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Annals of Operations Research

, Volume 262, Issue 2, pp 547–578 | Cite as

Risk-budgeting multi-portfolio optimization with portfolio and marginal risk constraints

  • Ran JiEmail author
  • Miguel A. Lejeune
S.I.: Financial Economics

Abstract

Multi-portfolio optimization problems and the incorporation of marginal risk contribution constraints have recently received a sustained interest from academia and financial practitioners. We propose a class of new stochastic risk budgeting multi-portfolio optimization models that impose portfolio as well as marginal risk constraints. The models permit the simultaneous and integrated optimization of multiple sub-portfolios in which the marginal risk contribution of each individual security is accounted for. A risk budget defined with a downside risk measure is allocated to each security. We consider the two cases in which the asset universes of the sub-portfolios are either disjoint (diversification of style) or overlap (diversification of judgment). The proposed models take the form of stochastic programming problems and include each a probabilistic constraint with multi-row random technology matrix. We expand a combinatorial modeling framework to represent the feasible set of the chance constraints first as a set of mixed-integer linear inequalities. The new reformulation proposed in this paper is much sparser than previously presented reformulations and allows the efficient solution of problem instances that could not be solved otherwise. We evaluate the efficiency and scalability of the proposed method that is general enough to be applied to general chance-constrained optimization problems. We conduct a cross-validation study via a rolling-horizon procedure to assess the performance of the models, and understand the impact of the parameters and diversification types on the portfolios.

Keywords

Multi-portfolio optimization Marginal risk contribution Downside risk Stochastic programming Risk budgeting 

