With the recent announcement that Bill Miller will be stepping down from running Legg Mason Value Trust fund, a number of people have used this as an opportunity to re-examine his incredible fifteen year streak of beating the S&P 500. Running from 1991 to 2006, this record has never been matched.

However, in recent years, Miller’s performance fared worse than the market’s, with some years losing up to 50%. Many are therefore now saying that Miller’s streak was nothing more than luck. If there are enough mutual fund managers competing to beat the market, surely at least a handful must have performance streaks like Miller’s.

Happily, we needn’t speculate. We can actually subject this sort of statement to rigorous mathematical analysis. I happen to be partial to using math to understand performance streaks, having examined the math behind 1941 Joe DiMaggio’s hitting streak, as well as streaks in mutual funds.

Unlike what others have done, a more subtle approach than multiplying simple probabilities is required. Specifically, we need to recognize that the probability distribution of beating the market can vary from year to year.

As described here (and in more detail here), I worked with Andrew Mauboussin to compare the performance streaks in the real world to ones in a computationally-generated null model. Looking over many decades (1962-2008), our model used the same numbers of funds each year and the fraction of funds that beat the market as the weights for our Bernoulli trials (weighted coin flips), in order to create as realistic a null model as possible, but one where skill would play no part. Running this model 10,000 times, we then checked to see the distribution of long performance streaks.

Unlike the real world, we found nothing that approached Bill Miller’s streak. In fact, streaks of fifteen years only occurred 30 times out of 10,000 runs, far below a reasonable expectation based on luck. In fact, the next longest streaks in the real world were only eleven years long (these occurred twice). Now this doesn’t sound like much of a difference. But remember, these are streaks. To beat the market year after year isn’t a little bit harder, it’s geometrically harder. This can be seen by looking at how often an eleven-year streak occurred in the null model. While still somewhat unlikely, these occurred in nearly a third of our simulations.

The impressiveness of Bill Miller’s streak is magnified by looking at its timing. Our analysis shows that the streak, begun in the early Nineties, occurred during an unlikely period when compared to the Seventies or early Two Thousands. Surprisingly, despite the market’s good overall performance in the Nineties, it was one of the worst decades for active managers. For example, only about one in ten funds beat the market in 1995 and 1997, demonstrating that a long streak during this time is far from inevitable.

While we can never say with certainty that Miller’s streak was due to skill, such a streak is not possible on the strength of luck alone.

*Image from Wikimedia Commons | Katrina.Tuliao*