Why Active Investing Is a Negative Sum Game
William F. Sharpe has a great article in the January/February 1991 issue of The Financial Analysts Journal (Vol. 47, No.1, pages 7-9). The title is “The Arithmetic of Active Management.” It should be required reading for academics and investment professionals alike.
The paper makes a simple point that we call equilibrium accounting. Consider the portfolio of U.S. common stocks with each stock weighted according to its market capitalization. There is another way to define this cap-weight market portfolio. It is the portfolio that combines the U.S. equity portfolios of investors, with each investor’s portfolio of U.S. equities weighted by that investor’s share of the total market cap of U.S. equities.
Suppose we define a passive investor as anyone whose portfolio of U.S. equities is the cap-weight market portfolio described above. Likewise, define an active investor as anyone whose portfolio of U.S. equities is the not the cap-weight market portfolio. It is nevertheless true that the aggregate portfolio of active investors (with each investor’s portfolio weighted by that investor’s share of the total value of the U.S. equities held by active investors) has to be the market portfolio. Since the aggregate portfolio of all investors (active plus passive) is the market portfolio and the aggregate for all passive investors is the market portfolio, the aggregate for all active investors must be the market portfolio.
All this is obvious. It is just the arithmetic of the fact that all U.S. equities are always held by investors. (That is why we call it equilibrium accounting.) Its implications, however, are often overlooked.
For example, a question commonly asked by academics and practitioners is whether active investors as a group produce superior returns. In other words, do active investors in aggregate earn a higher expected return than passive investors? The arithmetic of equilibrium accounting says we do not need empirical tests to answer this question. Since we are assuming they all hold the market portfolio, passive investors earn the return on the market minus their fees and expenses. In aggregate, active investors also hold the market portfolio, so they also earn the market return minus their fees and expenses. If the fees and expenses of active investors are higher than those of passive investors, active investors must in aggregate lose to passive investors. This is the unavoidable arithmetic of equilibrium accounting. And notice that this is not a statement about expected returns or about long-term average returns. In aggregate active investors lose to passive investors every instant.
It is, of course, possible that individual active investors add value. But if they do, it’s at the expense of other active investors. Again assume that passive investors always hold the cap-weight market portfolio. The arithmetic of equilibrium accounting then implies that the deviations from cap weights in one active investor’s portfolio must be absorbed by other active investors who take offsetting positions. In aggregate active investors hold the market portfolio, so if some skilled active investors overweight an undervalued stock, other active investors must underweight it. This means that, before fees and expenses, trading is a zero sum game. Ignoring costs, the gains of the skilled investors are, dollar for dollar, at the expense of other investors. But real investors cannot ignore costs. If some active investors win, others must lose, and they all pay to place their bets. (French 2008 provides dollar estimates of the massive costs borne by active investors in U.S. public equity).
Another common argument is that active investing is more likely to add value for small stocks than for the market portfolio in which big stocks dominate. Again, however, the arithmetic of equilibrium accounting plays a sobering role. Define a passive investor in small stocks as anyone who holds small stocks in cap-weight proportions, and define an active investor in small stocks as anyone who doesn’t hold small stocks in cap-weight proportions. (Both groups can hold big stocks in whatever proportions they see fit.) Since passive investors in small stocks in aggregate hold the cap-weight market portfolio of small stocks, in aggregate active small stock investors must also hold the cap-weight market portfolio of small stocks. This again means that skilled active small stock investors must win at the expense of unskilled active small stock investors. If there are more opportunities for skilled investors to add value in choosing among small stocks, it is because unskilled active investors in small stocks are worse than unskilled active investors in big stocks.
The same arguments apply whatever one takes to be the market, for example, value stocks, growth stocks, industry sector portfolios, portfolios of foreign stocks, emerging markets, etc. If passive investors in any market hold a cap-weight market portfolio, active investors must in aggregate do the same. Since passive investors do not deviate from cap weights, active investors can only win at the expense of other active investors. In short, active investing in any sector is always a zero sum game - before costs. After costs, active investing is a negative sum game.
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Republished July 26, 2023.
Eugene Fama and Ken French are members of the Board of Directors of the general partner of, and provide consulting services to, Dimensional Fund Advisors LP.
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