References

  1. Basak, S., & Shapiro, A. (2001). Value-at-risk-based risk management: Optimal policies and asset prices. Review of Financial Studies, 14(2), 371–405.CrossRefGoogle Scholar
  2. Benati, S., & Rizzi, R. (2007). A mixed integer linear programming formulation of the optimal mean/value-at-risk portfolio problem. European Journal of Operational Research, 176(1), 423–434.CrossRefGoogle Scholar
  3. Bertsimas, D., Darnell, C., & Soucy, R. (1999). Portfolio construction through mixed-integer programming at Grantham, Mayo, Van Otterloo and Company. Interfaces, 29(1), 49–66.CrossRefGoogle Scholar
  4. Blake, D., Rossi, A. G., Timmermann, A., Tonks, I., & Wermers, R. (2013). Decentralized investment management: Evidence from the pension fund industry. The Journal of Finance, 68(3), 1133–1178.CrossRefGoogle Scholar
  5. Bonami, P., & Lejeune, M. A. (2009). An exact solution approach for portfolio optimization problems under stochastic and integer constraints. Operations Research, 57, 650–670.CrossRefGoogle Scholar
  6. Boros, E., Hammer, P. L., Ibaraki, T., & Kogan, A. (1997). Logical analysis of numerical data. Mathematical Programming, 79, 163–190.Google Scholar
  7. Boudt, K., Peterson, B., & Croux, C. (2008). Estimation and decomposition of downside risk for portfolios with non-normal returns. Journal of Risk, 11(2), 79–103.CrossRefGoogle Scholar
  8. Boudt, K., Carl, P., & Peterson, B. (2013). Asset allocation with conditional value-at-risk budgets. Journal of Risk, 15(3), 39–68.CrossRefGoogle Scholar
  9. Bruder, B., & Roncalli, T. (2013). Managing risk exposures using the risk parity approach. Tech. rep., LYXOR Research Report.Google Scholar
  10. Charnes, A., & Cooper, W. W. (1959). Chance-constrained programming. Management Science, 6(1), 73–79.CrossRefGoogle Scholar
  11. Chopra, V. K., & Ziemba, W. T. (1993). The effect of errors in means, variances, and covariances on optimal portfolio choice. The Journal of Portfolio Management, 19(2), 6–11.CrossRefGoogle Scholar
  12. Cornuejóls, G., & Tútúncú, R. (2007). Optimization methods in finance. New York: Cambridge University Press.Google Scholar
  13. Crama, Y., & Hammer, P. L. (2011). Boolean functions—Theory, algorithms and applications. New York: Cambridge Press University.CrossRefGoogle Scholar
  14. DeMiguel, V., Garlappi, L., Nogales, F. J., & Uppal, R. (2009). A generalized approach to portfolio optimization: Improving performance by constraining portfolio norms. Management Science, 55(5), 798–812.CrossRefGoogle Scholar
  15. Elton, E. J., & Gruber, M. J. (2004). Optimum centralized portfolio construction with decentralized portfolio management. Journal of Financial and Quantitative Analysis, 39(3), 481–494.CrossRefGoogle Scholar
  16. Fabozzi, F., Huang, D., & Zhou, G. (2010). Robust portfolios: Contributions from operations research and finance. Annals of Operations Research, 176, 191–220.CrossRefGoogle Scholar
  17. Filomena, T. P., & Lejeune, M. A. (2012). Stochastic portfolio optimization with proportional transaction costs: Convex reformulations and computational experiments. Operations Research Letters, 40(3), 212–217.CrossRefGoogle Scholar
  18. Filomena, T. P., & Lejeune, M. A. (2014). Warm-start heuristic for stochastic portfolio optimization with fixed and proportional transaction costs. Journal of Optimization Theory and Applications, 161(1), 308–329.CrossRefGoogle Scholar
  19. Gaivoronski, A. A., & Pflug, G. (2005). Value-at-risk in portfolio optimization: Properties and computational approach. Journal of Risk, 7(2), 1–31.CrossRefGoogle Scholar
  20. Goh, J. W., Lim, K. G., Sim, M., & Zhang, W. (2012). Portfolio value-at-risk optimization for asymmetrically distributed asset returns. European Journal of Operational Research, 221(2), 397–406.CrossRefGoogle Scholar
  21. Hsia, Y., Wu, B., & Li, D. (2014). New reformulations for probabilistically constrained quadratic programs. European Journal of Operational Research, 233(3), 550–556.CrossRefGoogle Scholar
  22. Jaeger, R. A., Rausch, M. A., & Foley, M. (2010). Multi-horizon investing: A new paradigm for endowments and other long-term investors. The Journal of Wealth Management, 13(1), 32–42.CrossRefGoogle Scholar
  23. Jorion, P. (2001). Value at risk: The new benchmark for managing financial risk (2nd ed.). New York: McGraw-Hill.Google Scholar
  24. Kataoka, S. (1963). A stochastic programming model. Econometrica, 31(12), 181–196.CrossRefGoogle Scholar
  25. Kogan, A., & Lejeune, M. A. (2014). Threshold Boolean form for joint probabilistic constraints with random technology matrix. Mathematical Programming, 147(1–2), 391–427.CrossRefGoogle Scholar
