# American Institute of Mathematical Sciences

April  2018, 5(2): 61-94. doi: 10.3934/jdg.2018006

## Robust portfolio decisions for financial institutions

 1 Department of Financial and Management Engineering, University of the Aegean, 41 Kountouriotou Street, GR-82100, Chios, Greece 2 Department of International and European Economic Studies, Athens University of Economics and Business, 76 Patission Street, GR-10434, Athens, Greece 3 Department of Statistics & Laboratory of Stochastic Modeling and Applications, Athens University of Economics and Business, 76 Patission Street, GR-10434, Athens, Greece

The authors would like to thank the editor and the two anonymous referees for their careful reading and helpful comments.

Received  September 2017 Revised  December 2017 Published  February 2018

The present paper aims to study a robust-entropic optimal control problem arising in the management of financial institutions. More precisely, we consider an economic agent who manages the portfolio of a financial firm. The manager has the possibility to invest part of the firm's wealth in a classical Black-Scholes type financial market, and also, as the firm is exposed to a stochastic cash flow of liabilities, to proportionally transfer part of its liabilities to a third party as a means of reducing risk. However, model uncertainty aspects are introduced as the manager does not fully trust the model she faces, hence she decides to make her decision robust. By employing robust control and dynamic programming techniques, we provide closed form solutions for the cases of the (ⅰ) logarithmic; (ⅱ) exponential and (ⅲ) power utility functions. Moreover, we provide a detailed study of the limiting behavior, of the associated stochastic differential game at hand, which, in a special case, leads to break down of the solution of the resulting Hamilton-Jacobi-Bellman-Isaacs equation. Finally, we present a detailed numerical study that elucidates the effect of robustness on the optimal decisions of both players.

