Better Beta Is No Monkey Business

By AllianceBernstein

The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type a given text, such as the complete works of William Shakespeare. This makes perfect sense to me, but says more about infinity than it does about monkeys.

In his seminal book, A Random Walk Down Wall Street, Burton Malkiel argued that a blindfolded monkey throwing darts at a newspaper’s financial pages could select a portfolio that would do just as well as one carefully selected by the experts. This makes little sense to me: why does the monkey have to be blindfolded? Monkeys can’t read.

Unblindfolded, however, Malkiel’s monkey may make perfect sense. In a recent paper,1 Rob Arnott et al show that the legendary dart-throwing monkey would produce a portfolio with a substantial value and small-cap bias that would have historically outperformed the cap-weighted index. Such a portfolio, therefore, might have done just as well as one built by non-monkey experts.

Leaving monkeys (blindfolded or not) aside, the research conclusion is an important one. What it shows is the limitation of cap-weighted indices where the size of a constituent is a function of share price. Such indices by construction put more emphasis on stocks with high prices and less emphasis on stocks with low prices. They will favor components whose prices have risen the most.

This concentration risk is often unintended. And it creates risks that can be bad for your wealth when investors stampede out of crowded positions, causing violent market swings. As my colleague, Dave Barnard, points out in a recent paper,2 the technology sector ballooned to more than 29% of the S&P 500 in 2000 (Display). Over the next two years, the sector lost more than half its value. Similarly, Japanese stocks lost about a third of their value in the two years after their weight in the MSCI World Index peaked at 44% in late 1989. Similar trends played out in the energy sector in 1980 and in financials in 2007, at the peak of the credit bubble.



Today’s favorite market theme lies in so-called safety stocks—particularly, in the US, those with high dividend yields. At their peak in September 2012, stocks with high dividend yields had a 44% weight in the S&P 500, their largest weight since 1970 and far above their 35% average.

Cap-weighted index funds have their virtues. They can provide low-cost exposure to an asset class and, typically, they are easy to buy and sell.

But we believe that any approach which loosens the connection between weight and price is likely to have a performance edge. For example, investors could permit some increase in tracking error or create smarter-beta benchmarks based on equal-, value- or risk-weighted components, and with explicit mechanisms designed to avoid concentration risk. These solutions might be slightly more expensive than a typical passive index, but we think it’s a price worth paying to avoid the risks of a pure, cap-weighted approach. And it’s probably a better idea than giving a monkey some darts and a copy of the FT.

1. “The Surprising ‘Alpha’ from Malkiel’s Monkey and Upside-Down Strategies,” Arnott, Hsu, Kalesnik and Tindall in Journal of Portfolio Management, Summer 2013, Vol. 39, No. 4: pp. 91-105

2. “The Case for Integrated Wealth Management,” David Barnard, AllianceBernstein, July 2013

The views expressed herein do not constitute research, investment advice or trade recommendations and do not necessarily represent the views of all AllianceBernstein portfolio-management teams.

Patrick Rudden is Portfolio Manager, Dynamic Diversified Portfolio at AllianceBernstein.


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AllianceBernstein is a research-driven asset management firm that is global in scope and client-centered in its mission.

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  • marketfowl

    There is no “amount” of infinity; it can’t be measured. And the conditional probability of the next letter the monkey typed approaches lottery winning odds; integrate that over the many words Shakespeare wrote and you have a divergent series not a book of plays.

  • Jeff Dewhurst

    The “Infinite Monkey with a Typewriter” theory is demonstrably false and I’m surprised that anyone even refers to it. It’s false by definition because the monkeys produce a “RANDOM” sequence of characters which by definition requires that the characters have an equal probability of occurrence regardless of the preceding sequence. Any expression of ideas in any symbolic language such as Shakespeare’s Hamlet requires characters and symbols arranged in specific sequences conforming to the applicable rules of spelling and syntax which are by definition not RANDOM.
    I’d favor the “Diversify & Rebalance” portfolio management process over the Dart throwing Monkey as a more effective way handle “The slings and arrows of outrageous fortune and by opposing END THEM!”

  • Ubu


    Your understanding of randomness and infinity is incomplete. The presence of seemingly non-random character sequences, including a sequence that duplicates Hamlet or any other sequence of any length, in an infinite random series of characters is not an indication of non-randomness of the infinite series.

    For an infinite series of characters to be ‘completely random’, it must somewhere contain every possible permutation of characters, including: Hamlet, the contents of the entire Library of Congress in Dewey decimal order, the contents of the entire Library of Congress in pig Latin, etc., ad infinitum.

    The ‘randomness’ of a finite subset of an infinite series is also a tricky thing. A series of 10 coin flips can be the result of a random process, even if heads comes up 10 times in a row. Randomness tests of a finite series, such as the chi-squared test, measure characteristics that can be used to evaluate the suitability of the series for a particular purpose, such as simulations, or detect differences between multiple series that indicate possible differences in the process or processes that produced the series. Randomness tests cannot detect whether a finite series is a subset of a random infinite series.

    Mistaken perceptions of randomness and non-randomness lead to mistaken conclusions about causality. The consequences of causal fallacies, large and small, have plagued the affairs of humans always.