Active management is not zero-sum

Pedersen vs Sharpe, Buffett and Bogle

Let’s imagine that the market consists of 100 investors, each of whom initially owns one-hundredth of all the companies in the world. Fifty of them are passive investors who simply do nothing, while the other fifty are active investors who trade shares among themselves and pay high fees to advisors and managers of 2% per year. Now let’s wait a year. The two groups will have the same return before costs since, as a group, they start with the same portfolio. However, the passive group will have, on average, a 2% higher return after costs.

This thought experiment illustrates The Arithmetic of Active Management (1991), William Sharpe’s seminal article praised by Nobel laureates such as Fama and French (Why active investing is a negative sum game). Warren Buffett later popularized it in his 2005 annual letter (p. 18) with the parable of the Gotrocks family, and John Bogle included this parable in Chapter 1 of The Little Book of Common Sense Investing. The conclusion, in Buffett’s own words, is “For investors as a whole, returns decrease as motion increases”.

This thought experiment highlights key characteristics of passive management. Investors replicating a capitalization-weighted index are immune to price movements and are not forced to trade: if a company rises, it automatically weighs more heavily on both the index and their portfolio. It is also epistemologically neutral: If one active investor deviates from the index in a certain direction, another is deviating in the opposite direction. Together, they form the index. The capitalization-weighted index reflects the consensus of all active investors, and replicating it means assuming you do not know who is right.

However, something intuitively seems wrong. Sharpe’s framework describes a non-equilibrium world: if it were correct, active management would eventually disappear. Moreover, gathering information and analyzing it to make better decisions is a useful function that the market usually rewards in any micro context, but which it would not be compensating in this case.

Grossman and Stiglitz had already demonstrated 10 years earlier that the market rewards those who incorporate information into prices in On the Impossibility of Informationally Efficient Markets (1980) with their “equilibrium degree of disequilibrium”, but it seems that Sharpe’s arithmetic doesn’t die if you don’t point out the specific mechanism.

Sharpe argued in his article that:

1. before costs, returns of active and passive management are the same since, on an aggregate basis, both groups have the same portfolio;
2. after costs, returns of active management are lower than passive management since they have incurred higher costs.

This means active management is a zero-sum game, becoming negative-sum after costs. Furthermore, Sharpe made some statements in the article that, in retrospect, I find quite controversial. For example, he says that if data contradicted him, data is wrong: “Empirical analyses that appear to refute this principle are guilty of improper measurement”. Crucially, he also stated in footnote3 that some corporate actions “require more complex calculations but do not affect the basic principles”.

And it’s precisely this footnote that destroys his arithmetic. He likely considered it a minor detail — how could a few corporate actions change such a robust conclusion? Well, it may seem minor, but it’s the heart of his arithmetic. Sharpe compares whether the costs of gathering and analyzing information make sense, yet it completely omits the potential positive effects those costs can generate in aggregate. It is like evaluating insurance coverage in a world without accidents, or assessing R&D spending utility in a world where nothing remains to be invented. It makes no sense.

Active management incurs costs to find assets more profitable than the index; that is, it tries to change the index, to improve it (if the market could not change or we knew that it was not going to change, active management would indeed be a waste of resources). If this activity involves only active managers swapping shares among themselves, the net effect is zero: one active manager thinks one thing and another thinks the opposite. This may result in a more accurate price (which could be useful), but it yields no real economic change to justify the costs incurred.

However, when the opinions of active managers don’t fully offset each other, there is a net effect that moves asset prices, and this ultimately translates into changes in the index’s composition. In other words, active management as a whole, beyond the fact that some of its conclusions may cancel each other out, fulfills the function of finding and maintaining the productive organization that best satisfies our consumption desires. And the specific mechanism through which these changes in the index manifest themselves are precisely corporate actions: issues, buybacks, inclusions, and deletions.

And this is precisely what Lasse Heje Pedersen points out in his article titled Sharpening the Arithmetic of Active Management (2018). Sharpe’s arithmetic is based on the implicit assumption that the market never changes. But the real world isn’t like that; the composition of the market is not static. The average annual turnover reaches 7.6% for all US-listed stocks from 1926–2015 (and for bonds it would be even higher ~20%). Because of this, even passive investors must regularly trade to maintain their market portfolio. This graph shows how a passive investor who never trades would drift from the market:

Pedersen doesn’t just tell a story; he also provides empirical support and attempts to estimate the impact of these corporate actions. He concludes that active management plays a crucial economic role aiding efficient capital allocation, and that active and passive investors coexist in an “efficiently inefficient” market. This is, in fact, the title of his latest book (Efficiently Inefficient: How Smart Money Invests and Market Prices Are Determined), which leaves us with quotes like these:

If markets are fully efficient, there is no point in active investing because the prices already reflect as much information as you could hope to collect. But without active investors, who would make the market efficient in the first place?

Further, given that investors are paying billions of dollars in fees to active managers, either the securities markets are inefficient (so active managers can outperform) or the market for asset management is inefficient (because investors would pay fees for nothing) — it is logically impossible that all these markets are fully efficient.

Efficiently inefficient: just inefficient enough that money managers can be compensated for their costs and risks through superior performance and just efficient enough that the rewards to money management after all costs do not encourage entry of new managers or additional capital.