Citation: Ioannis Baltas, Anastasios Xepapadeas, Athanasios N. Yannacopoulos. Robust portfolio decisions for financial institutions. Journal of Dynamics and Games, 2018, 5 (2) : 61-94. doi: 10.3934/jdg.2018006
##### References:
 [1] E. Anderson, L. Hansen and T. Sargent, A quartet of semigroups for model specification, robustness, prices of risk, and model detection, Journal of the European Economic Association, 1 (2003), 68-123. [2] E. Anderson, E. Ghysels and J. Juergens, The impact of risk and uncertainty on expected returns, Journal of Financial Economics, 94 (2009), 233-263.  doi: 10.1016/j.jfineco.2008.11.001. [3] L. Bai and G. Guo, Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance: Mathematics and Economics, 42 (2008), 968-975.  doi: 10.1016/j.insmatheco.2007.11.002. [4] I. D. Baltas, N. E. Frangos and A. N. Yannacopoulos, Optimal investment and reinsurance policies in insurance markets under the effect of inside information, Applied Stochastic Models in Business and Industry, 28 (2012), 506-528.  doi: 10.1002/asmb.925. [5] I. D. Baltas and A. N. Yannacopoulos, Uncertainty and inside information, Journal of Dynamics and Games, 3 (2016), 1-24.  doi: 10.3934/jdg.2016001. [6] E. Bayraktar and S. Yao, Doubly reflected BSDEs with integrable parameters and related Dynkin games, Stochastic Processes and their Applications, 125 (2015), 4489-4542.  doi: 10.1016/j.spa.2015.07.007. [7] S. Biagini and M. Pinar, The Robust Merton Problem of an Ambiguity-Averse Investor, Mathematics and Financial Economics, 11 (2017), 1-24.  doi: 10.1007/s11579-016-0168-6. [8] N. Branger, L. Larsen and C. Munk, Robust portfolio choice with ambiguity and learning predictability, Journal of Banking and Finance, 37 (2013), 1397-1411.  doi: 10.2139/ssrn.1859916. [9] W. A. Brock, A. Xepapadeas and A. N. Yannacopoulos, Robust control and hot spots in spatiotemporal economic systems, Dyn. Games Appl., 4 (2014), 257-289.  doi: 10.1007/s13235-014-0109-z. [10] W. A. Brock, A. Xepapadeas and A. N. Yannacopoulos, Robust control of a spatially distributed commercial fishery, in Dynamic Optimization in Environmental Economics, (eds. E. Moser, W. Semmler, G. Tragler, V. Veliov), Springer-Verlag, Heidelberg, 15 (2014), 215-241. doi: 10.1007/978-3-642-54086-8_10. [11] S. Browne, Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin, Mathematics of Operations Research, 20 (1995), 937-958.  doi: 10.1287/moor.20.4.937. [12] R. Buckdahn and J. Li, Stochastic differential games with reflection and related obstacle problems for Isaacs equations, Acta Mathematicae Applicatae Sinica, English Series, 27 (2011), 647-678.  doi: 10.1007/s10255-011-0068-8. [13] R. Buckdahn and J. Li, Stochastic differential games and viscosity solutions of Hamilton-Jacobi-Bellman-Isaacs equations, SIAM Journal on Control and Optimization, 47 (2008), 444-475.  doi: 10.1137/060671954. [14] A. Cairns, A discussion of parameter and model uncertainty in insurance, Insurance: Mathematics and Economics, 27 (2000), 313-330.  doi: 10.1016/S0167-6687(00)00055-X. [15] R. Cont, Model uncertainty and its impact on the pricing of derivative instruments, Mathematical Finance, 16 (2004), 519-547.  doi: 10.1111/j.1467-9965.2006.00281.x. [16] M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5. [17] W. Fleming and P. Souganidis, On the existence of value functions of two player zero sum stochastic differential games, Indiana University Mathematics Journal, 38 (1989), 293-314.  doi: 10.1512/iumj.1989.38.38015. [18] C. Flor and L. Larsen, Robust portfolio choice with stochastic interest rates, Annals of Finance, 10 (2014), 243-265.  