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November  2020, 16(6): 2857-2890. doi: 10.3934/jimo.2019084

## Multi-period optimal investment choice post-retirement with inter-temporal restrictions in a defined contribution pension plan

 1 China Institute for Actuarial Science, Central University of Finance and Economics, Beijing 100081, China 2 School of Statistics and Mathematics, Central University of Finance and Economics, Beijing 100081, China 3 Beijing Branch, China Minsheng Banking Corporation Limited, Beijing 100031, China 4 School of Mathematics and Statistics, Guangdong University of Foreign Studies, Guangzhou 510006, China

* Corresponding author: zengli@gdufs.edu.cn

Received  October 2018 Revised  March 2019 Published  November 2020 Early access  July 2019

Fund Project: The first author is supported by grants from National Natural Science Foundation of China (Nos. 11671411, 11771465)

This paper studies a multi-period portfolio selection problem during the post-retirement phase of a defined contribution pension plan. The retiree is allowed to defer the purchase of the annuity until the time of compulsory annuitization. A series of investment targets over time are set, and restrictions on the inter-temporal expected values of the portfolio are considered. We aim to minimize the accumulated variances from the time of retirement to the time of compulsory annuitization. Using the Lagrange multiplier technique and dynamic programming, we study in detail the existence of the optimal strategy and derive its closed-form expression. For comparison purposes, the explicit solution of the classical target-based model is also provided. The properties of the optimal investment strategy, the probabilities of achieving a worse or better pension at the time of compulsory annuitization and the bankruptcy probability are compared in detail under two models. The comparison shows that our model can greatly decrease the probability of achieving a worse pension at the compulsory time and can significantly increase the probability of achieving a better pension.

Citation: Huiling Wu, Xiuguo Wang, Yuanyuan Liu, Li Zeng. Multi-period optimal investment choice post-retirement with inter-temporal restrictions in a defined contribution pension plan. Journal of Industrial and Management Optimization, 2020, 16 (6) : 2857-2890. doi: 10.3934/jimo.2019084
##### References:
 [1] M. Akian, J. Menaldi and A. Sulem, On an investment-consumption model with transaction costs, SIAM Journal on Control and Optimization, 34 (1996), 329-364.  doi: 10.1137/S0363012993247159. [2] P. Albrecht and R. Maurer, Self-annuitization, consumption shortfall in retirement and asset allocation: The annuity benchmark, Journal of Pension Economics and Finance, 1 (2002), 269-288.  doi: 10.1017/S1474747202001117. [3] P. Battocchio, F. Menoncin and O. Scaillet, Optimal asset allocation for pension funds under mortality risk during the accumulation and decumulation phases, Annals of Operations Research, 152 (2007), 141-165.  doi: 10.1007/s10479-006-0144-2. [4] E. Bayraktar and V. R. Young, Minimizing the probability of lifetime ruin with deferred life annuities, North American Actuarial Journal, 13 (2009), 141-154.  doi: 10.1080/10920277.2009.10597543. [5] M. S. Bazaraa, H. D. Sherali and C. M. Shetty, Nonlinear Programming: Theory and Algorithms, Wiley Interscience, New Jersey, 2006. doi: 10.1002/0471787779. [6] D. Blake, A. J. G. Cairns and K. Dowd, Pensionmetrics 2: Stochastic pension plan design during the distribution phase, Insurance: Mathematics and Economics, 33 (2003), 29-47.  doi: 10.1016/S0167-6687(03)00141-0. [7] J. R. Brown, Private pensions, mortality risk, and the decision to annuitize, Journal of Public Economics, 82 (2001), 29-62.  doi: 10.3386/w7191. [8] O. L. V. Costa and R. B. Nabholz, Multiperiod mean-variance optimization with intertemporal restrictions, Journal of Optimization Theory and Applications, 134 (2007), 257-274.  doi: 10.1007/s10957-007-9233-x. [9] H. Dadashi, Optimal investment-consumption problem post-retirement with a minimum guarantee, Working paper, arXiv: 1803.00611, 2018. doi: 10.1016/j.cam.2019.02.027. [10] M. Di Giacinto, S. Federico, F. Gozzi and E. Vigna, Income drawdown option with minimum guarantee, European Journal of Operational Research, 234 (2014), 610-624.  