| Title:
|
Expected utility maximization and conditional value-at-risk deviation-based Sharpe ratio in dynamic stochastic portfolio optimization (English) |
| Author:
|
Kilianová, Soňa |
| Author:
|
Ševčovič, Daniel |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 (print) |
| ISSN:
|
1805-949X (online) |
| Volume:
|
54 |
| Issue:
|
6 |
| Year:
|
2018 |
| Pages:
|
1167-1183 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we investigate the expected terminal utility maximization approach for a dynamic stochastic portfolio optimization problem. We solve it numerically by solving an evolutionary Hamilton-Jacobi-Bellman equation which is transformed by means of the Riccati transformation. We examine the dependence of the results on the shape of a chosen utility function in regard to the associated risk aversion level. We define the Conditional value-at-risk deviation ($CVaRD$) based Sharpe ratio for measuring risk-adjusted performance of a dynamic portfolio. We compute optimal strategies for a portfolio investment problem motivated by the German DAX 30 Index and we evaluate and analyze the dependence of the $CVaRD$-based Sharpe ratio on the utility function and the associated risk aversion level. (English) |
| Keyword:
|
dynamic stochastic portfolio optimization |
| Keyword:
|
Hamilton-Jacobi-Bellman equation |
| Keyword:
|
Conditional value-at-risk |
| Keyword:
|
$CVaRD$-based Sharpe ratio |
| MSC:
|
34E05 |
| MSC:
|
35K55 |
| MSC:
|
70H20 |
| MSC:
|
90C15 |
| MSC:
|
91B16 |
| MSC:
|
91B70 |
| idZBL:
|
Zbl 07031767 |
| idMR:
|
MR3902627 |
| DOI:
|
10.14736/kyb-2018-6-1167 |
| . |
| Date available:
|
2019-02-18T14:47:34Z |
| Last updated:
|
2020-01-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147603 |
| . |
| Reference:
|
[1] Abe, R., Ishimura, N.: Existence of solutions for the nonlinear partial differential equation arising in the optimal investment problem..Proc. Japan Acad. Ser. A 84 (2008), 1, 11-14. MR 2381178, 10.3792/pjaa.84.11 |
| Reference:
|
[2] Agarwal, V., Naik, N. Y.: Risk and portfolio decisions involving hedge funds..Rev. Financ. Stud. 17 (2004), 1, 63-98. 10.1093/rfs/hhg044 |
| Reference:
|
[3] Andrieu, L., Lara, M. De, Seck, B.: Conditional Value-at-Risk Constraint and Loss Aversion Utility Functions..https://arxiv.org/pdf/0906.3425.pdf |
| Reference:
|
[4] Arrow, K. J.: Aspects of the theory of risk bearing..In: The Theory of Risk Aversion. Helsinki: Yrjo Jahnssonin Saatio. (Reprinted in: Essays in the Theory of Risk Bearing, Markham Publ. Co., Chicago, 1971), (1965), pp. 90-109. MR 0363427 |
| Reference:
|
[5] Aubin, J. P.: Lipschitz behavior of solutions to convex minimization problems..Math. Oper. Res. 9 (1984), 87-111. MR 0736641, 10.1287/moor.9.1.87 |
| Reference:
|
[6] Bank, B., Guddat, J., Klatte, D., Kummer, B., Tammer, K.: Non-linear Parametric Optimization. Licensed ed..Birkhauser Verlag, Basel-Boston, Mass., 1983. MR 0701243, 10.1007/978-3-0348-6328-5 |
| Reference:
|
[7] Bertsekas, D. P.: Dynamic Programming and Stochastic Control..Academic Press, New York 1976. MR 0688509, 10.1016/s0076-5392(08)x6050-3 |
| Reference:
|
[8] Biglova, A., Ortobelli, S., Rachev, S., Stoyanov, S.: Different Approaches to Risk Estimation in Portfolio Theory..J. Portfolio Management 31 (2004), 1, 103-112. 10.3905/jpm.2004.443328 |
| Reference:
|
[9] Browne, S.: Risk-constrained dynamic active portfolio management..Management Sci. 46 (2000), 9, 1188-1199. 10.1287/mnsc.46.9.1188.12233 |
