Sharpe’s Arithmetic, Revisited: When ‘Average Active’ Might Beat ‘Average Passive’ Before Costs

William Sharpe’s ‘Arithmetic of Active Management’ is one of those ideas that feels too theoretically neat to be true, and yet it remains one of the most robust starting points for thinking about active versus passive. Sharpe’s premise is deliberately simple: the market is the sum of all investors, so if some investors are ‘passive’ and hold the market portfolio in market-cap weights, then everyone else, collectively, must hold the same market portfolio as well. That is not an opinion about skill, it is an accounting identity. From that identity, Sharpe draws the familiar conclusion: before costs, the return on the average actively managed pound must equal the return on the average passively managed pound, because they are collectively holding the same aggregate portfolio. After costs, active must underperform passive in aggregate, because active bears higher fees, turnover, and implementation frictions (Sharpe 1991).

The part that often gets lost is how specific the conditions are under which that equality holds exactly at each point in time. Sharpe’s proof is cleanest in a world where the market portfolio is effectively static aside from price moves, and where ‘passive’ can be thought of as a buy-and-hold position in the market portfolio. The real market is not like that. Shares are constantly being created and destroyed through corporate actions: new issuance, secondary offerings, mergers, spin-offs, bankruptcies, share-based compensation and, crucially, buybacks. Those events change the investable opportunity set and the relative weights of securities for reasons that are not just price fluctuations. Once you admit that the market portfolio itself is moving because the underlying claims are being reshaped, it becomes meaningful to ask a different question: do passive investors always hold the market portfolio, continuously and perfectly synchronised?

In practice, index-tracking funds adjust discretely, not continuously. They have to follow index methodologies, reconstitutions, and rebalancing schedules. Even where indices incorporate corporate actions promptly, index trackers still face operational lags, trading costs, and the need to trade around known implementation dates. That creates short windows where the holdings of index trackers can differ from the instantaneous market composition. In those windows, the complement, what the rest of the market holds, is not forced to be exactly the market portfolio at each moment, because passive is not exactly ‘the market’ in real time. This is the core insight formalised by Pedersen, who shows that once the market is considered to be dynamic, the pre-cost equality between average active and average passive does not need to hold with the same knife-edge precision as in Sharpe’s original static framing (Pedersen 2018).

The Buyback Timing Intuition

Buybacks are a simple way to see the mechanism. A repurchase reduces shares outstanding, mechanically increasing each remaining share’s claim on the firm’s future cashflows. Markets often respond at announcement, and there can be a discrete price jump. If a passive strategy’s exposure to that firm adjusts only as the index and the fund rebalance through time, whilst active investors can reposition immediately or pre-position ahead of predictable corporate actions, then it is possible for active investors, in aggregate, to earn a slightly higher gross return than passive investors, even without anyone displaying forecasting brilliance. The source of the advantage is not ‘skill’ in the usual sense; it is that passive portfolios, by design, trade on rules and schedules, and those rules can create predictable patterns of demand and supply that other investors can intermediate (Pedersen 2018).

A two-stock example captures this cleanly:

• Two stocks A and B, each have a £1,000 market cap and each is 50% of the index.

• Passive holds 50% A and 50% B.

• Active holds 70% A and 30% B.

• A announces a buyback and jumps +10%.

The passive fund ends at a value of £1,050, or a +5.0% return. The active fund ends at £1,070, or a +7.0% return. The difference is not magic. It is simply that the active portfolio had more exposure to the stock whose weight and price moved for a corporate-action reason. If active investors as a group tend to be overweight the kinds of firms whose weights are about to rise mechanically, and underweight those about to be diluted via share issuance, then average active can beat average passive before costs, even though Sharpe’s static arithmetic would say this cannot happen (Pedersen 2018; Sharpe 1991).

Index Rebalancing

This broader idea also shows up around index additions, deletions and reconstitutions, where index trackers are forced buyers and sellers at predictable times. A large literature documents price and volume effects around S&P 500 index changes, consistent with some combination of price pressure, downwards-sloping demand curves for individual stocks, and limits to arbitrage (Harris and Gurel 1986; Shleifer 1986; Wurgler and Zhuravskaya 2002; Chen, Noronha, and Singal 2004). The details vary across eras and studies, but the conceptual point is stable: if passive money must trade in a predictable way, other investors can compete to provide liquidity around those flows and earn excess risk-adjusted returns. However, competition tends to compress the easy profits over time, which is exactly what you would expect if the gross ‘edge’ available from intermediating passive rebalancing is real but contested (Greenwood and Sammon 2025).

Are Average Gross Returns from Active Management Higher than Those from Passive Management?

Not really, but it does refine the story in a useful way. Sharpe’s theory is still the right place to start thinking about investing: the investment industry as a whole cannot collectively beat itself. And, indeed, higher costs still drag aggregate active performance below aggregate passive performance for end investors (Sharpe 1991). Pedersen’s extension is not a free lunch for active investors; it is a clarification that the world is not perfectly synchronised, and the market portfolio is not a static object. In that more realistic setting, it is entirely possible for average active to outperform average passive before costs by a small amount, simply because someone has to intermediate the market’s continuous corporate actions and the discrete trading of index trackers (Pedersen 2018). However, the practical punchline remains: after fees and trading frictions, most investors should still expect passive to win on average, but the reason is less ‘active is always zero-sum’ and more ‘any small structural gross edge is likely to be competed away and overwhelmed by costs’.

References

Chen, Honghui, Gregory Noronha, and Vijay Singal. 2004. ‘The Price Response to S&P 500 Index Additions and Deletions: Evidence of Asymmetry and a New Explanation’. Journal of Finance 59 (4): 1901–1929. doi:10.1111/j.1540-6261.2004.00683.x.

Greenwood, Robin, and Marco Sammon. 2025. ‘The Disappearing Index Effect’. Journal of Finance 80 (2): 657–698. doi:10.1111/jofi.13410.

Harris, Lawrence, and Eitan Gurel. 1986. ‘Price and Volume Effects Associated with Changes in the S&P 500 List: New Evidence for the Existence of Price Pressures’. Journal of Finance 41 (4): 815–829. doi:10.1111/j.1540-6261.1986.tb04550.x.

Pedersen, Lasse Heje. 2018. ‘Sharpening the Arithmetic of Active Management’. Financial Analysts Journal 74 (1): 21–36. doi:10.2469/faj.v74.n1.4.

Sharpe, William F. 1991. ‘The Arithmetic of Active Management’. Financial Analysts Journal 47 (1): 7–9.

Shleifer, Andrei. 1986. ‘Do Demand Curves for Stocks Slope Down?’ Journal of Finance 41 (3): 579–590. doi:10.1111/j.1540-6261.1986.tb04518.x.

Wurgler, Jeffrey, and Ekaterina Zhuravskaya. 2002. ‘Does Arbitrage Flatten Demand Curves for Stocks?’ Journal of Business 75 (4): 583–608. doi:10.1086/341636.