Perhaps median returns tell an even less-pleasant story.

**Three Stocks**

Assume a three-stock market. Each of the three companies is worth $1 billion. If one stock makes no money on the year, the second gains 10%, and the third jumps 50%, an index fund that owns the market portfolio will rise 20%. (Since each company is the same size, the market’s capitalization-weighted return matches its equal-weighted return, so the math is indeed as simple as (0+10+50)/3.)

Three actively managed mutual funds exist. Each fund holds but a single stock, and has such miserly fees that its expenses equal that of the sole index fund that tracks this market. One fund owns the first company, another holds the second, and the final fund possesses the third. Collectively, these three actively run funds earn the index fund’s return. The first fund makes nothing, the second appreciates 10%, and the third fund gains 50%. The average is 20%, before expenses that match those of the index fund.

“Active management keeps pace with indexers!,” read the headlines. Yet investors in the actively run funds beg to differ. Each of the three funds has the same number of shareholders and the same amount of assets. Thus, although the average gain for active funds did indeed match that of the index fund, two thirds of all investors in actively run funds trailed the index fund for the calendar year. So did two thirds of all actively run assets. “Active management fails again!,” investors respond.

This, of course, is a parable of the difference between an average and a median. The average looks just fine—but the median does not.

**Many Stocks**

The unlikely combination of a lawyer, a Chicago Booth business school professor, and an Oxford mathematician have co-authored a short paper called “Why Indexing Works” (Heaton, Polson, Witte) that relates this tale more formally. (As it turns out, the lawyer also has a Ph.D. in financial economics—smart fellow, he.) After presenting the equations and running simulations, the authors conclude, “While randomly selecting a subset of securities from the index increases the chance of outperforming the index, it also increases the chance of underperforming the index, with the frequency of underperformance being larger than the frequency of overperformance.”

That is, all things being equal, most active stock funds will trail an index over a given time period. However, when an active fund does beat the index, the size of the victory will generally be larger than the size of its defeat. That very much was the case in the three-stock world. The sole winning fund beat the index by 30 points, while the two losing funds trailed by 10 and 20 points. Of course that is an extreme example; in this living, breathing world, the U.S. stock market has thousands of securities rather than three, and mutual funds have a minimum of several dozen. The effect therefore is much diluted.

(Note: My three-stock example is rigged. Imagine instead that the market had a down year, and the three companies returned negative 50%, negative 10%, and 0%. Had that occurred, the average would be been *worse *than the median, with the average being negative 20% and the median at negative 10%. The discrepancy occurs because my initial example has positive skew—meaning that the tail of the distribution drifts to the right—while the bear-market example has negative skew. For the most part, actual stock returns have positive skew, so the assumption that underlies the initial example is warranted, if once again exaggerated for instructional purposes.)

**Messy, Messy**

While the average-median theory is certainly sound, it is difficult to measure in practice because all things are *not* equal. Both my example and the authors’ test assume that a fund manager’s investment universe precisely corresponds with a benchmark. That is, the fund’s neutral position *is* the benchmark, such that every deviation represents a conscious choice by the investment manager to add value. In reality, the manager’s universe is almost certainly quite different than that of an index. (What large-cap stock-fund manager starts the investment process by considering only those stocks in the S&P 500, weighted by their size?) This complication doesn’t change the mathematical intuition, but it can distort the picture, if the performance for the securities that lie outside the benchmark differs markedly from those that are inside it.