# American Institute of Mathematical Sciences

November  2022, 18(6): 4183-4190. doi: 10.3934/jimo.2021153

## Solving a fractional programming problem in a commercial bank

 1 School of Business, National University of Mongolia, Ulaanbaatar, PC 14200, Mongolia 2 Institute of Mathematics and Digital Technology, Mongolian Academy of Sciences, Ulaanbaatar, PC 13330, Mongolia

* Corresponding author: Ankhbayar Chuluunbaatar

Received  March 2021 Revised  June 2021 Published  November 2022 Early access  September 2021

We formulate a new optimization problem which arises in the Bank Asset and Liability Management (ALM). The problem is a fractional programming which belongs to a class of global optimization. Most of optimization problems in the Bank Asset and Liability Management are return maximization or risk minimization problems. For solving the fractional programming problem, we propose curvilinear multi-start algorithm which finds the best local solutions to the problem. Numerical results are given based on the balance sheets of 5 commercial banks of Mongolia.

Citation: Ankhbayar Chuluunbaatar, Enkhbat Rentsen. Solving a fractional programming problem in a commercial bank. Journal of Industrial and Management Optimization, 2022, 18 (6) : 4183-4190. doi: 10.3934/jimo.2021153
##### References:
 [1] C. Ankhbayar and R. Enkhbat, A fractional programming problem for bank asset and liability management, IBusiness, 10 (2018), 119-127. [2] J. R. Birge, The value of the stochastic solution in stochastic linear programs with fixed recourse, Math. Programming, 24 (1982), 314-325.  doi: 10.1007/BF01585113. [3] D. P. Bertsekas, Nonlinear Programming, 2$^{nd}$ edition, Athena Scientific, 1999. [4] K. J. Cohen and S. Thore, Programming bank portfolio under uncertainty, Journal of Bank Research, 2 (1970), 28-40. [5] D. Chambers and A. Charnes, Inter-temporal analysis and optimization of bank portfolio, Management Science, 7 (1961), 393-410.  doi: 10.1287/mnsc.7.4.393. [6] G. D. Eppen and E. F. Fama, Three asset cash balance and dynamic portfolio problems, Management Science, 17 (1971), 311-319.  doi: 10.1287/mnsc.17.5.311. [7] R. Enkhbat, S. Batbileg, N. Tungalag, A. Anikin and A. Gornov, A global optimization approach to nonzero sum six-person game, Frontiers in Games and Dynamic Games, (eds. Y. David, L. Shravan and K.L. Chee), Academic Press, (2020), 219–227. [8] L. F. Escudero and A. Garin, On multistage stochastic integer programming for incorporating logical constraints in asset and liability management under uncertainty, Computer Management Science, 6 (2009), 307-327.  doi: 10.1007/s10287-006-0035-7. [9] L. Eatman and J. Sealey, A multi-objective linear programming model for commercial bank balance sheet management, Journal of Bank Research, 9 (1979), 227-236. [10] D. Giokas and M. Vassiloglou, A goal programming model for bank assets and liabilities management, European Journal of Operations Research, 50 (1991), 48-60.  doi: 10.1016/0377-2217(91)90038-W. [11] K. Kosmidou and C. Zopounidis, Generating interest rate scenarios for bank asset liability management, Optimization Letters, 2 (2008), 157-169.  doi: 10.1007/s11590-007-0050-9. [12] M. I. Kusy and W. T. Ziemba, A bank asset and liability management model, Operations Research, 34 (1986), 356-376.  doi: 10.1287/opre.34.3.356. [13] K. Kosmidou and C. Zopounidis, A multi-objective methodology for bank asset and liability management, Financial Engineering, 7 (2002), 139-151. [14] R. Kouwenberg and S. Zenios, Stochastic programming models for asset and liability management, Handbook of Asset and Liability Management, (eds. S.A. Zenios and W.T. Ziemba), Academic Press, (2006), 253–303. [15] H. M. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91. [16] R. C. Merton, Lifetime portfolio selection under certainty: Continuous time case, Review of Economics and Statistics, 5 (1969), 373-413. [17] R. Mohammandi and M. Sherafati, Optimization of bank liquidity management using goal programming and fuzzy AHP, Research Journal of Recent Sciences, 4 (2015), 53-61. [18] J. D. Pinter, Global Optimization in Action, Kluwer Academic Publishers, 1996. [19] P. Samuelson, Lifetime portfolio selection by dynamic stochastic programming, Review of Economics and Statistics, 8 (1969), 239-246. [20] R. Schultz and S. Tiedemann, Risk aversion via excess probabilities in stochastic programs with mixed-integers recourse, SIAM J. Optim., 14 (2004), 115-138.  doi: 10.1137/S1052623402410855. [21] F. V. Vasiliev, Numerical Methods of Extremal Problems, Nauka, Moskow, 1998. [22] Y. Zeng and Z. Li, Asset and liability management under benchmark and mean-variance criteria in a jump diffusion market, Journal of Systems Science and Complex, 24 (2011), 317-327.  doi: 10.1007/s11424-011-9105-1.

