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Multi-period portfolio selection with drawdown control

  • S.I.: Application of O. R. to Financial Markets
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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.

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Notes

  1. When \(H=1\), the multi-period problem (2) with risk function (3) reduces to the single-period mean–variance problem studied by Markowitz (1952).

  2. If the underlying return distribution is Gaussian with known parameters, then the portfolio that minimizes expected shortfall for a given expected return is equivalent to the portfolio that minimizes variance with the same expected return (Rockafellar and Uryasev 2000).

  3. Price impact is the price movement against the trader that tends to occur when a large order is executed.

  4. A quantitative manifestation of this fact is that while returns themselves are uncorrelated, absolute and squared returns display a positive, significant, and slowly decaying autocorrelation function.

  5. See also the survey by Khreich et al. (2012).

  6. The eight indices are MSCI World, MSCI Emerging Markets, FTSE EPRA/NAREIT Developed Real Estate, BofA Merrill Lynch U.S. High Yield, S&P GSCI Crude Oil (funded futures roll), LBMA Gold Price, Barclays U.S. Aggregate Corporate Bonds, and Bloomberg Barclays U.S. Government Bonds.

  7. Days on which more than half of the indices had zero price change (27 days in total) have been removed. In the few months where only monthly prices are available for DM high-yield bonds, linear interpolation with Gaussian noise has been used to fill the gaps.

  8. The ten indices are MSCI World, MSCI Emerging Markets, FTSE EPRA/NAREIT Developed Real Estate, BofA Merrill Lynch U.S. High Yield, Barclays Emerging Markets High Yield, S&P GSCI Crude Oil (funded futures roll), LBMA Gold Price, Barclays U.S. Aggregate Corporate Bonds, Barclays World Inflation-Linked Bonds (hedged to USD), and Citi G7 Government Bonds (hedged to USD).

  9. Days on which more than half of the indices had zero price change (19 days in total) have been removed.

  10. The Sharpe ratio is the excess return divided by the excess risk (Sharpe 1966, 1994).

  11. The maximum drawdown is the largest relative decline from a historical peak in the index value, as defined in Sect. 2.4.

  12. The Calmar ratio is the annualized excess return divided by the maximum drawdown.

  13. The adjustment leads to the reported excess risks being higher than had they been annualized under the assumption of independence, as most of the indices display positive autocorrelation. The largest impact was on the excess risk of EM stocks that went from 0.20 to 0.28 and the excess risk of DM high-yield bonds that went from 0.05 to 0.12.

  14. A transaction cost of 10 basis points is within the range of values estimated in empirical studies (see Pedersen 2015, Chapter 5). It could be argued that transaction costs should be lower for some indices and higher for others. This could easily be implemented as the elements of \(\kappa _{1}\) and \(\kappa _{2}\) in (7) need not all be the same.

  15. See Grinold (2006) and Boyd et al. (2017) for more on amortization of transaction and holding costs.

  16. Note that all hyperparameters were selected in sample based on a daily update frequency (Sect. 4.2). When these parameters are used with a lower update frequency, as expected, the results are worse.

  17. Results from the experiments with weekly portfolio adjustments are not reported in the article but are available upon request.

References

  • Almgren, R., & Chriss, N. (2001). Optimal execution of portfolio transactions. Journal of Risk, 3(2), 5–39.

    Google Scholar 

  • Ang, A., & Bekaert, G. (2004). How regimes affect asset allocation. Financial Analysts Journal, 60(2), 86–99.

    Google Scholar 

  • Ang, A., & Timmermann, A. (2012). Regime changes and financial markets. Annual Review of Financial Economics, 4(1), 313–337.

    Google Scholar 

  • 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 

  • Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9(3), 203–228.

    Google Scholar 

  • 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 

  • Bell, D. E. (1982). Regret in decision making under uncertainty. Operations Research, 30(5), 961–981.

    Google Scholar 

  • Bellman, R. (1956). Dynamic programming and Lagrange multipliers. Proceedings of the National Academy of Sciences, 42(10), 767–769.

    Google Scholar 

  • 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 

  • 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 

  • Black, F., & Jones, R. W. (1987). Simplifying portfolio insurance. Journal of Portfolio Management, 14(1), 48–51.

