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Multi-period portfolio selection with drawdown control

  • Peter Nystrup
  • Stephen Boyd
  • Erik Lindström
  • Henrik Madsen
S.I.: Application of O. R. to Financial Markets

Abstract

In this article, model predictive control is used to dynamically optimize an investment portfolio and control drawdowns. The control is based on multi-period forecasts of the mean and covariance of financial returns from a multivariate hidden Markov model with time-varying parameters. There are computational advantages to using model predictive control when estimates of future returns are updated every time new observations become available, because the optimal control actions are reconsidered anyway. Transaction and holding costs are discussed as a means to address estimation error and regularize the optimization problem. The proposed approach to multi-period portfolio selection is tested out of sample over two decades based on available market indices chosen to mimic the major liquid asset classes typically considered by institutional investors. By adjusting the risk aversion based on realized drawdown, it successfully controls drawdowns with little or no sacrifice of mean–variance efficiency. Using leverage it is possible to further increase the return without increasing the maximum drawdown.

Keywords

Risk management Maximum drawdown Dynamic asset allocation Model predictive control Regime switching Forecasting 

Notes

Acknowledgements

The authors are thankful for the helpful comments from the responsible editor Stavros A. Zenios and two anonymous referees.

References

  1. Almgren, R., & Chriss, N. (2001). Optimal execution of portfolio transactions. Journal of Risk, 3(2), 5–39.Google Scholar
  2. Ang, A., & Bekaert, G. (2004). How regimes affect asset allocation. Financial Analysts Journal, 60(2), 86–99.Google Scholar
  3. Ang, A., & Timmermann, A. (2012). Regime changes and financial markets. Annual Review of Financial Economics, 4(1), 313–337.Google Scholar
  4. Ardia, D., Bolliger, G., Boudt, K., & Gagnon-Fleury, J. P. (2017). The impact of covariance misspecification in risk-based portfolios. Annals of Operations Research, 254(1–2), 1–16.Google Scholar
  5. Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9(3), 203–228.Google Scholar
  6. Bae, G. I., Kim, W. C., & Mulvey, J. M. (2014). Dynamic asset allocation for varied financial markets under regime switching framework. European Journal of Operational Research, 234(2), 450–458.Google Scholar
  7. Bell, D. E. (1982). Regret in decision making under uncertainty. Operations Research, 30(5), 961–981.Google Scholar
  8. Bellman, R. (1956). Dynamic programming and Lagrange multipliers. Proceedings of the National Academy of Sciences, 42(10), 767–769.Google Scholar
  9. Bemporad, A., Bellucci, L., & Gabbriellini, T. (2014). Dynamic option hedging via stochastic model predictive control based on scenario simulation. Quantitative Finance, 14(10), 1739–1751.Google Scholar
  10. Bertsimas, D., Lauprete, G. J., & Samarov, A. (2004). Shortfall as a risk measure: Properties, optimization and applications. Journal of Economic Dynamics & Control, 28(7), 1353–1381.Google Scholar
  11. Black, F., & Jones, R. W. (1987). Simplifying portfolio insurance. Journal of Portfolio Management, 14(1), 48–51.Google Scholar
  12. Black, F., & Litterman, R. (1992). Global portfolio optimization. Financial Analysts Journal, 48(5), 28–43.Google Scholar
  13. Black, F., & Perold, A. F. (1992). Theory of constant proportion portfolio insurance. Journal of Economic Dynamics & Control, 16(3–4), 403–426.Google Scholar
  14. Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81(3), 637–654.Google Scholar