doi: 10.1007/s10436-013-0234-5. [19] I. Girsanov, On transforming a certain class of stochastic processes by absolutely continuous substitution of measures, (Russian)Teor. Verojatnost. i Primenen., 5 (1960), 314–330. doi: 10.1137/1105027. [20] L. Hansen and T. Sargent, Robust control and model uncertainty, Uncertainty within Economic Models, 6 (2014), 145-154. Available from: http://www.jstor.org/stable/2677734 doi: 10.1142/9789814578127_0005. [21] R. Isaacs, Differential Games, Dover, 1999. [22] R. Korn, Worst case scenario investment for insurers, Insurance: Mathematics and economics, 36 (2005), 1-11.  doi: 10.1016/j.insmatheco.2004.10.004. [23] A. Lioui and P. Pocet, On model ambiguity and money neutrality, Journal of Macroeconomics, 34 (2012), 1020-1033.  doi: 10.1016/j.jmacro.2012.08.003. [24] H. Liu, Robust consumption and portfolio choice for time varying investment, Annals of Finance, 6 (2010), 435-454.  doi: 10.1007/s10436-010-0164-4. [25] P. Maenhout, Robust portfolio rules and asset pricing, The Review of Financial Studies, 17 (2004), 951-983.  doi: 10.1093/rfs/hhh003. [26] S. Mataramvura and B. Øksendal, Risk minimizing portfolios and HJBI equations for stochastic differential games, Stochastics: An International Journal of Probability and Stochastic Processes, 80 (2008), 317-337.  doi: 10.1080/17442500701655408. [27] C. McMillan and R. Triggiani, Min-Max Game Theory and Algebraic Riccati Equations for Boundary Control Problems with Continuous Input-Solution Map. Part Ⅱ: the General Case, Appl Math Optim, 29 (1994), 1-65.  doi: 10.1007/BF01191106. [28] R. Merton, Lifetime portfolio selection under uncertainty: The continuous case, Rev. Econ. Stat., 51 (1969), 247-257.  doi: 10.2307/1926560. [29] H. Nikaidô, On Von Neumann's minimax theorem, Pacific Journal of Mathematics, 4 (1954), 65-72.  doi: 10.2140/pjm.1954.4.65. [30] M. Nisio, Stochastic differential games and viscosity solutions of Isaacs equations, Nagoya Mathematical Journal, 110 (1988), 163-184.  doi: 10.1017/S0027763000002932. [31] A. A. Novikov, On conditions for uniform integrability for continuous exponential martingales, Stochastic Differential Systems, Proc. IFIP-WG 7/1 Work. Conf., Vilnius/Lith. 1978, Lect. Notes Control Inf. Sci., 25 (1980), 304-310.  doi: 10.1093/rfs/hhh003. [32] M. Pinar, On Robust Mean-Variance Portfolios, Optimization, 65 (2016), 1039-1048.  doi: 10.1080/02331934.2015.1132216. [33] S. D. Promislow and V. Young, Minimizing the probability of ruin when claims follow Brownian motion with drift, North American Actuarial Journal, 9 (2005), 109-128.  doi: 10.1080/10920277.2005.10596214. [34] U. Rieder and C. Wopperer, Robust consumption-investment problems with random market coefficients, Math Finan Econ, 6 (2012), 295-311.  doi: 10.1007/s11579-012-0073-6. [35] H. Schmidli, Diffusion approximations for a risk process with the possibility of borrowing and investment, Communications in Statistics, Stochastic Models, 10 (1994), 365-388.  doi: 10.1080/15326349408807300. [36] M. Sion, On general minimax theorems, Pacific Journal of Mathematics, 8 (1958), 171-176.  doi: 10.2140/pjm.1958.8.171. [37] C. Skiadas, Robust control and recursive utility, Finance and Stochastics, 7 (2003), 475-489.  doi: 10.1007/s007800300100. [38] R. Uppal and T. Wang, Model misspecification and underdiversification, The Journal of Finance, 58 (2003), 2465-2486.  doi: 10.1046/j.1540-6261.2003.00612.x. [39] H. Wang and S. Hou, Robust consumption and portfolio choice with habit formation, the spirit of capitalism and recursive utility, Annals of Economics and Finance, 16 (2015), 393-416. [40] D. Zawisza, Robust portfolio selection under exponential preferences, Applicationes Mathematicae, 37 (2010), 215-230.  doi: 10.4064/am37-2-6. [41] D. Zawisza, Robust consumption-investment problem on infinite horizon, Appl. Math. Optim, 72 (2015), 469-491.  doi: 10.1007/s00245-014-9287-8.

