3. Numerical Results
3.1 Results for S1 (Buy-and-Hold Strategy)
For this strategy, we consider 10 allocation mixes: (0, 0, 1), (0, 0.25, 0.75), (0, 0.5, 0.5), (0.06, 0.63, 0.31), (0.25, 0.5, 0.25), (0.435, 0.38, 0.185), (0.62, 0.26, 0.12), (0.81, 0.13, 0.06), (1, 0, 0), and (1/3, 1/3, 1/3). The (0, 0, 1) mix represents the case when the
6 The simulation procedure is described in Appendix (B).
7 We describe how to derive the estimates of , , and r in Appendix (C). We follow the approach proposed by Scott (1996) to simulate short rates in the CIR model. The details appear in Glasserman (2004).
8 The values of G, L, and f are arbitrary in the standard scenario. We analyze how average guarantee costs and retirement benefits change with different G, L, and f subsequently.
participant always allocates contributions to portfolio C; (1, 0, 0) indicates investment only in portfolio A; and (1/3, 1/3, 1/3) means that the participant adopts the framing 1/N strategy. We derive the first nine mixes from the efficient portfolio frontier (Figure 2) and order them according to their expected annual rates of return.
The average guarantee costs and average retirement benefits for strategy S1 with a type-I guarantee appear in Figure 3; those for the type-II guarantee are in Figure 4. In addition, Tables 1 and 2 respectively list the type-I and type-II guarantee costs for S1 with different allocation proportion mixes. As Figure 3 and Table 1 show, in the standard scenario, the lowest average costs for the type-I guarantee emerge from the (0, 0, 1) mix. The high expected rate of return from this mix leads to high actual rates of return, so the guarantee provider rarely must pay out significant compensations because of this guarantee. The highest average costs for the type-I guarantee exist for the (1, 0, 0) mix, because portfolio A features the lowest expected rate of return, and the actual benefits are very low. The harm that portfolio A causes to the type-I guarantee provider is emphasized by the risk measures in Table 1. Allocating all contributions to portfolio A results in the highest 99% and 95% quantile risk measures of type-I guarantee costs.
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
0.00 0.05 0.10 0.15 0.20 0.25
Annualized Volatility
ExpectedAnnualRateofReturn
Figure 2. Efficient Frontier Generated by Portfolios A, B, and C
Notes: The efficient frontier is generated by assuming that the weight of each portfolio is not negative.
In contrast, Figure 4 and Table 2 reveal that the highest average and risk measures of type-II guarantee costs come from the (0, 0, 1) mix. Therefore, the portfolio with the highest expected rate of return and volatility appears to cause the highest type-II guarantee costs. We explain this finding on the basis of the payoff structure. From the multi-period binding property and equation (8), we know that high actual rates of return in some periods will magnify the guarantee obligation. Even though only one record date exists such that exp(R(t,t))
exp( R
t(i))
, the total guarantee costs will be very great if the otherR
t(i)’s are very high. Hence, portfolio C, with its high expected rates of return, usually leads to high guarantee costs. Such a magnifying effect does not exist in the type-I guarantee. Equation (6) indicates that low(0, 0, 1)
(0, 0.25, 0.75) (0, 0.5, 0.5)
(0.06, 0.63, 0.31)
(0.25, 0.5, 0.25)
(0.435, 0.38, 0.185) (0.62, 0.26, 0.12) (0.81, 0.13, 0.06) (1, 0, 0)
guaranteed rates in some periods diminish the effect of high guaranteed rates in other periods, and high actual rates of return reduce the shortfall made up for by the guarantee provider. Thus, the (0, 0, 1) mix does not produce high type-I guarantee costs.
Tables 3 and 4 report the statistics related to retirement benefits and income replacement ratios for S1. The (0, 0, 1) mix has the highest average retirement benefits and (1, 0, 0) has the lowest, regardless of the type of guarantee. The guarantee mechanism transfers the downside risk of poor investment performance to the guarantee provider, so investing all contributions in portfolio C takes advantage of the high expected rate of return and seldom involves downside risk. Portfolio A, with the lowest expected rate of return, counteracts wealth accumulation. The last rows in Tables 3 and 4 show that the framing 1/N strategy is not necessarily good for the participant, because it cannot create the highest retirement benefits.