  15. Boyd, S., & Vandenberghe, L. (2004). Convex optimization. New York: Cambridge University Press.Google Scholar
  16. Boyd, S., Mueller, M. T., O’Donoghue, B., & Wang, Y. (2014). Performance bounds and suboptimal policies for multi-period investment. Foundations and Trends in Optimization, 1(1), 1–72.Google Scholar
  17. Boyd, S., Busseti, E., Diamond, S., Kahn, R. N., Koh, K., Nystrup, P., et al. (2017). Multi-period trading via convex optimization. Foundations and Trends in Optimization, 3(1), 1–76.Google Scholar
  18. Broadie, M. (1993). Computing efficient frontiers using estimated parameters. Annals of Operations Research, 45(1), 21–58.Google Scholar
  19. Brodie, J., Daubechies, I., Mol, C. D., Giannone, D., & Loris, I. (2009). Sparse and stable Markowitz portfolios. Proceedings of the National Academy of Sciences of the United States of America, 106(30), 12267–12272.Google Scholar
  20. Bulla, J., Mergner, S., Bulla, I., Sesboüé, A., & Chesneau, C. (2011). Markov-switching asset allocation: Do profitable strategies exist? Journal of Asset Management, 12(5), 310–321.Google Scholar
  21. Chaudhuri, S. E., & Lo, A. W. (2016). Spectral portfolio theory. Available at SSRN, 2788999, 1–44.Google Scholar
  22. Chopra, V. K., & Ziemba, W. T. (1993). The effect of errors in means, variances, and covariances on optimal portfolio choice. Journal of Portfolio Management, 19(2), 6–11.Google Scholar
  23. Cui, X., Gao, J., Li, X., & Li, D. (2014). Optimal multi-period mean-variance policy under no-shorting constraint. European Journal of Operational Research, 234(2), 459–468.Google Scholar
  24. Dai, M., Xu, Z. Q., & Zhou, X. Y. (2010). Continuous-time Markowitz’s model with transaction costs. SIAM Journal on Financial Mathematics, 1(1), 96–125.Google Scholar
  25. Dantzig, G. B., & Infanger, G. (1993). Multi-stage stochastic linear programs for portfolio optimization. Annals of Operations Research, 45(1), 59–76.Google Scholar
  26. DeMiguel, V., Garlappi, L., Nogales, F., & Uppal, R. (2009a). A generalized approach to portfolio optimization: Improving performance by constraining portfolio norms. Management Science, 55(5), 798–812.Google Scholar
  27. DeMiguel, V., Garlappi, L., & Uppal, R. (2009b). Optimal versus naive diversification: How inefficient is the $1/N$ portfolio strategy? Review of Financial Studies, 22(5), 1915–1953.Google Scholar
  28. Diamond, S., & Boyd, S. (2016). CVXPY: A Python-embedded modeling language for convex optimization. Journal of Machine Learning Research, 17(83), 1–5.Google Scholar
  29. Dias, J. G., Vermunt, J. K., & Ramos, S. (2015). Clustering financial time series: New insights from an extended hidden Markov model. European Journal of Operational Research, 243(3), 852–864.Google Scholar
  30. Dohi, T., & Osaki, S. (1993). A note on portfolio optimization with path-dependent utility. Annals of Operations Research, 45(1), 77–90.Google Scholar
  31. Domahidi, A., Chu, E., & Boyd, S. (2013). ECOS: An SOCP solver for embedded systems. In Proceedings of the 12th European control conference (pp. 3071–3076).Google Scholar
  32. Downing, C., Madhavan, A., Ulitsky, A., & Singh, A. (2015). Portfolio construction and tail risk. Journal of Portfolio Management, 42(1), 85–102.Google Scholar
  33. Fabozzi, F. J., Huang, D., & Zhou, G. (2010). Robust portfolios: Contributions from operations research and finance. Annals of Operations Research, 176(1), 191–220.Google Scholar
  34. Fastrich, B., Paterlini, S., & Winker, P. (2015). Constructing optimal sparse portfolios using regularization methods. Computational Management Science, 12(3), 417–434.Google Scholar
