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In "The Drunkard's Walk"?, Caltech physicist Leonard Mlodinow's book about how people misunderstand the amount of randomness in their lives, thereâ??s a short but fascinating passage discussing Bill Miller's 15-year streak, beginning in 1991, of beating the S&P 500.
Mlodinowâ??s book is what first came to mind when we heard the news last week that Miller was stepping down from the Legg Mason Value Trust fund, which heâ??d managed for thirty years. In the five years since the streak ended, Millerâ??s fund lost 9 per cent annually and ranked dead last out of the 840 funds in its category, according to Lipper.
Predictably, most of the commentary weâ??ve seen has focussed on his spectacular crash after his equally spectacular run. The obvious reason most people considered the streak so extraordinary â?? and Miller to be such an impressive stock-picker â?? was that the odds of any single mutual fund generating such a run were so infinitesimally small.
Perhaps luck could account for a few good bets or a couple of good years, the thinking went, but surely it couldn't account for such extraordinary and sustained outperformance. A newsletter published by Credit Suisse-First Boston in 2003, a few years before the streak ended, calculated the odds of a manager outperforming the market on chance alone for 12 straight years to be one in 2.2 billion.
But what if Millerâ??s streak wasnâ??t so remarkable to begin with?
The statisticians like those from CSFB were considering the odds that a specific fund would outperform for 12 straight years if the fund begins investing at a specific time. But as Mlodinow explained, maybe the better question to ask is actually this: given the number of mutual funds that have existed in the modern era, what are the odds that any of them would have beaten the market over any 15-year period of time on chance alone?
Answer: 3 out of 4.
Hereâ??s the relevant excerpt from the book (weâ??ve broken up the paragraphs to make it more readable):
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 period of fifteen years or longer, 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 â?? certainly an underestimate â?? 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 wouldnâ??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 per cent.
Finally, there is no reason to consider only those who started tossing coins in 1991 â?? 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 Credit Suisse newsletter] 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 period of fifteen years or longer. That latitude increased the odds again, to the probability I quoted earlier, almost 3 out of 4.
So rather than being surprised by Millerâ??s streak, I would say if no one had achieved a streak like Millerâ??s, you could have legitimately complained that all those highly paid managers were performing worse than they would have have by blind chance.
The coin-toss experiment is imperfect, of course. For it to apply precisely to the mutual fund universe, youâ??d have to assume that on chance alone a given fund manager would have 50 percent odds of beating the market. That seems unknowable.
Thereâ??s no way to discern, for instance, whether Bill Miller was a phenomenally talented stock-picker who simply made a disastrous decision to double-down on financials at a terrible time and near the end of his glittering career â?? or if randomness, in a sense, had just â??selectedâ? him to have an unprecedented 15-year streak.
All the same, we think there are some profound, if obvious, lessons here â?? mostly about hero worship in the investment world and how we fail to understand, or try to avoid understanding, the role of luck in our lives and in the lives of those we emulate.
It's a behavioural quirk that we feel the need to explain what we don't really understand, and because we're not good at understanding randomness, we tend to think that excellent outcomes are mostly the result of great decisions made by wise individuals "â? even in situations where we can't verify if that's the case.
That wonâ??t change, and weâ??re a forgetful species. Eventually â?? maybe not for years, but eventually â?? the economy and the markets will be healed, and new fund managers will grace multiple Barronâ??s covers and be elevated in the minds of their peers to plains higher than what is justified or reasonable.
The cautionary tale of Bill Miller will be soon forgotten. Given the way his tenure as head of the Legg Mason Value Trust has ended, perhaps he can take solace in that.
Related links: From the alpha to theâ?¦ whatever this is â?? FT Alphaville Are We Coins? â?? NPRâ??s Radiolab
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