doi: 10.1016/j.ejor.2013.10.026. [11] P. Emms, Relative choice models for income drawdown in a defined contribution pension scheme, North American Actuarial Journal, 14 (2010), 176-197.  doi: 10.1080/10920277.2010.10597584. [12] P. Emms and S. Haberman, Income drawdown schemes for a defined-contribution pension plan, The Journal of Risk and Insurance, 75 (2008), 739-761.  doi: 10.1080/10920277.2010.10597584. [13] A. Finkelstein and J. Poterba, Selection effects in the United Kingdom individual annuities market, Economic Journal, 112 (2002), 28-50.  doi: 10.1111/1468-0297.0j672. [14] A. M. Geoffrion, Duality in nonlinear programming: A simplified application-oriented development, SIAM Review, 13 (1971), 1-37.  doi: 10.1137/1013001. [15] R. Gerrard, S. Haberman and E. Vigna, Optimal investment choices post-retirement in a defined contribution pension scheme, Insurance: Mathematics and Economics, 35 (2004), 321-342.  doi: 10.1016/j.insmatheco.2004.06.002. [16] R. Gerrard, S. Haberman and E. Vigna, The management of decumulation risks in a defined contribution pension plan, North American Actuarial Journal, 10 (2006), 84-110.  doi: 10.1080/10920277.2006.10596241. [17] R. Gerrard, B. Høgaard and E. Vigna, Choosing the optimal annuitization time post retirement, Quantitative Finance, 12 (2012), 1143-1159.  doi: 10.1080/14697680903358248. [18] J. Inkmann, P. Lopes and A. Michaelides, How deep is the annuity market participation puzzle?, The Review of Financial Studies, 24 (2011), 279-319. [19] C. W. Lin, L. Zeng and H. L. Wu, Multi-period portfolio optimization in a defined contribution pension plan during the decumulation phase, J. Ind. Manag. Optim., 15 (2019), 401-427.  doi: 10.3934/jimo.2018059. [20] L. M. Lockwood, Bequest motives and the annuity puzzle, Review of Economic Dynamics, 15 (2012), 226-243.  doi: 10.1016/j.red.2011.03.001. [21] F. Menoncin and E. Vigna, Mean-variance target-based optimisation for defined contribution pension schemes in a stochastic framework, Insurance: Mathmatics and Economics, 76 (2017), 172-184.  doi: 10.1016/j.insmatheco.2017.08.002. [22] M. A. Milevsky, Optimal annuitization policies: Analysis of the options, North American Actuarial Journal, 5 (2001), 57-69.  doi: 10.1080/10920277.2001.10595953. [23] M. A. Milevsky and C. Robinson, Self-annuitization and ruin in retirement, North American Actuarial Journal, 4 (2000), 113-129.  doi: 10.1080/10920277.2000.10595940. [24] M. A. Milevsky, K. S. Moore and V. R. Young, Optimal asset allocation and ruin minimization annuitization strategies, Mathematical Finance, 16 (2006), 647-671.  doi: 10.1016/j.jedc.2006.11.003. [25] M. A. Milevsky and V. R. Young, Annuitization and asset allocation, Journal of Economic Dynamics and Control, 31 (2007), 3138-3177.  doi: 10.1016/j.jedc.2006.11.003. [26] G. Stabile, Optimal timing of the annuity purchase: combined stochastic control and optimal stopping problem, International Journal of Theoretical and Applied Finance, 9 (2006), 151-170.  doi: 10.1142/S0219024906003524. [27] B. Z. Temocin, R. Korn and A. S. Selcuk-Kestel, Constant proportion portfolio insurance in defined contribution pension plan management under discrete-time trading, Annals of Operations Research, 266 (2018), 329-348.  doi: 10.1007/s10479-017-2449-8. [28] E. Vigna, On efficiency of mean-variance based portfolio selection in defined contribution pension schemes, Quantitative Finance, 14 (2014), 237-258.  doi: 10.1080/14697688.2012.708778. [29] H. L. Wu and Y. Zeng, Equilibrium investment strategy for defined-contribution pension schemes with generalized mean-variance criterion and mortality risk, Insurance: Mathematics and Economics, 64 (2015), 396-408.  doi: 10.1016/j.insmatheco.2015.07.007. [30] X. Y. Zhang and J. Y. Guo, The role of inflation-indexed bond in optimal management of defined contribution pension plan during the decumulation phase, Risks, 24 (2018).  doi: 10.3390/risks6020024. [31] S. S. Zhu, D. Li and S. Y. Wang, Risk control over bankruptcy in dynamic portfolio selection: A generalized mean-variance formulation, IEEE Transactions on Automatic Control, 49 (2004), 447-457.  doi: 10.1109/TAC.2004.824474.