  35. Fiecas, M., Franke, J., von Sachs, R., & Kamgaing, J. T. (2017). Shrinkage estimation for multivariate hidden Markov models. Journal of the American Statistical Association, 112(517), 424–435.Google Scholar
  36. Fleming, J., Kirby, C., & Ostdiek, B. (2001). The economic value of volatility timing. Journal of Finance, 56(1), 329–352.Google Scholar
  37. Frühwirth-Schnatter, S. (2006). Finite mixture and Markov switching models. New York: Springer.Google Scholar
  38. Garlappi, L., Uppal, R., & Wang, T. (2006). Portfolio selection with parameter and model uncertainty: A multi-prior approach. Review of Financial Studies, 20(1), 41–81.Google Scholar
  39. Gârleanu, N., & Pedersen, L. H. (2013). Dynamic trading with predictable returns and transaction costs. Journal of Finance, 68(6), 2309–2340.Google Scholar
  40. Goltz, F., Martellini, L., & Simsek, K. D. (2008). Optimal static allocation decisions in the presence of portfolio insurance. Journal of Investment Management, 6(2), 37–56.Google Scholar
  41. Grinold, R. C. (2006). A dynamic model of portfolio management. Journal of Investment Management, 4(2), 5–22.Google Scholar
  42. Grinold, R. C., & Kahn, R. N. (2000). Active portfolio management: A quantitative approach for providing superior returns and controlling risk (2nd ed.). New York: McGraw-Hill.Google Scholar
  43. Grossman, S. J., & Zhou, Z. (1993). Optimal investment strategies for controlling drawdowns. Mathematical Finance, 3(3), 241–276.Google Scholar
  44. Guidolin, M., & Timmermann, A. (2007). Asset allocation under multivariate regime switching. Journal of Economic Dynamics and Control, 31(11), 3503–3544.Google Scholar
  45. Gülpınar, N., & Rustem, B. (2007). Worst-case robust decisions for multi-period mean-variance portfolio optimization. European Journal of Operational Research, 183(3), 981–1000.Google Scholar
  46. Herzog, F., Dondi, G., & Geering, H. P. (2007). Stochastic model predictive control and portfolio optimization. International Journal of Theoretical and Applied Finance, 10(2), 203–233.Google Scholar
  47. Ho, M., Sun, Z., & Xin, J. (2015). Weighted elastic net penalized mean-variance portfolio design and computation. SIAM Journal on Financial Mathematics, 6(1), 1220–1244.Google Scholar
  48. Ibragimov, R., Jaffee, D., & Walden, J. (2011). Diversification disasters. Journal of Financial Economics, 99(2), 333–348.Google Scholar
  49. Ilmanen, A. (2012). Do financial markets reward buying or selling insurance and lottery tickets? Financial Analysts Journal, 68(5), 26–36.Google Scholar
  50. Jagannathan, R., & Ma, T. (2003). Risk reduction in large portfolios: Why imposing the wrong constraints helps. Journal of Finance, 58(4), 1651–1683.Google Scholar
  51. Jorion, P. (1985). International portfolio diversification with estimation risk. Journal of Business, 58(3), 259–278.Google Scholar
  52. Kan, R., & Zhou, G. (2007). Optimal portfolio choice with parameter uncertainty. Journal of Financial and Quantitative Analysis, 42(3), 621–656.Google Scholar
  53. Khreich, W., Granger, E., Miri, A., & Sabourin, R. (2012). A survey of techniques for incremental learning of HMM parameters. Information Sciences, 197, 105–130.Google Scholar
  54. Kinlaw, W., Kritzman, M., & Turkington, D. (2014). The divergence of high- and low-frequency estimation: Causes and consequences. Journal of Portfolio Management, 40(5), 156–168.Google Scholar
  55. Kinlaw, W., Kritzman, M., & Turkington, D. (2015). The divergence of high- and low-frequency estimation: Implications for performance measurement. Journal of Portfolio Management, 41(3), 14–21.Google Scholar
  56. Kolm, P., Tütüncü, R., & Fabozzi, F. (2014). 60 years of portfolio optimization: Practical challenges and current trends. European Journal of Operational Research, 234(2), 356–371.Google Scholar
