the CEC measure for different bequest and RRA coefficients. All CEC values are calculated at age 65.
In Table 3.8, we compare the results for CEC more directly, by considering the change relative to the no annuity case, Case 3.1. Thus, Table 3.8 shows the percentage changes calculated using the formula
different cases and different pensioner’s preferences towards risk and bequest. Assumed interest rate during the year prior to retirement is 2.00%. Initial pension wealth is 200,000 money units.
Bequest and for the values of CEC measure in amounts given in Table 3.7.
Table 3.9 shows the REW measures for one set of parameters such that the pension wealth values in a particular row give the pensioner the same expected discounted utility derived from future consumption. Benchmark wealth is in Case 3.1 and it is 200,000. Again, all the calculations assume that the pensioner’s age is 65.
Similarly to the case for CEC measure, we can compare the REW measure with negative sign in order to get positive percentages.
Bequest and
Case shown in the column to obtain the same utility as 200,000 in Case 3.1. for the values in the Cases given in Table 3.9.
In Tables 3.7–3.10 we clearly see the importance of using annuities. Each case with annuities is more favourable in terms of both the CEC and REW measure.
If we compare Tables 3.8 and 3.10, we find that the ratio of the values in the cells in Table 3.10 and the value in the relevant cells in Table 3.8 is between 2.5 and 2.8. So, we find that for a given pensioner’s preferences, both CEC and REW measures give similar results in terms of ratios between any two values in a single row in Table 3.8, and the ratio between the values in the relevant cells in Table 3.10. In percentage terms, the values obtained using REW measure are higher than the relevant values obtained using CEC measure. The results of CEC and REW measures in percentages give us relative gains and losses for the pensioner due to the introduction of a certain class of annuities. However as we noted earlier, CEC measure includes the utility from consumption and does not include the utility from a bequest. CEC measure does not give us the appropriate result in the cases with a bequest motive.
Thus, one should observe Tables 3.9 and 3.10 when investigating rows 4–6.
Depending on the pensioner’s preferences, the importance of access to annuities varies significantly. The biggest difference is between rows 3 and 4. We can say that these two parameter combinations are the two most extreme investigated as all of the other results are somewhere between these results. It is interesting to see from Figures 3.13 and 3.15 that all of the mean optimal consumption paths for γ = − and 9 bt = 0 apart from Case 1 are almost the same, and for γ = − and 1 bt = almost all are the 1 same up to about age 80. However, we see in Tables 3.7–3.10 that the results lead to much larger differences in terms of both CEC and REW measures for γ = − and 9
t 0
b = than for γ = − and 1 bt = . For 1 γ = − and 9 bt = , it is optimal to annuitise 0 more than 97% of pension wealth in Case 3.2, about 84% in Case 3.3, and in Cases 3.4, 3.5 and 3.6 mean optimal annuitisation is about 65% at age 65 and then about 7%–15% afterwards. For γ = − and 1 bt = , it is optimal not to annuitise at all in Case 1 3.2, to annuitise about 4% in Case 3.3, and in Cases 3.4, 3.5 and 3.6 it is optimal to start annuitisation at age 70 and to annuitise mostly 5% afterwards. Due to the lower differences amongst cases regarding optimal annuitisation for γ = − and 1 bt = than 1 for γ = − and 9 bt = , we observe the lower differences in the mean optimal 0 consumption paths and consequently the lower differences in CEC and REW measures.
Case 3.4 is always more advantageous than Case 3.2 because there are weaker constraints on the annuitisation in Case 3.4 than in Case 3.2. Similarly Case 3.5 is always more advantageous than Case 3.3. In Case 3.6, the pensioner has no
constraints on the annuitisation strategy and obviously, this is the most favourable case for the pensioner. The results in Tables 3.8 and 3.10 confirm this as we observe that for the lower constraints on the annuitisation the higher are the values of CEC and REW measures.
From Tables 3.7–3.10, we can say that generally Cases 3.4–3.6 result in similar values of CEC and REW measures. From detailed results not presented here, we also observe that the pensioners with different preferences will behave similarly in these cases regarding mean optimal asset allocation. Also, if we observe the sum of the mean optimal nominal and real annuitisation in Case 3.6, the mean optimal nominal annuitisation in Case 3.4, and the mean optimal real annuitisation in Case 3.5, then we find a similar optimal annuitisation strategy.
In terms of both CEC and REW measures, Case 3.6 and Case 3.5 are the best cases for the pensioner. For γ = − and 1 bt = , the pensioner purchases real annuities only 1 and is indifferent between Cases 3.5 and 3.6. For the other combinations of the parameters γ and bt, the losses of Case 3.5 compared to Case 3.6 are almost negligible. In almost all of our examples, the pensioner in Case 3.6 optimally purchases significantly more real annuities than the nominal ones and real annuities are optimally bought earlier in the retirement than the nominal ones. These are the reason that the differences between Cases 3.5 and 3.6 are so small.
Case 3.4 is always inferior compared to Cases 3.5. We find the largest differences for γ = − and 9 bt = where the losses are about 0.55% according to the CEC measure 0 and 1.03% according to the REW measure. With the introduction of the bequest motive the losses decrease. Mean optimal consumption in Cases 3.4 and 3.5 seems to be very similar in the early years of retirement. However, we observe that mean optimal consumption in Case 3.4 decreases below the values of mean optimal consumption in Case 3.5 in the later years of the retirement due to the effects of inflation on the income from nominal annuities. The more income is received in nominal terms the stronger is the effect of the erosion of the income in real terms, and consequently optimal consumption decreases in the later years of retirement due to this effect. Thus, if the pensioner in Case 3.4 optimally purchases more nominal annuities and if he purchases them earlier in retirement then his income will decrease more in real terms, and the difference between Case 3.4 and Case 3.5 will be larger in terms of CEC and REW measures.
