# American Institute of Mathematical Sciences

October  2018, 14(4): 1617-1649. doi: 10.3934/jimo.2018024

## Tunneling behaviors of two mutual funds

 1 China Financial Policy Research Center, School of Finance, Renmin University of China, Beijing 100872, China 2 Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

*Corresponding author

Received  June 2017 Revised  October 2017 Published  January 2018

Fund Project: The first author is supported by National Natural Science Foundation of China grant No.71501178; The second author is supported by National Natural Science Foundation of China grant No.11471183.

In practice, the mutual fund manager charges asset based management fee as the incentives. Meanwhile, we suppose that the investor could sustainedly obtain the fixed proportions of the fund values as the rewards. In this perspective, the objectives of the investor and the manager seem to be consistent. Unfortunately, it is a common situation that the fund managers have private relations, and they transfer the assets illegally. In this paper, we study the optimal tunneling behaviors of the two fund managers to maximize the overall performance criterions. It is the first time to use two prototypes whether the management fee rates are consistent with the investment returns to study the impacts of the two factors on the tunneling behaviors. We firstly study the problem without transaction cost between funds, and it is formalized as a two-dimensional stochastic optimal control problem, whose semi-analytical solution is derived by the dynamic programming methods. Furthermore, the transaction cost is considered, and we explore the penalty method and the finite difference method to establish the numerical solutions. The results show that the well performed and high rewarded fund manager obtains most of the total assets by tunneling, and only keep the other fund at the brink of maximal withdraws for the liquidity considerations. Moreover, the well performed and low rewarded fund manager obtains most of the total assets. Being inconsistent with the instinct, the high management fee rate could neither make the fund managers work efficiently, nor induce the beneficial tunneling behaviors.

Citation: Lin He, Zongxia Liang, Xiaoyang Zhao. Tunneling behaviors of two mutual funds. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1617-1649. doi: 10.3934/jimo.2018024
##### References:
 [1] H. Albrecher, P. Azcue and N. Muler, Optimal dividend strategies for two collaborating insurance companies, Applied Probability, 49 (2017), 515-548.  doi: 10.1017/apr.2017.11.  Google Scholar [2] B. Avanz, Strategies for dividend distribution: A review, North American Actuarial Journal, 13 (2009), 217-251.  doi: 10.1080/10920277.2009.10597549.  Google Scholar [3] F. Avram, Z. Palmowski and M. R. Pistorius, On the optimal dividend problem for a spectrally negative l$\acute{e}$vy process, The Annals of Applied Probability, 17 (2007), 156-180.  doi: 10.1214/105051606000000709.  Google Scholar [4] P. Azcue and N. Muler, Minimizing the ruin probability allowing investments in two assets: A two-dimensional problem, Mathematical Methods of Operations Research, 77 (2013), 177-206.  doi: 10.1007/s00186-012-0424-3.  Google Scholar [5] K. C. Brown, W. V. Harlow and L. T. Starks, Of tournaments and temptations: an analysis of managerial incentives in the mutual fund industry, Journal of Finance, 51 (1996), 85-110.   Google Scholar [6] J. Chevalier and G. Ellison, Risk taking by mutual funds as a response to incentives, Journal of Political Economy, 105 (1997), 1167-1199.   