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<reference>
Medium

This article compares Entropy Pooling (EP) to the Black-Litterman (BL) model and concludes that EP is superior for subjective views and stress-testing.

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Entropy Pooling
Black-Litterman
Conclusion
Anton Vorobets
Founder & CEO of Fortitudo Technologies. See
https://antonvorobets.substack.com
and
https://github.com/fortitudo-tech/fortitudo.tech
Black Litterman
Entropy Pooling
Python
Cvar
Monte Carlo
Entropy Pooling vs Black-Litterman
Anton Vorobets
4 min read
·
Apr 11, 2024
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An overview of the Entropy Pooling and Black-Litterman equations.
This article compares Entropy Pooling (EP) to the Black-Litterman (BL) model. In short, Entropy Pooling is a views and stress-testing method for fully general Monte Carlo distributions with flexible joint scenario probabilities. Black-Litterman is a views method that relies on the CAPM model, the normal distribution, and other unrealistic assumptions.
Entropy Pooling
When you work with a method for an extended period of time, you tend to forget that not all people know and understand its basics. I experienced this recently when someone gave me feedback on the Sequential Entropy Pooling Heuristics article, see
Vorobets (2021)
¹, saying that it wasn’t clear what the purpose of EP was.
At first, I couldn’t understand how it wasn’t clear. After all, the recap gives a very short and sweet description of what happens when you solve the EP problem, but then I realized that the issue was that this person didn’t know why one would solve the particular mathematical problem in the first place. As a consequence, I made this video explaining the Entropy Pooling intuition:
You can also find this
YouTube video
² that gives more perspectives and a simple example of sequential EP with views on mean and volatility of S&P 500 and STOXX 50. For more details about the sequential EP algorithms, see
this YouTube video
³.
For an overall introduction to EP and aspects related to view confidences, it is recommended to read the original article by
Meucci (2008)
⁴, while the notation is more concise and to the point in
Vorobets (2021)
¹, formally introducing sequential EP that usually gives significantly better results. Entropy Pooling can also be combined with Bayesian networks for causal and predictive analysis, see
Vorobets (2023)
⁵.
For a collection of Entropy Pooling resources,
see this Substack article
⁶.
Black-Litterman
I am by no means an expert on the BL model and its extensions. I decided to discard it due to its many highly unrealistic assumptions and duct tape engineering more than 10 years ago when I learned about it for the first time while writing my master’s thesis on Bayesian portfolio selection. In this article, I focus on the original BL model, while several adjustments have been proposed, see
Meucci (2008)
⁷. However, I view these adjustments not as solutions to the fundamental problems but additional duct taping on a poor foundation.
The BL model makes many highly unrealistic assumptions regarding markets, for example, that the CAPM holds, and that markets are normally distributed. There is an additional need to specify a risk aversion parameter to determine the “equilibrium expected returns”, and a tau parameter that needs to be specified somehow. As I do not recommend using the BL model, I will not go into details with these aspects, while they can be found in
Meucci (2008)
⁷.
I have seen people write that EP and BL give the same results when markets are normally distributed, but I am not aware of any material showing that this is actually the case. The BL model has an extra tau parameter and some logical inconsistencies with view confidences, see
Meucci (2008)
⁷, while EP handles view confidences in a natural and logically consistent way. Hence, I do not see how the two methods will give the same results in the normally distributed case.
Conclusion
There is absolutely no reason to continue using BL when EP is now well-known with fast and stable Python implementations freely available online, see the
entropy-pooling
⁸ package. If you want to combine EP with CVaR optimization, you can use the
fortitudo.tech
⁹ package. For an introduction to the combination between EP and CVaR, see the
Entropy Pooling and CVaR Optimization in Python
¹⁰ article.
Entropy Pooling is a core element of the next generation investment analysis framework that also utilizes fully general Monte Carlo distributions and CVaR analysis. For a video introduction to some of the things that you can do with the next generation framework, see
this YouTube video
¹¹.
If you want to build a really deep understanding of the next generation investment framework, there is currently a project to write a
Portfolio Construction and Risk Management book
¹² that you can support and receive perks for.
[1]: Vorobets, Anton, Sequential Entropy Pooling Heuristics (October 5, 2021). Available at SSRN:
https://ssrn.com/abstract=3936392
[2]: Sequential Entropy Pooling video:
https://www.youtube.com/watch?v=DK1Pv5tuLgo
[3]: Entropy Pooling video:
https://www.youtube.com/watch?v=hDt103zEML8
[4]: Meucci, Attilio, Fully Flexible Views: Theory and Practice (August 8, 2008). Fully Flexible Views: Theory and Practice, Risk, Vol. 21, №10, pp. 97–102, October 2008, Available at SSRN:
https://ssrn.com/abstract=1213325
[5]: Vorobets, Anton, Causal and Predictive Market Views and Stress-Testing (May 11, 2023). Available at SSRN:
https://ssrn.com/abstract=4444291
[6]: Entropy Pooling Collection Substack post:
https://antonvorobets.substack.com/p/entropy-pooling-collection
[7]: Meucci, Attilio, The Black-Litterman Approach: Original Model and Extensions (August 1, 2008). Available at SSRN:
https://ssrn.com/abstract=1117574
[8]: entropy-pooling Python package. Available at:
https://github.com/fortitudo-tech/entropy-pooling
[9]: fortitudo.tech Python package. Available at:
https://github.com/fortitudo-tech/entropy-pooling
[10]: Entropy Pooling and CVaR Optimization in Python:
https://medium.com/@ft_anvo/entropy-pooling-and-cvar-portfolio-optimization-in-python-ffed736a8347
[11]: Anton Vorobets, Next Generation Investment Analysis @ The London Quant Club:
https://www.youtube.com/watch?v=4ESigySdGf8
[12]: Portfolio Construction and Risk Management book crowdfunding:
https://igg.me/at/pcrm-book
Black Litterman
Entropy Pooling
Python
Cvar
Monte Carlo
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Written by
Anton Vorobets
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Founder & CEO of Fortitudo Technologies. See
https://antonvorobets.substack.com
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https://github.com/fortitudo-tech/fortitudo.tech
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</reference>

<statements>
1. Nevertheless, the framework remains constrained by its foundational reliance on Gaussian distributions, meaning it cannot natively evaluate asymmetric tail dependence or non-linear downside exposures
2. In the Black-Litterman Framework, Tail Risk & Non-Linearity is restricted by conditional multivariate Gaussian assumptions
3. In the Integrated Modern Hybrid Framework, Tail Risk & Non-Linearity is fully non-linear through Entropy Pooling and Monte Carlo scenario generation
4. The calibrated views \((P, Q, \Omega)\) are integrated using Attilio Meucci’s Entropy Pooling framework to accommodate non-normal return distributions
5. Accommodates fat tails, skewness, and non-linear views without normal assumptions
6. Non-parametric Bayesian extensions—most notably Meucci’s Entropy Pooling—integrate these views into full Monte Carlo scenario distributions, capturing tail risk, skewness, and non-linear market co-movements
</statements>

Begin the assessment now. Output only the JSON list, without any conversational text or explanations.