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By Root 504 0

the S&P trounced Miller that year, but actually he counted 2004 in his “win” column. That is because the S&P 500 is not the simple average of the prices of the stocks it comprises; it is a weighted average in which stocks exert influence proportional to each company’s capitalization. Miller’s fund did worse than the simple average of S&P stocks but better than that weighted average. Actually, there were more than thirty twelve-month periods during his streak in which he lost to the weighted average, but they weren’t calendar years, and the streak was based on the intervals from January 1 to December 31.27 So the streak in a sense was an artificial one to start with, one that by chance was defined in a manner that worked for Miller.

But how can we reconcile these revelations with those 372,529-to-1 odds against him? In discussing Miller’s streak in 2003, writers for The Consilient Observer newsletter (published by Credit Suisse–First Boston) said that “no other fund has ever outperformed the market for a dozen consecutive years in the last 40 years.” They raised the question of the probability of a fund’s accomplishing that by chance and went on to give three estimates of that probability (the year being 2003, they referred to the chances of a fund’s beating the market for only twelve consecutive years): 1 in 4,096, 1 in 477,000, and 1 in 2.2 billion.28 To paraphrase Einstein, if their estimates were correct, they would have needed only one. What were the actual chances? Roughly 3 out of 4, or 75 percent. That’s quite a discrepancy, so I’d better explain.

Those who quoted the low odds were right in one sense: if you had singled out Bill Miller in particular at the start of 1991 in particular and calculated the odds that by pure chance the specific person you selected would beat the market for precisely the next fifteen years, then those odds would indeed have been astronomically low. You would have had the same odds against you if you had flipped a coin once a year for fifteen years with the goal of having it land heads up each time. But as in the Roger Maris home run analysis, those are not the relevant odds because there are thousands of mutual fund managers (over 6,000 currently), and there were many fifteen-year periods in which the feat could have been accomplished. So the relevant question is, if thousands of people are tossing coins once a year and have been doing so for decades, what are the chances that one of them, for some fifteen-year period, will toss all heads? That probability is far, far higher than the odds of simply tossing fifteen heads in a row.

To make this explanation concrete, suppose 1,000 fund managers—certainly an underestimate—had each tossed a coin once a year starting in 1991 (the year Miller began his streak). After the first year about half of them would have tossed heads; after two years about one-quarter of them would have tossed two heads; after the third year one-eighth of them would have tossed three heads; and so on. By then some who had tossed tails would have started to drop out of the game, but that wouldn’t affect the analysis because they had already failed. The chances that, after fifteen years, a particular coin tosser would have tossed all heads are then 1 in 32,768. But the chances that someone among the 1,000 who had started tossing coins in 1991 would have tossed all heads are much higher, about 3 percent. Finally, there is no reason to consider only those who started tossing coins in 1991—the fund managers could have started in 1990 or 1970 or any other year in the era of modern mutual funds. Since the writers for The Consilient Observer used forty years in their discussion, I calculated the odds that by chance some manager in the last four decades would beat the market each year for some fifteen-year period. That latitude increased the odds again, to the probability I quoted earlier, almost 3 out of 4. So rather than being surprised by Miller’s streak, I would say that if no one had achieved a streak like Miller’s, you could have legitimately complained that all those highly