show all references

##### References:
 [1] M. Akian, J. Menaldi and A. Sulem, On an investment-consumption model with transaction costs, SIAM Journal on Control and Optimization, 34 (1996), 329-364.  doi: 10.1137/S0363012993247159. [2] P. Albrecht and R. Maurer, Self-annuitization, consumption shortfall in retirement and asset allocation: The annuity benchmark, Journal of Pension Economics and Finance, 1 (2002), 269-288.  doi: 10.1017/S1474747202001117. [3] P. Battocchio, F. Menoncin and O. Scaillet, Optimal asset allocation for pension funds under mortality risk during the accumulation and decumulation phases, Annals of Operations Research, 152 (2007), 141-165.  doi: 10.1007/s10479-006-0144-2. [4] E. Bayraktar and V. R. Young, Minimizing the probability of lifetime ruin with deferred life annuities, North American Actuarial Journal, 13 (2009), 141-154.  doi: 10.1080/10920277.2009.10597543. [5] M. S. Bazaraa, H. D. Sherali and C. M. Shetty, Nonlinear Programming: Theory and Algorithms, Wiley Interscience, New Jersey, 2006. doi: 10.1002/0471787779. [6] D. Blake, A. J. G. Cairns and K. Dowd, Pensionmetrics 2: Stochastic pension plan design during the distribution phase, Insurance: Mathematics and Economics, 33 (2003), 29-47.  doi: 10.1016/S0167-6687(03)00141-0. [7] J. R. Brown, Private pensions, mortality risk, and the decision to annuitize, Journal of Public Economics, 82 (2001), 29-62.  doi: 10.3386/w7191. [8] O. L. V. Costa and R. B. Nabholz, Multiperiod mean-variance optimization with intertemporal restrictions, Journal of Optimization Theory and Applications, 134 (2007), 257-274.  doi: 10.1007/s10957-007-9233-x. [9] H. Dadashi, Optimal investment-consumption problem post-retirement with a minimum guarantee, Working paper, arXiv: 1803.00611, 2018. doi: 10.1016/j.cam.2019.02.027. [10] M. Di Giacinto, S. Federico, F. Gozzi and E. Vigna, Income drawdown option with minimum guarantee, European Journal of Operational Research, 234 (2014), 610-624.  doi: 10.1016/j.ejor.2013.10.026. [11] P. Emms, Relative choice models for income drawdown in a defined contribution pension scheme, North American Actuarial Journal, 14 (2010), 176-197.  doi: 10.1080/10920277.2010.10597584. [12] P. Emms and S. Haberman, Income drawdown schemes for a defined-contribution pension plan, The Journal of Risk and Insurance, 75 (2008), 739-761.  doi: 10.1080/10920277.2010.10597584. [13] A. Finkelstein and J. Poterba, Selection effects in the United Kingdom individual annuities market, Economic Journal, 112 (2002), 28-50.  doi: 10.1111/1468-0297.0j672. [14] A. M. Geoffrion, Duality in nonlinear programming: A simplified application-oriented development, SIAM Review, 13 (1971), 1-37.  doi: 10.1137/1013001. [15] R. Gerrard, S. Haberman and E. Vigna, Optimal investment choices post-retirement in a defined contribution pension scheme, Insurance: Mathematics and Economics, 35 (2004), 321-342.  doi: 10.1016/j.insmatheco.2004.06.002. [16] R. Gerrard, S. Haberman and E. Vigna, The management of decumulation risks in a defined contribution pension plan, North American Actuarial Journal, 10 (2006), 84-110.  