    Google Scholar 

  • Black, F., & Litterman, R. (1992). Global portfolio optimization. Financial Analysts Journal, 48(5), 28–43.

    Google Scholar 

  • Black, F., & Perold, A. F. (1992). Theory of constant proportion portfolio insurance. Journal of Economic Dynamics & Control, 16(3–4), 403–426.

    Google Scholar 

  • Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81(3), 637–654.

    Google Scholar 

  • Boyd, S., & Vandenberghe, L. (2004). Convex optimization. New York: Cambridge University Press.

    Google Scholar 

  • 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 

  • 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 

  • Broadie, M. (1993). Computing efficient frontiers using estimated parameters. Annals of Operations Research, 45(1), 21–58.

    Google Scholar 

  • 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 

  • 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 

  • Chaudhuri, S. E., & Lo, A. W. (2016). Spectral portfolio theory. Available at SSRN, 2788999, 1–44.

  • 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 

  • 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 

  • 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 

  • Dantzig, G. B., & Infanger, G. (1993). Multi-stage stochastic linear programs for portfolio optimization. Annals of Operations Research, 45(1), 59–76.

    Google Scholar 

  • 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 

  • 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 

  • 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 

  • 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 

  • Dohi, T., & Osaki, S. (1993). A note on portfolio optimization with path-dependent utility. Annals of Operations Research, 45(1), 77–90.

    Google Scholar 

  • 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).

  • Downing, C., Madhavan, A., Ulitsky, A., & Singh, A. (2015). Portfolio construction and tail risk. Journal of Portfolio Management, 42(1), 85–102.

    Google Scholar 

  • 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 

  • Fastrich, B., Paterlini, S., & Winker, P. (2015). Constructing optimal sparse portfolios using regularization methods. Computational Management Science, 12(3), 417–434.

    Google Scholar 

  • 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 

  • Fleming, J., Kirby, C., & Ostdiek, B. (2001). The economic value of volatility timing. Journal of Finance, 56(1), 329–352.

    Google Scholar 

  • Frühwirth-Schnatter, S. (2006). Finite mixture and Markov switching models. New York: Springer.

    Google Scholar 

  • 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 

  • Gârleanu, N., & Pedersen, L. H. (2013). Dynamic trading with predictable returns and transaction costs. Journal of Finance, 68(6), 2309–2340.

    Google Scholar 

  • 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 

  • Grinold, R. C. (2006). A dynamic model of portfolio management. Journal of Investment Management, 4(2), 5–22.

    Google Scholar 

  • 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 

  • Grossman, S. J., & Zhou, Z. (1993). Optimal investment strategies for controlling drawdowns. Mathematical Finance, 3(3), 241–276.

    Google Scholar 

  • Guidolin, M., & Timmermann, A. (2007). Asset allocation under multivariate regime switching. Journal of Economic Dynamics and Control, 31(11), 3503–3544.

    Google Scholar 

  • 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 

  • 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 

  • 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 

  • Ibragimov, R., Jaffee, D., & Walden, J. (2011). Diversification disasters. Journal of Financial Economics, 99(2), 333–348.

    Google Scholar 

  • Ilmanen, A. (2012). Do financial markets reward buying or selling insurance and lottery tickets? Financial Analysts Journal, 68(5), 26–36.

    Google Scholar 

  • 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 

  • Jorion, P. (1985). International portfolio diversification with estimation risk. Journal of Business, 58(3), 259–278.

    Google Scholar 

  • Kan, R., & Zhou, G. (2007). Optimal portfolio choice with parameter uncertainty. Journal of Financial and Quantitative Analysis, 42(3), 621–656.

    Google Scholar 

  • 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 

  • 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 

  • 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 

  • 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 

  • Kritzman, M., & Li, Y. (2010). Skulls, financial turbulence, and risk management. Financial Analysts Journal, 66(5), 30–41.

    Google Scholar 

  • Kritzman, M., Page, S., & Turkington, D. (2012). Regime shifts: Implications for dynamic strategies. Financial Analysts Journal, 68(3), 22–39.

    Google Scholar 

  • 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 

  • Ledoit, O., & Wolf, M. (2004). A well-conditioned estimator for large-dimensional covariance matrices. Journal of Multivariate Analysis, 88(2), 365–411.

