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A generalized approach to sparse and stable portfolio optimization problem
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 
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 twodimensional stochastic optimal control problem, whose semianalytical 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.
References:
[1] 
H. Albrecher, P. Azcue and N. Muler, Optimal dividend strategies for two collaborating insurance companies, Applied Probability, 49 (2017), 515548. doi: 10.1017/apr.2017.11. 
[2] 
B. Avanz, Strategies for dividend distribution: A review, North American Actuarial Journal, 13 (2009), 217251. doi: 10.1080/10920277.2009.10597549. 
[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), 156180. doi: 10.1214/105051606000000709. 
[4] 
P. Azcue and N. Muler, Minimizing the ruin probability allowing investments in two assets: A twodimensional problem, Mathematical Methods of Operations Research, 77 (2013), 177206. doi: 10.1007/s0018601204243. 
[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), 85110. 
[6] 
J. Chevalier and G. Ellison, Risk taking by mutual funds as a response to incentives, Journal of Political Economy, 105 (1997), 11671199. 
[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), 319334. 
[8] 
J. S. Demski and G. A. Feltham, Econiomic incentives in budgetary control systems, Accounting Review, 53 (1978), 336360. 
[9] 
K. M. Eisenhardt, Agency theory: An assessment and review, Academy of Management Review, 14 (1989), 5774. 
[10] 
L. F. Fant and E. S. O'Neal, Temporal changes in the determinants of mutual fund flows, Journal of Financial Research, 23 (2000), 353371. 
[11] 
D. P. Foster and H. Peyton Young, Gaming performance fees by portfolio managers, The Quarterly Journal of Economics, 125 (2010), 14351458. 
[12] 
J. GilBazo and P. RuizVerd$\acute{u}$, The relation between price and performance in the mutual fund industry, The Journal of Finance, 64 (2009), 21532183. 
[13] 
W. N. Goetzmann and R. G. Ibbotson, Do winners repeat, Journal of Portfolio Management, 20 (1994), 918. 
[14] 
L. GomezMejia and R. M. Wiseman, Refraining executive compensation: An assessment and out look, Journal of Management, 23 (1997), 291374. 
[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), 523557. 
[16] 
T. Houge and J. Wellman, Fallout from the mutual fund trading scandal, Journal of Business Ethics, 62 (2005), 129139. 
[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), 221238. doi: 10.1007/s1025501706536. 
[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), 279293. doi: 10.1007/s1095700995597. 
[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), 365389. doi: 10.3934/jimo.2013.9.365. 
[20] 
P. L. Lions and A. S. Sznitman, Stochastic differential equations with reflecting boundary conditions, Communications on Pure and Applied Mathematics, 37 (1984), 511537. doi: 10.1002/cpa.3160370408. 
[21] 
B. Maiden, SEC faces critics over mutual funds scandal, International Financial Law Review, 22 (2003), 2123. 
[22] 
B. Øksendal and A. Sulem, Applied Stochastic Control of Jump Diffusions, 2^{nd} edition, Springer, 2007. 
[23] 
S. Reynolds, F. Schultz and D. Hekman, Stakeholder theory and managerial decisionmaking: Constraints and implications of balancing stakeholder interests, Journal of Business Ethics, 64 (2006), 285301. 
[24] 
E. R. Sirri and P. Tufano, Costly search and mutual fund flows, Journal of Finance, 53 (1998), 15891622. 
[25] 
N. Stoughton, Moral hazard and the portfolio management problem, Journal of Finance, 48 (1993), 20092028. 
[26] 
J. H. Witte and C. Reisinger, A penalty method for the numerical solution of HamiltonJacobiBellman (HJB) equations in finance, SIAM Journal on Numerical Analysis, 49 (2011), 213231. doi: 10.1137/100797606. 
[27] 
J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer Science and Business Media, 1999. 
show all references
References:
[1] 
H. Albrecher, P. Azcue and N. Muler, Optimal dividend strategies for two collaborating insurance companies, Applied Probability, 49 (2017), 515548. doi: 10.1017/apr.2017.11. 
[2] 
B. Avanz, Strategies for dividend distribution: A review, North American Actuarial Journal, 13 (2009), 217251. doi: 10.1080/10920277.2009.10597549. 
[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), 156180. doi: 10.1214/105051606000000709. 
[4] 
P. Azcue and N. Muler, Minimizing the ruin probability allowing investments in two assets: A twodimensional problem, Mathematical Methods of Operations Research, 77 (2013), 177206. doi: 10.1007/s0018601204243. 
[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), 85110. 
[6] 
J. Chevalier and G. Ellison, Risk taking by mutual funds as a response to incentives, Journal of Political Economy, 105 (1997), 11671199. 
[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), 319334. 
[8] 
J. S. Demski and G. A. Feltham, Econiomic incentives in budgetary control systems, Accounting Review, 53 (1978), 336360. 
[9] 
K. M. Eisenhardt, Agency theory: An assessment and review, Academy of Management Review, 14 (1989), 5774. 
[10] 
L. F. Fant and E. S. O'Neal, Temporal changes in the determinants of mutual fund flows, Journal of Financial Research, 23 (2000), 353371. 
[11] 
D. P. Foster and H. Peyton Young, Gaming performance fees by portfolio managers, The Quarterly Journal of Economics, 125 (2010), 14351458. 
