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The Black-Litterman model has gained popularity in applications in the area of quantitative equity portfolio management. Unfortunately, many recent applications of the Black-Litterman to novel aspects of quantitative portfolio management have neglected the rigor of the original Black-Litterman modelling. In this article, we critically examine some of these applications from a Bayesian perspective. We identify three reasons why these applications may create losses to investors. These three reasons are: 1) Using a prior without “anchoring” the prior to an equilibrium model; 2) Using a prior and an equilibrium model that conflict with one another; and 3) Ignoring the implications of the estimation error of the variance-covariance matrix. We also quantify the loss first analytically and also numerically based on historical data on 10 major world stock market indices. Our conservative estimate of the loss is around a 1% reduction in the annualized return of the portfolio.

The Black-Litterman model is a powerful tool in the portfolio construction process. It has gained popularity among practitioners for the past two decades, and its applications to various aspects of the portfolio construction process have been discussed in the literature. We believe, however, that some of the applications adopted by practitioners and discussed in the literature deserve more critical examination. In this paper, we will highlight the uses of the Black Litterman model that we find problematic, or at least lacking the vigor of the original formulation of the Black-Litterman model. We will present numerical examples to make our case stronger and, where appropriate, will propose alternative approaches.

One can quantify the loss generated in a portfolio construction process in terms of the Sharpe ratio, the information ratio, or an investor’s utility. For the purpose at hand, the most natural choice is to use the investor’s utility implicit in the Black Litterman approach, which is

^{1}The Black Litterman model allows an expression of subjective views such as “the sum of asset A return and asset B return will be positive.” That is, it allows the investor to make a statement on a linear combination of many asset returns. If this type of statement is not allowed, one may have to be more explicit. An example is: “asset a return is likely to be around 10% while asset B return is likely to be around 5%.”

No doubt the Black-Litterman model brought a new tool to aid asset allocators, portfolio managers, and traders with the construction of optimal portfolios. In particular, it provided a theoretical framework upon which an investor’s prior views about asset markets or stocks could be combined with the actual historical data in order to construct optimal investments. The main application of the Black-Litterman model has been to asset allocation across broad asset classes. Recent research has tried to integrate the Black-Litterman model in the trading framework and the quantitative equity portfolio framework. These advances are inspiring as they improve the tool set for quantitative managers, unfortunately some of the applications have side effects which we must be aware of. In this paper, we have discussed some of the potential side effects of these applications of the Black-Litterman model. Our estimates of the losses associated with these side effects are large from a portfolio management perspective. We also suggest a straightforward way to apply the Black-Litterman model without this bias in the portfolio construction process.

The Black-Litterman (BL) model is a widely used asset allocation model in the financial industry. Introduced in , the model uses an equilibrium analysis to estimate the returns of uncertain investments and employs a Bayesian methodology to “blend” these equilibrium estimates with an investor's private information, or *views*, about the investments. Computational experience has shown that the portfolios constructed by this method are more stable and better diversified than those constructed from the conventional mean-variance approach. Consequently, the model has found much favor with practitioners. The U.S. investment bank Goldman Sachs regularly publishes recommendations for investor allocations based on the BL model and has issued reports describing the firm's experience using the model (). A host of other firms (Zephyr Analytics, BlackRock, Neuberger Berman, etc.) also use the BL model at the core of many of their investment analytics.

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The Black-Litterman (BL) model is a widely used asset allocation model in the financial industry. In this paper, we provide a new perspective. The key insight is to replace the statistical framework in the original approach with ideas from inverse optimization. This insight allows us to significantly expand the scope and applicability of the BL model. We provide a richer formulation that, unlike the original model, is flexible enough to incorporate investor information on volatility and market dynamics. Equally importantly, our approach allows us to move beyond the traditional mean-variance paradigm of the original model and construct “BL”-type estimators for more general notions of risk such as coherent risk measures. Computationally, we introduce and study two new “BL”-type estimators and their corresponding portfolios: a Mean Variance Inverse Optimization (MV-IO) portfolio and a Robust Mean Variance Inverse Optimization (RMV-IO) portfolio. These two approaches are motivated by ideas from arbitrage pricing theory and volatility uncertainty. Using numerical simulation and historical backtesting, we show that both methods often demonstrate a better risk-reward tradeoff than their BL counterparts and are more robust to incorrect investor views.

By including in the variance of, we are assuming that the overall uncertainty of the factor premium is proportional to the overall market volatility.1^{4} This is the simplest prior one can adopt, and is general enough for our purpose. Satchell and Scowcroft (2000) also use this prior when they interpret the Black Litterman model. Note also that this prior allows for the noninformative prior as a special case, as will be shown in the next section.

Fabozzi, Forcardi, and Kolm (2006; henceafter FFK) suggested that one could incorporate a trading strategy, possibly based on a factor model, as a prior in the Black Litterman approach. The portfolio construction process might look as follows.^{24}

If a portfolio construction process combines the model and the view efficiently, then by construction the loss is zero. This is true in the original Black Litterman approach. It is not possible to improve the utility given the model and the view. In the JLZ procedure, however, it is possible to improve the utility given the model and the view. It is possible to do so by adding the mean component of the model, as we will show below.

This procedure takes care of the problem identified in the previous section. The Black Litterman alpha is derived from an equilibrium model as well as from a prior. Information on the mean returns is not discarded, removing the potential for the type of loss identified in the previous section.

The JLZ procedure calculates the Black Litterman alpha, , based on a view and a model. The view is summarized in and the model is summarized in an estimate of. As mentioned above, however, the view and the model are not combined in an efficient way. In particular, the mean component of the model is ignored. This creates some loss, the magnitude of which we will quantify in this subsection.

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