    Google Scholar 

  • Leland, H. E. (1980). Who should buy portfolio insurance? Journal of Finance, 35(2), 581–594.

    Google Scholar 

  • Li, J. (2015). Sparse and stable portfolio selection with parameter uncertainty. Journal of Business & Economic Statistics, 33(3), 381–392.

    Google Scholar 

  • 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 

  • López de Prado, M. (2016). Building diversified portfolios that outperform out of sample. Journal of Portfolio Management, 42(4), 59–69.

    Google Scholar 

  • Mandelbrot, B. (1963). The variation of certain speculative prices. Journal of Business, 36(4), 394–419.

    Google Scholar 

  • Markowitz, H. (1952). Portfolio selection. Journal of Finance, 7(1), 77–91.

    Google Scholar 

  • Markowitz, H. (2014). Mean-variance approximations to expected utility. European Journal of Operational Research, 234(2), 346–355.

    Google Scholar 

  • Mattingley, J., & Boyd, S. (2012). CVXGEN: a code generator for embedded convex optimization. Optimization and Engineering, 13(1), 1–27.

    Google Scholar 

  • 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 

  • 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 

  • Merton, R. C. (1969). Lifetime portfolio selection under uncertainty: The continuous-time case. Review of Economics and Statistics, 51(3), 247–257.

    Google Scholar 

  • Merton, R. C. (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4(1), 141–183.

    Google Scholar 

  • 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 

  • Michaud, R. O. (1989). The Markowitz optimization Enigma: Is ’optimized’ optimal? Financial Analysts Journal, 45(1), 31–42.

    Google Scholar 

  • Moreira, A., & Muir, T. (2017). Volatility-managed portfolios. Journal of Finance, 72(4), 1611–1644.

    Google Scholar 

  • Mossin, J. (1968). Optimal multiperiod portfolio policies. Journal of Business, 41(2), 215–229.

    Google Scholar 

  • Mulvey, J. M., & Shetty, B. (2004). Financial planning via multi-stage stochastic optimization. Computers & Operations Research, 31(1), 1–20.

    Google Scholar 

  • 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 

  • 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 

  • 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 

  • 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 

  • Nystrup, P., Madsen, H., & Lindström, E. (2018). Dynamic portfolio optimization across hidden market regimes. Quantitative Finance, 18(1), 83–95.

    Google Scholar 

  • Pedersen, L. H. (2009). When everyone runs for the exit. International Journal of Central Banking, 5(4), 177–199.

    Google Scholar 

  • Pedersen, L. H. (2015). Efficiently inefficient: how smart money invests and market prices are determined. Princeton: Princeton University Press.

    Google Scholar 

  • 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 

  • Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of Risk, 2(3), 21–42.

    Google Scholar 

  • Rubinstein, M., & Leland, H. E. (1981). Replicating options with positions in stock and cash. Financial Analysts Journal, 37(4), 63–72.

    Google Scholar 

  • 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 

  • Samuelson, P. A. (1969). Lifetime portfolio selection by dynamic stochastic programming. Review of Economics and Statistics, 51(3), 239–246.

    Google Scholar 

  • Scutellà, M. G., & Recchia, R. (2013). Robust portfolio asset allocation and risk measures. Annals of Operations Research, 204(1), 145–169.

    Google Scholar 

  • Sharpe, W. F. (1966). Mutual fund performance. Journal of Business, 39(1), 119–138.

    Google Scholar 

  • Sharpe, W. F. (1994). The Sharpe ratio. Journal of Portfolio Management, 21(1), 49–58.

    Google Scholar 

  • 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).

  • 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.

  • 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).

  • 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 

  • von Neumann, J., & Morgenstern, O. (1953). Theory of games and economic behavior (3rd ed.). Princeton: Princeton University Press.

    Google Scholar 

  • Zenios, S. A. (2007). Practical financial optimization: Decision making for financial engineers. Malden: Blackwell.

    Google Scholar 

  • 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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Acknowledgements

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

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Correspondence to Peter Nystrup.

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This work was supported by Sampension and Innovation Fund Denmark under Grant No. 4135-00077B.

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Nystrup, P., Boyd, S., Lindström, E. et al. Multi-period portfolio selection with drawdown control. Ann Oper Res 282, 245–271 (2019). https://doi.org/10.1007/s10479-018-2947-3

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