The Korean Journal of Applied Statistics (응용통계연구)

The Korean Statistical Society (한국통계학회)
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Linear programming models using a Dantzig type risk for portfolio optimization

Dantzig 위험을 사용한 포트폴리오 최적화 선형계획법 모형

  • Ahn, Dayoung (Department of Statistics, Sungkyunkwan University) ;
  • Park, Seyoung (Department of Statistics, Sungkyunkwan University)
  • 안다영 (성균관대학교 통계학과) ;
  • 박세영 (성균관대학교 통계학과)
  • Received : 2021.10.29
  • Accepted : 2021.12.30
  • Published : 2022.04.30
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Abstract

Since the publication of Markowitz's (1952) mean-variance portfolio model, research on portfolio optimization has been conducted in many fields. The existing mean-variance portfolio model forms a nonlinear convex problem. Applying Dantzig's linear programming method, it was converted to a linear form, which can effectively reduce the algorithm computation time. In this paper, we proposed a Dantzig perturbation portfolio model that can reduce management costs and transaction costs by constructing a portfolio with stable and small (sparse) assets. The average return and risk were adjusted according to the purpose by applying a perturbation method in which a certain part is invested in the existing benchmark and the rest is invested in the assets proposed as a portfolio optimization model. For a covariance estimation, we proposed a Gaussian kernel weight covariance that considers time-dependent weights by reflecting time-series data characteristics. The performance of the proposed model was evaluated by comparing it with the benchmark portfolio with 5 real data sets. Empirical results show that the proposed portfolios provide higher expected returns or lower risks than the benchmark. Further, sparse and stable asset selection was obtained in the proposed portfolios.

포트폴리오 최적화 이론의 초석인 Markowitz의 평균-분산 포트폴리오 모형 (1952)이 발표된 이후로 많은 분야에서 포트폴리오 최적화에 대한 다양한 연구가 진행되었다. 기존의 평균-분산 포트폴리오 모형은 주로 목적함수나 제약식에 비선형 볼록 형태를 포함한다. 이를 Dantzig의 선형계획법을 적용하여 선형으로 변환시켜 알고리즘 계산 시간을 효율적으로 감소시켰다. 또한 시계열 데이터 특성을 반영하여 시간에 따른 가중치를 고려하는 가우시안 커널 가중치 공분산을 제안하였다. 여기에 일정 부분은 벤치마크에 투자하고 나머지는 포트폴리오 최적화 모형으로 제안된 자산들에 투자하는 퍼터베이션 방법을 적용하여 평균 수익률과 위험도를 목적에 맞게 조절하도록 하였다. 또한, 본 논문에서는 안정적이면서도 적은 자산을 보유하게 포트폴리오를 구성하여 관리비용(management costs)과 거래비용(transaction costs)를 낮출 수 있는 Dantzig-type 퍼터베이션 포트폴리오 모형을 제안하였다. 제안된 모형의 성능은 5개의 실제 데이터 세트로 벤치마크 포트폴리오와 비교 분석하여 평가하였다. 최종적으로 제안한 최적화 모형은 벤치마크보다 높은 기대수익률이나 낮은 위험도를 갖는 포트폴리오를 구성하여 퍼터베이션 목적을 만족하며, 투자한 자산의 수와 시간에 따른 자산 구성 변화를 일정 수준 이하로 조절하는 희소하며 안정적인 결과를 얻었다.

