Here we provide a taxonomy of the models used by various authors. This table is not meant to be complete, but to be a summary of important papers on the model. The information is organized historically and by expression of the model.
At the highest level we can separate the expression of the model used into Bayesian and non-Bayesian. The Bayesian expression of the model is the canonical model presented by Black and Litterman (1991, 1992), He and Litterman (1999) and Litterman et al, (2003). Under non-Bayesian we can further separate the models into a hybrid model which is non-Bayesian but continues to use the τ parameter, and the alternative reference model which eliminates τ.
This list is not complete, though I expect to continually add information to it as time is available.
|Black and Litterman (1991, 1992)||Yes||Yes||Provides descriptions of the model, but does not include all formulas.|
|Bevan and Winkelmann (1998)||Yes||Yes|
|He and Litterman (1999)||Yes||Yes||Best summary of the mathematics of the Black-Litterman model by original authors.|
|Drobetz (2001)||Yes||Yes||One of the first papers on the canonical form not by an original author of the model.|
|Litterman et al (2003)||Yes||Yes||Overview of Goldman Sachs asset allocation process using BL, little detail on the BL model itself.|
|Blamont and Firoozy (2003)||Yes||Yes||Provide insight into the parameter τ.|
|Beach and Orlov (2006)||Yes||Yes||Quantitative views from EGARCH models|
|Cheung (2009)||Yes||Yes||Integrating factor models|
|Satchell and Scowcroft (2000)||No||Yes||τ usually set to 1|
|Qian and Gorman (2001)||No||Yes||Adds views on covariance of returns|
|Herold (2003)||No||Yes||Applies to Active Management vs a benchmark|
|Idzorek (2005)||No||Yes||New expression of ω|
|Braga and Natale (2007)||No||Yes||TEV sensitivity to views|
|Martellini and Ziemann (2007)||No||Yes||VaR, CVaR as objectives, inclusion of higher moments in identifying neutral portfolio|
|Bertsimas, et al (2013)||No||Yes||Replace reverse optimization and bayes formula with various optimization models. Allows for arbitrary distributions.|
|Michaud et al (2013)||No||Yes||Provides arguments against using hybird and alternative variants of the model. Ignores modern econometrics and Bayesian statistics in the process.|
|Fusai and Meucci (2003)||No||No||Uses α to tune mixing|
|Krishnan and Mains (2005)||No||No||Introduce unpriced factor|
|Giacometti et al (2006)||No||No||Student T and alternative portfolio choice, CVAR|
|Giacometti, et al (2007)||No||No||VaR, CVaR objectives and use of stable distributions rather than normal distributions.|
Description of the various attributes in the table
Bayes - the authors use a Bayesian interpretation of the model expressing uncertainty in the prior and in their estimates. No means the author uses a non-Bayesian or frequentist approach to the model.
τ - The authors use the variable τ in their formulas.
For a more thorough discussion see my paper on the Black-Litterman model.