For three decades, William Sharpe’s Arithmetic of active managementpublished in the Journal for financial analysts in 1991 was considered a near-scripture for passive investing. The Nobel Prize winner, protégé of Harry Markowitz and creator of the Capital Asset Pricing Model (CAPM), applied a clean, elegant logic that has shaped investment thinking ever since.
Sharpe’s statement was blunt: higher fees cause active portfolios to lag behind passive portfolios. Before costs, both groups earn the same market returns; after fees, active investing becomes a zero-sum and ultimately a negative-sum game. Sharpe’s 1991 paper was among those recognized as having a lasting impact on the investment industry as part of the CFA Institute Research and Policy Center’s annual celebration of the 80e Anniversary of the Journal for financial analysts.
It’s a message that has fueled the rise of index funds and haunted generations of investors. Why bother paying for skills when the average return of the market is right there, free for the taking? Sharpe’s logic was groundbreaking, but it described a closed, static market. Later thinkers, especially Lasse Heje Pedersenhave shown how active management contributes to market evolution rather than simply redistributing returns.
This article follows that development and shows how Pedersen’s refinement completes Sharpe’s arithmetic and restores the constructive role of active management in market efficiency.
Sharpe’s theorem captures what passive management really is: effortless exposure to the collective wisdom of the market. In a capitalization-weighted index, the portfolio weights automatically adjust to price movements. No trading is required. For every active bet there is an equal and opposite. The index is that equilibrium point, the distilled consensus of all investors. Following it means letting the market decide who is right.
Source: Diego Costa
Yet something about this logic feels incomplete. If Sharpe’s world were completely accurate, active management would eventually disappear and the markets would stop functioning altogether.
Sanford J. Grossman and Josephe Stiglitz had already shown ten years earlier, in 1980, that the market rewards those who incorporate information into prices. On the impossibility of informationally efficient markets with their ‘equilibrium degree of disequilibrium’.
As a result, Sharpe’s arithmetic only works if you ignore the economic mechanisms that allow markets to function. So what if markets are not static and active management not only redistributes wealth, but actually creates it?
The simple elegance of Sharpe’s arithmetic
Imagine a world with a hundred investors, each owning a hundredth of every company. Fifty is passive, fifty is active. Passive investors are in a pinch. The active companies trade among themselves and pay managers and advisors 2% in annual fees.
After a year, active investors earn less overall because of the fees. The logic impressed Nobel Prize winners Eugene Fama and Ken French. Warren Buffett later retold it as The Gotrocks family similaritywarning that “returns decrease as movement increases.” John Bogle built an empire around it The little book about investing with common sense.
The message was clear: markets are a closed system. Every winner has a loser. So why play a negative sum game?
Mathematics is not “math”
The problem is that Sharpe’s arithmetic describes a world that does not exist.
He made a number of statements in his article that, in retrospect, I find quite controversial. For example, he says that if data contradict him, the data is wrong: “Empirical analyzes that seem to refute this principle are guilty of incorrect measurements.” Crucially, in one of the footnotes he also states that corporate actions “require more complex calculations, but do not affect the basic principles.”
That footnote, often overlooked, turns out to be a crack in the foundation.
Sharpe’s model assumes a static market, a snapshot in time in which no new companies are born, none die, and nothing changes except existing stocks. But in the real world, everything moves. Companies issue new shares, buy back old ones, merge, spin off or go bankrupt. Markets are living, breathing entities that reflect human behavior and trends. Indices evolve and rebalance to reflect the changing structure of the economy.
Active management attempts to do just that: change the index to improve it. Sharpe’s analysis fails because it ignores the potential positive effects that its total costs enable. It’s like assessing R&D expenditure in a world where nothing needs to be invented anymore.
Sharpening the arithmetic
This is exactly what Pedersen noted in his 2018 article Sharpening the arithmetic of active managementalso published in the Journal for financial analysts. His insight was simple but profound: markets evolve, and active managers play a crucial role in that evolution.
Pedersen compiled data showing that the average annual turnover of U.S. stocks is about 7.6%. Bonds were trading closer to 20%. Even if every investor stopped trading, the market would still change. And even passive investors must trade regularly to maintain and rebalance their portfolio to maintain market weight: selling what leaves an index and buying what enters.