| Reference:
|
[10] Denuit, M., Dhaene, J., Goovaerts, M., Kaas, R., Laeven, R.: Risk measurement with equivalent utility principles..Statist. Decisions 24 (2006), 1-25. MR 2323186, 10.1524/stnd.2006.24.1.1 |
| Reference:
|
[11] Farinelli, S., Ferreira, M., Rosselloc, D., Thoeny, M., Tibiletti, L.: Beyond Sharpe ratio: Optimal asset allocation using different performance ratios..J. Banking Finance 32 (2008), 10, 2057-2063. 10.1016/j.jbankfin.2007.12.026 |
| Reference:
|
[12] Huang, Y., Forsyth, P. A., Labahn, G.: Combined fixed point and policy iteration for Hamilton-Jacobi-Bellman equations in finance..SIAM J. Numer. Anal. 50 (2012), 4, 1861-1882. MR 3022201, 10.1137/100812641 |
| Reference:
|
[13] Ishimura, N., Koleva, M. N., Vulkov, L. G.: Numerical solution via transformation methods of nonlinear models in option pricing..AIP Conference Proceedings 1301 (2010), 1, 387-394. 10.1063/1.3526637 |
| Reference:
|
[14] Ishimura, N., Ševčovič, D.: On traveling wave solutions to a Hamilton-Jacobi-Bellman equation with inequality constraints..Japan J. Ind. Appl. Math. 30 (2013), 1, 51-67. MR 3022806, 10.1007/s13160-012-0087-8 |
| Reference:
|
[15] Karatzas, I., Lehoczky, J. P., Sethi, S. P., Shreve, S.: Explicit solution of a general consumption/investment problem..Math. Oper. Res. 11 (1986), 2, 261-294. MR 0844005, 10.1287/moor.11.2.261 |
| Reference:
|
[16] Kilianová, S., Ševčovič, D.: A transformation method for solving the Hamilton-Jacobi-Bellman equation for a constrained dynamic stochastic optimal allocation problem..ANZIAM J. 55 (2013), 14-38. MR 3144202, 10.21914/anziamj.v55i0.6816 |
| Reference:
|
[17] Kilianová, S., Trnovská, M.: Robust portfolio optimization via solution to the Hamilton-Jacobi-Bellman equation..Int. J. Comput. Math. 93 (2016), 725-734. MR 3483306, 10.1080/00207160.2013.871542 |
| Reference:
|
[18] Klatte, D.: On the {L}ipschitz behavior of optimal solutions in parametric problems of quadratic optimization and linear complementarity..Optim. J. Math. Program. Oper. Res. 16 (1985), 6, 819-831. MR 0814211, 10.1080/02331938508843080 |
| Reference:
|
[19] Koleva, M. N.: Iterative methods for solving nonlinear parabolic problem in pension saving management..AIP Confer. Proc. 1404 (2011), 1, 457-463. 10.1063/1.3659948 |
| Reference:
|
[20] Koleva, M. N., Vulkov, L.: Quasilinearization numerical scheme for fully nonlinear parabolic problems with applications in models of mathematical finance..Math. Comput. Modell. 57 (2013), 2564-2575. MR 3068748, 10.1016/j.mcm.2013.01.008 |
| Reference:
|
[21] Kútik, P., Mikula, K.: Finite volume schemes for solving nonlinear partial differential equations in financial mathematics..In: Finite Volumes for Complex Applications VI, Problems and Perspectives (J. Fořt, J. Fürst, J. Halama, R. Herbin, and F. Hubert, eds.), Springer Proc. Math. 4 (2011), pp. 643-651. MR 2882342, 10.1007/978-3-642-20671-9_68 |
| Reference:
|
[22] LeVeque, R. J.: Finite Volume Methods for Hyperbolic Problems..Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge 2002. MR 1925043, 10.1017/cbo9780511791253 |
| Reference:
|
[23] Lin, S., Ohnishi, M.: Optimal portfolio selection by CVaR based Sharpe ratio genetic algorithm approach..Sci. Math. Japon. Online e-2006 (2006), 1229-1251. MR 2300341 |
| Reference:
|
[24] Macová, Z., Ševčovič, D.: Weakly nonlinear analysis of the Hamilton-Jacobi-Bellman equation arising from pension savings management..Int. J. Numer. Anal. Model. 7 (2010), 4, 619-638. MR 2644295 |
