Abstract
In this work we propose the construction of optimized forecast-portfolios where analysts are thought of as “assets” with specific characteristics that may be combined in portfolios. The analysts’ forecasts were made about the German stock market index DAX on a 6-month horizon as provided by the ZEW Financial Market Survey. A Differential Evolution algorithm is applied that is flexible enough to work with the holey structure of the survey dataset, as it allows for the introduction of a weights-shifting scheme that prevents the exclusion of analysts with missing data and the resulting loss of information. The method is implemented for each of three objective functions that translate ideas from financial management into the forecast-portfolio framework: the mean–variance, the Value-at-Risk, and an asymmetric target strategy. With a backtest we show that weighting several individual forecasts with a forecast-portfolio can indeed improve the forecast quality.
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Notes
Detailed information about the survey can be found on the ZEW homepage. The survey questionnaire can be downloaded in the German language from the internet address http://www.zew.de/de/publikationen/Konjunkturerwartungen/Fragebogen_Deutsch_Muster.pdf.
In fact, the methodology we propose does not technically require the reduction of the dataset and the choice of 75 % is rather ad hoc. However, it alleviates the complexity of the empirical analyses.
An alternative could be to use the average daily closing calls in month \(t\). However, preliminary studies have shown that the differences are negligible compared to the magnitude of the forecast errors that will be presented in the empirical results.
Note that it is the absolute mean forecast error instead of the mean absolute forecast error. The latter results from the asymmetric target approach when appropriately parameterized.
The parenthesis \(\langle \cdot \rangle \) indicate an index number that corresponds to an ascending order while operator \([\cdot ]_{G}\) rounds down to the next smallest integer.
The superscript \(c\) indicates the current population. Analogously, \(n\) will denote the new population.
In Sect. 4 we assumed that all data are available so that \(\mathbf{e}_{p} = \mathbf{Fw}-\mathbf{I}\). When the assumption is dropped, \(\mathbf{Fw}-\mathbf{I}\) in all objective functions must simply be replaced by \((\mathbf{F}\odot \widehat{\mathbf{W}})\mathbf{1}_{K}-\mathbf{I}\).
In Appendix 1 we show results also for a specification with \(K_{\max } = 10\).
The parameter choices for the DE were found to work well for these problems in preliminary experiments.
Note that the allowed shorting of at most \(S_{\max } = 0.20\) is apparently not sufficient to compensate for the collective over- or underestimation. However, we also carried out experiments with larger values for \(S_{\max }\) which rather impaired the results. We believe, then, that the gained flexibility with which the portfolio can be fit to the in-sample data leads to overfitting, i.e. the portfolio is fitted to sample specific characteristicts that will not persist out-of-sample and deteriorate the performance.
The corresponding results are presented in Appendix 2 by the familiar table.
The small differences only arise by the varying in-sample window length and forecast horizons.
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Fastrich, B., Winker, P. Combining Forecasts with Missing Data: Making Use of Portfolio Theory. Comput Econ 44, 127–152 (2014). https://doi.org/10.1007/s10614-013-9401-z
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DOI: https://doi.org/10.1007/s10614-013-9401-z