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In Grinold & Kahn (2000), the authors emphasized the separation of stock selection and benchmark timing in active portfolio management. So if we avoid benchmark timing, the optimal portfolio's beta should be 1 and we need to add this constraint to our portfolio optimization problem:

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where h_P is the vector of portfolio weights. The objective is the portfolio's residual return adjusted by residual risk [see p.139-p.142 in Grinold & Kahn (2000)].

But I can't find this constraint in other books on quantitative portfolio management such as Chincarini & Kim (2007) and Qian, Hua & Sorensen (2007). In these two books, the portfolio optimization problem is constructed as the maximization of portfolio's active return adjusted by tracking error without any constraint on portfolio's beta [see p.281 in Chincarini & Kim (2007) and p.35 in Qian, Hua & Sorensen (2007)]: enter image description here

where f is forecasts of excess return, f_{PA} is portfolio's active return, h_{PA} is active weights, i.e. h_{PA} = h_P - h_B, and h_B is benchmark weights. (Solve this problem we can get h_{PA}, then h_P = h_{PA} + h_B)

Questions: What's the relationship between these two kinds of portfolio optimization problems? Why doesn't the "maximization of active return adjusted by tracking error" consider benchmark timing and constraint on portfolio's beta? How should we deal with benchmark timing in the framework of Chincarini & Kim (2007) and Qian, Hua & Sorensen (2007)?

Reference

Grinold & Kahn, 2000, Active Portfolio Management: A Quantitative Approach for Producing Superior Returns and Controlling Risk

Qian, Hua & Sorensen, 2007, Quantitative Equity Portfolio Management: Modern Techniques and Applications

Chincarini & Kim 2007, Quantitative Equity Portfolio Management: An Active Approach to Portfolio Construction and Management

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