Keywords

References

  1. Andersen ED, Gondzio J, Meszaros C, and Xu X (1996). Implementation of interior point methods for large scale linear programming, HEC/Universite de Geneve.
  2. Best MJ and Grauer RR (1991). On the sensitivity of mean-variance-efficient portfolios to changes in asset means: some analytical and computational results, The review of financial studies, 4, 315-342. https://doi.org/10.1093/rfs/4.2.315
  3. Bixby RE (2002). Solving real-world linear programs: A decade and more of progress, Operations research, 50, 3-15. https://doi.org/10.1287/opre.50.1.3.17780
  4. Brodie J, Daubechies I, De Mol C, Giannone D, and Loris I (2009). Sparse and stable Markowitz portfolios. In Proceedings of the National Academy of Sciences, 106, 12267-12272. https://doi.org/10.1073/pnas.0904287106
  5. Cai T, Liu W, and Luo X (2011). A constrained ℓ 1 minimization approach to sparse precision matrix estimation, Journal of the American Statistical Association, 106, 594-607. https://doi.org/10.1198/jasa.2011.tm10155
  6. Candes E and Tao T (2007). The Dantzig selector: Statistical estimation when p is much larger than n, The annals of Statistics, 35, 2313-2351. https://doi.org/10.1214/009053606000001523
  7. Cesarone F, Scozzari A, and Tardella F (2011). Portfolio Selection Problems in Practice: A Comparison between Linear and Quadratic Optimization Models, arXiv preprint arXiv:1105.3594
  8. Chen C, Li X, Tolman C, Wang S, and Ye Y (2013). Sparse Portfolio Selection via Quasi-Norm Regularization, arXiv preprint arXiv:1312.6350
  9. DeMiguel V, Garlappi L, and Uppal R (2009). Optimal versus naive diversification: How inefficient is the 1/N portfolio strategy?, The Review of Financial Studies, 22, 1915-1953. https://doi.org/10.1093/rfs/hhm075
  10. Dantzig GB and Infanger G (1993). Multi-stage stochastic linear programs for portfolio optimization, Annals of Operations Research, 45, 59-76. https://doi.org/10.1007/BF02282041
  11. Fan J, Liao Y, and Mincheva M (2013). Large covariance estimation by thresholding principal orthogonal complements, Journal of the Royal Statistical Society. Series B, Statistical methodology, 75.
  12. Fastrich B, Paterlini S, and Winker P (2015). Constructing optimal sparse portfolios using regularization methods, Computational Management Science, 12, 417-434. https://doi.org/10.1007/s10287-014-0227-5
  13. Koberstein A (2005). The dual simplex method, techniques for a fast and stable implementation, Unpublished doctoral thesis, Universitat Paderborn, Paderborn, Germany.
  14. Konno H and Wijayanayake A (1999). Mean-absolute deviation portfolio optimization model under transaction costs, Journal of the Operations Research Society of Japan, 42, 422-435. https://doi.org/10.1016/S0453-4514(00)87111-2
  15. Konno H and Yamazaki H (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market, Management science, 37, 519-531. https://doi.org/10.1287/mnsc.37.5.519
  16. Ledoit O and Wolf M (2003). Improved estimation of the covariance matrix of stock returns with an application to portfolio selection, Journal of Empirical Finance, 10, 603-621. https://doi.org/10.1016/S0927-5398(03)00007-0
  17. Lee TH and Seregina E (2020). Optimal Portfolio Using Factor Graphical Lasso, arXiv preprint arXiv:2011.00435
  18. Li J (2015). Sparse and stable portfolio selection with parameter uncertainty, Journal of Business & Economic Statistics, 33, 381-392. https://doi.org/10.1080/07350015.2014.954708
  19. Lintner J (1965). The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets, In Stochastic Optimization Models in Finance, 131-155.
  20. Mansini R, Ogryczak W, and Speranza MG (2014). Twenty years of linear programming based portfolio optimization, European Journal of Operational Research, 234, 518-535. https://doi.org/10.1016/j.ejor.2013.08.035
  21. Markowitz H (1952). Portfolio Selection, The Journal of Finance, 7, 77-91. https://doi.org/10.1111/j.1540-6261.1952.tb01525.x
  22. Millington T and Niranjan M (2017). Robust portfolio risk minimization using the graphical lasso. In International Conference on Neural Information Processing, 863-872, Springer.
  23. Murillo JML and Rodr'iguez AA (2008). Algorithms for Gaussian bandwidth selection in kernel density estimators. Advances in Neural Information Processing Systems.
  24. Pantaleo E, Tumminello M, Lillo F, and Mantegna RN (2011). When do improved covariance matrix estimators enhance portfolio optimization? An empirical comparative study of nine estimators, Quantitative Finance, 11, 1067-1080. https://doi.org/10.1080/14697688.2010.534813
  25. Park S, Lee ER, Lee S, and Kim G (2019a). Dantzig type optimization method with applications to portfolio selection, Sustainability, 11, 3216. https://doi.org/10.3390/su11113216
  26. Park S, Song H, and Lee S (2019b). Linear programing models for portfolio optimization using a benchmark, The European Journal of Finance, 25, 435-457. https://doi.org/10.1080/1351847x.2018.1536070
  27. Pun CS and Wong HY (2015). High-dimensional static and dynamic portfolio selection problems via l1 minimization, Working paper of the Chinese University of Hong Kong.
  28. Rockafellar RT and Uryasev S (2000). Optimization of conditional value-at-risk, Journal of risk, 2, 21-42. https://doi.org/10.21314/JOR.2000.038
  29. Ross SA (1976). The arbitrage theory of capital asset pricing, Handbook of the Fundamentals of Financial Decision Making: Part I, 11-30.
  30. Sharpe WF (1964). Capital asset prices: A theory of market equilibrium under conditions of risk, The Journal of Finance, 19, 425-442. https://doi.org/10.1111/j.1540-6261.1964.tb02865.x
  31. Shen W, Wang J, and Ma S (2014). Doubly regularized portfolio with risk minimization, Twenty-Eighth AAAI Conference on Artificial Intelligence, 28, 1286-1292.
  32. Zhou S, Lafferty J, and Wasserman L (2010). Time varying undirected graphs, Machine Learning, 80, 295-319. https://doi.org/10.1007/s10994-010-5180-0