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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 $\Pi$?

  1. should I use only the stocks in my portfolio
  2. 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

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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:

  1. 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)
  2. 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)

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  • $\begingroup$ Yes but what if I have a portfolio which no benchmark covers entirely? $\endgroup$ Mar 9, 2022 at 22:05
  • $\begingroup$ I am not sure I get it, can you make an example of what you mean? $\endgroup$
    – Dark2018
    Mar 11, 2022 at 7:58
  • $\begingroup$ Let's say my portfolio is all AAPL, MSFT and a european stock. What weights should I use to calculate PI? You say: look at the contribution in a market index. OK. AAPL contribute 5% to the S&P500, MSFT 3%, and the european stock contribute 1% to the MSCI Europe. So my weights are now 5%, 3% and 1%? This doesnt make sense, because they dont add up to 100%. $\endgroup$ Mar 11, 2022 at 9:19
  • $\begingroup$ in that case I would reweigh the ptf..so 5%,3%,1% is my 100% therefore 5/9=x/100 and so on for the others..in that case you will have a benchmark that will only contain your stocks and sum up to 100% $\endgroup$
    – Dark2018
    Mar 16, 2022 at 7:53
  • $\begingroup$ What about directly calculating the implied returns as beta*market returns? isn't it easier? It should be approximately the same thing $\endgroup$ Mar 17, 2022 at 18:55

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