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Combining Forecasts with Missing Data: Making Use of Portfolio Theory

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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

  1. 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.

  2. 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.

  3. 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.

  4. 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.

  5. 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.

  6. The superscript \(c\) indicates the current population. Analogously, \(n\) will denote the new population.

  7. 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 \((\mathbf{F}\odot \widehat{\mathbf{W}})\mathbf{1}_{K}-\mathbf{I}\).

  8. In Appendix 1 we show results also for a specification with \(K_{\max } = 10\).

  9. The parameter choices for the DE were found to work well for these problems in preliminary experiments.

  10. 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.

    Fig. 4
    figure 4

    Graphical representation of Panel A in Table 1. ac Forecast errors (left axis) of the forecast-portfolio (bold line), the most successful analyst (asterisk), the equally weighted portfolio 10 K (circles), and the no-change prognosis (squares). The shaded bars indicate periods when the most successful analysts (out of all) did not provide a forecast. The shaded area shows the number of available analysts (right axis). a corresponds to the mean–variance framework, b to the value-at-risk framework, and c to the asymmetric target framework. d shows the number of analysts that underestimate (below the baseline) and overestimate (above the baseline) the true index value

  11. The corresponding results are presented in Appendix 2 by the familiar table.

  12. The small differences only arise by the varying in-sample window length and forecast horizons.

References

  • Bates, J., & Granger, C. (1969). The combination of forecasts. Operations Research Quarterly, 20(4), 451–468.

    Article  Google Scholar 

  • Brodie, J., Daubechies, I., DeMol, C., Giannone, D., Loris, D., (2009) Sparse and stable Markowitz portfolios. Proceedings of the National Academy of Science USA , vol. 106(30) (pp. 12267–12272).

    Google Scholar 

  • Capistran, C., & Timmermann, A. (2009). Forecast combination with entry and exit of experts. Journal of Business and Economic Statsitics, 27(4), 428–440.

    Article  Google Scholar 

  • Clemen, R. (1989). Combining forecasts: A review and annotated bibliography. International Journal of Forecasting, 5(4), 559–583.

    Article  Google Scholar 

  • Clemen, R., & Winkler, R. (1986). Combining economic forecasts. Journal of Business and Economic Statistics, 4(1), 39–46.

    Google Scholar 

  • Diebold, F., & Mariano, R. (1995). Comparing predictive accuracy. Journal of Business and Economic Statistics, 13(3), 253–263.

    Google Scholar 

  • Diebold, J., & Pauly, P. (1987). Structural change and the combination of forecasts. Journal of Forecasting, 6(1), 21–40.

    Article  Google Scholar 

  • Elliott, G., & Timmermann, A. (2004). Optimal forecast combinations under general loss functions andforecast error distributions. Journal of Econometrics, 122(1), 39–46.

    Article  Google Scholar 

  • Genre, V., Kenny, G., Mayler, A., & Timmermann, A. (2013). Combining expert forecasts: Can anything beat the simple avarage? International Journal of Forecasting, 29(1), 108–121.

    Article  Google Scholar 

  • Granger, C., & Ramanathan, R. (1984). Improved methods of combining forecasts. Journal of Forecasting, 3(2), 197–204.

    Article  Google Scholar 

  • Gupta, S., & Wilton, P. (1987). Combination of forecasts: An extension. Management Science, 33(3), 356–372.

    Article  Google Scholar 

  • Harvey, D., Leybourne, S., & Newbold, P. (1997). Testing the equality of prediction mean squared errors. International Journal of Forecasting, 13(2), 281–291.

    Article  Google Scholar 

  • Jagannathan, R., & Ma, T. (2003). Risk reduction in large portfolios: Why imposing the wrong constraints helps. The Journal of Finance, 58(4), 1651–1683.

    Article  Google Scholar 

  • Jorion, P. (2001). Value at risk. The new benchmark for managing financial risk. New York: McGraw-Hill.

    Google Scholar 

  • Krink, T., Paterlini, S. (2010). Multiobjective optimization using differential evolution for real-world portfolio optimization. Computational Management Science (in press).

  • Krink, T., Mittnik, S., & Paterlini, S. (2009). Differential evolution and combinatorial search for constrained index tracking. Annals of Operations Research. doi:10.1007/s10479-009-0552-1.

  • 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, 603–621.

    Article  Google Scholar 

  • Ledoit, O., & Wolf, M. (2004). Honey, i shrunk the sample covariance matrix: Problems in mean–variance optimization. Journal of Portfolio Management, 30, 110–119.

    Article  Google Scholar 

  • Maringer, D., & Oyewumi, O. (2007). Index tracking with constrained portfolios. Intelligent Systems in Accounting, Finance and Management, 15, 57–71.

    Article  Google Scholar 

  • Maringer, D., & Parpas, P. (2009). Global optimization of higher order moments in portfolio selection. Journal of Global Optimization, 43(2–3), 219–230.

    Article  Google Scholar 

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

    Google Scholar 

  • Pierdzioch, C., & Rülke, J. (2012). Forecasting stock prices: Do forecasters herd? Economics Letters, 116(3), 326–329.

    Article  Google Scholar 

  • Price, K., Storn, R., & Lampinen, J. (2005). Differential evolution. A practical guide to global optimization. Natural computing series. Berlin: Springer.

    Google Scholar 

  • Ramnath, S., Rock, S., & Shane, P. (2008). The financial analyst forecasting literature: A taxonomy with suggestions for further research. International Journal of Forecasting, 24(1), 34–75.

    Article  Google Scholar 

  • Söderlind, P. (2010). Predicting stock price movements: Regression versus economists. Applied Economics Letters, 17(8), 869–874.

    Article  Google Scholar 

  • Stock, J., & Watson, M. (1999). Forecasting inflation. Journal of Monetary Economics, 44(2), 293–335.

    Article  Google Scholar 

  • Stock, J., & Watson, M. (2001). A comparison of linear and nonlinear univariate models for forecasting macroeconomic time series. In R. Engle & H. White (Eds.), Cointegration, causality and forecasting: A Festschrift for Clive W.J. Granger (pp. 1–44). Oxford: Oxford University Press. http://www.economics.harvard.edu/faculty/stock/files/Granger_Festshrift_StockandWatson.pdf

  • Storn, R., & Price, K. (1995). Differential evolution: A simple and efficient heuristic for global optimization over continuous spaces. Technical Report. International Computer Science Institute, Berkeley, CA

  • Storn, R., & Price, K. (1997). Differential evolution: A simple and efficient heuristic for global optimization over continuous spaces. Journal of Global Optimization, 11(4), 341–359.

    Article  Google Scholar 

  • Timmermann, A. (2006). Forecast combinations, Chap. 4. In G. Elliot, C. Granger, & A. Timmermann (Eds.), Handbook of economic forecastin (Vol. 1, pp. 135–196). North-Holland: Handbooks in Economics.

    Google Scholar 

  • Yang, Y. (2004). Combining forecast procedures: Some theoretical results. Econometric Theory, 20(1), 176–190.

    Article  Google Scholar 

  • Zhang, J., & Maringer, D. (2009). Improving sharpe ratios and stability of portfolios by using a clustering technique. Proceedings of the World Congress on Engineering (pp. 1–6).

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Correspondence to Björn Fastrich.

Appendices

Appendix 1: Portfolio with Only Few Analysts

See Table 5.

Table 5 Empirical results when \(\tau = 12\) in-sample observations are used to make a forecast

Appendix 2: No-Change Prognosis as a Constituent

See Table 6.

Table 6 Empirical results when \(\tau = 12\) in-sample observations are used to make a forecast

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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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