doi: 10.1080/10920277.2006.10596241. [17] R. Gerrard, B. Høgaard and E. Vigna, Choosing the optimal annuitization time post retirement, Quantitative Finance, 12 (2012), 1143-1159.  doi: 10.1080/14697680903358248. [18] J. Inkmann, P. Lopes and A. Michaelides, How deep is the annuity market participation puzzle?, The Review of Financial Studies, 24 (2011), 279-319. [19] C. W. Lin, L. Zeng and H. L. Wu, Multi-period portfolio optimization in a defined contribution pension plan during the decumulation phase, J. Ind. Manag. Optim., 15 (2019), 401-427.  doi: 10.3934/jimo.2018059. [20] L. M. Lockwood, Bequest motives and the annuity puzzle, Review of Economic Dynamics, 15 (2012), 226-243.  doi: 10.1016/j.red.2011.03.001. [21] F. Menoncin and E. Vigna, Mean-variance target-based optimisation for defined contribution pension schemes in a stochastic framework, Insurance: Mathmatics and Economics, 76 (2017), 172-184.  doi: 10.1016/j.insmatheco.2017.08.002. [22] M. A. Milevsky, Optimal annuitization policies: Analysis of the options, North American Actuarial Journal, 5 (2001), 57-69.  doi: 10.1080/10920277.2001.10595953. [23] M. A. Milevsky and C. Robinson, Self-annuitization and ruin in retirement, North American Actuarial Journal, 4 (2000), 113-129.  doi: 10.1080/10920277.2000.10595940. [24] M. A. Milevsky, K. S. Moore and V. R. Young, Optimal asset allocation and ruin minimization annuitization strategies, Mathematical Finance, 16 (2006), 647-671.  doi: 10.1016/j.jedc.2006.11.003. [25] M. A. Milevsky and V. R. Young, Annuitization and asset allocation, Journal of Economic Dynamics and Control, 31 (2007), 3138-3177.  doi: 10.1016/j.jedc.2006.11.003. [26] G. Stabile, Optimal timing of the annuity purchase: combined stochastic control and optimal stopping problem, International Journal of Theoretical and Applied Finance, 9 (2006), 151-170.  doi: 10.1142/S0219024906003524. [27] B. Z. Temocin, R. Korn and A. S. Selcuk-Kestel, Constant proportion portfolio insurance in defined contribution pension plan management under discrete-time trading, Annals of Operations Research, 266 (2018), 329-348.  doi: 10.1007/s10479-017-2449-8. [28] E. Vigna, On efficiency of mean-variance based portfolio selection in defined contribution pension schemes, Quantitative Finance, 14 (2014), 237-258.  doi: 10.1080/14697688.2012.708778. [29] H. L. Wu and Y. Zeng, Equilibrium investment strategy for defined-contribution pension schemes with generalized mean-variance criterion and mortality risk, Insurance: Mathematics and Economics, 64 (2015), 396-408.  doi: 10.1016/j.insmatheco.2015.07.007. [30] X. Y. Zhang and J. Y. Guo, The role of inflation-indexed bond in optimal management of defined contribution pension plan during the decumulation phase, Risks, 24 (2018).  doi: 10.3390/risks6020024. [31] S. S. Zhu, D. Li and S. Y. Wang, Risk control over bankruptcy in dynamic portfolio selection: A generalized mean-variance formulation, IEEE Transactions on Automatic Control, 49 (2004), 447-457.  doi: 10.1109/TAC.2004.824474.
