Research Articles

Mild Debunking of Sharpe’s Logic

The following is an abstract. Please contact me to request a copy of the full article.

A popular argument exposing issues with active management was presented by William F. Sharpe in his “The Arithmetic of Active Management” article published in The Financial Analyst’s Journal approximately 25 years ago. Many passive investing advocates have relied on this as unassailable proof active management is a “loser’s game”.

Sharpe assumes one can divide the market into two subsets: one held by passive investors and the other held by active investors. Because both hold shares in the same percentages, they will both ultimately achieve the same “market” returns. However, the higher turnover and trading costs for the active side will lower their net return in aggregate.

Some have argued Sharpe’s claims are not valid in practice by challenging his assumptions. In particular, they dispute both the frame of reference within which his argument holds and his definition of passive. We agree and briefly discuss (and extend) these challenges. However, the primary purpose of this article is to put forth new and stronger challenge to his claims without questioning these assumptions.

There is a larger hole in Sharpe’s logic that invalidates his conclusions even if we look past the above issues. His argument hinges on the notion that shares held by the passive and active subsets exist in mutually exclusive, closed systems. In other words, passive investors do not trade with active investors. On the surface, this appears to be a fair assumption and his conclusions follow with little more than simple arithmetic. However, this market model is flawed; passive investors may not explicitly trade with active investors, but the companies they own do via issuance and buyback programs. Given the buy-and-hold nature of the passive group, the corporations must be trading with the active managers. This creates an asymmetry in holdings and thus performance between the passive and active sides. We suspect this works out in the favor of active managers.

To be fair, buybacks were not nearly as significant a phenomenon when Sharpe originally penned his article. However, given the extraordinary growth of buybacks over the years, the buyback-imposed asymmetry between passive and active investors is bigger than ever.

While we believe this further weakens Sharpe’s claims regarding active management necessarily underperforming passive, we are neither condemning passive nor endorsing active management. For example, we acknowledge evidence indicating active returns (net of fees) are lower than their passive counterparts.