Google Scholar [7] J. L. Davis, G. Tyge Payne and G. C. McMahan, A few bad apples? scandalous behavior of mutual fund managers, Journal of Business Ethics, 76 (2007), 319-334.   Google Scholar [8] J. S. Demski and G. A. Feltham, Econiomic incentives in budgetary control systems, Accounting Review, 53 (1978), 336-360.   Google Scholar [9] K. M. Eisenhardt, Agency theory: An assessment and review, Academy of Management Review, 14 (1989), 57-74.   Google Scholar [10] L. F. Fant and E. S. O'Neal, Temporal changes in the determinants of mutual fund flows, Journal of Financial Research, 23 (2000), 353-371.   Google Scholar [11] D. P. Foster and H. Peyton Young, Gaming performance fees by portfolio managers, The Quarterly Journal of Economics, 125 (2010), 1435-1458.   Google Scholar [12] J. Gil-Bazo and P. Ruiz-Verd$\acute{u}$, The relation between price and performance in the mutual fund industry, The Journal of Finance, 64 (2009), 2153-2183.   Google Scholar [13] W. N. Goetzmann and R. G. Ibbotson, Do winners repeat, Journal of Portfolio Management, 20 (1994), 9-18.   Google Scholar [14] L. Gomez-Mejia and R. M. Wiseman, Refraining executive compensation: An assessment and out look, Journal of Management, 23 (1997), 291-374.   Google Scholar [15] D. Guercio and P. A. Tkac, The determinants of the flow of funds of managed portfolios: Mutual funds versus pension funds, Journal of Financial and Quantitative Analysis, 37 (2002), 523-557.   Google Scholar [16] T. Houge and J. Wellman, Fallout from the mutual fund trading scandal, Journal of Business Ethics, 62 (2005), 129-139.   Google Scholar [17] Z. Jin, H. L. Yang and G. Yin, A numerical approach to optimal dividend policies with capital injections and transaction costs, Acta Mathematicae Applicatae Sinica, 33 (2017), 221-238.  doi: 10.1007/s10255-017-0653-6.  Google Scholar [18] W. Li and S. Wang, A penalty approach to the hjb equation arising in european stock option pricing with proportional transation costs, Journal of Optimization Theory and Applications, 143 (2009), 279-293.  doi: 10.1007/s10957-009-9559-7.  Google Scholar [19] W. Li and S. Wang, Pricing American options under proportional transaction costs using a penalty approach and a finite difference scheme, Journal of Industrial and Management Optimization, 9 (2013), 365-389.  doi: 10.3934/jimo.2013.9.365.  Google Scholar [20] P. L. Lions and A. S. Sznitman, Stochastic differential equations with reflecting boundary conditions, Communications on Pure and Applied Mathematics, 37 (1984), 511-537.  doi: 10.1002/cpa.3160370408.  Google Scholar [21] B. Maiden, SEC faces critics over mutual funds scandal, International Financial Law Review, 22 (2003), 21-23.   Google Scholar [22] B. Øksendal and A. Sulem, Applied Stochastic Control of Jump Diffusions, 2nd edition, Springer, 2007.  Google Scholar [23] S. Reynolds, F. Schultz and D. Hekman, Stakeholder theory and managerial decision-making: Constraints and implications of balancing stakeholder interests, Journal of Business Ethics, 64 (2006), 285-301.   Google Scholar [24] E. R. Sirri and P. Tufano, Costly search and mutual fund flows, Journal of Finance, 53 (1998), 1589-1622.   Google Scholar [25] N. Stoughton, Moral hazard and the portfolio management problem, Journal of Finance, 48 (1993), 2009-2028.   Google Scholar [26] J. H. Witte and C. Reisinger, A penalty method for the numerical solution of Hamilton-Jacobi-Bellman (HJB) equations in finance, SIAM Journal on Numerical Analysis, 49 (2011), 213-231.  doi: 10.1137/100797606.  Google Scholar [27] J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer Science and Business Media, 1999.  Google Scholar

