Alpha-Beta and the Heisenberg Uncertainty Principle

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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, the concept of "yīn yang" is used to describe how polar or seemingly contrary forces are interconnected and interdependent 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 raising the question of "how to define the expected risk of the manager’s investment position". In other words, when marketing "alpha" portfolio 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 a meaningful 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 its meaning is that it is impossible to determine simultaneously both properties with any great degree of accuracy or certainty. However, according to Ballentine and others 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

References:
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).

The CAPM Debate and the Search for "True Beta"

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Conventional investment theory states that when an investor constructs a well-diversified portfolio, the unsystematic sources of risk are diversified away leaving the systematic or non-diversifiable source of risk as the relevant risks. The capital asset pricing model (CAPM), developed by Sharpe (1964), Lintner (1965) and Black (1972) [zero-beta version], asserts that the correct measure of this riskiness is its measure known as the "beta coefficient" or just "beta". Effectively, beta is a measure of an asset’s correlated volatility relative to the volatility of the overall market. Consequently, given the beta of an asset and the risk-free rate, the CAPM should be able to predict the expected return for that asset, and correspondingly the expected risk premium as well.

This explanation is textbook. However, unbeknownst to most investors, there has been a long running argument in academic circles on the CAPM and other pricing models, even within the milieu of traditional investments. Without going into the details of this debate, certain empirical studies have revealed "cross-sectional variations" in the CAPM questioning the validity of the model. In direct response to the challenge by Fama and French (1992), Jagannathan and Wang (1996) theorized that “…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. Such an analysis must include a measure of the return on the aggregate wealth portfolio of all agents in the economy.”

Financial institutions have not been left behind by these evolving academic theories. Index creation and benchmarking has become standard fare, and since the introduction of 'exchange traded funds' (ETFs), a veritable industry has developed around the "multiple beta" concept. But by no means has the plethora of these instruments captured every aspect of the aggregate wealth portfolio of all agents in the global economy, although at the current pace of ETF development it would seem that this is the undeclared objective.

Such backdrop is the principal context which gives impetus to the notion of "exotic betas". The term, a recent addition to the investment lexicon which evolved from ideas advanced by proponents of alternative investments, suggests that certain alternative investment assets and/or strategies, representing commonly pursued market paradigms, can be identified, tracked and replicated employing a predefined passive approach/model similar to traditional index construction.

This leaves open the question as to whether institutions, through sophisticated financial engineering, can truly capture in a passive way all possible sources of return in the global economy. Or, does some aspect which the industry loosely calls alpha (i.e., skill-based returns) always remain outside the grasp of such institutions’ arbitrary models of beta?

Jagannathan - The CAPM Debate

References:
Black, Fischer (1972). “Capital Market Equilibrium with Restricted Borrowing” Journal of Business 45, July, pp. 444-455.

Fama, Eugene F.; French, Kenneth R. (1992). “The Cross-Section of Expected Stock Returns” Journal of Finance 47, June, pp. 427-465.

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.

The Mysterious Case of an Enigma within a Paradox

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In 1995, Cambridge University Press published in its Journal of Economic History an article by John R. Garrett titled, "Monetary Policy and Expectations: Market-Control Techniques and the Bank of England, 1925-1931". Garrett begins by referencing Richard Sayers who, as the official Bank of England historian with full access to confidential archives, revealed in a 1976 book titled, "The Bank of England 1891-1944," that the Bank had influenced interest rates by manipulating reported gold flows between 1925-1931. Based on such enigmatic evidence, Garrett (1995) "models expectations manipulation as a monetary policy channel," and shows that "gold flow falsification was over two-thirds as effective as an open-market operation."

As Garrett noted, "These results contradict accepted new classical models and suggest that credibility benefits from new classical policy are small." The enigma within a paradox relates to the concept of 'rational expectations' as described below:

Our working paper investigated various models which deal with the potential sources of return to speculators in the futures market, including one of our own which exemplifies the complexity of these markets. Admittedly, models are only an abstraction from reality. Expecting such models to be exactly right is unreasonable, and it is generally understood that neoclassical economic theory has inherent limitations related to the analysis of markets within the context of “rational equilibrium systems.” Such systems are based on perfect competition, and assume markets naturally return to equilibrium after a disturbance. Hence, modern finance seeks to maximize utility and/or profits in a world of constraints based on the choices of “rational” economic agents. By definition then, these models relegate speculators to the role of that very agent which maintains equilibrium. Yet a survey of real-life speculators reveals that these practitioners do not as a general rule use academic models in their day-to-day trading decisions. Paradoxically, this same group plays a key influence upon the selfsame futures data from which such models are constructed. So if the data series is assumed to represent equilibrium and “the future is merely the statistical reflection of the past,” then one could inversely argue that perfect competition and rational expectations minimize these models’ usefulness as a mechanism from which to make speculative decisions. In other words, rational expectations compel such models to simply validate that current price data is equal to equilibrium, unless the opposite is true—that markets are in fact imperfect and rational expectations is untenable, which in turn undermines the veracity of these models. [Source: The Search for the Beta of Commodity Futures]

Following the twists and turns in this "riddle, wrapped in a mystery, inside an enigma" is the implication that the 'rational expectations equilibrium' paradox is being turned back on itself by the central banks themselves. It is not just that central banks are using game theory to manage expectations (and by implication models based on rational expectations), the 1925-1931 Bank of England episode illustrates a situation where economic reality was changed by a central bank despite it being based on a lie. In Garret's words:

Markets can not tell when a central bank is lying. They then have the option to accept all or reject all forecast information emanating from the central bank. Under such circumstances the credibility model asserts that private financial markets reject all central bank information. This is possible because the financial markets' private information is assumed to be almost com- plete. However, the results presented here contradict this assumption and lend support to the opposite case: the markets' private information is so incomplete that they can not dispense with central-bank sources. The implication for the credibility model is devastating because perva- sive ignorance and uncertainty allow the central bank to maintain its position as a disseminator of forecast information even if the central bank is guilty of extreme dishonesty, as under Norman. Under these circumstances monetary policy will be an effective instrument to stabilize the economy against both money demand and real shocks, which contradicts the core result found in the large and influential credibility-model literature.

The article "Priceless: How The Federal Reserve Bought The Economics Profession" adds support to this idea of an enigma within a paradox, and lends credibility to concerns of economic/market groupthink. In the words of Morpheus: "I imagine that right now, you're feeling a bit like Alice. Hmm? Tumbling down the rabbit hole?"

John Garrett - Monetary Policy and Expectations

References:
Garrett, John R. (1995) "Monetary Policy and Expectations: Market-Control Techniques and the Bank of England, 1925-1931" The Journal of Economic History, Cambridge University Press, Sept. 1995, pp. 612-636

Frankfurter, Mack and Accomazzo, Davide (2007) "Is Managed Futures an Asset Class? The Search for the Beta of Commodity Futures" (December 31, 2007). Available at SSRN: http://ssrn.com/abstract=1029243