Generalized risk parity portfolio optimization: an ADMM approach

Abstract

The risk parity solution to the asset allocation problem yields portfolios where the risk contribution from each asset is made equal. We consider a generalized approach to this problem. First, we set an objective that seeks to maximize the portfolio expected return while minimizing portfolio risk. Second, we relax the risk parity condition and instead bound the risk dispersion of the constituents within a predefined limit. This allows an investor to prescribe a desired risk dispersion range, yielding a portfolio with an optimal risk–return profile that is still well-diversified from a risk-based standpoint. We add robustness to our framework by introducing an ellipsoidal uncertainty structure around our estimated asset expected returns to mitigate estimation error. Our proposed framework does not impose any restrictions on short selling. A limitation of risk parity is that allowing of short sales leads to a non-convex problem. However, we propose an approach that relaxes our generalized risk parity model into a convex semidefinite program. We proceed to tighten this relaxation sequentially through the alternating direction method of multipliers. This procedure iterates between the convex optimization problem and the non-convex problem with a rank constraint. In addition, we can exploit this structure to solve the non-convex problem analytically and efficiently during every iteration. Numerical results suggest that this algorithm converges to a higher quality optimal solution when compared to the competing non-convex problem, and can also yield a higher ex post risk-adjusted rate of return.

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Notes

  1. 1.

    Short selling pertains to taking negative positions on our investments. ‘Long-only’ refers to portfolios where we can only take non-negative positions.

  2. 2.

    An estimated covariance matrix of financial assets has the useful property of being symmetric and positive semidefinite, provided sufficient data is used for estimation.

References

  1. 1.

    Bai, X., Scheinberg, K., Tütüncü, R.H.: Least-squares approach to risk parity in portfolio selection. Quantitative Finance 16(3), 357–376 (2016)

    MathSciNet  Google Scholar 

  2. 2.

    Best, M.J., Grauer, R.R.: On the sensitivity of mean-variance-efficient portfolios to changes in asset means: some analytical and computational results. Rev. Financ. Stud. 4(2), 315–342 (1991)

    Google Scholar 

  3. 3.

    Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J., et al.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends ® in Mach. Learn. 3(1), 1–122 (2011)

    MATH  Google Scholar 

  4. 4.

    Broadie, M.: Computing efficient frontiers using estimated parameters. Ann. Oper. Res. 45(1), 21–58 (1993)

    MATH  Google Scholar 

  5. 5.

    Bruder, B., Roncalli, T., et al.: Managing Risk Exposures Using the Risk Budgeting Approach. Tech. rep., University Library of Munich, Germany (2012)

    Google Scholar 

  6. 6.

    Chen, G., Teboulle, M.: A proximal-based decomposition method for convex minimization problems. Math. Program. 64(1–3), 81–101 (1994)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Chopra, V.K., Ziemba, W.T.: The effect of errors in means, variances, and covariances on optimal portfolio choice. J. Portf. Manag. 19(2), 6–11 (1993)

  8. 8.

    Costa, G., Kwon, R.H.: Risk parity portfolio optimization under a markov regime-switching framework. Quant. Finance 19(3), 453–471 (2019)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Delage, E., Ye, Y.: Distributionally robust optimization under moment uncertainty with application to data-driven problems. Oper. Res. 58(3), 595–612 (2010)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Dunning, I., Huchette, J., Lubin, M.: Jump: a modeling language for mathematical optimization. Soc. Ind. Appl. Math. 59(2), 295–320 (2017). https://doi.org/10.1137/15M1020575

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Eckstein, J., Bertsekas, D.P.: On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55(1–3), 293–318 (1992)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Eckstein, J., Fukushima, M.: Some reformulations and applications of the alternating direction method of multipliers. In: Hager, W.W., Hearn, D.W., Pardalos, P.M. (eds.) Large Scale Optimization, pp. 115–134. Springer, Boston (1994)

    Google Scholar 

  13. 13.

    Fabozzi, F.J., Kolm, P.N., Pachamanova, D.A., Focardi, S.M.: Robust portfolio optimization. J. Portf. Manag. 33(3), 40 (2007)

    Google Scholar 

  14. 14.

    Fama, E.F., French, K.R.: Common risk factors in the returns on stocks and bonds. J. Financ. Econ. 33(1), 3–56 (1993)

    MATH  Google Scholar 

  15. 15.

    Feng, Y., Palomar, D.P.: SCRIP: Successive convex optimization methods for risk parity portfolio design. IEEE Trans. Signal Process. 63(19), 5285–5300 (2015)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Fortin, M., Glowinski, R.: On decomposition-coordination methods using an augmented Lagrangian. In: Fortin, M., Glowinski, R. (eds.) Augmented Lagrangian Methods: Applications to the Solution of Boundary-Value Problems, vol. 15. Elsevier, Amsterdam (1983)

    Google Scholar 

  17. 17.

    Fukushima, M.: Application of the alternating direction method of multipliers to separable convex programming problems. Comput. Optim. Appl. 1(1), 93–111 (1992)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Gabay, D.: Applications of the method of multipliers to variational inequalities. In: Fortin, M., Glowinski, R. (eds.) Augmented Lagrangian Methods: Applications to the Solution of Boundary-Value Problems, vol. 15, pp. 299–331. Elsevier, Amsterdam (1983)

    Google Scholar 

  19. 19.

    Gabay, D., Mercier, B.: A dual algorithm for the solution of non linear variational problems via finite element approximation. Institut de recherche d’informatique et d’automatique 2, 17–40 (1975)

    MATH  Google Scholar 

  20. 20.