show all references

##### References:
 [1] H. Albrecher, P. Azcue and N. Muler, Optimal dividend strategies for two collaborating insurance companies, Applied Probability, 49 (2017), 515-548.  doi: 10.1017/apr.2017.11.  Google Scholar [2] B. Avanz, Strategies for dividend distribution: A review, North American Actuarial Journal, 13 (2009), 217-251.  doi: 10.1080/10920277.2009.10597549.  Google Scholar [3] F. Avram, Z. Palmowski and M. R. Pistorius, On the optimal dividend problem for a spectrally negative l$\acute{e}$vy process, The Annals of Applied Probability, 17 (2007), 156-180.  doi: 10.1214/105051606000000709.  Google Scholar [4] P. Azcue and N. Muler, Minimizing the ruin probability allowing investments in two assets: A two-dimensional problem, Mathematical Methods of Operations Research, 77 (2013), 177-206.  doi: 10.1007/s00186-012-0424-3.  Google Scholar [5] K. C. Brown, W. V. Harlow and L. T. Starks, Of tournaments and temptations: an analysis of managerial incentives in the mutual fund industry, Journal of Finance, 51 (1996), 85-110.   Google Scholar [6] J. Chevalier and G. Ellison, Risk taking by mutual funds as a response to incentives, Journal of Political Economy, 105 (1997), 1167-1199.   Google Scholar [7] J. L. Davis, G. Tyge Payne and G. C. McMahan, A few bad apples? scandalous behavior of mutual fund managers, Journal of Business Ethics, 76 (2007), 319-334.   Google Scholar [8] J. S. Demski and G. A. Feltham, Econiomic incentives in budgetary control systems, Accounting Review, 53 (1978), 336-360.   Google Scholar [9] K. M. Eisenhardt, Agency theory: An assessment and review, Academy of Management Review, 14 (1989), 57-74.   Google Scholar [10] L. F. Fant and E. S. O'Neal, Temporal changes in the determinants of mutual fund flows, Journal of Financial Research, 23 (2000), 353-371.   Google Scholar [11] D. P. Foster and H. Peyton Young, Gaming performance fees by portfolio managers, The Quarterly Journal of Economics, 125 (2010), 1435-1458.   Google Scholar [12] J. Gil-Bazo and P. Ruiz-Verd$\acute{u}$, The relation between price and performance in the mutual fund industry, The Journal of Finance, 64 (2009), 2153-2183.   Google Scholar [13] W. N. Goetzmann and R. G. Ibbotson, Do winners repeat, Journal of Portfolio Management, 20 (1994), 9-18.   Google Scholar [14] L. Gomez-Mejia and R. M. Wiseman, Refraining executive compensation: An assessment and out look, Journal of Management, 23 (1997), 291-374.   Google Scholar [15] D. Guercio and P. A. Tkac, The determinants of the flow of funds of managed portfolios: Mutual funds versus pension funds, Journal of Financial and Quantitative Analysis, 37 (2002), 523-557.   Google Scholar [16] T. Houge and J. Wellman, Fallout from the mutual fund trading scandal, Journal of Business Ethics, 62 (2005), 129-139.   Google Scholar [17] Z. Jin, H. L. Yang and G. Yin, A numerical approach to optimal dividend policies with capital injections and transaction costs, Acta Mathematicae Applicatae Sinica, 33 (2017), 221-238.  doi: 10.1007/s10255-017-0653-6.  Google Scholar [18] W. Li and S. Wang, A penalty approach to the hjb equation arising in european stock option pricing with proportional transation costs, Journal of Optimization Theory and Applications, 143 (2009), 279-293.  doi: 10.1007/s10957-009-9559-7.  Google Scholar [19] W. Li and S. Wang, Pricing American options under proportional transaction costs using a penalty approach and a finite difference scheme, Journal of Industrial and Management Optimization, 9 (2013), 365-389.  doi: 10.3934/jimo.2013.9.365.  Google Scholar [20] P. L. Lions and A. S. Sznitman, Stochastic differential equations with reflecting boundary conditions, Communications on Pure and Applied Mathematics, 37 (1984), 511-537.  doi: 10.1002/cpa.3160370408.  Google Scholar [21] B. Maiden, SEC faces critics over mutual funds scandal, International Financial Law Review, 22 (2003), 21-23.   Google Scholar [22] B. Øksendal and A. Sulem, Applied Stochastic Control of Jump Diffusions, 2nd edition, Springer, 2007.  Google Scholar [23] S. Reynolds, F. Schultz and D. Hekman, Stakeholder theory and managerial decision-making: Constraints and implications of balancing stakeholder interests, Journal of Business Ethics, 64 (2006), 285-301.   Google Scholar [24] E. R. Sirri and P. Tufano, Costly search and mutual fund flows, Journal of Finance, 53 (1998), 1589-1622.   Google Scholar [25] N. Stoughton, Moral hazard and the portfolio management problem, Journal of Finance, 48 (1993), 2009-2028.   Google Scholar [26] J. H. Witte and C. Reisinger, A penalty method for the numerical solution of Hamilton-Jacobi-Bellman (HJB) equations in finance, SIAM Journal on Numerical Analysis, 49 (2011), 213-231.  doi: 10.1137/100797606.  Google Scholar [27] J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer Science and Business Media, 1999.  Google Scholar
