### Resources

This section includes links to the papers I found most useful in understanding the Black-Litterman Model, along with minor comments on each reference.

The Black-Litterman Model: A Detailed Exploration is a paper that I maintain which provides a comprehensive discussion of the model include derivations of thevarious formulas and the theory behind the model. It provides a detailed discussion of how to actually implement the Black-Litterman model as well as worked examples from several of the papers. This paper is updated on a regular basis.

The Factor Tau in the Black-Litterman Model is a paper which lays out what the factor tau is and why you might want to use it. It also shows that most implementors can ignore tau and can manage the blending process totally through the value of Omega. This paper is updated on a regular basis.

Global Portfolio Optimization, Black and Litterman, Financial Analysts Journal, 1992.This is the original paper that started it all. Black and Litterman describe their model and provide some details on how it is used. They definitely don't show all the formulas, or derive much of what they do show. They also use a complicated enough example that it's not easy to reproduce exactly what they do.

The Intuition Behind the Black-Litterman Model Portfolios by He and Litterman, 1999. For me, this is one of the two key papers in the literature providing details on how Goldman Sachs uses the Black-Litterman model from one of the authors of the original paper. There are a few versions of this paper floating around. This link is to a version that doesn't have the colorful charts, but instead has more detailed formulas, explanations and tables of results. He and Litterman provide the formulas they used and enough data to reproduce their results.

A Step-By-Step Guide to the Black-Litterman Model by Thomas Idzorek, 2004. This paper is the second key papers in terms of understanding the Black-Litterman model. Idzorek makes a good effort to explain the basic Black-Litterman model, and has then added a very valuable extension to the model for allowing the specification of the view confidence as a simple percentage. He does provide enough data to reproduce his results, but doesn't provide all the formulas you need to implement the basic Black-Litterman model. He also does not work his example of his extension through to completion.

Consistent Asset Return Estimates: The Black-Litterman Approach, Werner Koch, April 2005.This is a set of slides from a presentation on Black-Litterman. It is included in this list because it includes a nice derivation of the core Black-Litterman formulas.

Using the Black-Litterman Global Asset Allocation Model: Three Years of Practical Experience, Bevan and Winkelmann, June 1998.This paper summarizes information on how Goldman Sachs uses Black-Litterman in the Fixed Income group, and what data they use for input and how they calibrate the model. It does not offer enough details to back out specific implementation details of Black-Litterman itself.

The Black-Litterman Model - Mathematical and Behavioral Finance Approaches Towards its Use in Practice, Mankert, 2006. Licentiate Thesis. This masters thesis was written by a student who implemented Black-Litterman for a Swedish Bank and noted that there wasn't a lot of good literature on the subject. She has some good detail on derivations of the core formulas of the Black-Litterman model, and provides new insights into the model by a novel derivation using sampling theory. She doesn't provide all the formulas needed and doesn't provide much in the way of worked examples. Still a very good source of information on the Black-Litterman model.

Krishnan, Hari and Norman Mains (2005): “The Two-Factor Black Litterman Model,”

Risk Magazine, July, pp. 69-73. This is an interesting article from Risk Magazine which shows how to add an additional factor to the Black-Litterman model, and how to compute new equilibrium returns. The factor they add to the model is recession risk as proxied by the Altman index of high yield bonds. I previously found this on the internet, but can't find the links anymore. I am hoping to find them again and include them here.

In Bayesian Asset Allocation: Black- Litterman by Daniel Blamont and Nick Firoozye they work through an example in global fixed income illustrating tactical asset allocation versus a benchmark. Thanks to Daniel Blamont of Deutsche Bank for providing me a copy and for permission from Deutsche Bank to post it for easier access by students of the Black-Litterman model.

Assessing Views by Fusai and Meucci, 2003. This paper from Risk Magazine provides a quick tour of the Black-Litterman related content which ends up in Attilio Meucci's book, Asset Allocation and Risk, Springer Finance, 2005. He provides the alternate formulation of the posterior variance and also a new measure for determining whether one views are extreme. Attilio Meucci has several more papers and some MATLAB code for implementing his ideas on his website.