Rui Ding
ABSTRACT
In this paper, we draw connections between second-order stochastic dominance (SSD), CVaR risk, and weak sub-majorization for empirical observations of random variables, where many other known equivalences and extensions can be made. As an application, we formulate and solve portfolio optimization problems based on SSD constraints. Based on an alternative characterization of the SSD rules for empirical distributions through weak sub-majorization conditions, we propose an alternative formulation of the SSD-constrained portfolio optimization problems with linear constraints using doubly substochastic matrices. The same technique is also applied to controlling drawdowns of a portfolio, where we formulate and solve drawdown SSD-constrained portfolio optimization problems with linear constraints. These convex programs result in an optimal portfolio that demonstrates more controlled drawdown behaviors than the benchmark portfolio chosen as the reference in SSD constraints.
- [1] Antoniadis, A., Barcelo, N., Nugent, M., Pruhs, K., Schewior, K., and Scquizzato, M. (2016). Chasing Convex Bodies and Functions. In: Kranakis, E., Navarro, G., Chávez, E. (eds) LATIN 2016: Theoretical Informatics, LATIN 2016, Lecture Notes in Computer Science, vol. 9644, Springer, Berlin, Heidelberg.Google Scholar
Cross Ref
- [2] Argue, C. J., Gupta, A., and Guruganesh, G. (2020). Dimension-Free Bounds for Chasing Convex Functions. Proceedings of Machine Learning Research, 125: 1-23.Google Scholar
- [3] Argue, C. J., Gupta, A., Guruganesh, G., and Tang, Z. (2021). Chasing Convex Bodies with Linear Competitive Ratio. J. ACM, 68, 5, Article 32 (October 2021).Google ScholarDigital Library
- [4] Bäuerle, N., and Müller A. (2006). Stochastic Orders and Risk Measures: Consistency and Bounds. Insurance: Mathematics and Economics, 38(1): 132-148.Google Scholar
- [5] Bubeck, S., Lee, Y. T., Li, Y., and Sellke, M. (2019). Competitively Chasing Convex Bodies. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (STOC 2019), Association for Computing Machinery, New York, NY, USA, 861–868.Google ScholarDigital Library
- [6] Chen, N., Goel, G., and Wierman, A. (2018). Smoothed Online Convex Optimization in High Dimensions via Online Balanced Descent. ACM SIGMETRICS Performance Evaluation Review., 46.Google Scholar
- [7] Christianson, N. H., Handina, T., and Wierman, A. (2022). Chasing Convex Bodies and Functions with Black-Box Advice. Proceedings of Machine Learning Research, 178: 1–42.Google Scholar
- [8] Dentcheva, D., and Ruszczyński, A. (2004). Convexification of Stochastic Ordering. Comptes Rendus de l’Academie Bulgare des Sciences., 57(4): 11.Google Scholar
- [9] Dentcheva, D., and Ruszczyński, A. (2006). Portfolio Optimization with Stochastic Dominance Constraints. Journal of Banking & Finance, 30(2): 433-451.Google ScholarCross Ref
- [10] Ding, R., and Uryasev, S. (2022). Drawdown Beta and Portfolio Optimization. Quantitative Finance, 22(7): 1265-1276.Google ScholarCross Ref
- [11] Ding, R. (2023). f-Betas and Portfolio Optimization with f-Divergence induced Risk Measures. arXiv:2302.00452.Google Scholar
- [12] Dommel, P., and Pichler, A. (2020). Convex Risk Measures based on Divergence. arXiv:2003.07648.Google Scholar
- [13] Domingo-Enricha, C., Schiff, Y., and Mroueh, Y. (2022). Learning with Stochastic Orders. arXiv:2205.13684.Google Scholar
- [14] Fishburn, P. (1976). Continua of Stochastic Dominance Relations for Bounded Probability Distributions. Journal of Mathematical Economics, 3(3): 295-311.Google ScholarCross Ref
- [15] Fishburn, P. (1980). Continua of Stochastic Dominance Relations for Unbounded Probability Distributions. Journal of Mathematical Economics, 7(3): 271-285.Google ScholarCross Ref
- [16] Huang, R. J., Tzeng, L. Y., and Zhao, L. (2020). Fractional Degree Stochastic Dominance. Management Science, 66(10): 4630-4647.Google ScholarDigital Library
- [17] Kamihigashi, T., and Stachurski, J. (2014). An Axiomatic Approach to Measuring Degree of Stochastic Dominance. Discussion Paper Series DP2014-36, Research Institute for Economics & Business Administration, Kobe University.Google Scholar
- [18] Krokhmal, P., Uryasev, S., and Zrazhevsky, G. (2005). Numerical Comparison of Conditional Value-at-Risk and Conditional Drawdown-at-Risk Approaches: Application to Hedge Funds. In Applications of Stochastic Programming, pp. 609-631, Society for Industrial and Applied Mathematics.Google ScholarCross Ref
- [19] Levy, H. (2006). Stochastic Dominance: Investment Decision Making Under Uncertainty, Springer, Berlin.Google ScholarCross Ref
- [20] Marshall, A. W., and Olkin, I. (1979). Inequalities: Theory of Majorization and its Applications. Academic Press, New York.Google Scholar
- [21] Marcus, M., and Ree, R. (1959). Diagonals of Doubly Stochastic Matrices. Quarterly Journal of Mathematics, 10(1959): 296-302.Google ScholarCross Ref
- [22] Ok, E. (2007). Real Analysis with Economics Applications. Princeton University Press.Google ScholarCross Ref
- [23] Przeslawski, K., and Yost, D. (1989). Continuity Properties of Selectors. Michigan Math J., 36(1): 13.Google Scholar
- [24] Sellke, M. (2020). Chasing Convex Bodies Optimally. In Proceedings of the Thirty-First Annual ACM-SIAM Symposium on Discrete Algorithms (SODA ’20), Society for Industrial and Applied Mathematics, USA, 1509–1518.Google ScholarDigital Library
- [25] Stoyanov, S., Rachev, S. T., and Fabozzi, F. J. (2012). Metrization of Stochastic Dominance Rules. International Journal of Theoretical and Applied Finance (IJTAF), 15, 1250017-1.Google ScholarCross Ref
- [26] Schmid, F., and Trede, M. (1996). Nonparametric Inference for Second Order Stochastic Dominance. Discussion Papers in Econometrics and Statistics 2/96, University of Cologne, Institute of Econometrics and Statistics.Google Scholar
- [27] Xue, M., Shi, Y., and Sun, H. (2020). Portfolio Optimization with Relaxation of Stochastic Second Order Dominance Constraints via Conditional Value at Risk. Journal of Industrial and Management Optimization, 13, 1.Google Scholar
Index Terms
- Stochastic Dominance, Risk, and Weak Sub-majorization with Applications to Portfolio Optimization
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