doi: 10.3934/jimo.2022074
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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. BhattacharyyaA. 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. ChenY. 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. KlibanoffM. 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. LiJ. 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. MulveyG. 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. BhattacharyyaA. 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. ChenY. 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. KlibanoffM. 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. LiJ. 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. MulveyG. 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..

Figure 1.  Stackelberg Competition
Table 1.  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 $3$ $A_{3}$ $0.05$ $0.5$ $0$
Table 2.  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 $2$ $P_{2}$ ${\mathcal{L}}(0.08,0.11)$ $40\%$ $1.1$
Table 3.  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) $134.8$ $(0, 1100)$ $(165, 0,275)$
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