The graph below is from Pedersen’s 2018 article and shows what happens to an investor who invests money but then never trades again.
Source: Lasse Heje Pedersen, Sharpening the arithmetic of active managementpg 9, Financial Analysts Journal, 2018. “The solid blue line shows an investor who bought the entire US stock market in 1926 and did not participate in IPOs, SEOs or stock buybacks and did not reinvest dividends….” Therefore, the investor’s market share deteriorates over time. The other lines indicate the same for investors who started investing in 1946, 1966, 1986 and 2006.
Active management as an economic engine
Pedersen’s revision not only corrects Sharpe’s mathematics; it reframes the purpose of active management. When active managers identify capital misallocations – companies that waste resources or those that could lead to higher productivity – they are not just dealing in paper. They redistribute capital to its most productive use.
Through involvement, voting and investment decisions, active managers influence which companies issue stock, which buy back stock, which expand and which shrink. These actions shape the real economy: which technologies are financed, which innovations reach the market, and which industries shrink to make room for more efficient industries. In effect, it creates price discovery, that elusive measure of a company’s value at a given point in time within a given economic framework.
The active manager fees that investors pay are not just transaction costs. Active management fulfills a social function: it discovers and maintains the productive organization that best meets our collective consumption desires.
Unlike Sharpe, Pedersen’s model provides an equilibrium: there is an optimal point in the number of resources the market should devote to analysis. Below this point, active managers will earn extraordinary profits, and above this point they will fail to cover costs. Moreover, there is a stable equilibrium: every disturbance endogenously generates incentives to return to equilibrium.
The market must match the planned production (companies) and the desired consumption (investors). Active management can create value by influencing production flows (corporate actions) and consumption flows (subscriptions and redemptions).
It took 30 years, but we can finally sleep well.
The Gotrocks revisited
Warren Buffett’s “Gotrocks” parable became a classic defense of passive investing because it showed that trading activities and fees erode returns rather than create value. The story went like this: the Gotrocks family owned every company in the world. Over time, they became rich together. Then, some family members hired managers to trade stocks among themselves, paying fees. The family’s total wealth began to grow more slowly. “For investors as a whole,” Buffett warned, “returns decline as movement increases.”
But Pedersen’s model suggests a movie sequel.
Suppose a cousin of Gotrocks notices that a company is burning money on unprofitable projects. He is selling his interest in buybacks to free up capital. Another cousin sees a company with a high-return investment opportunity and participates in the new stock issue.
Capital has now shifted from wasteful to productive hands. The family’s total wealth has increased, not decreased. Later, when the index funds rebalance to meet these new weights, they indirectly benefit from the price discovery that active managers paid to discover. In this sense, active managers have acted as a catalyst that makes the passive portfolio possible.
An “efficiently inefficient” market
Pedersen’s model achieves an intuitive balance. If markets were completely efficient, there would be no incentive for active managers to research or trade; the prices would already reflect all the information. But if no one traded, the markets couldn’t become efficient in the first place.
There must therefore be a middle ground: a market that is just inefficient enough to reward those who expose information, but efficient enough to prevent the profits from lasting too long.
Pedersen quantifies this balance. Active managers achieve extraordinary profits by exploiting mispricing. As more capital flows into active strategies, those profits shrink. Ultimately, expected returns drop to the level of costs. That’s equilibrium: the market allocates just enough resources to research and analysis to keep prices largely stable.
It’s not a perfect system, but it corrects itself.
Why it matters
Pedersen does not reverse Sharpe’s arithmetic, he actually completes it. Sharpe took a static snapshot; Pedersen adds movement and shows markets as developing systems, and not as a sum at a given point in time.
The conclusion is clear: active and passive management are not opponents, but partners in an ecosystem. Active managers create value by absorbing information and using capital in the most productive way, and by efficiently bridging temporary liquidity gaps between passive investors.
Passive investors enhance efficiency by keeping costs low and anchoring the market to fundamentals. Too much activity causes noise; too much passivity softens price signals. In a living market, where companies spend, buy back and evolve, active management goes from zero-sum to positive. That’s not arithmetic; it is progress.
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