Sharpening the Arithmetic of Active Management

Abstract

I challenge William F. Sharpe’s famous equality that “before costs, the return on the average actively managed dollar will equal the return on the average passively managed dollar.” This equality is based on the implicit assumption that the market portfolio never changes, which does not hold in the real world because new shares are issued, others are repurchased, and indexes are reconstituted—so even “passive” investors must regularly trade. Therefore, active managers can be worth positive fees in aggregate, allowing them to play an important economic role: helping allocate resources efficiently. Passive investing also plays a useful economic role: creating low-cost access to markets.

Disclosure: The author is a principal at AQR Capital Management, a global investment management firm, which may or may not apply similar investment techniques or methods of analysis as described herein. The views expressed here are those of the author and not necessarily those of AQR.

Editor’s Note

Submitted 10 January 2017

Accepted 4 July 2017 by Stephen J. Brown

Acknowledgment

I am grateful for helpful comments from Yakov Amihud, Cliff Asness, Jatin Bhatia, David Blitz, Executive Editor Stephen Brown, Darrell Duffie, Jens Dick-Nielsen, Nicolae Gârleanu, Niels Joachim Gormsen, Sanford Grossman, Søren Hvidkjær, Antti Ilmanen, Ronen Israel, David Lando, John Liew, Robert C. Merton, Toby Moskowitz, Lukasz Pomorski, Jesper Rangvid, Scott Richardson, William Sharpe, and Rodney Sullivan, CFA, as well as seminar participants at CFA Society Denmark, the University of Chicago’s Investment Committee of the Board of Trustees, and the NBER New Developments in Long-Term Asset Management Conference.

Notes

¹ Berkshire Hathaway Inc., “2016 Annual Report,” pp. 24–25.

² Footnote 4 in Sharpe (1991, p. 8) states, “We assume here that passive managers purchase their securities before the beginning of the period in question and do not sell them until after the period ends. When passive managers do buy or sell, they may have to trade with active managers; at such times, the active managers may gain from the passive managers, because of the active managers’ willingness to provide desired liquidity (at a price).”

³ Event-driven hedge funds specialize in trading around such corporate events as mergers, new issues, seasoned equity offerings, spinoffs, and so on.

⁴ For an exploration of leverage constraints and betting-against-beta strategies, see Frazzini and Pedersen (2014).

⁵ Ljungqvist (2008) documented an average IPO underpricing of 10%–20%. The IPO underpricing is needed to compensate passive investors for adverse selection and give active investors an incentive to take part in the underwriting process and spend resources in determining the value of the securities (Rock 1986).

⁶ Furthermore, the passive trades in connection with IPOs and index reconstitutions discussed previously are more complex than many investors realize. When a passive investor buys shares in an IPO, where does the money for these shares come from? If he does not want to use cash, he needs to sell parts of all his other security holdings.

⁷ If passive investors hold only publicly traded securities and these securities are mispriced, passive investors may also be exploited through companies’ decisions to list on exchanges or delist. That is, the effects discussed in Example 1 may be exacerbated by endogenous decisions to add and subtract securities from the public market at opportune times.

⁸ Figure 1 reports the total market value of the buy-and-hold strategy as a fraction of the total market value of all shares, adjusted for stock splits by assuming that the passive investor is treated like other investors in any stock split. Some readers may be surprised to learn that doing nothing is not enough to be passive, whereas others may be surprised that the investor from 1926 continues to hold almost 10% of the market today (this is due to old giants like Standard Oil, GE, Chevron, and Coca-Cola).

⁹ Turnover is calculated using the CRSP database for 1926–2015 for US common stocks (Share Codes 10 and 11). The fixed-income issuance is calculated as annual issuance divided by bonds outstanding, averaged over 1996–2015, based on data from the Securities Industry and Financial Markets Association, “US Bond Market Issuance and Outstanding” (www.sifma.org/resources/archive/research/). The S&P 500 and Russell 2000 numbers are for 1990–2005 and were taken from Petajisto (2011), supplemented by SEO and repurchase data from CRSP for the S&P 500 over the same period. The Bank of America Merrill Lynch (BAML) data are calculated using the BAML database for 2000–2016.

¹⁰ These turnover rates can be computed by looking at the separate numbers for purchases and sales in the footnotes to the annual reports. Turnover could be higher still because certain types of trades are not included in these statistics—for example, derivatives trading.

¹¹ See also Chen, Noronha, and Singal (2006, p. 45), who found that the “loss to an investor in the Russell 2000 may be about 130 bps a year and can be as high as 184 bps a year, and S&P 500 investors may lose as much as 12 bps a year. Consistent with this finding, we found that the Russell 2000 underperformed other small-cap indices by more than 3 percentage points a year in the 1995–2002 period, even though comparable indices did not entail greater risk.”

¹² This debate has been intense ever since passive investing was introduced (see, e.g., Langbein and Posner 1977).

¹³ Pástor and Stambaugh (2012) studied the size of the active management industry.

¹⁴ Small investors are likely to perform better with low-cost, passive investing, whereas large investors are more likely to benefit from being active. The marginal investor should be indifferent between passive investing and spending resources to find an active manager who is worth more than his fees. See the formal model of asset management by Gârleanu and Pedersen (forthcoming 2017) and practical examples in Pedersen (2015).

¹⁵ Example 4 from earlier in the article corresponds to a time-varying premium x (and a corresponding time-varying θ in the equilibrium model), leading to additional dynamic effects not considered here.

¹⁶ I assume that securities have independently and identically distributed dividend risk, I consider a steady-state equilibrium with constant prices, and I assume that risk associated with inclusions and deletions is fully diversifiable. Hence, the variance of a diversified portfolio is ${\text{var}}_t\left\{{\mathrm{\pi }}^{\text{'}}\left[D_{t+1}^{}+P_{t+1}^{}-\left(1+r^f\right)P_t^{}\right]\right\}={\mathrm{\sigma }}^2{\mathrm{\pi }}^{\mathrm{\text{'}}}\mathrm{\pi }$; that is, the variance from dividend risk ${\mathrm{\sigma }}^2$ times the squared portfolio weights. In other words, in each period, a given fraction $s^d$ of securities are deleted from the market so that there is no aggregate deletion risk—and similarly for additions.

¹⁷ For an estimate of the size of passive investors as a group, see, for example, Morningstar (2017, Exhibit 10).

¹⁸ The choice of I depends on the setting. For US exchange-traded equities, S&P 500 stocks constitute about 80% of the market value of listed US stocks. But historically, that number is lower, and it is even lower if you assume that active investors can also hold some foreign equities, convertibles, private equities, and other assets.

¹⁹ If the price impact of inclusion/deletion is temporary, included securities continue to suffer from a low return because of the price drop associated with deletions and nonincluded securities continue to benefit from a price jump upon inclusion. But several other effects arise. Newly included securities have a low return because their recent price increase reverses, recently deleted securities have a high return because their price drop reverses, and the dividend-yield effect discussed previously disappears if price levels are equal for the two groups.