# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2022074
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## A bi-level optimization model for the asset-liability management of insurance companies

 1 School of Finance, Nankai University, Tianjin, China 2 Institute of Insurance and Risk Management, Department of Finance and Insurance, Lingnan University, Hong Kong, China 3 Department of Mathematical Sciences, University of Cincinnati, United States of America

*Corresponding author: Xiaowei Chen

Received  December 2021 Revised  February 2022 Early access May 2022

Fund Project: The first author is supported by National Natural Science Foundation of China No. 61673225

Different from traditional asset-liability management where only investment allocation is considered, this paper introduces policy product allocation into asset-liability management of insurance companies. In order to balance product allocation and investment allocation, a bi-level optimization model is employed. Since the decision-making environment of the two allocation processes is full of indeterminacy, the imprecise information of the model is measured by uncertain variables in order to deal with the lack of enough historical data. To solve this bi-level Optimization problem containing uncertain variables, an uncertain bilevel programming model is used. Furthermore, we simulate a scenario to compare the bi-level optimization approach with other approaches by virtue of hybrid intelligent algorithms.

Citation: Xiaowei Chen, Qianlong Liu, Dan A. Ralescu. A bi-level optimization model for the asset-liability management of insurance companies. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022074
##### References:
 [1] N. Al Najjar and J. Weinstein, The ambiguity aversion literature: A critical assessment, Econ. Philos., 25 (2009), 249-284. [2] R. Bhattacharyya, A. Chatterjee and S. Kar, Uncertainty theory based novel multi-objective optimization technique using embedding theorem with application to R & D project portfolio selection, Appl. Math., 1 (2010), 189-199. [3] J. Bracken and J. McGill, Mathematical programs with optimization problems in the constraints, Oper. Res., 21 (1973), 37-44.  doi: 10.1287/opre.21.1.37. [4] D. R. Cariño and W. T. Ziemba, Formulation of the Russell-Yasuda Kasai financial planning model, Oper. Res., 46 (1998), 433-449.  doi: 10.1287/opre.46.4.433. [5] X. Chen, Y. Liu and D. A. Ralescu, Uncertain stock model with periodic dividends, Fuzzy Optim. Decis. Mak., 12 (2013), 111-123.  doi: 10.1007/s10700-012-9141-x. [6] M. I. Cusy and W. T. Ziemba, A Bank asset and liability management model, Oper. Res., 34 (1986), 356-376. [7] H. G. Deallenbach and S. A. Archer, Optimal bank liquidity: A multi-period stochastic model, J. Financ. Quant. Anal., 4 (1969), 329-343. [8] G. D. Eppen and E. F. Fama, Three assets cash balance and dynamic portfolio problems, Manage. Sci., 17 (1971), 311-319. [9] L. Epstein, A definition of uncertainty aversion, Rev. Econ. Stud., 66 (1999), 579-608.  doi: 10.1111/1467-937X.00099. [10] J. J. Judice and A. Faustion, The linear-quadratic bilevel programming problem, INFOR, 32 (1994), 87-98. [11] P. Klibanoff, M. Marinacci and S. Mukerji, A smooth model of decision making under uncertainy, Econometrica, 73 (2005), 1849-1892.  doi: 10.1111/j.1468-0262.2005.00640.x. [12] S. Li, J. Peng and B. Zhang, The uncertain premium principle based on the distortion function, Insur. Math. Econ., 53 (2013), 317-324.  doi: 10.1016/j.insmatheco.2013.06.005. [13] J. Lintner, The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets, Rev. Econ. Stat., 47 (1965), 317-324. [14] B. Liu, Stackelberg-Nash equilibrium for multilevel programming with multiple followers using genetic algorithms, Comput. Math. Appl., 36 (1998), 79-89.  doi: 10.1016/S0898-1221(98)00174-6. [15] B. Liu, Uncertainty Theory, 2$^{nd}$ edition, Springer-Verlag, Berlin, 2004. [16] B. Liu, Some research problems in uncertainty theory, Journal of Uncertain Systems, 3 (2009), 3-10. [17] B. Liu, Theory and Practice of Uncertain Programming, 2$^{nd}$ edition, Springer-Verlag, Berlin, 2009. [18] B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 2010. [19] B. Liu, Why is there a need for uncertainty theory, J. Uncertain Systems, 6 (2012), 3-10. [20] Y. Liu and M. Ha, Expected value of function of uncertain variables, J. Uncertain Systems, 4 (2010), 181-186. [21] H. Markowitz, Portfolio Selection, New York; Chapman & Hall, Ltd., London 1959 [22] J. M. Mulvey, G. Gould and C. Morgan, Asset and liability management system for Towers Perrin-Tillinghast, Interfaces, 7 (30), 96-114. [23] J. Mossin, Equilibrium in a capital asset market, Econometrica, 34 (1966), 768-783. [24] D. H. Pyle, On the theory of financial intermedian, J. Financ., 26 (1971), 737-746. [25] G. Savard and J. Gauvin, The steepest descent direction for the nonlinear bilevel programming problem, Oper. Res. Lett., 15 (1994), 265-272.  doi: 10.1016/0167-6377(94)90086-8. [26] W. Sharpe, Capital asset prices: A theory of market equilibrium under conditions of risk, J. Financ., 19 (1964), 425-442. [27] H. V. Stackelberg, Marktform und Gleichgewicht, Springer-Verlag, Wien & Berlin, 1934. [28] J. L. Treynor and K. K. Mazuy, Can mutual funds outguess the market, Harvard Bus. Rev., 8 (1966), 131-136. [29] C. Wang, Y. Ni and X. Yang, The inventory replenishment policy in an uncertain production-inventory-routing system, J. Ind. Manag. Optim..

