The Passive vs. Active Conundrum: A New Perspective on the Arithmetic of Index Investing

Abstract

The rise in passive investment through indexes has been significant in recent decades and is causing a drastic reduction in actively managed assets. In addition to the empirical evidence that index funds outperform actively managed funds, net of fees, the logical basis for index investing, as famously articulated by Professor William Sharpe in The Arithmetic of Active Investing and in other publications, has provided proponents with an unchallenged rationale for indexing — that the average active manager cannot beat the index and actually tends to underperform it after deducting fees. This logical argument has persuaded many investors to increase allocations to index funds given their admitted inability to accurately choose the active outperformers from the active underperformers in advance. It has also reduced active manager’s credibility given their inability to add value (in terms of returns vs. an index) as a group. This paper reexamines the “arithmetic of active investing” as explained by Professor Sharpe and attempts to show that its assumptions do not necessarily hold in practice. If active buyers and active sellers have a roughly equal probability of buying or selling each security in the index, liquidity is available on both sides of each trade and index investors are able to execute orders for all securities at proper market-capitalization weightings. However, once active investors concentrate their orders on one side of a trade, it becomes difficult for index investors to execute the proper orders to keep their weightings in-line with the index. This idea of failed order execution (or costly order execution) has been shown by previous authors in the context of how “informed” traders trade with “uninformed” traders. The same concept can be applied to active vs. passive investors as long as active investors are “informed” on average (i.e., possess skill). By illustrating how index investors are required to trade in the underlying shares of the index (or equivalently that ETF Trusts are forced to issue and redeem shares on a continuous basis) it can be shown that order concentration by active investors can cause index investors to experience tracking difference (commonly referred to as tracking error). By including tracking difference in the arithmetic of index investing, active management is no longer a zero-sum game. As long as active managers as a group possess skill, they can beat the index at the expense of index investors who underperform by an amount equal to tracking difference. This framework also indicates that tracking difference is not necessarily a fixed cost for index investors and could potentially increase to undesirable levels in the future.