    Ghadimi, E., Teixeira, A., Shames, I., Johansson, M.: Optimal parameter selection for the alternating direction method of multipliers (ADMM): quadratic problems. IEEE Trans. Autom. Control 60(3), 644–658 (2015)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Glowinski, R., Marroco, A.: Sur l’approximation, par éléments finis d’ordre un, et la résolution, par pénalisation-dualité d’une classe de problèmes de dirichlet non linéaires. Revue française d’automatique, informatique, recherche opérationnelle. Analyse numérique 9, 41–76 (1975)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Goldfarb, D., Iyengar, G.: Robust portfolio selection problems. Math. Oper. Res. 28(1), 1–38 (2003)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Goldstein, T., O’Donoghue, B., Setzer, S., Baraniuk, R.: Fast alternating direction optimization methods. SIAM J. Imaging Sci. 7(3), 1588–1623 (2014)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Haugh, M., Iyengar, G., Song, I.: A generalized risk budgeting approach to portfolio construction. J. Comput. Finance 21(2), 29–60 (2017)

    Google Scholar 

  25. 25.

    He, B., Yang, H., Wang, S.: Alternating direction method with self-adaptive penalty parameters for monotone variational inequalities. J. Optim. Theory Appl. 106(2), 337–356 (2000)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Jorion, P.: Bayes-Stein estimation for portfolio analysis. J. Financ. Quant. Anal. 21(3), 279–292 (1986)

    Google Scholar 

  27. 27.

    Kapsos, M., Christofides, N., Rustem, B.: Robust risk budgeting. Ann. Oper. Res. 266, 1–23 (2017)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Ledoit, O., Wolf, M.: Improved estimation of the covariance matrix of stock returns with an application to portfolio selection. J. Empir. Finance 10(5), 603–621 (2003)

    Google Scholar 

  29. 29.

    Lintner, J.: The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets. Rev. Econ. Stat. 47, 13–37 (1965)

    Google Scholar 

  30. 30.

    Lobo, M.S., Boyd, S.: The worst-case risk of a portfolio. Technical report. Available from http://web.stanford.edu/~boyd/papers/pdf/risk_bnd.pdf (2000)

  31. 31.

    Maillard, S., Roncalli, T., Teïletche, J.: The properties of equally weighted risk contribution portfolios. J. Portf. Manag. 36(4), 60–70 (2010)

    Google Scholar 

  32. 32.

    Markowitz, H.: Portfolio selection. J. Finance 7(1), 77–91 (1952)

    Google Scholar 

  33. 33.

    Merton, R.C.: On estimating the expected return on the market: an exploratory investigation. J. Financ. Econ. 8(4), 323–361 (1980)

    Google Scholar 

  34. 34.

    Mossin, J.: Equilibrium in a capital asset market. Econom. J. Econom. Soc. 34, 768–783 (1966)

    Google Scholar 

  35. 35.

    Nesterov, Y.: Introductory Lectures on Convex Optimization: A Basic Course, vol. 87. Springer, New York (2013)

    Google Scholar 

  36. 36.

    Qian, E.: Risk parity portfolios: Efficient portfolios through true diversification. Panagora Asset Management White Paper (2005)

  37. 37.

    Qian, E.: On the financial interpretation of risk contribution: risk budgets do add up. J. Invest. Manag. 4(4), 41–51 (2006)

    Google Scholar 

  38. 38.

    Quandl.com: Wiki—various end-of-day stock prices (2017). https://www.quandl.com/databases/WIKIP/usage/export. [Online; Accessed 07-Nov-2017]

  39. 39.

    Raghunathan, A.U., Di Cairano, S.: Alternating direction method of multipliers for strictly convex quadratic programs: Optimal parameter selection. In: American Control Conference (ACC), pp. 4324–4329. IEEE (2014)

  40. 40.

    Sharpe, W.F.: Capital asset prices: a theory of market equilibrium under conditions of risk. J. Finance 19(3), 425–442 (1964)

    Google Scholar 

  41. 41.

    Sharpe, W.F.: The sharpe ratio. J. Portf. Manag. 21(1), 49–58 (1994)

    Google Scholar 

  42. 42.

    Tseng, P.: Applications of a splitting algorithm to decomposition in convex programming and variational inequalities. SIAM J. Control Optim. 29(1), 119–138 (1991)

    MathSciNet  MATH  Google Scholar 

  43. 43.

    Tütüncü, R.H., Koenig, M.: Robust asset allocation. Ann. Oper. Res. 132(1–4), 157–187 (2004)

    MathSciNet  MATH  Google Scholar 

  44. 44.

    Wang, S., Liao, L.: Decomposition method with a variable parameter for a class of monotone variational inequality problems. J. Optim. Theory Appl. 109(2), 415–429 (2001)

    MathSciNet  MATH  Google Scholar 

  45. 45.

    You, S., Peng, Q.: A non-convex alternating direction method of multipliers heuristic for optimal power flow. In: 2014 IEEE International Conference on Smart Grid Communications (SmartGridComm), pp. 788–793. IEEE (2014)

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Funding

Funding was provided by Natural Sciences and Engineering Research Council of Canada (Grant Number 455963).

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Correspondence to Roy H. Kwon.

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Costa, G., Kwon, R.H. Generalized risk parity portfolio optimization: an ADMM approach. J Glob Optim (2020). https://doi.org/10.1007/s10898-020-00915-x

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Keywords

  • Non-convex optimization
  • Robust optimization
  • ADMM
  • Risk parity
  • Asset allocation