In fact, only 2 out of 2,862 broad domestic stock funds were able to outperform their peers consistently over five years, according to one measure: performance in the top quartile of funds over five consecutive 12-month periods ended in March 2014. That translates to just 0.07 percent of the funds, which means that more than 99.9 percent of funds fell short of that feat.
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Repeat those double flips five times and you’ll find the probability of a mutual fund ending up in the top quartile five times in a row through chance: 0.098 percent. (We’re flipping the coin twice for each year of mutual fund performance.) That’s a bigger probability than the 0.07 percent scored by the actual funds. This means that if mutual fund managers had just flipped coins, roughly three of them — not two — would have been expected to end up in the top quartile for five years in a row.
Indeed:
>>> 100 * .25 ** 5
0.09765625However the success rate of 2 / 2862 is not 0.07% but is instead 0.02–0.25%:
>>> 100 * beta.ppf([.025, .975], 1 + 2, 1 + 2862 - 2)
array([ 0.02161444, 0.25211661])And 0.09% falls squarely in that range. So the market did not underperform random chance by this measure. It is rather indistinguishable. Further to calculate the expected range of outcomes:
>>> binom.ppf([.025, .975], 2862, .25 ** 5)
array([ 0., 6.])With a group size of 2,862 we'd expect anywhere between 0 and 6 funds to be in a given quartile for each of the five years.
This in turn means that it is impossible to underperform random chance on this particular measure.