show all references

##### References:
 [1] N. Al Najjar and J. Weinstein, The ambiguity aversion literature: A critical assessment, Econ. Philos., 25 (2009), 249-284. [2] R. Bhattacharyya, A. Chatterjee and S. Kar, Uncertainty theory based novel multi-objective optimization technique using embedding theorem with application to R & D project portfolio selection, Appl. Math., 1 (2010), 189-199. [3] J. Bracken and J. McGill, Mathematical programs with optimization problems in the constraints, Oper. Res., 21 (1973), 37-44.  doi: 10.1287/opre.21.1.37. [4] D. R. Cariño and W. T. Ziemba, Formulation of the Russell-Yasuda Kasai financial planning model, Oper. Res., 46 (1998), 433-449.  doi: 10.1287/opre.46.4.433. [5] X. Chen, Y. Liu and D. A. Ralescu, Uncertain stock model with periodic dividends, Fuzzy Optim. Decis. Mak., 12 (2013), 111-123.  doi: 10.1007/s10700-012-9141-x. [6] M. I. Cusy and W. T. Ziemba, A Bank asset and liability management model, Oper. Res., 34 (1986), 356-376. [7] H. G. Deallenbach and S. A. Archer, Optimal bank liquidity: A multi-period stochastic model, J. Financ. Quant. Anal., 4 (1969), 329-343. [8] G. D. Eppen and E. F. Fama, Three assets cash balance and dynamic portfolio problems, Manage. Sci., 17 (1971), 311-319. [9] L. Epstein, A definition of uncertainty aversion, Rev. Econ. Stud., 66 (1999), 579-608.  doi: 10.1111/1467-937X.00099. [10] J. J. Judice and A. Faustion, The linear-quadratic bilevel programming problem, INFOR, 32 (1994), 87-98. [11] P. Klibanoff, M. Marinacci and S. Mukerji, A smooth model of decision making under uncertainy, Econometrica, 73 (2005), 1849-1892.  doi: 10.1111/j.1468-0262.2005.00640.x. [12] S. Li, J. Peng and B. Zhang, The uncertain premium principle based on the distortion function, Insur. Math. Econ., 53 (2013), 317-324.  doi: 10.1016/j.insmatheco.2013.06.005. [13] J. Lintner, The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets, Rev. Econ. Stat., 47 (1965), 317-324. [14] B. Liu, Stackelberg-Nash equilibrium for multilevel programming with multiple followers using genetic algorithms, Comput. Math. Appl., 36 (1998), 79-89.  doi: 10.1016/S0898-1221(98)00174-6. [15] B. Liu, Uncertainty Theory, 2$^{nd}$ edition, Springer-Verlag, Berlin, 2004. [16] B. Liu, Some research problems in uncertainty theory, Journal of Uncertain Systems, 3 (2009), 3-10. [17] B. Liu, Theory and Practice of Uncertain Programming, 2$^{nd}$ edition, Springer-Verlag, Berlin, 2009. [18] B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 2010. [19] B. Liu, Why is there a need for uncertainty theory, J. Uncertain Systems, 6 (2012), 3-10. [20] Y. Liu and M. Ha, Expected value of function of uncertain variables, J. Uncertain Systems, 4 (2010), 181-186. [21] H. Markowitz, Portfolio Selection, New York; Chapman & Hall, Ltd., London 1959 [22] J. M. Mulvey, G. Gould and C. Morgan, Asset and liability management system for Towers Perrin-Tillinghast, Interfaces, 7 (30), 96-114. [23] J. Mossin, Equilibrium in a capital asset market, Econometrica, 34 (1966), 768-783. [24] D. H. Pyle, On the theory of financial intermedian, J. Financ., 26 (1971), 737-746. [25] G. Savard and J. Gauvin, The steepest descent direction for the nonlinear bilevel programming problem, Oper. Res. Lett., 15 (1994), 265-272.  doi: 10.1016/0167-6377(94)90086-8. [26] W. Sharpe, Capital asset prices: A theory of market equilibrium under conditions of risk, J. Financ., 19 (1964), 425-442. [27] H. V. Stackelberg, Marktform und Gleichgewicht, Springer-Verlag, Wien & Berlin, 1934. [28] J. L. Treynor and K. K. Mazuy, Can mutual funds outguess the market, Harvard Bus. Rev., 8 (1966), 131-136. [29] C. Wang, Y. Ni and X. Yang, The inventory replenishment policy in an uncertain production-inventory-routing system, J. Ind. Manag. Optim..