$\eta_n- {{\rm{E}}}_{0,x_0}\left(X^{\tilde\pi}_{n}\right)$, $n = 1,2,\ldots,T$
The targets over time
 $\eta_1=198747$ $\eta_2=197494$ $\eta_3=196241$ $\eta_4=194988$ $\eta_5=193735$ $\eta_6=192482$ $\eta_7=191229$ $\eta_8=189976$ $\eta_9=188723$ $\eta_{10}=187470$ $\eta_{11}=186217$ $\eta_{12}=184964$ $\eta_{13}=183711$ $\eta_{14}=182458$ $\eta_{15}=181205$
 $\eta_1=198747$ $\eta_2=197494$ $\eta_3=196241$ $\eta_4=194988$ $\eta_5=193735$ $\eta_6=192482$ $\eta_7=191229$ $\eta_8=189976$ $\eta_9=188723$ $\eta_{10}=187470$ $\eta_{11}=186217$ $\eta_{12}=184964$ $\eta_{13}=183711$ $\eta_{14}=182458$ $\eta_{15}=181205$
The targets over time
 $\eta_1=201356$ $\eta_2=202333$ $\eta_3=202942$ $\eta_4=203160$ $\eta_5=202960$ $\eta_6=202362$ $\eta_7=201386$ $\eta_8=200057$ $\eta_9=198330$ $\eta_{10}=196284$ $\eta_{11}=193886$ $\eta_{12}=191191$ $\eta_{13}=188231$ $\eta_{14}=184908$ $\eta_{15}=181193$
 $\eta_1=201356$ $\eta_2=202333$ $\eta_3=202942$ $\eta_4=203160$ $\eta_5=202960$ $\eta_6=202362$ $\eta_7=201386$ $\eta_8=200057$ $\eta_9=198330$ $\eta_{10}=196284$ $\eta_{11}=193886$ $\eta_{12}=191191$ $\eta_{13}=188231$ $\eta_{14}=184908$ $\eta_{15}=181193$
The frequencies that $X_{T}/$ä$_{75} $$<\zeta_0  The first decision mechanism of targets Final target 1.5\zeta_0 ä _{75} 1.7\zeta_0 ä _{75} 2\zeta_0 ä _{75} {{\rm{Var}}}(R_n)=0.4 p_1\; \; \text{Our model} 0.2774 0.2661 0.2535 p_2\; \; \text{Target-based model} 0.5394 0.4597 0.3975 {{\rm{Var}}}(R_n)=0.6 p_1\; \; \text{Our model} 0.3300 0.3151 0.3024 p_2\; \; \text{Target-based model} 0.6624 0.5736 0.4957 {{\rm{Var}}}(R_n)=0.8 p_1\; \; \text{Our model} 0.3661 0.3507 0.3360 p_2\; \; \text{Target-based model} 0.7428 0.6532 0.5672 The second decision mechanism of targets Final target 1.5\zeta_0 ä _{75} 1.7\zeta_0 ä _{75} 2\zeta_0 ä _{75} {{\rm{Var}}}(R_n)=0.4 p_1\; \; \text{Our model} 0.2759 0.2611 0.2523 p_2\; \; \text{Target-based model} 0.5125 0.4391 0.3760 {{\rm{Var}}}(R_n)=0.6 p_1\; \; \text{Our model} 0.3302 0.3167 0.3090 p_2\; \; \text{Target-based model} 0.6395 0.5450 0.4763 {{\rm{Var}}}(R_n)=0.8 p_1\; \; \text{Our model} 0.3641 0.3545 0.3431 p_2\; \; \text{Target-based model} 0.7178 0.6313 0.5508  The first decision mechanism of targets Final target 1.5\zeta_0 ä _{75} 1.7\zeta_0 ä _{75} 2\zeta_0 ä _{75} {{\rm{Var}}}(R_n)=0.4 p_1\; \; \text{Our model} 0.2774 0.2661 0.2535 p_2\; \; \text{Target-based model} 0.5394 0.4597 0.3975 {{\rm{Var}}}(R_n)=0.6 p_1\; \; \text{Our model} 0.3300 0.3151 0.3024 p_2\; \; \text{Target-based model} 0.6624 0.5736 0.4957 {{\rm{Var}}}(R_n)=0.8 p_1\; \; \text{Our model} 0.3661 0.3507 0.3360 p_2\; \; \text{Target-based model} 0.7428 0.6532 0.5672 The second decision mechanism of targets Final target 1.5\zeta_0 ä _{75} 1.7\zeta_0 ä _{75} 2\zeta_0 ä _{75} {{\rm{Var}}}(R_n)=0.4 