Monday, December 17, 2012

Alpha-Beta and Heisenberg's Uncertainty Principle

"The lack of empirical support for the CAPM may be due to the inappropriateness of some assumptions made to facilitate the empirical analysis of the model." ~Jagannathan and Wang (1996)

In physics, complementarity is a basic principle of quantum theory closely identified with the Copenhagen interpretation, and refers to effects such as "wave–particle duality," in which different measurements made on a system reveal it to have either particle-like or wave-like properties.

Niels Bohr is usually associated with this concept, which he developed at Copenhagen with Heisenberg as a philosophical adjunct to the mathematics of quantum mechanics and in particular the Heisenberg uncertainty principle. The Heisenberg uncertainty principle states that certain pairs of physical properties, like position and momentum, cannot both be known with precision. That is, the more precisely one property is known, the less precisely the other property can be known.

In Chinese philosophy, "yīn yang" is used to describe how polar or seemingly contrary forces are interconnected and inter-dependent in the natural world, and how they give rise to each other in turn. Yin yang are complementary opposites within a greater whole. Everything has both yin and yang aspects, although yin or yang elements may manifest more strongly in different objects or at different times. Yin yang constantly interacts, never existing in absolute stasis as symbolized by the Taijitu symbol.

A similar paradox exists within the CAPM paradigm involving the relationship between the concept of "beta," as determined by the market portfolio, and "alpha," which loosely represents "a proxy for manager skill". As is inferred by our prior posting, "The CAPM Debate and the Search for 'True Beta'", the yin yang "whole" relates to the "True Beta" concept which Jagannathan and Wang (1996) theorized must encompass "the aggregate wealth portfolio of all agents in the economy".

Moreover, one could apply aspects of "beta" to the symbology associated with "yin," which is usually characterized as slow, diffuse, tranquil, femininity and night; and apply aspects of "alpha" to the symbolism of "yang," which by contrast is characterized as fast, hard, focused, masculinity and day.

Schneeweis (1999) investigates this alpha-beta paradox in his article, "Alpha, Alpha, Whose got the Alpha?" wherein he writes about the problem of measuring "alpha" by questioning "how to define the expected risk of the manager’s investment position".

When marketing "alpha" managers often assume "the reference benchmark is the appropriate benchmark and that the strategy has the same leverage as the benchmark". Unfortunately, "[w]ith the exception of a strategy that is designed to replicate the returns of the benchmark, the alpha generated by this approach is essentially meaningless". Hence, investors often mistakenly rely on a single-index model as the benchmark from which to gauge the factors "driving the return of the strategy," when often a "multi-factor model should be used to describe the various market factors that drive the return strategy".

The problem is that statistically it is "better to over-specify a model… than to under-specify. If the model is over-specified, many of the betas will simply be zero. However, if under-specified, there is the possibility of significant bias".

Which brings us back to the Heisenberg uncertainty principle...

Just like the physical properties of position and momentum cannot both be known with precision, the properties of "alpha" and "beta" also cannot be measured precisely. This statement can be interpreted in two different ways:

According to Heisenberg it is impossible to determine simultaneously both properties with any great degree of accuracy or certainty. However, according to Ballentine this is not a statement about the limitations of a researcher's ability to measure particular quantities of a system, but it is a statement about the nature of the system itself as described by the equations.

Alpha Alpha Whose Got the Alpha - Schneeweis

Ballentine, L.E. The statistical interpretation of quantum mechanics, Rev. Mod. Phys. 42, 358–381 (1970).

Bohr, Niels. "Atomic Physics and Human Knowledge," p. 38.

Heisenberg, W. "Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik," In: Zeitschrift für Physik. 43 1927, S. 172–198.

Jagannathan, Ravi; McGrattan, Ellen R. (1995). "The CAPM Debate" Federal Reserve Bank of Minneapolis Quarterly Review, Vol. 19, No. 4, Fall 1995, pp. 2-17.

Jagannathan, Ravi; Wang, Zhenyu (1993). "The CAPM is Alive and Well" Research Department Staff Report 165. Federal Reserve Bank of Minneapolis.

Jagannathan, Ravi; Wang, Zhenyu (1996). "The Conditional CAPM and the Cross-Section of Expected Returns" Journal of Finance, Vol. 51, No. 1, March, pp. 3-53.

Schneeweis, Thomas (1999). "Alpha, Alpha, Whose got the Alpha?" University of Massachusetts, School of Management (October 5, 1999).

No comments: