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The Black–Litterman model and views from a reverse optimization procedure: an out-of-sample performance evaluation

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Abstract

The Black–Litterman (BL) model has been proposed as a valid solution to the problem of the estimation error in the mean–variance (MV) model. However, very little research has been done in order to empirically test the performance of the model. The paper contributes to the existing literature by empirically examining the out-of-sample performance of the BL model with respect to other asset allocation strategies. As another contribution of the paper, we suggest a novel approach to specify the investor’s views in the BL model. Overall our results suggest that the BL model is a valid asset allocation strategy.

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Notes

  1. For a detailed description on the BL model see He (1999), Idzorek (2005), Meucci (2010), Bertsimas et al. (2012) and Allaj (2013).

  2. The BL alternate formulas are derived by applying the Woodbury matrix identity to Eqs. (19) and (20). They are given respectively by \(\hat{\varvec{\mu }}_{p}=\varvec{\pi }+(\tau \varvec{\Sigma })\mathbf{P} '[\mathbf{P} (\tau \varvec{\Sigma })\mathbf{P} '+\varvec{\Omega }]^{-1}(\mathbf{v} -\mathbf{P} \varvec{\pi })\) and \(\hat{\varvec{\Sigma }}_{p}=(\tau \varvec{\Sigma })-(\tau \varvec{\Sigma })\mathbf{P} '[\mathbf{P} (\tau \varvec{\Sigma })\mathbf{P} '+\varvec{\Omega }]^{-1}{} \mathbf{P} (\tau \varvec{\Sigma })\). See, for example, Meucci (2010) and Allaj (2013) for a proof.

  3. Black and Litterman (1991), Black and Litterman (1992), He (1999), and Idzorek (2005) suggest that tau should vary from a value of 0.025 to 0.5.

  4. This appendix is titled “Online Appendix” and is available at https://sites.google.com/site/erindiallaj/appendices?authuser=1.

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A Statistical characteristics of asset excess returns

A Statistical characteristics of asset excess returns

See Table 4.

Table 4 Descriptive statistics of asset excess returns

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Allaj, E. The Black–Litterman model and views from a reverse optimization procedure: an out-of-sample performance evaluation. Comput Manag Sci 17, 465–492 (2020). https://doi.org/10.1007/s10287-020-00373-6

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