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##### References:
 [1] C. Ankhbayar and R. Enkhbat, A fractional programming problem for bank asset and liability management, IBusiness, 10 (2018), 119-127. [2] J. R. Birge, The value of the stochastic solution in stochastic linear programs with fixed recourse, Math. Programming, 24 (1982), 314-325.  doi: 10.1007/BF01585113. [3] D. P. Bertsekas, Nonlinear Programming, 2$^{nd}$ edition, Athena Scientific, 1999. [4] K. J. Cohen and S. Thore, Programming bank portfolio under uncertainty, Journal of Bank Research, 2 (1970), 28-40. [5] D. Chambers and A. Charnes, Inter-temporal analysis and optimization of bank portfolio, Management Science, 7 (1961), 393-410.  doi: 10.1287/mnsc.7.4.393. [6] G. D. Eppen and E. F. Fama, Three asset cash balance and dynamic portfolio problems, Management Science, 17 (1971), 311-319.  doi: 10.1287/mnsc.17.5.311. [7] R. Enkhbat, S. Batbileg, N. Tungalag, A. Anikin and A. Gornov, A global optimization approach to nonzero sum six-person game, Frontiers in Games and Dynamic Games, (eds. Y. David, L. Shravan and K.L. Chee), Academic Press, (2020), 219–227. [8] L. F. Escudero and A. Garin, On multistage stochastic integer programming for incorporating logical constraints in asset and liability management under uncertainty, Computer Management Science, 6 (2009), 307-327.  doi: 10.1007/s10287-006-0035-7. [9] L. Eatman and J. Sealey, A multi-objective linear programming model for commercial bank balance sheet management, Journal of Bank Research, 9 (1979), 227-236. [10] D. Giokas and M. Vassiloglou, A goal programming model for bank assets and liabilities management, European Journal of Operations Research, 50 (1991), 48-60.  doi: 10.1016/0377-2217(91)90038-W. [11] K. Kosmidou and C. Zopounidis, Generating interest rate scenarios for bank asset liability management, Optimization Letters, 2 (2008), 157-169.  doi: 10.1007/s11590-007-0050-9. [12] M. I. Kusy and W. T. Ziemba, A bank asset and liability management model, Operations Research, 34 (1986), 356-376.  doi: 10.1287/opre.34.3.356. [13] K. Kosmidou and C. Zopounidis, A multi-objective methodology for bank asset and liability management, Financial Engineering, 7 (2002), 139-151. [14] R. Kouwenberg and S. Zenios, Stochastic programming models for asset and liability management, Handbook of Asset and Liability Management, (eds. S.A. Zenios and W.T. Ziemba), Academic Press, (2006), 253–303. [15] H. M. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91. [16] R. C. Merton, Lifetime portfolio selection under certainty: Continuous time case, Review of Economics and Statistics, 5 (1969), 373-413. [17] R. Mohammandi and M. Sherafati, Optimization of bank liquidity management using goal programming and fuzzy AHP, Research Journal of Recent Sciences, 4 (2015), 53-61. [18] J. D. Pinter, Global Optimization in Action, Kluwer Academic Publishers, 1996. [19] P. Samuelson, Lifetime portfolio selection by dynamic stochastic programming, Review of Economics and Statistics, 8 (1969), 239-246. [20] R. Schultz and S. Tiedemann, Risk aversion via excess probabilities in stochastic programs with mixed-integers recourse, SIAM J. Optim., 14 (2004), 115-138.  doi: 10.1137/S1052623402410855. [21] F. V. Vasiliev, Numerical Methods of Extremal Problems, Nauka, Moskow, 1998. [22] Y. Zeng and Z. Li, Asset and liability management under benchmark and mean-variance criteria in a jump diffusion market, Journal of Systems Science and Complex, 24 (2011), 317-327.  doi: 10.1007/s11424-011-9105-1.