0 5 10 15 20 25 30
(0, 0, 1) (0, 0.25, 0.75)
(0, 0.5, 0.5)
(0.06, 0.63, 0.31)
(0.25, 0.5, 0.25)
(0.435, 0.38, 0.185)
(0.62, 0.26, 0.12)
(0.81, 0.13, 0.06)
(1, 0, 0)
Allocation M ix (A, B, C) Type-IGuaranteeCostsor RetirementBenefits
Average Retirement Benefits Average Guarantee Costs
Figure 3. Average Retirement Benefits and Guarantee Costs under a Type-I Guarantee in S1 and the Standard Scenario
0 10 20 30 40 50 60 70 80
(0, 0, 1) (0, 0.25, 0.75)
(0, 0.5, 0.5)
(0.06, 0.63, 0.31)
(0.25, 0.5, 0.25)
(0.435, 0.38, 0.185)
(0.62, 0.26, 0.12)
(0.81, 0.13, 0.06)
(1, 0, 0)
Allocation M ix (A, B, C) Type-IIGuaranteeCostsor RetirementBenefits
Average Retirement Benefits Average Guarantee Costs
Figure 4. Average Retirement Benefits and Guarantee Costs
under a Type-II Guarantee in S1 and the Standard Scenario
Table 1.
Summary Statistics of Type-I Guarantee Costs in S1 and the Standard Scenario
Allocation Mix
(A, B, C) Average S.E. VaR(99%) VaR(95%) CTE(99%) CTE(95%)
(0, 0, 1) 1.106 0.043 17.604 4.128 50.084 16.349
(0, 0.25, 0.75) 1.165 0.043 17.971 4.131 50.892 16.570
(0, 0.5, 0.5) 1.223 0.043 18.693 4.470 52.105 17.269
(0.06, 0.63, 0.31) 1.330 0.044 20.251 5.130 53.660 18.291
(0.25, 0.5, 0.25) 1.541 0.044 21.448 5.959 55.013 19.336
(0.435, 0.38, 0.185) 1.749 0.045 22.522 6.843 56.411 20.430
(0.62, 0.26, 0.12) 1.956 0.045 23.783 7.759 57.847 21.563
(0.81, 0.13, 0.06) 2.167 0.046 25.032 8.656 59.317 22.733
(1, 0, 0) 2.379 0.047 26.460 9.582 60.845 23.929
(1/3, 1/3, 1/3) 1.608 0.044 21.175 6.098 54.808 19.341
Table 2.
Summary Statistics of Type-II Guarantee Costs in S1 and the Standard Scenario
Allocation Mix
(A, B, C) Average S.E. VaR(99%) VaR(95%) CTE(99%) CTE(95%)
(0, 0, 1) 49.198 0.219 220.085 124.101 351.732 195.763
(0, 0.25, 0.75) 39.222 0.175 172.582 97.204 280.732 154.576 (0, 0.5, 0.5) 29.245 0.132 126.767 70.274 211.536 113.945 (0.06, 0.63, 0.31) 21.300 0.101 92.973 49.983 159.914 83.136
(0.25, 0.5, 0.25) 17.758 0.089 79.106 41.981 139.545 71.083 (0.435, 0.38, 0.185) 14.046 0.077 65.541 33.607 118.465 58.500 (0.62, 0.26, 0.12) 10.334 0.066 51.816 25.115 98.052 46.237 (0.81, 0.13, 0.06) 6.791 0.056 39.882 17.460 79.575 35.190
(1, 0, 0) 3.248 0.048 27.840 10.789 62.785 25.321
(1/3, 1/3, 1/3) 20.580 0.100 92.543 49.588 158.451 82.724
Table 3.
Summary Statistics of Retirement Benefits and IRR under a Type-I Guarantee in S1 and the Standard Scenario
Retirement Benefits Income Replacement Ratio Allocation Mix
(A, B, C) Average S.E. VaR (10%) Average S.E.
(0, 0, 1) 24.734 0.118 7.407 0.684 0.004
(0, 0.25, 0.75) 21.509 0.093 8.146 0.596 0.003
(0, 0.5, 0.5) 18.284 0.070 8.511 0.509 0.003
(0.06, 0.63, 0.31) 15.575 0.056 8.307 0.436 0.002
(0.25, 0.5, 0.25) 13.984 0.053 7.781 0.393 0.002
(0.435, 0.38, 0.185) 12.351 0.050 7.221 0.350 0.002
(0.62, 0.26, 0.12) 10.718 0.048 6.621 0.306 0.002
(0.81, 0.13, 0.06) 9.127 0.047 5.946 0.263 0.002
(1, 0, 0) 7.537 0.047 4.964 0.221 0.002
(1/3, 1/3, 1/3) 14.701 0.058 7.576 0.413 0.002
Table 4.