show all references

The authors would like to thank the editor and the two anonymous referees for their careful reading and helpful comments.

##### References:
 [1] E. Anderson, L. Hansen and T. Sargent, A quartet of semigroups for model specification, robustness, prices of risk, and model detection, Journal of the European Economic Association, 1 (2003), 68-123. [2] E. Anderson, E. Ghysels and J. Juergens, The impact of risk and uncertainty on expected returns, Journal of Financial Economics, 94 (2009), 233-263.  doi: 10.1016/j.jfineco.2008.11.001. [3] L. Bai and G. Guo, Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance: Mathematics and Economics, 42 (2008), 968-975.  doi: 10.1016/j.insmatheco.2007.11.002. [4] I. D. Baltas, N. E. Frangos and A. N. Yannacopoulos, Optimal investment and reinsurance policies in insurance markets under the effect of inside information, Applied Stochastic Models in Business and Industry, 28 (2012), 506-528.  doi: 10.1002/asmb.925. [5] I. D. Baltas and A. N. Yannacopoulos, Uncertainty and inside information, Journal of Dynamics and Games, 3 (2016), 1-24.  doi: 10.3934/jdg.2016001. [6] E. Bayraktar and S. Yao, Doubly reflected BSDEs with integrable parameters and related Dynkin games, Stochastic Processes and their Applications, 125 (2015), 4489-4542.  doi: 10.1016/j.spa.2015.07.007. [7] S. Biagini and M. Pinar, The Robust Merton Problem of an Ambiguity-Averse Investor, Mathematics and Financial Economics, 11 (2017), 1-24.  doi: 10.1007/s11579-016-0168-6. [8] N. Branger, L. Larsen and C. Munk, Robust portfolio choice with ambiguity and learning predictability, Journal of Banking and Finance, 37 (2013), 1397-1411.  doi: 10.2139/ssrn.1859916. [9] W. A. Brock, A. Xepapadeas and A. N. Yannacopoulos, Robust control and hot spots in spatiotemporal economic systems, Dyn. Games Appl., 4 (2014), 257-289.  doi: 10.1007/s13235-014-0109-z. [10] W. A. Brock, A. Xepapadeas and A. N. Yannacopoulos, Robust control of a spatially distributed commercial fishery, in Dynamic Optimization in Environmental Economics, (eds. E. Moser, W. Semmler, G. Tragler, V. Veliov), Springer-Verlag, Heidelberg, 15 (2014), 215-241. doi: 10.1007/978-3-642-54086-8_10. [11] S. Browne, Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin, Mathematics of Operations Research, 20 (1995), 937-958.  doi: 10.1287/moor.20.4.937. [12] R. Buckdahn and J. Li, Stochastic differential games with reflection and related obstacle problems for Isaacs equations, Acta Mathematicae Applicatae Sinica, English Series, 27 (2011), 647-678.  doi: 10.1007/s10255-011-0068-8. [13] R. Buckdahn and J. Li, Stochastic differential games and viscosity solutions of Hamilton-Jacobi-Bellman-Isaacs equations, SIAM Journal on Control and Optimization, 47 (2008), 444-475.  doi: 10.1137/060671954. [14] A. Cairns, A discussion of parameter and model uncertainty in insurance, Insurance: Mathematics and Economics, 27 (2000), 313-330.  doi: 10.1016/S0167-6687(00)00055-X. [15] R. Cont, Model uncertainty and its impact on the pricing of derivative instruments, Mathematical Finance, 16 (2004), 519-547.  doi: 10.1111/j.1467-9965.2006.00281.x. [16] M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5. [17] W. Fleming and P. Souganidis, On the existence of value functions of two player zero sum stochastic differential games, Indiana University Mathematics Journal, 38 (1989), 293-314.  doi: 10.1512/iumj.1989.38.38015. [18] C. Flor and L. Larsen, Robust portfolio choice with stochastic interest rates, Annals of Finance, 10 (2014), 243-265.  doi: 10.1007/s10436-013-0234-5. [19] I. Girsanov, On transforming a certain class of stochastic processes by absolutely continuous substitution of measures, (Russian)Teor. Verojatnost. i Primenen., 5 (1960), 314–330. doi: 10.1137/1105027. [20] L. Hansen and T. Sargent, Robust control and model uncertainty, Uncertainty within Economic Models, 6 (2014), 145-154. Available from: http://www.jstor.org/stable/2677734 doi: 10.1142/9789814578127_0005. [21] R. Isaacs, Differential Games, Dover, 1999. [22] R. Korn, Worst case scenario investment for insurers, Insurance: Mathematics and economics, 36 (2005), 1-11.  