  57. Kritzman, M., & Li, Y. (2010). Skulls, financial turbulence, and risk management. Financial Analysts Journal, 66(5), 30–41.Google Scholar
  58. Kritzman, M., Page, S., & Turkington, D. (2012). Regime shifts: Implications for dynamic strategies. Financial Analysts Journal, 68(3), 22–39.Google Scholar
  59. Ledoit, O., & Wolf, M. (2003). Improved estimation of the covariance matrix of stock returns with an application to portfolio selection. Journal of Empirical Finance, 10(5), 603–621.Google Scholar
  60. Ledoit, O., & Wolf, M. (2004). A well-conditioned estimator for large-dimensional covariance matrices. Journal of Multivariate Analysis, 88(2), 365–411.Google Scholar
  61. Leland, H. E. (1980). Who should buy portfolio insurance? Journal of Finance, 35(2), 581–594.Google Scholar
  62. Li, J. (2015). Sparse and stable portfolio selection with parameter uncertainty. Journal of Business & Economic Statistics, 33(3), 381–392.Google Scholar
  63. Lim, A. E., Shanthikumar, J. G., & Vahn, G. Y. (2011). Conditional value-at-risk in portfolio optimization: Coherent but fragile. Operations Research Letters, 39(3), 163–171.Google Scholar
  64. López de Prado, M. (2016). Building diversified portfolios that outperform out of sample. Journal of Portfolio Management, 42(4), 59–69.Google Scholar
  65. Mandelbrot, B. (1963). The variation of certain speculative prices. Journal of Business, 36(4), 394–419.Google Scholar
  66. Markowitz, H. (1952). Portfolio selection. Journal of Finance, 7(1), 77–91.Google Scholar
  67. Markowitz, H. (2014). Mean-variance approximations to expected utility. European Journal of Operational Research, 234(2), 346–355.Google Scholar
  68. Mattingley, J., & Boyd, S. (2012). CVXGEN: a code generator for embedded convex optimization. Optimization and Engineering, 13(1), 1–27.Google Scholar
  69. Mei, X., DeMiguel, V., & Nogales, F. J. (2016). Multiperiod portfolio optimization with multiple risky assets and general transaction costs. Journal of Banking & Finance, 69, 108–120.Google Scholar
  70. Meindl, P. J., & Primbs, J. A. (2008). Dynamic hedging of single and multi-dimensional options with transaction costs: A generalized utility maximization approach. Quantitative Finance, 8(3), 299–312.Google Scholar
  71. Merton, R. C. (1969). Lifetime portfolio selection under uncertainty: The continuous-time case. Review of Economics and Statistics, 51(3), 247–257.Google Scholar
  72. Merton, R. C. (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4(1), 141–183.Google Scholar
  73. Merton, R. C. (1980). On estimating the expected return on the market: An exploratory investigation. Journal of Financial Economics, 8(4), 323–361.Google Scholar
  74. Michaud, R. O. (1989). The Markowitz optimization Enigma: Is ’optimized’ optimal? Financial Analysts Journal, 45(1), 31–42.Google Scholar
  75. Moreira, A., & Muir, T. (2017). Volatility-managed portfolios. Journal of Finance, 72(4), 1611–1644.Google Scholar
  76. Mossin, J. (1968). Optimal multiperiod portfolio policies. Journal of Business, 41(2), 215–229.Google Scholar
  77. Mulvey, J. M., & Shetty, B. (2004). Financial planning via multi-stage stochastic optimization. Computers & Operations Research, 31(1), 1–20.Google Scholar
  78. Nystrup, P., Hansen, B. W., Madsen, H., & Lindström, E. (2015a). Regime-based versus static asset allocation: Letting the data speak. Journal of Portfolio Management, 42(1), 103–109.Google Scholar
  79. Nystrup, P., Madsen, H., & Lindström, E. (2015b). Stylised facts of financial time series and hidden Markov models in continuous time. Quantitative Finance, 15(9), 1531–1541.Google Scholar