Unlike Sharpe, Pedersen’s model offers an equilibrium: there is an optimal point in the amount of resources the market should dedicate to analysis, below this point active managers will obtain extraordinary profits, and above this point they will fail to cover costs. Furthermore, this is a stable equilibrium: any disturbance endogenously generates incentives to return to equilibrium.

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The market must match planned production (companies) and desired consumption (investors). Active management can create value by influencing production flows (corporate actions) and consumption flows (subscriptions and redemptions).

It has taken 30 years, but we can finally sleep soundly.

The True Story of the Gotrocks Family

Imagine for a moment that all the companies in the market are, and always will be, owned by a single family: the Gotrocks. Generation after generation, this family becomes richer by the aggregate amount earned by its companies. In the Gotrocks household everyone grows wealthier at the same pace, and all is harmonious. Each family member owns, proportionally, an identical share of all the companies in the market.

Now suppose that an active manager approaches a family member, let’s call him Mr. Buffit Gotrocks. The manager convinces Mr. Buffit to try to outsmart his relatives by buying some shares and selling others. The manager, for a fee of course, obligingly agrees to handle these transactions.

But who is he going to trade with? The entire market (except him) is passive. We might think there would be no prices or transactions, but there are for at least three reasons:

1. The active manager, even if he finds no sellers or buyers, will have a mental purchase and sale price: a bid and an ask that reflect his internal valuation of each company.
2. The passive relatives, even if they replicate the index, will have net flows from subscriptions and redemptions, and therefore will need to go to the market.
3. And above all, the companies themselves also have a bid and an ask: they are willing to buy back their own shares or issue new ones depending on the price and the expected profitability of their projects.

Over time, the active manager discovers that one of the companies is misusing resources and participates in its share buyback program, selling all its shares for 1M. Shortly after, it finds another company with potentially very profitable expansion projects and participates in a capital increase, buying new shares for 1M.

After a while, when the outcome of these corporate actions becomes public, the passive relatives realize that the market has escaped them. To maintain the index portfolio, they must sell shares in the wasteful company and buy shares in the promising company. The active manager is happy to act as their counterparty, of course at somewhat less attractive prices than the ones at which they initially traded.

Some time later, the active manager tells Mr. Buffit that he sees no more opportunities, that the “market portfolio” is perfect. Mr. Buffit tells him it doesn’t matter, he should keep trying to estimate how much his companies will earn. Over the next few weeks, he meets with his relatives and inquires about their stories: how much they earn, how much they spend, when they retire… He discovers that next Monday his brother-in-law will receive a 400k bonus that he predictably wants to invest. But if no one offers him any counterparty, prices will skyrocket. The following month, his cousin wants to buy a house and needs to repay 1M. But if no one offers any counterparty, prices will drop to zero. He decides, in exchange for an acceptable spread, to cover the gap himself: sell shares to his brother-in-law next Monday and buy them from his cousin in a month. The remaining $600k gap is precisely the amount the manager estimates will be covered by dividends collected the following month.

At the end of the period, Mr. Buffit has been able to beat his relatives even after adjusting for costs. Why? Now the market is presumably more valuable than at the beginning of the period, and not only because companies may have created value internally, but also because productive factors have been reorganized across companies: the same resources, organized differently, now produce more or more valuable things. The wasteful company, for example, had to sell some machinery and lay off some workers; with that money, it bought back shares. And where have those factors gone? Well, to the promising company, which bought them with the money it raised by issuing new shares. Furthermore, Mr. Buffit has helped match, at the margin, production flows (dividends) with consumption flows (subscriptions and redemptions), thereby smoothing the price- or time-mismatch problem his passive relatives faced. All this value creation is directly attributable to Mr. Buffit and his active manager, and the market rewards them with higher returns.

In the following years, Mr. Buffit’s extraordinary profit attracts more relatives to active management, which, due to the usual dynamics of any sector, erodes his profits. The opportunities to be found (both wasteful and promising companies) and the liquidity gaps to be filled (net subscriptions and redemptions) are the same, but more and more people are competing for them. There comes a point where the profit they earn above the index decreases to equal their costs. At that point, no family member has any incentive to change strategy. And finally, Buffett’s law comes true: whoever moves loses.

Footnotes

1. Sharpe’s footnote3 points out the first hole, and footnote4 the second: “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, because of the active managers’ willingness to provide desired liquidity (at a price).”
2. Why can’t this value creation by moving factors between companies be developed by the companies themselves? They can, and in fact they do, through their mergers and acquisitions departments, but the fact that *only* companies do so exposes investors to agency problems that will ultimately push them toward activism. In any case, analyzing these opportunities is valuable, and the investor will bear the cost of this function, whether it is performed within the company, outside of it, or a mix of both.
3. The Treynor model used by Perry Mehrling seems very suggestive to illustrate the situation: active management would act as a dealer, creating a market for passive investors within the outside spread generated by the true dealer of last resort, the companies themselves.
4. If the wasteful company bought back its own shares from Mr. Buffit at a certain price, why doesn’t it step in to provide a counterparty to his relatives when the latter sells them to Mr. Buffit at a lower price? I believe the answer lies in marginal returns.