The decision variables
 $Assets$ $Liabilities$ $A_1: \text{Cash and cash equivalents}$ $L_1: \text{Current account}$ $A_2: \text{Deposits to the Bank of Mongolia}$ $L_2: \text{Time deposit}$ $A_3: \text{Deposits at other banks}$ $L_3: \text{Demand deposit}$ $A_4: \text{Financial investments}$ $L_4: \text{Placements by other banks}$ $A_5: \text{Loans and advances}$ $L_5: \text{Other deposits}$ $A_6: \text{Other financial assets}$ $L_6: \text{Other liabilities}$ $A_7: \text{Fixed assets}$ $E: \text{Equity}$ $A: \text{Total assets}$ $L+E = A: \text{Total assets}$
 $Assets$ $Liabilities$ $A_1: \text{Cash and cash equivalents}$ $L_1: \text{Current account}$ $A_2: \text{Deposits to the Bank of Mongolia}$ $L_2: \text{Time deposit}$ $A_3: \text{Deposits at other banks}$ $L_3: \text{Demand deposit}$ $A_4: \text{Financial investments}$ $L_4: \text{Placements by other banks}$ $A_5: \text{Loans and advances}$ $L_5: \text{Other deposits}$ $A_6: \text{Other financial assets}$ $L_6: \text{Other liabilities}$ $A_7: \text{Fixed assets}$ $E: \text{Equity}$ $A: \text{Total assets}$ $L+E = A: \text{Total assets}$
nitial values of $v$ for banks
 $Ratio$ $Khan$ $TDB$ $XAC$ $State$ $Golomt$ $Mean$ $Stdev$ $v_1$ $0.564$ $0.520$ $0.545$ $0.409$ $0.528$ $0.513$ $0.061$ $v_2$ $0.457$ $0.403$ $0.442$ $0.599$ $0.445$ $0.469$ $0.075$ $v_3$ $0.405$ $0.253$ $0.358$ $0.496$ $0.412$ $0.385$ $0.089$ $v_4$ $0.477$ $0.452$ $0.483$ $0.362$ $0.472$ $0.449$ $0.050$ $v_5$ $0.202$ $0.203$ $0.413$ $0.181$ $0.186$ $0.237$ $0.099$ $v_6$ $0.998$ $1.648$ $1.522$ $0.988$ $0.859$ $1.203$ $0.356$ $v_7$ $0.150$ $0.296$ $0.206$ $0.064$ $0.118$ $0.166$ $0.089$ $v_8$ $0.300$ $0.232$ $0.116$ $0.251$ $0.311$ $0.242$ $0.078$ $v_9$ $0.228$ $0.232$ $0.446$ $0.197$ $0.201$ $0.260$ $0.103$ $v_{10}$ $0.129$ $0.140$ $0.070$ $0.089$ $0.081$ $0.102$ $0.031$ Source: Audited report of individual commercial bank
 $Ratio$ $Khan$ $TDB$ $XAC$ $State$ $Golomt$ $Mean$ $Stdev$ $v_1$ $0.564$ $0.520$ $0.545$ $0.409$ $0.528$ $0.513$ $0.061$ $v_2$ $0.457$ $0.403$ $0.442$ $0.599$ $0.445$ $0.469$ $0.075$ $v_3$ $0.405$ $0.253$ $0.358$ $0.496$ $0.412$ $0.385$ $0.089$ $v_4$ $0.477$ $0.452$ $0.483$ $0.362$ $0.472$ $0.449$ $0.050$ $v_5$ $0.202$ $0.203$ $0.413$ $0.181$ $0.186$ $0.237$ $0.099$ $v_6$ $0.998$ $1.648$ $1.522$ $0.988$ $0.859$ $1.203$ $0.356$ $v_7$ $0.150$ $0.296$ $0.206$ $0.064$ $0.118$ $0.166$ $0.089$ $v_8$ $0.300$ $0.232$ $0.116$ $0.251$ $0.311$ $0.242$ $0.078$ $v_9$ $0.228$ $0.232$ $0.446$ $0.197$ $0.201$ $0.260$ $0.103$ $v_{10}$ $0.129$ $0.140$ $0.070$ $0.089$ $0.081$ $0.102$ $0.031$ Source: Audited report of individual commercial bank
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