| Reference:
|
[25] McNeil, A. J., Frey, R., Embrechts, P.: Quantitattive Risk Management..Princeton Series in Finance, Princeton University Press, 2005. MR 2175089, 10.1007/s11408-006-0016-4 |
| Reference:
|
[26] Merton, R. C.: Optimal consumption and portfolio rules in a continuous time model..J. Economic Theory 71 (1971), 373-413. MR 0456373, 10.1016/0022-0531(71)90038-x |
| Reference:
|
[27] Milgrom, P., Segal, I.: Envelope theorems for arbitrary choice sets..Econometrica 70 (2002), 2, 583-601. MR 1913824, 10.1111/1468-0262.00296 |
| Reference:
|
[28] Musiela, M., Zariphopoulou, T.: An example of indifference prices under exponential preferences..Finance Stochast. 8 (2004), 2, 229-239. MR 2048829, 10.1007/s00780-003-0112-5 |
| Reference:
|
[29] Muthuraman, K., Kumar, S.: Multidimensional portfolio optimization with proportional transaction costs..Math. Finance 16 (2006), 2, 301-335. MR 2212268, 10.1111/j.1467-9965.2006.00273.x |
| Reference:
|
[30] Pflug, G. Ch., Römisch, W.: Modeling, Measuring and Managing Risk..World Scientific Publushing, 2007. MR 2424523, 10.1142/6478 |
| Reference:
|
[31] Post, T., Fang, Y., Kopa, M.: Linear tests for DARA stochastic dominance..Management Sci. 61 (2015), 1615-1629. MR 0668272, 10.1287/mnsc.2014.1960 |
| Reference:
|
[32] Pratt, J. W.: Risk aversion in the small and in the large..Econometrica. 32 (1964), 1-2, 122-136. Zbl 0267.90010, 10.2307/1913738 |
| Reference:
|
[33] Protter, M. H., Weinberger, H. F.: Maximum Principles in Differential Equations..Springer-Verlag, New York 1984. MR 0762825, 10.1007/978-1-4612-5282-5 |
| Reference:
|
[34] Reisinger, C., Witte, J. H.: On the use of policy iteration as an easy way of pricing American options..SIAM J. Financ. Math. 3 (2012), 459-478. MR 2968042, 10.1137/110823328 |
| Reference:
|
[35] Seck, B., Andrieu, L., Lara, M. De: Parametric multi-attribute utility functions for optimal profit under risk constraints..Theory Decision. 72 (2012), 2, 257-271. MR 2878102, 10.1007/s11238-011-9255-6 |
| Reference:
|
[36] Sharpe, W. F.: The Sharpe ratio..J. Portfolio Management 21 (1994), 1, 49-58. 10.3905/jpm.1994.409501 |
| Reference:
|
[37] Ševčovič, D., Stehlíková, B., Mikula, K.: Analytical and Numerical Methods for Pricing Financial Derivatives..Nova Science Publishers, Inc., Hauppauge 2011. |
| Reference:
|
[38] Tourin, A., Zariphopoulou, T.: Numerical schemes for investment models with singular transactions..Comput. Econ. 7 (1994), 4, 287-307. MR 1318095, 10.1007/bf01299457 |
| Reference:
|
[39] Vickson, R. G.: Stochastic dominance for decreasing absolute risk aversion..J. Financial Quantitative Analysis 10 (1975), 799-811. 10.2307/2330272 |
| Reference:
|
[40] Wiesinger, A.: Risk-Adjusted Performance Measurement State of the Art..Bachelor Thesis of the University of St. Gallen School of Business Administration, Economics, Law and Social Sciences (HSG), 2010. |
| Reference:
|
[41] Xia, J.: Risk aversion and portfolio selection in a continuous-time model..J. Control Optim. 49 (2011), 5, 1916-1937. MR 2837505, 10.1137/10080871x |
| Reference:
|
[42] Zariphopoulou, T.: Consumption-investment models with constraints..SIAM J. Control Optim. 32 (1994), 1, 59-85. MR 1255960, 10.1137/s0363012991218827 |
| Reference:
|
[43] Zheng, H.: Efficient frontier of utility and CVaR..Math. Meth. Oper. Res. 70 (2009), 1, 129-148. MR 2529428, 10.1007/s00186-008-0234-9 |
| . |