p_1\; \; \text{Our model} 0.2759 0.2611 0.2523 p_2\; \; \text{Target-based model} 0.5125 0.4391 0.3760 {{\rm{Var}}}(R_n)=0.6 p_1\; \; \text{Our model} 0.3302 0.3167 0.3090 p_2\; \; \text{Target-based model} 0.6395 0.5450 0.4763 {{\rm{Var}}}(R_n)=0.8 p_1\; \; \text{Our model} 0.3641 0.3545 0.3431 p_2\; \; \text{Target-based model} 0.7178 0.6313 0.5508 The gap between p_1 and p_2  The first decision mechanism of targets Final target 1.5\zeta_0 ä _{75} 1.7\zeta_0 ä _{75} 2\zeta_0 ä _{75} {{\rm{Var}}}(R_n)=0.4 p_2-p_1 0.2620 0.1936 0.1440 {{\rm{Var}}}(R_n)=0.6 p_2-p_1 0.3324 0.2585 0.1933 {{\rm{Var}}}(R_n)=0.8 p_2-p_1 0.3767 0.3025 0.2312 The second decision mechanism of targets Final target 1.5\zeta_0 ä _{75} 1.7\zeta_0 ä _{75} 2\zeta_0 ä _{75} {{\rm{Var}}}(R_n)=0.4 p_2-p_1 0.2366 0.1780 0.1237 {{\rm{Var}}}(R_n)=0.6 p_2-p_1 0.3093 0.2283 0.1673 {{\rm{Var}}}(R_n)=0.8 p_2-p_1 0.3537 0.2768 0.2077  The first decision mechanism of targets Final target 1.5\zeta_0 ä _{75} 1.7\zeta_0 ä _{75} 2\zeta_0 ä _{75} {{\rm{Var}}}(R_n)=0.4 p_2-p_1 0.2620 0.1936 0.1440 {{\rm{Var}}}(R_n)=0.6 p_2-p_1 0.3324 0.2585 0.1933 {{\rm{Var}}}(R_n)=0.8 p_2-p_1 0.3767 0.3025 0.2312 The second decision mechanism of targets Final target 1.5\zeta_0 ä _{75} 1.7\zeta_0 ä _{75} 2\zeta_0 ä _{75} {{\rm{Var}}}(R_n)=0.4 p_2-p_1 0.2366 0.1780 0.1237 {{\rm{Var}}}(R_n)=0.6 p_2-p_1 0.3093 0.2283 0.1673 {{\rm{Var}}}(R_n)=0.8 p_2-p_1 0.3537 0.2768 0.2077 The frequencies that X_{T}/ ä _{75}$$ \ge k\zeta_0$
 The first decision mechanism of targets Final target $1.5\zeta_0$ä$_{75}$ $1.7\zeta_0$ä$_{75}$ $2\zeta_0$ä$_{75}$ ${{\rm{Var}}}(R_n) = 0.4$ $g_1\; \; \text{Our model}$ 0.5717 0.5725 0.5762 $g_2\; \; \text{Target-based model}$ 0.0038 0.0045 0.0040 ${{\rm{Var}}}(R_n) = 0.6$ $g_1\; \; \text{Our model}$ 0.5458 0.5466 0.5493 $g_2\; \; \text{Target-based model}$ 0.0036 0.0028 0.0027 ${{\rm{Var}}}(R_n) = 0.8$ $g_1\; \; \text{Our model}$ 0.5198 0.5167 0.5250 $g_2\; \; \text{Target-based model}$ 0.0037 0.0029 0.0027 The second decision mechanism of targets Final target $1.5\zeta_0$ä$_{75}$ $1.7\zeta_0$ä$_{75}$ $2\zeta_0$ä$_{75}$ ${{\rm{Var}}}(R_n) = 0.4$ $g_1\; \; \text{Our model}$ 0.3042 0.4329 0.5427 $g_2\; \; \text{Target-based model}$ 0.0001 0.0004 0.0018 ${{\rm{Var}}}(R_n) = 0.6$ $g_1\; \; \text{Our model}$ 0.3530 0.4405 0.5220 $g_2\; \; \text{Target-based model}$ 0.0003 0.00045 0.0019 ${{\rm{Var}}}(R_n) = 0.8$ $g_1\; \; \text{Our model}$ 0.3617 0.4293 0.5033 $g_2\; \; \text{Target-based model}$ 0.00005 0.0003 0.0013