As we have already noted Cases 3.3 is inferior compared to Case 3.5 due to the stronger constraints in Case 3.3. Optimal asset allocation and annuitisation strategies differ significantly. The common conclusions for any pensioner’s preferences investigated are the following. The pensioner purchases more annuities in Case 3.3 than in Case 3.5 because he uses the only opportunity to purchase annuities at age 65.
As more annuitisation is done at age 65, the pensioner has a lower pension wealth available for investment in retirement in Case 3.3 than in Case 3.5. Consequently, because annuities are a kind of riskless investment, the pensioner invests the higher proportion of the available pension wealth in equities in Case 3.3 than in Case 3.5.
However, the optimal asset allocation and annuitisation strategy in Case 3.3 provides the pensioner with a lower gains than in Case 3.5 losses in terms of CEC and REW measures in all of the examples we investigated. The gains in Case 3.5 compared to Case 3.3 are higher if the pensioner has no bequest motive and if the pensioner is less risk averse. For example, the pensioner in Case 3.3 and with preferences γ = − and 1
t 0
b = optimally annuitises about 31% of his pension wealth at age 65, and in Case 3.5 he optimally starts annuitisation at age 69 and optimally annuitises up to 40% of his available pension wealth in a single year during the rest of the retirement. The difference between Case 3.3 and 3.5 for this pensioner in terms of CEC measure is 2.6%. The pensioner in Case 3.3 and with preferences γ = − and 1 bt = optimally 1 annuities about 4% of his pension wealth at age 65, and in Case 3.5 he optimally starts annuitisation at age 70 and optimally annuitises up to 6% of available pension wealth in a single year during the rest of the retirement. The difference between Case 3.3 and 3.5 for this pensioner in terms of CEC measure is 1.19%. If the pensioner in Case 3.3 has the preferences γ = − and 9 bt = he optimally annuitises about 84% of his 0 pension wealth at age 65, and in Case 3.5 he optimally annuitises at age 65 about 65%
and much lower percentages of his available pension wealth afterwards. The difference between Case 3.3 and 3.5 are now 0.65%. We conclude that if annuitisation is less attractive for the pensioner, depending on his risk and bequest preferences, then the differences in terms of CEC and REW measures are larger.
In Cases 3.3 and 3.5 the pensioner is always better off than in Cases 3.2 and 3.4 respectively, and we can say that real annuities provide gains in terms of CEC and REW measures compared to nominal annuities. However, there is no simple conclusion if we compare Case 3.3 and Case 3.4. In the third row in Tables 3.8 and 3.10, where the pensioner has preferences γ = − and 9 bt = , Case 3.3 is more 0 preferable than Case 3.4. It means that the availability of optimal real annuities at age 65 only is more favourable for the pensioner than optimal nominal annuities at any age for this choice of γ and bt. However, for all other pensioner’s preferences, Case
3.4 is more favourable than Case 3.3. The pensioner with no bequest motive, for γ = − gains 0.38% in terms of REW measure in Case 3.3 compared to Case 3.4, for 9 γ = − he loses 0.62%, and for 4 γ = − he loses 2.6% in Case 3.3 compared to Case 1 3.4. If the pensioner has the bequest motive, he is always better off in Case 3.4 than in Case 3.3 and the range of the gains in terms of REW measure is from 0.37% for γ = − to 1.11% for 9 γ = − . Thus, the more risk averse the pensioner is the lower are 1 the gains in Case 3.4 compared to Case 3.3 in terms of REW measure, and if no bequest motive is present, the more risk averse pensioner experiences even lower gains in Case 3.4 compared to Case 3.3.
When comparing Cases 3.2 and 3.3, we see in Tables 3.8 and 3.10 that Case 3.3 is always more favourable than Case 3.2 in terms of CEC and REW measures. The less risk averse pensioner optimally converts a larger part of the pension wealth into annuities in Case 3.3 than in Case 3.2, while the more risk averse pensioner optimally converts smaller part of the pension wealth into annuities in Case 3.3 than in Case 3.2.
It the pensioner has no bequest motive and if we observe the values of RRA coefficient in the range from –1 to –9, then the optimal nominal annuitisation in Case 3.2 ranges from 9% to 97%, while in Case 3.3 the range of optimal real annuitisation is from 31% to 84%. It the pensioner has the bequest motive, then optimal nominal annuitisation in Case 3.2 ranges from 0% to 80 %, while in Case 3.3 the range of optimal real annuitisation is from 4% to 69%. At the same time, we observe the larger differences in terms of CEC and REW measures for the less risk averse pensioner.
This again shows that the right choice of the optimal asset allocation and optimal annuitisation, together with optimal consumption is crucially important for attaining the best results in terms of CEC and REW measures. We conclude here that the gains in Case 3.3 compared to Case 3.2 decreases with a decrease of the pensioner’s risk aversion and decreases with the introduction of the bequest motive.
Case 3.1 is inferior to any other case, apart for the value of parameters γ = − and 1
t 1
b = . The availability of any kind of annuity investigated in this thesis is always beneficial to the pensioner. For γ = − and 9 bt = , the pensioner gains 23.13% in 0 terms of REW measures in Case 3.6 compared to Case 3.1. For γ = − and 1 bt = , he 0 gains 3.71%, for γ = − and 1 bt = , he gains 1.25%. The more risk averse the 1 pensioner is, the larger are the gains in terms of CEC and REW measures in Case 3.6 compared to Case 3.1. The pensioner with the bequest motive obtains the lower gains compared to the pensioner with the same RRA coefficient and with no bequest motive.