Summary Statistics of Retirement Benefits and IRR under a Type-II Guarantee in S1 and the Standard Scenario
Retirement Benefits Income Replacement Ratio Allocation Mix
(A, B, C) Average S.E. VaR (10%) Average S.E.
(0, 0, 1) 72.826 0.288 25.377 2.050 0.010
(0, 0.25, 0.75) 59.566 0.224 23.183 1.679 0.008
(0, 0.5, 0.5) 46.306 0.162 20.706 1.307 0.006
(0.06, 0.63, 0.31) 35.545 0.118 17.894 1.005 0.005
(0.25, 0.5, 0.25) 30.201 0.101 15.542 0.855 0.004
(0.435, 0.38, 0.185) 24.648 0.085 13.161 0.700 0.004
(0.62, 0.26, 0.12) 19.095 0.070 10.746 0.545 0.003
(0.81, 0.13, 0.06) 13.751 0.057 8.294 0.395 0.003
(1, 0, 0) 8.407 0.048 5.444 0.245 0.002
(1/3, 1/3, 1/3) 33.673 0.118 15.862 0.953 0.005
Several implications emerge from Figures 3 and 4 and Tables 1-4. First, with the type-I guarantee, both the participant and the guarantee provider suffer from a conservative allocation strategy. Only investing in portfolio A, the (1, 0, 0) mix, creates low retirement benefits and high type-I guarantee costs. This strategy seems inadequate for the participant but may be adopted in the real world. Thaler and Johnson (1990) indicate that prior losses often reduce people’swillingness to take risks, so a participant who has experienced great losses by investing in volatile portfolios may tend to allocate contributions to safer portfolios and thus place a heavier financial burden on type-I guarantee providers. However, such a conservative allocation strategy entails the least type-II guarantee costs and does the least harm to the type-II guarantee provider. This finding clearly shows that whether an allocation strategy is good or bad for the guarantee provider depends on the guarantee structure.
Perhaps the (1, 0, 0) mix is too extreme. We now explain the possibility of the other proportion mixes in the context of a type-I guarantee. Shefrin (2000) points out that hope and fear affect how investors evaluate alternatives. The participant hopes for a comfortable retirement life but fears sudden downtrends that depreciate the accumulated benefits. If this participant invests all contributions in portfolio A, the low actual benefits result in inadequate retirement life. But if he or she invests all
contributions in portfolio C, the participant faces greater risk from a downtrend.
Therefore, to balance hope and fear, the participant may simultaneously select more than one portfolio, which should create sufficient retirement benefits and still protect the participant from a more negative situation. Table 3 demonstrates this scenario.
Although the (0, 0.5, 0.5) mix does not produce the highest average retirement benefits, its lower 10% quantile is the highest, which supports the possibility that the participant may not allocate all contributions to portfolio C.
Second, from the participant’s viewpoint, the allocation strategy of investing all contributions in portfolio C results in the highest expected retirement benefits. But for the type-II guarantee provider, the (1, 0, 0) mix entails the lowest guarantee costs, whereas (0, 0, 1) is the least favorite mix. If the guarantee does not get charged and the provider bears these costs, a conflict of interest arises between the participant and the type-II guarantee provider. To diminish guarantee costs, the type-II guarantee provider imposes limitations on the allocation proportion, such as that the participant may not allocate all savings to portfolio C, which prevents the participant from gaining the highest retirement benefits. Moreover, the type-II guarantee provider has a strong incentive to offer only portfolios with low expected rates of return and volatility to reduce risks.
We also report the average costs and retirement benefits under type-I and type-II guarantees with different and in Tables 5-12. All values increase with ther estimate of (or ), because high (orr ) tends to have high guaranteed rates.r Furthermore, the implications mentioned previously still hold for different and
. Tables 9 and 10 also indicate that the type-I guarantee provider and ther
participant both prefer portfolio C and have consistent interests only when isr high. When = 0.05, the lowest average costs and the highest average retirementr benefits under the type-I guarantee exist for the mixes of (0, 0.5, 0.5) and (0, 0, 1), respectively. From the expression of R(t,t), we know that low does notr usually produce extremely high or low guaranteed rates. The mix of (0. 0.5, 0.5) has a high expected rate of return and moderately high volatility; therefore, the actual rates of return are usually high, and savings do not often depreciate significantly. Although the (0, 0, 1) mix offers the highest expected rate of return, its high volatility increases the probability that actual rates of return will be significantly lower than guaranteed rates. This implies that the conflict of interest between the participant and the guarantee provider may emerge under some term structures.
Table 5.