 The first decision mechanism of targets Final target $1.5\zeta_0$ä$_{75}$ $1.7\zeta_0$ä$_{75}$ $2\zeta_0$ä$_{75}$ ${{\rm{Var}}}(R_n) = 0.4$ $g_1\; \; \text{Our model}$ 0.5717 0.5725 0.5762 $g_2\; \; \text{Target-based model}$ 0.0038 0.0045 0.0040 ${{\rm{Var}}}(R_n) = 0.6$ $g_1\; \; \text{Our model}$ 0.5458 0.5466 0.5493 $g_2\; \; \text{Target-based model}$ 0.0036 0.0028 0.0027 ${{\rm{Var}}}(R_n) = 0.8$ $g_1\; \; \text{Our model}$ 0.5198 0.5167 0.5250 $g_2\; \; \text{Target-based model}$ 0.0037 0.0029 0.0027 The second decision mechanism of targets Final target $1.5\zeta_0$ä$_{75}$ $1.7\zeta_0$ä$_{75}$ $2\zeta_0$ä$_{75}$ ${{\rm{Var}}}(R_n) = 0.4$ $g_1\; \; \text{Our model}$ 0.3042 0.4329 0.5427 $g_2\; \; \text{Target-based model}$ 0.0001 0.0004 0.0018 ${{\rm{Var}}}(R_n) = 0.6$ $g_1\; \; \text{Our model}$ 0.3530 0.4405 0.5220 $g_2\; \; \text{Target-based model}$ 0.0003 0.00045 0.0019 ${{\rm{Var}}}(R_n) = 0.8$ $g_1\; \; \text{Our model}$ 0.3617 0.4293 0.5033 $g_2\; \; \text{Target-based model}$ 0.00005 0.0003 0.0013
The frequencies of bankruptcies–20000 simulations
 The first decision mechanism of targets $\eta_T=1.5\zeta_0$ä$_{75}$ Events Our model Target-based model Number of simulations(Frequencies) Number of simulations(Frequencies) $S=0$ 16355(0.8177) 19774(0.9887) $S\neq 0$ 3645(0.1823) 226(0.0113) Our model Target-based model Mean of $S$ 0.6833 0.0181 $\eta_T=2.0\zeta_0$ä$_{75}$ Events Our model Target-based model Number of simulations(Frequencies) Number of simulations(Frequencies) $S=0$ 14424(0.7212) 18909(0.9455) $S\neq 0$ 5576(0.2788) 1091(0.0546) Our model Target-based model Mean of $S$ 1.3019 0.1135 The second decision mechanism of targets $\eta_T=1.5\zeta_0$ä$_{75}$ Events Our model Target-based model Number of simulations(Frequencies) Number of simulations(Frequencies) $S=0$ 16270(0.8135) 19667(0.9833) $S\neq 0$ 3730(0.1865) 333(0.0167) Our model Target-based model Mean of $S$ 0.8719 0.0297 $\eta_T=2.0\zeta_0$ä$_{75}$ Events Our model Target-based model Number of simulations(Frequencies) Number of simulations(Frequencies) $S=0$ 14118(0.7059) 18726(0.9363) $S\neq 0$ 5882(0.2941) 1274(0.0637) Our model Target-based model Mean of $S$ 1.6798 0.1457
 The first decision mechanism of targets $\eta_T=1.5\zeta_0$ä$_{75}$ Events Our model Target-based model Number of simulations(Frequencies) Number of simulations(Frequencies) $S=0$ 16355(0.8177) 19774(0.9887) $S\neq 0$ 3645(0.1823) 226(0.0113) Our model Target-based model Mean of $S$ 0.6833 0.0181 $\eta_T=2.0\zeta_0$ä$_{75}$ Events Our model Target-based model Number of simulations(Frequencies) Number of simulations(Frequencies) $S=0$ 14424(0.7212) 18909(0.9455) $S\neq 0$ 5576(0.2788) 1091(0.0546) Our model Target-based model Mean of $S$ 1.3019 0.1135 The second decision mechanism of targets $\eta_T=1.5\zeta_0$ä$_{75}$ Events Our model Target-based model Number of simulations(Frequencies) Number of simulations(Frequencies) $S=0$ 16270(0.8135) 19667(0.9833) $S\neq 0$ 3730(0.1865) 333(0.0167) Our model Target-based model Mean of $S$ 0.8719 0.0297 $\eta_T=2.0\zeta_0$ä$_{75}$ Events Our model Target-based model Number of simulations(Frequencies) Number of simulations(Frequencies) $S=0$ 14118(0.7059) 18726(0.9363) $S\neq 0$ 5882(0.2941) 1274(0.0637) Our model Target-based model Mean of $S$ 1.6798 0.1457