[12] 
J. GilBazo and P. RuizVerd$\acute{u}$, The relation between price and performance in the mutual fund industry, The Journal of Finance, 64 (2009), 21532183. 
[13] 
W. N. Goetzmann and R. G. Ibbotson, Do winners repeat, Journal of Portfolio Management, 20 (1994), 918. 
[14] 
L. GomezMejia and R. M. Wiseman, Refraining executive compensation: An assessment and out look, Journal of Management, 23 (1997), 291374. 
[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), 523557. 
[16] 
T. Houge and J. Wellman, Fallout from the mutual fund trading scandal, Journal of Business Ethics, 62 (2005), 129139. 
[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), 221238. doi: 10.1007/s1025501706536. 
[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), 279293. doi: 10.1007/s1095700995597. 
[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), 365389. doi: 10.3934/jimo.2013.9.365. 
[20] 
P. L. Lions and A. S. Sznitman, Stochastic differential equations with reflecting boundary conditions, Communications on Pure and Applied Mathematics, 37 (1984), 511537. doi: 10.1002/cpa.3160370408. 
[21] 
B. Maiden, SEC faces critics over mutual funds scandal, International Financial Law Review, 22 (2003), 2123. 
[22] 
B. Øksendal and A. Sulem, Applied Stochastic Control of Jump Diffusions, 2^{nd} edition, Springer, 2007. 
[23] 
S. Reynolds, F. Schultz and D. Hekman, Stakeholder theory and managerial decisionmaking: Constraints and implications of balancing stakeholder interests, Journal of Business Ethics, 64 (2006), 285301. 
[24] 
E. R. Sirri and P. Tufano, Costly search and mutual fund flows, Journal of Finance, 53 (1998), 15891622. 
[25] 
N. Stoughton, Moral hazard and the portfolio management problem, Journal of Finance, 48 (1993), 20092028. 
[26] 
J. H. Witte and C. Reisinger, A penalty method for the numerical solution of HamiltonJacobiBellman (HJB) equations in finance, SIAM Journal on Numerical Analysis, 49 (2011), 213231. doi: 10.1137/100797606. 
[27] 
J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer Science and Business Media, 1999. 
Parameter  Value 
 0.8 
 0.8 
 0.03 
 0.04 
 0.04 
 0.3 
 0.3 
 10.7 
 10.7 
Case 1.  
 0.1 
 0.05 
 0.08 
 0.03 
Case 2.  
 0.06 
 0.05 
 0.08 
 0.03 
Parameter  Value 
 0.8 
 0.8 
 0.03 
 0.04 
 0.04 
 0.3 
 0.3 
 10.7 
 10.7 
Case 1.  
 0.1 
 0.05 
 0.08 
 0.03 
Case 2.  
 0.06 
 0.05 
 0.08 
 0.03 
Keynote  Value 
The management fee of Fund One without tunneling:  15.55 
The management fee of Fund Two without tunneling:  9.65 
The total management fee without tunneling:  25.2 
The management fee of Fund One with tunneling:  31.02 
The management fee of Fund Two with tunneling:  0.46 
The total management fee with tunneling:  31.48 
The duration of the Fund One without tunneling(year):  47.14 
The duration of the Fund Two without tunneling(year):  49.28 
The duration of the Fund One with tunneling(year):  49.22 
The duration of the Fund Two with tunneling(year):  49.19 
The average exchange amount(Fund Two to Fund One):  0.012 
Keynote  Value 
The management fee of Fund One without tunneling:  15.55 
The management fee of Fund Two without tunneling:  9.65 
The total management fee without tunneling:  25.2 
The management fee of Fund One with tunneling:  31.02 
The management fee of Fund Two with tunneling:  0.46 
The total management fee with tunneling:  31.48 
The duration of the Fund One without tunneling(year):  47.14 
The duration of the Fund Two without tunneling(year):  49.28 
The duration of the Fund One with tunneling(year):  49.22 
The duration of the Fund Two with tunneling(year):  49.19 
The average exchange amount(Fund Two to Fund One):  0.012 
Keynote  Value 
The management fee of Fund One without tunneling:  3.05 
The management fee of Fund Two without tunneling:  9.39 
The total management fee without tunneling:  12.44 
The management fee of Fund One with tunneling:  0.71 
The management fee of Fund Two with tunneling:  17.41 
The total management fee with tunneling:  18.12 
The duration of the Fund One without tunneling(year):  6.84 
The duration of the Fund Two without tunneling(year):  45.87 
The duration of the Fund One with tunneling(year):  45.13 
The duration of the Fund Two with tunneling(year):  45.17 
The average exchange amount(Fund Two to Fund One):  0.013 
Keynote  Value 
The management fee of Fund One without tunneling:  3.05 
The management fee of Fund Two without tunneling:  9.39 
The total management fee without tunneling:  12.44 
The management fee of Fund One with tunneling:  0.71 
The management fee of Fund Two with tunneling:  17.41 
The total management fee with tunneling:  18.12 
The duration of the Fund One without tunneling(year):  6.84 
The duration of the Fund Two without tunneling(year):  45.87 
The duration of the Fund One with tunneling(year):  45.13 
The duration of the Fund Two with tunneling(year):  45.17 
The average exchange amount(Fund Two to Fund One):  0.013 
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