  80. Nystrup, P., Hansen, B. W., Larsen, H. O., Madsen, H., & Lindström, E. (2017a). Dynamic allocation or diversification: A regime-based approach to multiple assets. Journal of Portfolio Management, 44(2), 62–73.Google Scholar
  81. Nystrup, P., Madsen, H., & Lindström, E. (2017b). Long memory of financial time series and hidden Markov models with time-varying parameters. Journal of Forecasting, 36(8), 989–1002.Google Scholar
  82. Nystrup, P., Madsen, H., & Lindström, E. (2018). Dynamic portfolio optimization across hidden market regimes. Quantitative Finance, 18(1), 83–95.Google Scholar
  83. Pedersen, L. H. (2009). When everyone runs for the exit. International Journal of Central Banking, 5(4), 177–199.Google Scholar
  84. Pedersen, L. H. (2015). Efficiently inefficient: how smart money invests and market prices are determined. Princeton: Princeton University Press.Google Scholar
  85. Pınar, M. Ç. (2007). Robust scenario optimization based on downside-risk measure for multi-period portfolio selection. OR Spectrum, 29(2), 295–309.Google Scholar
  86. Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of Risk, 2(3), 21–42.Google Scholar
  87. Rubinstein, M., & Leland, H. E. (1981). Replicating options with positions in stock and cash. Financial Analysts Journal, 37(4), 63–72.Google Scholar
  88. Rydén, T., Teräsvirta, T., & Åsbrink, S. (1998). Stylized facts of daily return series and the hidden Markov model. Journal of Applied Econometrics, 13(3), 217–244.Google Scholar
  89. Samuelson, P. A. (1969). Lifetime portfolio selection by dynamic stochastic programming. Review of Economics and Statistics, 51(3), 239–246.Google Scholar
  90. Scutellà, M. G., & Recchia, R. (2013). Robust portfolio asset allocation and risk measures. Annals of Operations Research, 204(1), 145–169.Google Scholar
  91. Sharpe, W. F. (1966). Mutual fund performance. Journal of Business, 39(1), 119–138.Google Scholar
  92. Sharpe, W. F. (1994). The Sharpe ratio. Journal of Portfolio Management, 21(1), 49–58.Google Scholar
  93. Smidl, V., & Gustafsson, F. (2012) Bayesian estimation of forgetting factor in adaptive filtering and change detection. In Proceedings of the 2012 IEEE statistical signal processing workshop (pp 197–200).Google Scholar
  94. Stein, C. (1956). Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. Proceedings of the third Berkeley symposium on mathematical statistics and probability (Vol. 1, pp. 197–206), University of California Press, Berkeley.Google Scholar
  95. Stenger, B., Ramesh, V., Paragios, N., Coetzee, F., & Buhmann, J. M. (2001). Topology free hidden Markov models: Application to background modeling. Proceedings of the eighth IEEE international conference on computer vision (Vol. 1, pp. 294–301).Google Scholar
  96. Stoyanov, S. V., Rachev, S. T., & Fabozzi, F. J. (2012). Sensitivity of portfolio VaR and CVaR to portfolio return characteristics. Annals of Operations Research, 205(1), 169–187.Google Scholar
  97. von Neumann, J., & Morgenstern, O. (1953). Theory of games and economic behavior (3rd ed.). Princeton: Princeton University Press.Google Scholar
  98. Zenios, S. A. (2007). Practical financial optimization: Decision making for financial engineers. Malden: Blackwell.Google Scholar
  99. Zhou, G., & Zhu, Y. (2010). Is the recent financial crisis really a “once-in-a-century” event? Financial Analysts Journal, 66(1), 24–27.Google Scholar

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Authors and Affiliations

  1. 1.Department of Applied Mathematics and Computer ScienceTechnical University of DenmarkKgs. LyngbyDenmark
  2. 2.ANNOXHellerupDenmark
  3. 3.Department of Electrical EngineeringStanford UniversityStanfordUSA
  4. 4.Centre for Mathematical SciencesLund UniversityLundSweden

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