The amount of assets transferred from Fund Two to Fund One in Case 1(without transaction cost).
The amount of assets transferred from Fund Two to Fund One in Case 2 (without transaction cost).
The amount of assets transferred from Fund Two to Fund One in Case 1 (with transaction cost).
The amount of assets transferred from Fund Two to Fund One in Case 2(with transaction cost).
The amount of assets transferred from Fund Two to Fund One in Case 1(with correlation).
The amount of assets transferred from Fund Two to Fund One in Case 2 (with correlation).
Values of the parameters in the model
 Parameter Value $a_1$ 0.8 $a_2$ 0.8 $\beta$ 0.03 $l_1$ 0.04 $l_2$ 0.04 $\sigma_1$ 0.3 $\sigma_2$ 0.3 $x_{10}$ 10.7 $x_{20}$ 10.7 Case 1. $\mu_1$ 0.1 $c_1$ 0.05 $\mu_2$ 0.08 $c_2$ 0.03 Case 2. $\mu_1$ 0.06 $c_1$ 0.05 $\mu_2$ 0.08 $c_2$ 0.03
 Parameter Value $a_1$ 0.8 $a_2$ 0.8 $\beta$ 0.03 $l_1$ 0.04 $l_2$ 0.04 $\sigma_1$ 0.3 $\sigma_2$ 0.3 $x_{10}$ 10.7 $x_{20}$ 10.7 Case 1. $\mu_1$ 0.1 $c_1$ 0.05 $\mu_2$ 0.08 $c_2$ 0.03 Case 2. $\mu_1$ 0.06 $c_1$ 0.05 $\mu_2$ 0.08 $c_2$ 0.03
The impacts of the tunneling behaviors in Case 1.
 Keynote Value The management fee of Fund One without tunneling: $V^{x_{10}}_1$ 15.55 The management fee of Fund Two without tunneling: $V^{x_{20}}_2$ 9.65 The total management fee without tunneling: $V^{x_{10}}_1+V^{x_{20}}_2$ 25.2 The management fee of Fund One with tunneling: $\tilde{V}^{x_{10}}_1$ 31.02 The management fee of Fund Two with tunneling: $\tilde{V}^{x_{20}}_2$ 0.46 The total management fee with tunneling: $V^{x_{10},x_{20}}=\tilde{V}^{x_{10}}_1+\tilde{V}^{x_{20}}_2$ 31.48 The duration of the Fund One without tunneling(year): $\mathbf{E}_{x_{10}}\{T_1\}$ 47.14 The duration of the Fund Two without tunneling(year): $\mathbf{E}_{x_{20}}\{T_2\}$ 49.28 The duration of the Fund One with tunneling(year): $\mathbf{E}_{x_{10}}\{\tau_1\}$ 49.22 The duration of the Fund Two with tunneling(year): $\mathbf{E}_{x_{20}}\{\tau_2\}$ 49.19 The average exchange amount(Fund Two to Fund One): $\mathbf{E}_{x_{10},x_{20}}\frac{\int^{\tau^1}_0 \delta(s)\mathrm{d}s}{\tau^1}$ 0.012
 Keynote Value The management fee of Fund One without tunneling: $V^{x_{10}}_1$ 15.55 The management fee of Fund Two without tunneling: $V^{x_{20}}_2$ 9.65 The total management fee without tunneling: $V^{x_{10}}_1+V^{x_{20}}_2$ 25.2 The management fee of Fund One with tunneling: $\tilde{V}^{x_{10}}_1$ 31.02 The management fee of Fund Two with tunneling: $\tilde{V}^{x_{20}}_2$ 0.46 The total management fee with tunneling: $V^{x_{10},x_{20}}=\tilde{V}^{x_{10}}_1+\tilde{V}^{x_{20}}_2$ 31.48 The duration of the Fund One without tunneling(year): $\mathbf{E}_{x_{10}}\{T_1\}$ 47.14 The duration of the Fund Two without tunneling(year): $\mathbf{E}_{x_{20}}\{T_2\}$ 49.28 The duration of the Fund One with tunneling(year): $\mathbf{E}_{x_{10}}\{\tau_1\}$ 49.22 The duration of the Fund Two with tunneling(year): $\mathbf{E}_{x_{20}}\{\tau_2\}$ 49.19 The average exchange amount(Fund Two to Fund One): $\mathbf{E}_{x_{10},x_{20}}\frac{\int^{\tau^1}_0 \delta(s)\mathrm{d}s}{\tau^1}$ 0.012
The impacts of the tunneling behaviors in Case 2.
 Keynote Value The management fee of Fund One without tunneling: $V^{x_{10}}_1$ 3.05 The management fee of Fund Two without tunneling: $V^{x_{20}}_2$ 9.39 The total management fee without tunneling: $V^{x_{10}}_1+V^{x_{20}}_2$ 12.44 The management fee of Fund One with tunneling: $\tilde{V}^{x_{10}}_1$ 0.71 The management fee of Fund Two with tunneling: $\tilde{V}^{x_{20}}_2$ 17.41 The total management fee with tunneling: $V^{x_{10},x_{20}}=\tilde{V}^{x_{10}}_1+\tilde{V}^{x_{20}}_2$ 18.12 The duration of the Fund One without tunneling(year): $\mathbf{E}_{x_{10}}\{T_1\}$ 6.84 The duration of the Fund Two without tunneling(year): $\mathbf{E}_{x_{20}}\{T_2\}$ 45.87 The duration of the Fund One with tunneling(year): $\mathbf{E}_{x_{10}}\{\tau_1\}$ 45.13 The duration of the Fund Two with tunneling(year): $\mathbf{E}_{x_{20}}\{\tau_2\}$ 45.17 The average exchange amount(Fund Two to Fund One): $\mathbf{E}_{x_{10},x_{20}}\frac{\int^{\tau^1}_0 \delta(s)\mathrm{d}s}{\tau^1}$ -0.013