doi: 10.1016/j.insmatheco.2004.10.004. [23] A. Lioui and P. Pocet, On model ambiguity and money neutrality, Journal of Macroeconomics, 34 (2012), 1020-1033.  doi: 10.1016/j.jmacro.2012.08.003. [24] H. Liu, Robust consumption and portfolio choice for time varying investment, Annals of Finance, 6 (2010), 435-454.  doi: 10.1007/s10436-010-0164-4. [25] P. Maenhout, Robust portfolio rules and asset pricing, The Review of Financial Studies, 17 (2004), 951-983.  doi: 10.1093/rfs/hhh003. [26] S. Mataramvura and B. Øksendal, Risk minimizing portfolios and HJBI equations for stochastic differential games, Stochastics: An International Journal of Probability and Stochastic Processes, 80 (2008), 317-337.  doi: 10.1080/17442500701655408. [27] C. McMillan and R. Triggiani, Min-Max Game Theory and Algebraic Riccati Equations for Boundary Control Problems with Continuous Input-Solution Map. Part Ⅱ: the General Case, Appl Math Optim, 29 (1994), 1-65.  doi: 10.1007/BF01191106. [28] R. Merton, Lifetime portfolio selection under uncertainty: The continuous case, Rev. Econ. Stat., 51 (1969), 247-257.  doi: 10.2307/1926560. [29] H. Nikaidô, On Von Neumann's minimax theorem, Pacific Journal of Mathematics, 4 (1954), 65-72.  doi: 10.2140/pjm.1954.4.65. [30] M. Nisio, Stochastic differential games and viscosity solutions of Isaacs equations, Nagoya Mathematical Journal, 110 (1988), 163-184.  doi: 10.1017/S0027763000002932. [31] A. A. Novikov, On conditions for uniform integrability for continuous exponential martingales, Stochastic Differential Systems, Proc. IFIP-WG 7/1 Work. Conf., Vilnius/Lith. 1978, Lect. Notes Control Inf. Sci., 25 (1980), 304-310.  doi: 10.1093/rfs/hhh003. [32] M. Pinar, On Robust Mean-Variance Portfolios, Optimization, 65 (2016), 1039-1048.  doi: 10.1080/02331934.2015.1132216. [33] S. D. Promislow and V. Young, Minimizing the probability of ruin when claims follow Brownian motion with drift, North American Actuarial Journal, 9 (2005), 109-128.  doi: 10.1080/10920277.2005.10596214. [34] U. Rieder and C. Wopperer, Robust consumption-investment problems with random market coefficients, Math Finan Econ, 6 (2012), 295-311.  doi: 10.1007/s11579-012-0073-6. [35] H. Schmidli, Diffusion approximations for a risk process with the possibility of borrowing and investment, Communications in Statistics, Stochastic Models, 10 (1994), 365-388.  doi: 10.1080/15326349408807300. [36] M. Sion, On general minimax theorems, Pacific Journal of Mathematics, 8 (1958), 171-176.  doi: 10.2140/pjm.1958.8.171. [37] C. Skiadas, Robust control and recursive utility, Finance and Stochastics, 7 (2003), 475-489.  doi: 10.1007/s007800300100. [38] R. Uppal and T. Wang, Model misspecification and underdiversification, The Journal of Finance, 58 (2003), 2465-2486.  doi: 10.1046/j.1540-6261.2003.00612.x. [39] H. Wang and S. Hou, Robust consumption and portfolio choice with habit formation, the spirit of capitalism and recursive utility, Annals of Economics and Finance, 16 (2015), 393-416. [40] D. Zawisza, Robust portfolio selection under exponential preferences, Applicationes Mathematicae, 37 (2010), 215-230.  doi: 10.4064/am37-2-6. [41] D. Zawisza, Robust consumption-investment problem on infinite horizon, Appl. Math. Optim, 72 (2015), 469-491.  doi: 10.1007/s00245-014-9287-8.
Average of 6000 optimal investment strategy paths for various levels of the preference for the robustness parameter, in the case of the exponential utility function.
Average of 6000 optimal coverage strategy paths for various levels for the preference for the robustness parameter, in the case of the exponential utility function.
Average of 6000 optimal investment strategy paths for various levels of the initial wealth, in the case of the exponential utility function, with robustness.
Average of 6000 optimal worst-case strategy paths for various levels for the preference for the robustness parameter, in the case of the exponential utility function.
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