Average Type-I Guarantee Costs, Given Different
Allocation Mix
(A, B, C) 0.03 0.04 0.05 0.06 0.07
(0, 0, 1) 0.370 0.654 1.106 1.802 2.854
(0, 0.25, 0.75) 0.376 0.678 1.165 1.916 3.049
(0, 0.5, 0.5) 0.381 0.703 1.223 2.029 3.244
(0.06, 0.63, 0.31) 0.406 0.759 1.330 2.208 3.518
(0.25, 0.5, 0.25) 0.473 0.886 1.541 2.527 3.964
(0.435, 0.38, 0.185) 0.539 1.010 1.749 2.841 4.403
(0.62, 0.26, 0.12) 0.604 1.134 1.956 3.155 4.842
(0.81, 0.13, 0.06) 0.671 1.261 2.167 3.474 5.287
(1, 0, 0) 0.738 1.387 2.379 3.794 5.733
(1/3, 1/3, 1/3) 0.500 0.931 1.608 2.617 4.074
Notes: All parameter estimates except for are the same as those in the standard scenario.
Table 6.
Average Retirement Benefits under a Type-I Guarantee, Given Different
Allocation Mix
(A, B, C) 0.03 0.04 0.05 0.06 0.07
(0, 0, 1) 23.998 24.282 24.734 25.430 26.482
(0, 0.25, 0.75) 20.720 21.023 21.509 22.260 23.393
(0, 0.5, 0.5) 17.442 17.763 18.284 19.090 20.304
(0.06, 0.63, 0.31) 14.651 15.004 15.575 16.452 17.763 (0.25, 0.5, 0.25) 12.916 13.329 13.984 14.970 16.407 (0.435, 0.38, 0.185) 11.141 11.612 12.351 13.443 15.005 (0.62, 0.26, 0.12) 9.366 9.895 10.718 11.916 13.603
(0.81, 0.13, 0.06) 7.631 8.220 9.127 10.434 12.247
(1, 0, 0) 5.896 6.546 7.537 8.952 10.891
(1/3, 1/3, 1/3) 13.593 14.024 14.701 15.710 17.167
Notes: All parameter estimates except for are the same as those in the standard scenario.
Table 7.
Average Type-II Guarantee Costs, Given Different
Allocation Mix
(A, B, C) 0.03 0.04 0.05 0.06 0.07
(0, 0, 1) 37.200 42.724 49.198 56.769 65.748
(0, 0.25, 0.75) 29.376 33.898 39.222 45.478 52.933
(0, 0.5, 0.5) 21.551 25.072 29.245 34.187 40.118
(0.06, 0.63, 0.31) 15.331 18.049 21.300 25.192 29.906 (0.25, 0.5, 0.25) 12.588 14.930 17.758 21.169 25.334 (0.435, 0.38, 0.185) 9.711 11.662 14.046 16.956 20.545 (0.62, 0.26, 0.12) 6.834 8.393 10.334 12.742 15.756
(0.81, 0.13, 0.06) 4.091 5.275 6.791 8.720 11.184
(1, 0, 0) 1.348 2.156 3.248 4.697 6.612
(1/3, 1/3, 1/3) 14.817 17.434 20.580 24.357 28.949
Notes: All parameter estimates except for are the same as those in the standard scenario.
Table 8.
Average Retirement Benefits under a Type-II Guarantee, Given Different
Allocation Mix
(A, B, C) 0.03 0.04 0.05 0.06 0.07
(0, 0, 1) 60.828 66.352 72.826 80.397 89.376
(0, 0.25, 0.75) 49.720 54.242 59.566 65.822 73.277
(0, 0.5, 0.5) 38.612 42.132 46.306 51.247 57.178
(0.06, 0.63, 0.31) 29.576 32.293 35.545 39.436 44.151 (0.25, 0.5, 0.25) 25.031 27.373 30.201 33.612 37.777 (0.435, 0.38, 0.185) 20.314 22.264 24.648 27.558 31.147 (0.62, 0.26, 0.12) 15.596 17.155 19.095 21.504 24.518 (0.81, 0.13, 0.06) 11.051 12.235 13.751 15.680 18.144
(1, 0, 0) 6.506 7.315 8.407 9.856 11.770
(1/3, 1/3, 1/3) 27.910 30.527 33.673 37.450 42.042
Notes: All parameter estimates except for are the same as those in the standard scenario.
Table 9.
