Reading a paper by Black and Litterman, I'm having trouble understanding the set of valid allocations in which we're trying to optimize expected returns.

In Table III, the authors show two portfolios that are supposed to be optimal (for different shorting constraints). But not only none of the given portfolios sum to 1 (or 100), they actually sum to different values.

Also, standard deviations are different for each portfolio, while the authors claim that both are optimal for a given risk of 10.7%.

I'm new to finance in general. Is there some convention that I'm failing to apply? Are negative values supposed to be treated in a special manner? I've assumed that they simply mean "betting against" and that financial instruments that preserve the expected return and standard deviation exist for such bet.

The aforementioned paper: http://www.sef.hku.hk/tpg/econ6017/2011/black-litterman-1992.pdf

  • 2
    $\begingroup$ Can't comment on the specific figures and details of this paper - but maybe just some starting points for you: yes, negative portfolio weights usually means "short-selling". You can (but need not!) assume the weight vector must sum up to 1; that is essentially a "budget restriction" (i.e., you can only spend what you own; either using existing cash, or, if shortselling is allowed, then additionally the proceeds of shorting A can be used to buy B). Lastly, a total weight vector sum > 1 generally indicates leverage, i.e., the possibility for you to borrow funds. $\endgroup$
    – KevinT
    Jun 20, 2022 at 7:13
  • $\begingroup$ @KevinT thank you, that already helps a lot. $\endgroup$ Jun 20, 2022 at 8:19


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