  26. Krokhmal, P., Palmquist, J., & Uryasev, S. (2001). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of Risk, 4, 43–68.CrossRefGoogle Scholar
  27. Lai, T. L., Xing, H., & Chen, Z. (2011). Mean-variance portfolio optimization when means and covariances are unknown. The Annals of Applied Statistics, 5(2A), 798–823.CrossRefGoogle Scholar
  28. Lee, J., & Prépoka, A. (2012). Properties and calculation of multivariate risk measures: MVaR and MCVaR. Rutcor Research Report.Google Scholar
  29. Lejeune, M. A. (2012a). Pattern-based modeling and solution of probabilistically constrained optimization problems. Operations Research, 60(6), 1356–1372.CrossRefGoogle Scholar
  30. Lejeune, M. A. (2012b). Pattern definition of the \(p\)-efficiency concept. Annals of Operations Research, 200(1), 23–36.CrossRefGoogle Scholar
  31. Lejeune, M. A., & Shen, S. (2014). Boolean formulations for multi-objective probabilistically constrained programs with variable risk. Working Paper.Google Scholar
  32. Markowitz, H. M. (1952). Portfolio selection. Journal of Finance, 7, 77–91. March.Google Scholar
  33. Markowitz, H. M. (1959). Portfolio selection: Efficient diversification of investments. New York: Wiley.Google Scholar
  34. Morgan, J. P. (1996). RiskMetrics: Technical document. New York: Morgan Guaranty Trust Company of New York.Google Scholar
  35. Morgan, J. P. (1997). CreditMetrics: Technical Document. New Yok: JP Morgan & Co.Google Scholar
  36. O’Cinneide, C., Scherer, B., & Xu, X. (2006). Pooling trades in a quantitative investment process. The Jounral of Portfolio Management, 32(4), 33–43.CrossRefGoogle Scholar
  37. Ogryczak, W., & Ruszczyński, A. (1999). From stochastic dominance to mean-risk models: Semi-deviations as risk measures. European Journal of Operational Research, 116, 33–50.CrossRefGoogle Scholar
  38. Pagnoncelli, B. K., Ahmed, S., & Shapiro, A. (2009). Sample average approximation method for chance constrained programming: Theory and applications. Journal of Optimization Theory and Applications, 142(2), 399–416.CrossRefGoogle Scholar
  39. Prékopa, A. (2012). Multivariate value at risk and related topics. Annals of Operations Research, 193, 49–69.CrossRefGoogle Scholar
  40. Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of Risk, 2, 21–42.CrossRefGoogle Scholar
  41. Rockafellar, R. T., & Uryasev, S. (2002). Conditional value-at-risk for general loss distributions. Journal of Banking & Finance, 26(7), 1443–1471.CrossRefGoogle Scholar
  42. Roncalli, T. (2013). Introduction to risk parity and budgeting. Boca Raton: CRC Financial Mathematics Series.Google Scholar
  43. Roy, A. D. (1952). Safety first and the holding of assets. Econometrica: Journal of the Econometric Society, 20(3), 431–449.CrossRefGoogle Scholar
  44. Savelsbergh, M. W. P., Stubbs, R. A., & Vandenbussche, D. (2010). Multiportfolio optimization: A natural next step. In J. B. Guerard (Ed.), Handbook of portfolio construction (pp. 565–581). Berlin: Springer.CrossRefGoogle Scholar
  45. Sharpe, W. F. (1981). Decentralized investment management. Journal of Finance, 36, 217–234.CrossRefGoogle Scholar
  46. Standard & Poor’s (S&P) and Morgan Stanley Capital International (MSCI). (2002). Global industry classification standard—A guide to the GICS methodology.Google Scholar
  47. Stubbs, R. A., & Vandenbussche, D. (2009). Multi-portfolio optimization and fairness in allocation of trades.Tech. rep., Axioma Research Paper.Google Scholar
  48. Unger, A. (2015). The use of risk budgets in portfolio optimization. Berlin: Springer.CrossRefGoogle Scholar
  49. Vardharaj, R., & Fabozzi, F. J. (2007). Sector, style, region: Explaining stock allocation performance. Financial Analysts Journal, 63(3), 59–70.CrossRefGoogle Scholar
  50. Wang, M. Y. (1999). Multiple-benchmark and multiple-portfolio optimization. Financial Analyst Journal, 55(1), 63–72.CrossRefGoogle Scholar
  51. Wozabal, D. (2012). Value-at-risk optimization using the difference of convex algorithm. OR Spectrum, 34(4), 861–883.CrossRefGoogle Scholar
  52. Yoda, K., & Prékopa, A. (2010). Optimal portfolio selection based on multiple value at risk constraints. Rutcor Research Report.Google Scholar
  53. Zheng, X., Sun, X., Li, D., & Cui, X. (2012). Lagrangian decomposition and mixed-integer quadratic programming reformulations for probabilistically constrained quadratic programs. European Journal of Operational Research, 221(1), 38–48.CrossRefGoogle Scholar
  54. Zhu, S., Li, D., & Sun, X. (2010). Portfolio selection with marginal risk control. Journal of Computational Finance, 14(1), 3–28.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.WashingtonUSA
  2. 2.WashingtonUSA

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