OR Spectrum

, Volume 37, Issue 3, pp 761–786 | Cite as

Data-driven portfolio management with quantile constraints

Regular Article

Abstract

We investigate an iterative, data-driven approximation to a problem where the investor seeks to maximize the expected return of her portfolio subject to a quantile constraint, given historical realizations of the stock returns. The approach, which was developed independently from Calafiore (SIAM J Optim 20:3427–3464 2010) but uses a similar idea, involves solving a series of linear programming problems and thus can be solved quickly for problems of large scale. We compare its performance to that of methods commonly used in the finance literature, such as fitting a Gaussian distribution to the returns (Keisler, Decision Anal 1:177–189 2004; Rachev et al. Advanced stochastic models, risk assessment and portfolio optimization: the ideal risk, uncertainty and performance measures, Wiley, New York 2008). We also analyze the resulting efficient frontier and extend our approach to the case where portfolio risk is measured by the inter-quartile range of its return. Our main contribution is in the detail of the implementation, i.e., the choice of the constraints to be generated in the master problem, as well as the numerical simulations and empirical tests, and the application to the inter-quartile range as a risk measure.

Keywords

Data-driven optimization Quantile constraints Iterative algorithm 

References

  1. Artzner P, Delbaen F, Eber JM, Health D (1999) Coherent measures of risk. Math Finance 9(3):203–228CrossRefGoogle Scholar
  2. Beasley J (2013) Portfolio optimization: models and solution approaches. Tutor Oper Res 1:201–221Google Scholar
  3. Benati S, Rizzi R (2007) A mixed integer linear programming formulation of the optimal mean/value-at-risk portfolio problem. Eur J Oper Res 176:423–434CrossRefGoogle Scholar
  4. Benninga S, Wiener Z (1998) Value-at-risk (var). Math Educ Res 7(4)Google Scholar
  5. Calafiore GC (2010) Random convex programs. SIAM J Optim 20(6):3427–3464CrossRefGoogle Scholar
  6. Calafiore GC (2013) Direct data-driven portfolio optimization with guaranteed shortfall probability. Automatica 49:370–380CrossRefGoogle Scholar
  7. Calafiore GC, Monastero B (2012) Data-driven asset allocation with guaranteed short-fall probability. IEEE American Control Conference, pp 3687–3692Google Scholar
  8. Cetinkaya E (2014) Essays in robust and data-driven risk management. Ph.D. thesis, Lehigh University, BethlehemGoogle Scholar
  9. Colombo M (2007) Advances in interior point methods for large scale linear programming. Ph.D. thesis, Doctor of Philosophy University of EdinburghGoogle Scholar
  10. Cornuejols G, Tutuncu R (2007) Optimization methods in finance. Cambridge University Press, New YorkGoogle Scholar
  11. Dentcheva D, Ruszczynski A (2006) Portfolio optimization with stochastic dominance constraints. J Bank Finance 30(2):433–451CrossRefGoogle Scholar
  12. El-Ghaoui L, Oks L, Oustry F (2000) Worst-case value-at-risk and robust asset allocation: a semidefinite programming approach. Tech. Rep. M00/59, University of California, BerkeleyGoogle Scholar
  13. Fabozzi F, Kolm P, Pachamanova D, Focardi S (2007) Robust portfolio optimization and management. Wiley, New YorkGoogle Scholar
  14. Fenton LF (1969) The sum of lognormal probability distributions in scatter transmission systems. IRE Trans Commun Syst CS 8(3):57–67Google Scholar
  15. Gaivoronski A, Pflug G (2005) Value-at-risk in portfolio optimization: properties and computational approach. J Risk 7(2):1–31Google Scholar
  16. Goh J, Lim K, Sim M, Zhang W (2012) Portfolio value-at-risk optimization for asymmetrically distributed asset returns. Eur J Oper Res 221(2):397–406CrossRefGoogle Scholar
