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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.

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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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Authors and Affiliations

  1. Department of Applied Mathematics and Computer Science, Technical University of Denmark, Asmussens Allé, Building 303B, 2800, Kgs. Lyngby, Denmark

    Peter Nystrup & Henrik Madsen

  2. ANNOX, Svanemøllevej 41, 2900, Hellerup, Denmark

    Peter Nystrup

  3. Department of Electrical Engineering, Stanford University, 350 Serra Mall, Stanford, CA, 94305, USA

    Stephen Boyd

  4. Centre for Mathematical Sciences, Lund University, Box 118, 221 00, Lund, Sweden

    Erik Lindström

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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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