Stackelberg Competition
Investment Items Parameters
 Investment Items $i$ Decision Variables Yield rate $\xi_{i}$ Beta-Value $\beta_{i}$ Unsystematic risk $\sigma_{i}$ Risk asset 1 $1$ $A_{1}$ ${\mathcal{N}}(0.1,0.01)$ $0.9$ $0.01$ Risk asset 2 $2$ $A_{2}$ ${\mathcal{N}}(0.08,0.005)$ $0.6$ $0.005$ Risk-free asset $3$ $A_{3}$ $0.05$ $0.5$ $0$
 Investment Items $i$ Decision Variables Yield rate $\xi_{i}$ Beta-Value $\beta_{i}$ Unsystematic risk $\sigma_{i}$ Risk asset 1 $1$ $A_{1}$ ${\mathcal{N}}(0.1,0.01)$ $0.9$ $0.01$ Risk asset 2 $2$ $A_{2}$ ${\mathcal{N}}(0.08,0.005)$ $0.6$ $0.005$ Risk-free asset $3$ $A_{3}$ $0.05$ $0.5$ $0$
Policy Products Parameters
 Policy Products $j$ Decision Variables Profit Margin $r_{j}$ drawing coefficient $\theta_{j}$ sale coefficient $s_{j}$ Policy 1 $1$ $P_{1}$ ${\mathcal{L}}(0.1,0.12 )$ $30\%$ $1$ Policy 2 $2$ $P_{2}$ ${\mathcal{L}}(0.08,0.11)$ $40\%$ $1.1$
 Policy Products $j$ Decision Variables Profit Margin $r_{j}$ drawing coefficient $\theta_{j}$ sale coefficient $s_{j}$ Policy 1 $1$ $P_{1}$ ${\mathcal{L}}(0.1,0.12 )$ $30\%$ $1$ Policy 2 $2$ $P_{2}$ ${\mathcal{L}}(0.08,0.11)$ $40\%$ $1.1$
Optimal Solutions of Different Programmings6
 Approaches Programmings Gross Profit Policy Allocation $\bf{P}$ Investment Allocation $\bf{A}$ Stackelberg Competition (11) $137.2$ $(0, 1100)$ $(162, 0,277.3)$ Decentralized Decision Making (12), (13) $135$ $(1000, 0)$ $(50,250, 0)$ Centralized Decision Making7 (14) $134.8$ $(0, 1100)$ $(165, 0,275)$
 Approaches Programmings Gross Profit Policy Allocation $\bf{P}$ Investment Allocation $\bf{A}$ Stackelberg Competition (11) $137.2$ $(0, 1100)$ $(162, 0,277.3)$ Decentralized Decision Making (12), (13) $135$ $(1000, 0)$ $(50,250, 0)$ Centralized Decision Making7 (14) $134.8$ $(0, 1100)$ $(165, 0,275)$
 [1] Zhongbao Zhou, Ximei Zeng, Helu Xiao, Tiantian Ren, Wenbin Liu. Multiperiod portfolio optimization for asset-liability management with quadratic transaction costs. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1493-1515. doi: 10.3934/jimo.2018106 [2] Yu Yuan, Hui Mi. Robust optimal asset-liability management with penalization on ambiguity. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021121 [3] Huai-Nian Zhu, Cheng-Ke Zhang, Zhuo Jin. Continuous-time mean-variance asset-liability management with stochastic interest rates and inflation risks. Journal of Industrial and Management Optimization, 2020, 16 (2) : 813-834. doi: 10.3934/jimo.2018180 [4] Lan Yi, Zhongfei Li, Duan Li. Multi-period portfolio selection for asset-liability management with uncertain investment horizon. Journal of Industrial and Management Optimization, 2008, 4 (3) : 535-552. doi: 10.3934/jimo.2008.4.535 [5] Lihua Bian, Zhongfei Li, Haixiang Yao. Time-consistent strategy for a multi-period mean-variance asset-liability management problem with stochastic interest rate. Journal of Industrial and Management Optimization, 2021, 17 (3) : 1383-1410. doi: 10.3934/jimo.2020026 [6] Zhe Zhang, Jiuping Xu. Bi-level multiple mode resource-constrained project scheduling problems under hybrid uncertainty. Journal of Industrial and