Comparison summary
 The first decision mechanism of targets Comparison items Our model Target-based model Times Frequencies that $X_{T}/$ä$_{75} $$<\zeta_0 \eta_T=1.5\zeta_0 ä _{75} 0.2774 0.5394 0.5143 Frequencies that X_{T}/ ä _{75}$$ \geq k\zeta_0$ $\eta_T=1.5\zeta_0$ä$_{75}$ 0.5717 0.0038 150.4 Bankruptcy probabilities $\eta_T=1.5\zeta_0$ä$_{75}$ 0.1823 0.0113 16.63 Frequencies that $X_{T}/$ä$_{75} $$<\zeta_0 \eta_T=2.0\zeta_0 ä _{75} 0.2535 0.3975 0.6377 Frequencies that X_{T}/ ä _{75}$$ \geq k\zeta_0$ $\eta_T=2.0\zeta_0$ä$_{75}$ 0.5762 0.0040 144.05 Bankruptcy probabilities $\eta_T=2.0\zeta_0$ä$_{75}$ 0.2788 0.0546 5.106 The second decision mechanism of targets Comparison items Our model Target-based model Times Frequencies that $X_{T}/$ä$_{75} $$<\zeta_0 \eta_T=1.5\zeta_0 ä _{75} 0.2759 0.5125 0.5383 Frequencies that X_{T}/ ä _{75}$$ \geq k\zeta_0$ $\eta_T=1.5\zeta_0$ä$_{75}$ 0.3042 0.0001 3042 Bankruptcy probabilities $\eta_T=1.5\zeta_0$ä$_{75}$ 0.1865 0.0167 11.17 Frequencies that $X_{T}/$ä$_{75} $$<\zeta_0 \eta_T=2.0\zeta_0 ä _{75} 0.2523 0.3760 0.6710 Frequencies that X_{T}/ ä _{75}$$ \geq k\zeta_0$ $\eta_T=2.0\zeta_0$ä$_{75}$ 0.5427 0.0018 301.5 Bankruptcy probabilities $\eta_T=2.0\zeta_0$ä$_{75}$ 0.2941 0.0637 4.617
 The first decision mechanism of targets Comparison items Our model Target-based model Times Frequencies that $X_{T}/$ä$_{75} $$<\zeta_0 \eta_T=1.5\zeta_0 ä _{75} 0.2774 0.5394 0.5143 Frequencies that X_{T}/ ä _{75}$$ \geq k\zeta_0$ $\eta_T=1.5\zeta_0$ä$_{75}$ 0.5717 0.0038 150.4 Bankruptcy probabilities $\eta_T=1.5\zeta_0$ä$_{75}$ 0.1823 0.0113 16.63 Frequencies that $X_{T}/$ä$_{75} $$<\zeta_0 \eta_T=2.0\zeta_0 ä _{75} 0.2535 0.3975 0.6377 Frequencies that X_{T}/ ä _{75}$$ \geq k\zeta_0$ $\eta_T=2.0\zeta_0$ä$_{75}$ 0.5762 0.0040 144.05 Bankruptcy probabilities $\eta_T=2.0\zeta_0$ä$_{75}$ 0.2788 0.0546 5.106 The second decision mechanism of targets Comparison items Our model Target-based model Times Frequencies that $X_{T}/$ä$_{75} $$<\zeta_0 \eta_T=1.5\zeta_0 ä _{75} 0.2759 0.5125 0.5383 Frequencies that X_{T}/ ä _{75}$$ \geq k\zeta_0$ $\eta_T=1.5\zeta_0$ä$_{75}$ 0.3042 0.0001 3042 Bankruptcy probabilities $\eta_T=1.5\zeta_0$ä$_{75}$ 0.1865 0.0167 11.17 Frequencies that $X_{T}/$ä$_{75} $$<\zeta_0 \eta_T=2.0\zeta_0 ä _{75} 0.2523 0.3760 0.6710 Frequencies that X_{T}/ ä _{75}$$ \geq k\zeta_0$ $\eta_T=2.0\zeta_0$ä$_{75}$ 0.5427 0.0018 301.5 Bankruptcy probabilities $\eta_T=2.0\zeta_0$ä$_{75}$ 0.2941 0.0637 4.617
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