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Classic mean–variance optimization is very sensitive to expected returns. An alternative and more robust approach is to calculate the implied returns given the current portfolio allocation and risk profile. Portfolio managers can then do a reality check on the implied returns and find opportunities for better allocations. The most common implied return calculation assumes normal distribution and unlimited leverage, and use volatility as risk measure and covariance matrix as model input. However, practitioners usually have leverage constraints, often use non-parametric risk models, and care about portfolio tail risk. This paper presents a new approach to calculate expected returns with leverage constraints. This approach is flexible enough to alleviate normal distribution assumption, connect with non-parametric risk models, and use tail risk measures, such as conditional VaR. [ABSTRACT FROM AUTHOR]

There is apparent persistent confusion over certain aspects of Black-Litterman expected returns, with a number of publications offering various explanations, clarifications, and criticisms as to how the model works in practice. The parameter tau ( τ) has proved to be a particularly confounding feature of the model: A wide range of opinions and suggestions on how to interpret and quantify tau has accumulated in the literature and includes some harsh condemnation of Black-Litterman specifically related to tau. This article presents a simple interpretation of tau, shows that it is directly related to the level of active risk implicit in the Black-Litterman model, and is easily calibrated so that Black-Litterman expected returns produce portfolios with targeted levels of active risk. The main contribution is an alternative derivation of Black-Litterman that affords a direct way to target active risk without requiring a specific value for tau. This derivation reveals that portfolio construction using Black-Litterman is equivalent to creating a mean-variance optimal portfolio of active strategies that is then overlaid onto a benchmark portfolio, and by targeting active risk directly, users of the Black-Litterman model do not need to consider tau at all. [ABSTRACT FROM AUTHOR]

The Black-Litterman method is widely used in the investment management industry to incorporate views in investment portfolios. The method applies when views are expressed as expected returns over the horizon for which allocation decisions are made, i.e., the investment horizon. In practice, however, the investor's views are typically formulated for the near future while the investor's investment horizon is much longer. To incorporate such views, we introduce the time-dependent Black-Litterman method and show that, in a time-dependent setting, a distinction should be made between unconditional and conditional views. Furthermore, we demonstrate its use for buy and hold investors. In an example, we show that the investor's views have a plausible impact on resulting allocation decisions. [ABSTRACT FROM AUTHOR]