# The Elusive Market Portfolio, from Alex Castaldo

November 13, 2009 |

I have two investments A and B. If I regress B's return on A the intercept (alpha) will show whether B is preferable to A. But whether I regress A against B, or B against A, I get a positive intercept. It's as if A has alpha over B, and also B has alpha over A, which makes no sense. — A Reader

In Markowitz's theory, given two different investments A and B it is not in general possible to say "it has to be the case that either A outperforms B or B outperforms A, tell me which it is." There is no way to compare two investments and rank them in this way in general (as mathematicians would say there is not a "total order"). So it is not entirely surprising that the method of regressing A on B and B on A does not give a consistent answer as to which of A or B is better. No method will give such an answer in general. We have to live with that.

What Markowitz does say is that if you have \$1, you can allocate it across A and B in various proportions (which could include shorting one of the assets to buy more of the other) and thus generate a "portfolio frontier" of points with various risk/return. Whether, in the particular case the letter writer brings up, this exercise would yield worthwhile insights I do not know. For example, if the writer revealed to us the variance-covariance matrix of returns (i.e. three numbers c11, c12 and c22) we could compute the Global Minimum Variance Portfolio weights [which are w1= (c22-c12)/(c11-2*c12+c22) and w2=(c11-c12)/(c11-2*c12+c22)], and if we knew the returns as well we could trace out the frontier. We could plot it and look at it. Useful? I don't know.

The theoreticians who came after Markowitz (Sharpe, Jensen, et. al.) believed that there is a very special portfolio in the universe called the Market Portfolio and that everyone would want to hold that plus perhaps small amounts of "other stuff." Under some fairly restrictive assumptions the desirability of the "other stuff" could be gauged by regressing its return on the Market Portfolio, always assuming that you could identify what this Market Portfolio is. Only by convention, or approximation, is this portfolio identified with the Standard & Poor 500. In any case the situation is not symmetric and the Market Portfolio plays a very special role in the theory.

What I am trying to point out is that we are so used to regressing things against the S&P 500 or other indexes every day that we sometimes lose track of the fact that this procedure does not in general allow us to rank two arbitrary investments. To compute an alpha you have to do the regression against "the market factor(s)" — not just any investment.

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