Bagged Pretested Portfolio Selection

Abstract

This article exploits the idea of combining pretesting and bagging to choose between competing portfolio strategies. We propose an estimator for the portfolio weight vector, which optimally trades off Type I against Type II errors when choosing the best investment strategy. Furthermore, we accommodate the idea of bagging in the portfolio testing problem, which helps to avoid sharp thresholding and reduces turnover costs substantially. Our Bagged Pretested Portfolio Selection (BPPS) approach borrows from both the shrinkage and the forecast combination literature. The portfolio weights of our strategy are weighted averages of the portfolio weights from a set of stand-alone strategies. More specifically, the weights are generated from pseudo-out-of-sample portfolio pretesting, such that they reflect the probability that a given strategy will be overall best performing. The resulting strategy allows for a flexible and smooth switch between the underlying strategies and outperforms the corresponding stand-alone strategies. Besides yielding high point estimates of the portfolio performance measures, the BPPS approach performs exceptionally well in terms of precision and is robust against outliers resulting from the choice of the asset space.

KEYWORD:

• Adaptive learning
• Bagging
• Portfolio allocation
• Pretest estimation

Supplementary Materials

In the supplementary web appendix we provide explicit formulas for the standard errors of the pretest test statistic, proofs of the BPPS properties and additional empirical results. We also include MATLAB replication codes and the data we used.

Acknowledgments

The authors would like to thank Allan Timmermann and Yarema Okhrin for helpful comments. Early versions of the article have been presented at the SoFiE in Lugano, the Conference on Factor Investing in Lancaster, QFFE in Marseille, DAGStat in Munich and the Workshop on Financial Econometrics, Oerbro. All remaining errors are ours.

Notes

1 Note that in the literature the use of the term tangency portfolio is not unique. Here we follow, for example, Britten-Jones (1999) and define the tangency portfolio as the one which is tangent to the minimum-variance bound such that the weights add up to one.

2 The data is taken from the Kenneth R. French website and contains monthly excess returns from May 1964 until December 2015: $\_$library.html" class="url" >http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data{}$\_$library.html.

3 This statement strictly holds if the objective function and the performance measure coincide, for example, for the GMVP and the portfolio variance, respectively.

4 Note that all the conclusions of this section also hold for testing the difference in Sharpe ratios and portfolio variances, respectively.

5 In machine learning literature the first part would be typically referred to as a training set and the second part as a validation set.

6 Note, that monthly returns do not possess any significant SACF or SPACF patterns. Therefore, the iid bootstrap of Efron (1992) is sufficient to use. For the returns of higher frequencies the circular block bootstrap by Politis and Romano (1992) is recommended.

7 See Schanbacher (2015) for a discussion on the concavity of different performance measures. The CE is a strictly concave measure as long as different strategies generate different portfolio variances.

8 In machine learning literature such estimators are often referred to as ensemble methods.

9 The average estimated shrinkage parameter for the sample is $\widehat{\theta }=0.93$.

10 We like to thank the editor for suggesting to us the SPA test as a tool to compare the BPPS with the competing portfolio strategies.

11 Turnover of a strategy s is computed as $TO(s)=\frac{1}{H}{\sum }_{t=1}^H\left({\sum }_{j=1}^N|{\widehat{\omega }}_{j,t+1}(s)-{\widehat{\omega }}_{j,t^{+}}(s)|\right)$.

Additional information

Funding

Financial support by Graduate School of Decision Sciences (GSDS) and the German Science Foundation (DFG) is gratefully acknowledged.