Management Optimization, 2016, 12 (2) : 565-593. doi: 10.3934/jimo.2016.12.565 [7] Yufeng Zhou, Bin Zheng, Jiafu Su, Yufeng Li. The joint location-transportation model based on grey bi-level programming for early post-earthquake relief. Journal of Industrial and Management Optimization, 2022, 18 (1) : 45-73. doi: 10.3934/jimo.2020142 [8] Xianping Wu, Xun Li, Zhongfei Li. A mean-field formulation for multi-period asset-liability mean-variance portfolio selection with probability constraints. Journal of Industrial and Management Optimization, 2018, 14 (1) : 249-265. doi: 10.3934/jimo.2017045 [9] Yan Zhang, Yonghong Wu, Benchawan Wiwatanapataphee, Francisca Angkola. Asset liability management for an ordinary insurance system with proportional reinsurance in a CIR stochastic interest rate and Heston stochastic volatility framework. Journal of Industrial and Management Optimization, 2020, 16 (1) : 71-101. doi: 10.3934/jimo.2018141 [10] Gayatri Pany, Ram N. Mohapatra. A study on vector variational-like inequalities using convexificators and application to its bi-level form. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021161 [11] Mourad Nachaoui, Lekbir Afraites, Aissam Hadri, Amine Laghrib. A non-convex non-smooth bi-level parameter learning for impulse and Gaussian noise mixture removing. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1249-1291. doi: 10.3934/cpaa.2022018 [12] Lekbir Afraites, Aissam Hadri, Amine Laghrib, Mourad Nachaoui. A non-convex denoising model for impulse and Gaussian noise mixture removing using bi-level parameter identification. Inverse Problems and Imaging, 2022, 16 (4) : 827-870. doi: 10.3934/ipi.2022001 [13] Paul B. Hermanns, Nguyen Van Thoai. Global optimization algorithm for solving bilevel programming problems with quadratic lower levels. Journal of Industrial and Management Optimization, 2010, 6 (1) : 177-196. doi: 10.3934/jimo.2010.6.177 [14] Lan Luo, Zhe Zhang, Yong Yin. Simulated annealing and genetic algorithm based method for a bi-level seru loading problem with worker assignment in seru production systems. Journal of Industrial and Management Optimization, 2021, 17 (2) : 779-803. doi: 10.3934/jimo.2019134 [15] Xiaoni Chi, Zhongping Wan, Zijun Hao. Second order sufficient conditions for a class of bilevel programs with lower level second-order cone programming problem. Journal of Industrial and Management Optimization, 2015, 11 (4) : 1111-1125. doi: 10.3934/jimo.2015.11.1111 [16] Francesca Biagini, Jacopo Mancin. Financial asset price bubbles under model uncertainty. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 14-. doi: 10.1186/s41546-017-0026-3 [17] Yu Chen, Zixian Cui, Shihan Di, Peibiao Zhao. Capital asset pricing model under distribution uncertainty. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021113 [18] Fatima Fali, Mustapha Moulaï. Solving discrete linear fractional bilevel programs with multiple objectives at the upper level. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022059 [19] Yibing Lv, Zhongping Wan. Linear bilevel multiobjective optimization problem: Penalty approach. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1213-1223. doi: 10.3934/jimo.2018092 [20] Harald Held, Gabriela Martinez, Philipp Emanuel Stelzig. Stochastic programming approach for energy management in electric microgrids. Numerical Algebra, Control and Optimization, 2014, 4 (3) : 241-267. doi: 10.3934/naco.2014.4.241

2021 Impact Factor: 1.411