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Quant Investing: Robust Beta and Its Effectiveness in Single-index Model 中文

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本研究旨在驗證具穩健性的Beta估計方式是否能為Single-index Model帶來整體績效的提昇,並透過實驗發現其在各情境下對投資組合績效的影響。

Robust Beta and Its Effectiveness in Single-index Model


In this work, we try to perform a comprehensive analysis on how effective the robust linear model can enhance the performance of single-index model.

Single-index Model

The idea of single-index model is first introduced in Sharpe, W. F. (1963), and the detailed portfolio construction used in this work is presented in Elton, E. J., Gruber, M. J., Brown, S. J., & Goetzmann, W. N. (2009). Empirically, single-index model is often used as a good proxy for complex correlation structures among assets in portfolio.

Robust Beta Estimation

Single-index model requires estimates of the beta of each asset to determine whether a given asset is a potential candidate for inclusion in optimal portfolio or not. Elton, E. J., Gruber, M. J., Brown, S. J., & Goetzmann, W. N. (2009) summarized various classic way to improve beta estimation, including adjusted beta, Elton, E. J., Gruber, M. J., & Urich, T. J. (1978)’s beta forecastability and Beaver, W., Kettler, P., & Scholes, M. (1970)’s fundamental betas. The main purpose of these adjustments on beta is to make it much more accurate and to embed more factor information during estimation.


We are going to build up the sector-rotation portfolio by using single-index model with both OLS and RLM estimation. The following sector ETFs will be utilized as investment universe in our experiment starting from 1999-02-01 to 2022-02-28:

  • Consumer Discretionary Select Sector SPDR ETF (TICKER: XLY)
  • Consumer Staples Select Sector SPDR ETF (TICKER: XLP)
  • Energy Select Sector SPDR ETF (TICKER: XLE)
  • Financials Select Sector SPDR ETF (TICKER: XLF)
  • Health Care Select Sector SPDR ETF (TICKER: XLV)
  • Industrials Select Sector SPDR ETF (TICKER: XLI)
  • Materials Select Sector SPDR ETF (TICKER: XLB)
  • Technology Select Sector SPDR ETF (TICKER: XLK)
  • Utilities Select Sector SPDR ETF (TICKER: XLU)


[1] Sharpe, W. F. (1963). A simplified model for portfolio analysis. Management science, 9(2), 277-293.

[2] Elton, E. J., Gruber, M. J., Brown, S. J., & Goetzmann, W. N. (2009). Modern portfolio theory and investment analysis (pp. 126-203). John Wiley & Sons.

[3] Elton, E. J., Gruber, M. J., & Urich, T. J. (1978). Are betas best?. The Journal of Finance, 33(5), 1375-1384.

[4] Beaver, W., Kettler, P., & Scholes, M. (1970). The association between market determined and accounting determined risk measures. The Accounting Review, 45(4), 654-682.

[5] Genton, M. G., & Ronchetti, E. (2008). Robust prediction of beta. In Computational Methods in Financial Engineering (pp. 147-161). Springer, Berlin, Heidelberg.

[6] Wang, Y. G., Lin, X., Zhu, M., & Bai, Z. (2007). Robust estimation using the Huber function with a data-dependent tuning constant. Journal of Computational and Graphical Statistics, 16(2), 468-481.

[7] Street, J. O., Carroll, R. J., & Ruppert, D. (1988). A note on computing robust regression estimates via iteratively reweighted least squares. The American Statistician, 42(2), 152-154.

[8] Huber, P. J. (1996). Robust statistical procedures. Society for Industrial and Applied Mathematics.

[9] Lintner, J. (1965). Security prices, risk, and maximal gains from diversification. The journal of finance, 20(4), 587-615.

YC Lin
AVP Quantitative FinanceGamma Paradigm

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