  17. Harlow WV (1991) Asset allocation in a downside risk framework. Financial Anal J 47(5):28–40CrossRefGoogle Scholar
  18. Keisler J (2004) Value of information in portfolio decision analysis. Decision Anal 1(3):177–189CrossRefGoogle Scholar
  19. Kim JH, Powell WB (2011) Quantile optimization for heavy-tailed distribution using asymmetric signum functions. Princeton UniversityGoogle Scholar
  20. Larsen N, Mausser H, Uryasev S (2002) Algorithms for optimization of value-at-risk. In: Pardalos P, Tsitsiringos VK (eds) Financial Engineering, E-Commerce and Supply Chain. Applied Optimization, vol 126. Springer, pp 19–46Google Scholar
  21. Linsmeier TJ, Pearson ND (2000) Value at risk. Financial Anal J 56(2):47–67CrossRefGoogle Scholar
  22. Lobo M, Fazel M, Boyd S (2006) Portfolio optimization with linear and fixed transaction costs. Ann Oper Res 152(5)Google Scholar
  23. Markovitz HM (1952) Portfolio selection. J Finance 7(1):77–91Google Scholar
  24. Markovitz HM (1959) Portfolio selection. Wiley, New YorkGoogle Scholar
  25. Naumov AV, Kibzun AI (1992) Quantile optimization techniques with application to chance constrained problem for water-supply system design. Tech. Rep. 92–5, Department of Industrial and Operations Engineering at University of Michigan and Department of Applied Mathematics Moscow Aviation Institute, Ann Arbor. Mi 48109 and Moskiw, 127080, RussiaGoogle Scholar
  26. Oyama T (2007) Determinants of stock prices: the case of zimbabwe. A Working Paper of the International Monetary FundGoogle Scholar
  27. Pankov AR, Platonov EN, Semenikhin KV (2002) Minimax optimization of investment portfolio by quantile criterion. Autom Remote control 64(7):1122–1137CrossRefGoogle Scholar
  28. Pfaff B (2013) Financial risk modeling and portfolio optimization with R. Wiley, New YorkGoogle Scholar
  29. Rachev S, Stoyanov S, Fabozzi F (2008) Advanced stochastic models, risk assessment and portfolio optimization: the ideal risk, uncertainty and performance measures. Wiley, New YorkGoogle Scholar
  30. Rockafellar RT, Uryasev S (2000) Optimization of conditional value at risk. J Risk 2(3):21–41Google Scholar
  31. Rodriguez GJL (1999) Portfolio optimization with quantile-based risk measures. Ph.D. thesis, Massachusetts Institute of Technology, MassachusettsGoogle Scholar
  32. Roy A (1952) Safety first and the holding of assets. Econometrica 20(3):431–449CrossRefGoogle Scholar
  33. Ruszczynski A, Vanderbei R (2003) Frontiers of stochastically nondominated portfolios. Econometrica 71(4):1287–1297CrossRefGoogle Scholar
  34. Sharpe WF (1966) Mutual fund performance. J Bus 39:119–138CrossRefGoogle Scholar
  35. Sharpe WF (1971) Mean absolute deviation characteristic lines for securities and portfolios. Manag Sci 18(2):B1–B13CrossRefGoogle Scholar
  36. Sortino FA, Price LN (1994) Performance measurement in downside risk framework. J Invest 3:59–64CrossRefGoogle Scholar
  37. Szegö G (2002) Measures of risk. J Bank Finance 26:1253–1272CrossRefGoogle Scholar
  38. Uryasev S (ed) (2000) Probabilistic constrained optimization: methodology and applications. Springer, New YorkGoogle Scholar
  39. Wozabal D (2012) Value-at-risk optimization using the difference of convex algorithm. OR Spectrum 34:681–683CrossRefGoogle Scholar
  40. Yitzhaki S (1982) Stochastic dominance, mean variance, and gini’s mean difference. Am Econ Assoc 72(1):178–185Google Scholar
  41. Zymler S, Kuhn D, Rustem B (2013) Worst-case value-at-risk of nonlinear portfolios. Manag Sci 59(1):172–188CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Amazon.comSeattleUSA
  2. 2.Department of Industrial and Systems EngineeringLehigh UniversityBethlehemUSA

Personalised recommendations