The last term of the return equation is luck. It is the main source of short-term outcomes and averages zero over the long run, yet it is the term most easily mistaken for skill. The overall result of active management is fixed by simple arithmetic: before costs, active investors taken together exactly equal the market; after costs, they must underperform it. Whether an individual manager can outperform depends on how accurate each judgment is, how many independent judgments are made each year, and how much constraints discount those judgments; even for a good manager, the product of these three is small. Identifying skill from outcomes often takes decades of data. And for most ordinary investors, what is harder than beating the market is sticking with a method through volatility.
In 1991 Sharpe used simple arithmetic to show the overall result of active management. Divide all investors into two groups: passive investors hold the whole market at market-value weights, and active investors try to beat the market by buying and selling. The combined return of passive investors equals the market return. All investors together also hold the whole market, so their combined return equals the market return too. Active investors taken together can therefore, before costs, only equal the market. Active management costs more, so after costs active investors as a whole must underperform passive investors.
The addition and subtraction of active management
The market rises 8% in a year. Drag the share of active investors and compare the two groups' returns before and after costs.
Whatever their share, active investors together can only equal the market before costs; their costs are higher, so after costs they must fall behind.
This conclusion depends neither on whether markets are efficient nor on any model; it is just addition and subtraction. It does not deny that an individual active manager can outperform, but it shows that for one to outperform, another must underperform.
Sharpe's arithmetic shows that active investors as a whole cannot outperform, but it does not explain what might allow a few managers to do so. Grinold (1989) gave the answer: a manager's information ratio, the excess return earned per unit of tracking error, is approximately the information coefficient times the square root of breadth. The information coefficient measures how closely each judgment correlates with the actual outcome; breadth is the number of mutually independent judgments made each year. Clarke, de Silva and Thorley (2002) added the transfer coefficient: constraints such as no short selling, position limits and sector neutrality mean that only part of each judgment turns into actual holdings. Grinold and Kahn (2000) noted that an information coefficient of 0.05 is already good, and an information ratio of 0.5 is already top tier. The law says three things: a slight edge in judgment can only be amplified through a large number of independent judgments, and only by the square root; dozens of bets on the same sector or the same kind of factor contain far fewer effective independent judgments than their nominal number; and the more constraints there are, the smaller the share of judgment that turns into return.
Even if no fund manager had any skill at all, some would still post impressive records. Imagine 1,000 managers with no skill whatsoever, each with an even chance of beating the market every year. After five years, about 31 of them will have beaten the market five years in a row. In hindsight, their records look flawless.
Carhart (1997) found that good past performance by funds barely carries into the future; what does persist is mainly poor performance, mostly because of high fees.
Confirming skill from noisy performance takes a long time. Suppose a manager beats the benchmark by 2% a year, but performance relative to the benchmark fluctuates by 6% a year (the tracking error). To say with statistical confidence that the 2% is not luck takes about 36 years of data. The ratio of 2% to 6% is the information ratio, about 0.33; at 95% confidence, the number of years required is roughly (2 ÷ information ratio) squared, so the lower the information ratio, the longer it takes. Most funds do not survive that long, and most investors will not wait that long.
How many years before it is not just luck
Adjust how much the manager beats the benchmark each year and the tracking error (volatility relative to the benchmark) to see how many years of performance are needed to confirm skill statistically.
Fama and French (2010) compared the actual performance of US active equity funds with simulations of what performance would look like if no manager had any skill. They found that after fees, the distribution of actual performance was, as a whole, worse than the pure-luck simulations. This means that most funds have negative true excess returns after fees and that managers who reliably cover their costs are very rare; only in the two tails, before fees, can some genuine differences in ability be seen. S&P Dow Jones Indices' scorecard comparing active funds with their benchmarks likewise shows that, as of the end of 2025, about 90% of US active equity funds underperformed the S&P Composite 1500 over ten years, and about 93% over fifteen years.
After fees, actual performance is worse than pure luck
Grey is the distribution of t-statistics of fund excess returns if no manager had any skill; the orange line is the actual distribution after fees (illustrative). Switch to compare.
Since active management as a whole underperforms, buying and holding an index seems the best choice. But the author argued in a published article that for many ordinary investors, the author included, buy-and-hold does not exist unless there is a method for dealing with volatility that keeps them steady. Of the passive investors who claim simply to sit tight, few actually sit tight to the end and collect the average return.
The evidence supports this view. Dichev (2007) compared buy-and-hold returns on the New York and American stock exchanges from 1926 to 2002 with investors' returns calculated from their actual flows of money in and out, and found the latter about 1.3 percentage points a year lower: investors tend to put in more after prices rise and pull out after prices fall. Products display the fund's return, but investors receive their own.
The fund's return and the investor's return
The same index fund over 20 years. Compare two investors: one puts in a fixed 10,000 yuan every year; the other puts in more after rises and pulls out after falls.
In the same article the author argued further that every school that has succeeded in practice has its own way of dealing with volatility. Take Fisher's view as an example: valuation itself is complicated, and a share price anywhere between half and twice fair value can be regarded as reasonable. With an ample margin of safety, most trades happen within the reasonable range, and value investing is in substance a grid with very wide spacing. Often it pays to give up the fixation on beating buy-and-hold, and to settle for a limited share of the profit in exchange for safety.
Along the same line, grid trading and periodic rebalancing write "sell high, buy low" directly into rules, and in return the method becomes something one can stick with.
A higher theoretical return, or a method you can stick with
Each method is simulated over 2,000 twenty-year paths. Assume the investor gives up and moves the money into a deposit when the drawdown exceeds what they can bear. Drag the maximum tolerable drawdown and compare the typical outcomes of the two methods.
In the language of this book: a method with a higher theoretical return that the investor abandons at the point of maximum drawdown may end up with worse actual results than a method with a slightly lower theoretical return that the investor can stick with to the end. A method should be judged by whether it can be carried out within the user's real tolerance for risk.
Luck, costs and persistence determine the distance between the outcome an investor finally receives and the theory.
After fees, most active funds have negative true excess returns. An active product should state how much more it charges than a low-cost benchmark, and how much excess return it needs to cover that.
Performance displays should cover a long enough period, compare against a benchmark, show the information ratio, and say whether the result can be statistically distinguished from luck.
Let users see how much chasing rises and fleeing falls has cost them, not just the product's return.
Test a plan against the user's real tolerance: a plan that will be abandoned halfway is worthless, however high its theoretical return.