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How Does “The Arithmetic of Active Management” Add Up?

Updated: Apr 8, 2019


by Edward Lynch


Recently, I carved out some time to go through my archive of papers, studies and research accumulated in the day-to-day running of our practice in order to revisit some of the strongest arguments I can find in the active-versus-passive investment management debate. I culled twenty papers and studies that, in coming weeks, I will re-read and, in some cases, summarize and engage in this cloud space.


Even though I’ve been both actor in and observer of the management of investment portfolios for nearly 35 years, I still find the discussion fascinating and, for the reason stated immediately below, critical.

Revisiting this “debate” is appropriate for the blog of a firm devoted to the promulgation of fiduciary responsibility as the standard for the conduct of financial and investment services because, at the most basic level, we are dealing with people’s money…as well as their hopes and dreams; their security, their dignity and the legacy they will leave.



Because of the trust that recommending or selecting investments for others entails, it’s imperative, from time-to-time, to step back, listen – especially to contrary perspectives – and reflect. We must remain open to having our perspective modified because when it comes to the financial markets only change is certain. 


My reading began with a succinct, closely-reasoned piece entitled “The Arithmetic of Active Management.” Its author is Bill Sharpe, Stanford University professor emeritus, one originator of the capital asset pricing model (CAPM), creator of the Sharpe ratio and winner, along with Harry Markowitz and Merton Miller, of the 1990 Nobel Prize in Economics.


The thesis of “Arithmetic” is that the “after-cost return from active management must be lower than that from passive management.” To make his argument, Sharpe begins with two points:

If ‘active’ and ‘passive’ management styles are defined in sensible ways, it must be the case that before costs, the return on the average actively managed dollar will equal the return on the average passively managed dollar and after costs, the return on the average actively managed dollar will be less than the return on the average passively managed dollar.

These assertions will hold for any time period.


This is, clearly, the classic passive management argument. It cannot, I think, be refuted if - and this “if” is critical - the qualifiers Sharpe imports are accepted. However, before accepting Sharpe’s conclusion, it’s necessary to note just what those qualifiers are:


First, note that what he’s focused on is not all actively and passively managed dollars but on the average actively and passively managed dollarsThese are, I think, quite different things. What Sharpe is assuming here (and a moment of reflection will confirm to be true) is that in the aggregate the investment markets amount to a zero-sum game in which a win (or gain) by any participant is offset by an equal loss by another participant.

Sharpe further qualifies his position with the (empirically valid) assertion that the costs of actively managing a given number of dollars will exceed those of passive management.” Sharpe further specifies that “’active’ and ‘passive’ management [be] defined in sensible ways,” meaning the only investor who qualifies as “passive” is one who “always holds every security from the market, with each represented in the same manner as the market.” By extension, then, “[a]n active investor is one who is not passive.”


Given these qualifiers, the passive argument is compelling. However, I don’t think it ends here. In fact, for Sharpe, it clearly doesn’t. Although he goes on to underscore the passive case:

"Empirical analyses that appear to refute [the assertion that the average actively managed dollar must underperform the average passively managed dollar], net of costs…are guilty of improper measurement".


In the very next paragraph he allows:


"It is perfectly possible for some active managers to beat their passive brethren, even after costs [and it] is also possible for an investor…to choose a set of active managers that, collectively, provides a total return better than that of a passive alternative, even after costs".


Although at first this seems an odd turn-about in his argument, what it seems to me is that once terms are defined and qualified the active/passive “debate” may not be a debate at all.


The conclusion of “Arithmetic” is that the only appropriate way to measure the success of active investment management is to compare an active approach against a truly comparable passive alternative such as an index fund “identified in advance of the period over which performance is measured.” The key, in other words is to (1) properly delineate a passive baseline, such as an index and (2) establish the time period for evaluation. Once done, as Sharpe says at the outset, let the “laws of addition, subtraction, multiplication and division” be arbiter and judge.

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