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Asset allocation:combining investor views with market equilibriium

An 1990 article by F Black and R Litterman that can be found here and here.

Why is this article important?

This is one of the most often cited scientific paper on the mean variance optimization due to the domineering influence of the Black Litterman model in the 90s. The Capital Asset Pricing Model (CAPM) under the Efficient Market Hypothesis (EMH) can be used to imply market expected returns from

  • market equilibrium weights and
  • historical asset returns covariance matrix.

Note that the EMH is the result of a set of simplifying hypothesis and there is statistical evidence against its applicability in the real world.

The Black Litterman model can be described as a way to mix personal investor's view with CAPM implied weights. To do this, it uses a set of quantitative views:

  • manager's view on some asset absolute or relative returns
  • manager's view on his uncertainty of these views

While equity market weights are simply market capitalization for a given domestic market, the article starts as an international fixed income paper, making the question of what are equilibrium weights much more complex, and subject of this other paper by F Black.

Article Content

The article discusses:

  • the instability of Markovitz MVO approach, where changing an expected return by 0.1% leads to wildly different weights
  • the author's assumption that a change of 0.1% of expected return is actually big, this is not supported by an out of sample analysis
  • the problems associated with constraining weights or introducing high switching costs to stabilize the portfolio allocation
  • some tips about calculating daily covariance matrix for international financial time series

What it does not discuss or show:

  • the mathematical description of the input model, the view matrix $P$ and expectation $\nu$, and the view covariance $\Sigma_\epsilon$
  • how a manager could come up with a reasonable view covariance matrix
  • the mathematical derivation of the model
  • any evidence of the statistical robustness of the alleged equilibrium weights or the daily covariance obtained


This is a marketing piece rather than a scientific paper. It makes claims whose robustness is not verified and even does not methodology thus totally avoiding falsifiability.

The contribution of the Black Litterman model was to give a portfolio manager a position sizing methodology to "tilt" his position from a reference index portfolio according to his view.

It is the first approach that inverts the MVO problem to start from a reasonable point: the index weight. This avoids getting unreasonable weights and constraining them. However, this just hides rather than address the question of the sample covariance matrix inverse having arbitrarily high eigenvalues. This problem has not been solved here and it means that a timid manager view can lead to an arbitrarily large change in weights, solely because of a statistical estimator artifact.