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Choosing the Market Universe for Black–Litterman Implied Returns

Article Quant Q&A · Author: randomwalker
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Summary

The document discusses how to choose the market weights and asset universe when calculating the Black–Litterman implied return prior. One response recommends selecting a representative benchmark, then using the portfolio securities’ weights within that benchmark. It suggests matching the benchmark to the assets’ geographic coverage and notes that a risk-free rate may need to be included in the implied-return formula. Views and their confidence then shift the resulting returns away from the benchmark prior.

A contrasting response argues that the theoretical market portfolio should include all asset classes, since a narrow universe omits relationships with other assets. The answers therefore present a practical benchmark-based approach alongside a broader theoretical position, without resolving the disagreement. The exchange gives no empirical comparison or worked example, and it leaves implementation choices such as missing securities and estimation of covariance to the researcher. The central lesson is to state the benchmark assumption clearly, because the universe used changes the prior and the portfolio optimization that follows.

Key ideas

  • Black–Litterman implied returns depend on the chosen benchmark weights and covariance matrix.
  • One practical proposal uses a broad index and the selected securities’ contributions to it.
  • A broader theoretical view favors including all asset classes in the market portfolio.
  • The risk-free rate may be relevant when specifying the implied-return formula.
  • The discussion offers competing recommendations rather than evidence that settles the universe choice.

Tags

Portfolio constructionEquitiesMulti-assetStatistics
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# Black-Litterman and Implied Market Returns


# Black-Litterman and Implied Market Returns












The first step in the Black-Litterman method is to find the "implied market returns" (the prior). Usually this is calculated as: $\Pi = \lambda \Sigma w$, where $\Pi$ is the vector of returns "implied by the market", $w$ is the vector of market weights (each element = security market cap / total market cap), $\Sigma$ is the covariance matrix, $\lambda$ is the market risk aversion (a constant).

I would like to use Black-Litterman to optimise a portfolio of individual stocks (something like 20 securities). My question is on the calculation of $\Pi$. Which universe should I use to calculate the vector $\Pi$?

- should I use only the stocks in my portfolio

- should I use all the stocks in the "market"? I could use all the stocks in the "market" but usually if I take an index (like FTSE World or S&P500) some security in the portfolio might not be present in the index. This might be an issue.

Thanks

## Answer by Dark2018 (score 0)

https://quant.stackexchange.com/a/70108

The first thing I would say is that your formula for $\Pi$ is missing the risk free rate, but I guess you assumed it zero.

Second, as prior, one should build a benchmark. If you do not have one yourself, the benchmark will be the market neutral, so based on the stocks you picked you may select MSCI and from it take the composition of your stocks. That will give you the w.

As I said depends on the stocks, if you have only stocks from developed country you can select MSCI developed, otherwise MSCI all country.

Once your PI is built you can then set your views and confidence so that if you have no views or if your confidence is very low, the returns of Black-Litterman will converge towards the market neutral.

So to answer:

- yes you should use your stocks in portfolio but you need to look for their contribution in a market index (MSCI - go to iShares to look for an etf and its composition)

- no not all the stock in the market, you need to consider a market index yes (or multiple market indeces if you have a different variety of assets - for example if you have bonds you can check the FTSE WGB), but inside that index only look for the composition/contribution of your stocks to that index

In addition (some people do it, some others don't) you can then adjust your w (weights of market neutral) by adding a biased based on your "home" (see https://www.jstor.org/stable/3211582)

## Answer by Delta (score 0)

https://quant.stackexchange.com/a/82404

- No. The implied market return are calculated with the global market portfolio, invested in all assets classes, in proportion to market caps. If you use another portfolio, you have a bad estimator of Pi. (because you miss the linear correlations with the other assets in the world).

- You should use all assets (stocks, bonds, REITS, commodities, crypto, etc.). This is the weak point of this approach -> theoretical

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.