Average Type-I Guarantee Costs, Given Different
rAllocation Mix r
(A, B, C) 0.05 0.12 0.13 0.14 0.15 0.16
(0, 0, 1) 0.311 0.715 0.882 1.106 1.419 1.897
(0, 0.25, 0.75) 0.287 0.748 0.927 1.165 1.490 1.982
(0, 0.5, 0.5) 0.263 0.781 0.973 1.223 1.562 2.066
(0.06, 0.63, 0.31) 0.291 0.867 1.069 1.330 1.679 2.194
(0.25, 0.5, 0.25) 0.433 1.066 1.275 1.541 1.896 2.414
(0.435, 0.38, 0.185) 0.570 1.261 1.476 1.749 2.108 2.631
(0.62, 0.26, 0.12) 0.707 1.456 1.677 1.956 2.321 2.848
(0.81, 0.13, 0.06) 0.849 1.655 1.883 2.167 2.537 3.068
(1, 0, 0) 0.991 1.854 2.088 2.379 2.753 3.289
(1/3, 1/3, 1/3) 0.505 1.139 1.345 1.608 1.959 2.474
Notes: All parameter estimates except for r are the same as those in the standard scenario.
Table 10.
Average Retirement Benefits under a Type-I Guarantee, Given Different
rAllocation Mix r
(A, B, C) 0.05 0.12 0.13 0.14 0.15 0.16
(0, 0, 1) 23.939 24.343 24.510 24.734 25.047 25.525
(0, 0.25, 0.75) 20.631 21.092 21.272 21.509 21.835 22.326
(0, 0.5, 0.5) 17.323 17.842 18.033 18.284 18.622 19.127
(0.06, 0.63, 0.31) 14.536 15.111 15.314 15.575 15.924 16.438 (0.25, 0.5, 0.25) 12.876 13.509 13.717 13.984 14.339 14.857 (0.435, 0.38, 0.185) 11.172 11.863 12.078 12.351 12.710 13.233 (0.62, 0.26, 0.12) 9.469 10.217 10.439 10.718 11.082 11.609
(0.81, 0.13, 0.06) 7.809 8.615 8.843 9.127 9.497 10.028
(1, 0, 0) 6.149 7.013 7.247 7.537 7.911 8.447
(1/3, 1/3, 1/3) 13.598 14.232 14.438 14.701 15.052 15.567
Notes: All parameter estimates except for r are the same as those in the standard scenario.
Table 11.
Average Type-II Guarantee Costs, Given Different
rAllocation Mix r
(A, B, C) 0.05 0.12 0.13 0.14 0.15 0.16
(0, 0, 1) 44.050 47.230 48.114 49.198 50.500 52.246
(0, 0.25, 0.75) 34.819 37.531 38.289 39.222 40.349 41.867
(0, 0.5, 0.5) 25.588 27.832 28.465 29.245 30.199 31.488
(0.06, 0.63, 0.31) 18.236 20.112 20.644 21.300 22.111 23.213 (0.25, 0.5, 0.25) 14.956 16.682 17.163 17.758 18.493 19.495 (0.435, 0.38, 0.185) 11.519 13.086 13.516 14.046 14.703 15.602
(0.62, 0.26, 0.12) 8.083 9.490 9.869 10.334 10.913 11.709
(0.81, 0.13, 0.06) 4.802 6.060 6.389 6.791 7.295 7.991
(1, 0, 0) 1.522 2.629 2.908 3.248 3.677 4.273
(1/3, 1/3, 1/3) 17.566 19.431 19.946 20.580 21.358 22.416
Notes: All parameter estimates except for r are the same as those in the standard scenario.
Table 12.
Average Retirement Benefits under a Type-II Guarantee, Given Different
rAllocation Mix r
(A, B, C) 0.05 0.12 0.13 0.14 0.15 0.16
(0, 0, 1) 67.678 70.858 71.742 72.826 74.128 75.874
(0, 0.25, 0.75) 55.163 57.875 58.634 59.566 60.694 62.211
(0, 0.5, 0.5) 42.648 44.892 45.525 46.306 47.259 48.548
(0.06, 0.63, 0.31) 32.481 34.357 34.888 35.545 36.356 37.457 (0.25, 0.5, 0.25) 27.399 29.124 29.606 30.201 30.936 31.938 (0.435, 0.38, 0.185) 22.122 23.688 24.118 24.648 25.305 26.204 (0.62, 0.26, 0.12) 16.844 18.252 18.631 19.095 19.675 20.470 (0.81, 0.13, 0.06) 11.762 13.020 13.349 13.751 14.255 14.951
(1, 0, 0) 6.681 7.787 8.067 8.407 8.835 9.431
(1/3, 1/3, 1/3) 30.659 32.524 33.039 33.673 34.451 35.509
Notes: All parameter estimates except for r are the same as those in the standard scenario.