 Keynote Value The management fee of Fund One without tunneling: $V^{x_{10}}_1$ 3.05 The management fee of Fund Two without tunneling: $V^{x_{20}}_2$ 9.39 The total management fee without tunneling: $V^{x_{10}}_1+V^{x_{20}}_2$ 12.44 The management fee of Fund One with tunneling: $\tilde{V}^{x_{10}}_1$ 0.71 The management fee of Fund Two with tunneling: $\tilde{V}^{x_{20}}_2$ 17.41 The total management fee with tunneling: $V^{x_{10},x_{20}}=\tilde{V}^{x_{10}}_1+\tilde{V}^{x_{20}}_2$ 18.12 The duration of the Fund One without tunneling(year): $\mathbf{E}_{x_{10}}\{T_1\}$ 6.84 The duration of the Fund Two without tunneling(year): $\mathbf{E}_{x_{20}}\{T_2\}$ 45.87 The duration of the Fund One with tunneling(year): $\mathbf{E}_{x_{10}}\{\tau_1\}$ 45.13 The duration of the Fund Two with tunneling(year): $\mathbf{E}_{x_{20}}\{\tau_2\}$ 45.17 The average exchange amount(Fund Two to Fund One): $\mathbf{E}_{x_{10},x_{20}}\frac{\int^{\tau^1}_0 \delta(s)\mathrm{d}s}{\tau^1}$ -0.013
 [1] Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437 [2] Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521 [3] Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183 [4] Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399 [5] Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233 [6] Martin Bohner, Sabrina Streipert. Optimal harvesting policy for the Beverton--Holt model. Mathematical Biosciences & Engineering, 2016, 13 (4) : 673-695. doi: 10.3934/mbe.2016014 [7] Xingchun Wang, Yongjin Wang. Variance-optimal hedging for target volatility options. Journal of Industrial & Management Optimization, 2014, 10 (1) : 207-218. doi: 10.3934/jimo.2014.10.207 [8] Andrea Cianchi, Adele Ferone. Improving sharp Sobolev type inequalities by optimal remainder gradient norms. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1363-1386. doi: 10.3934/cpaa.2012.11.1363 [9] Chih-Chiang Fang. Bayesian decision making in determining optimal leased term and preventive maintenance scheme for leased facilities. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020127 [10] Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271 [11] J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008 [12] Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327 [13] Tao Wu, Yu Lei, Jiao Shi, Maoguo Gong. An evolutionary multiobjective method for low-rank and sparse matrix decomposition. Big Data & Information Analytics, 2017, 2 (1) : 23-37. doi: 10.3934/bdia.2017006 [14] Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017 [15] Boris Kramer, John R. Singler. A POD projection method for large-scale algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 413-435. doi: 10.3934/naco.2016018 [16] Petra Csomós, Hermann Mena. Fourier-splitting method for solving hyperbolic LQR problems. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 17-46. doi: 10.3934/naco.2018002 [17] Christina Surulescu, Nicolae Surulescu. Modeling and simulation of some cell dispersion problems by a nonparametric method. Mathematical Biosciences & Engineering, 2011, 8 (2) : 263-277. doi: 10.3934/mbe.2011.8.263 [18] Min Li. A three term Polak-Ribière-Polyak conjugate gradient method close to the memoryless BFGS quasi-Newton method. Journal of Industrial & Management Optimization, 2020, 16 (1) : 245-260. doi: 10.3934/jimo.2018149 [19] Manfred Einsiedler, Elon Lindenstrauss. On measures invariant under diagonalizable actions: the Rank-One case and the general Low-Entropy method. Journal of Modern Dynamics, 2008, 2 (1) : 83-128. doi: 10.3934/jmd.2008.2.83 [20] Xiaomao Deng, Xiao-Chuan Cai, Jun Zou. A parallel space-time domain decomposition method for unsteady source inversion problems. Inverse Problems & Imaging, 2015, 9 (4) : 1069-1091. doi: 10.3934/ipi.2015.9.1069

2019 Impact Factor: 1.366