As bloggers, we certainly understand the compelling motivation to write sensational headlines for posts. But sometimes writers go too far in their quest for clicks. Such was the case not so long ago when and article entitled “Junk Bond Funds Just Experienced A 6-Sigma Event” hit my inbox.
The article explores the substantially large outflow of money form high-yield bond funds in the first week of August. For clarity, in our practice we don’t pay attention to indicators like these since we’ve yet to see compelling evidence that they provide any reliable insights into future expected returns. Nonetheless, we always enjoy taking a look at interesting numbers, especially when they’re used to characterize incredible events.
The article starts out innocently enough by quantifying exactly what is meant by the term 6-sigma (emphasis mine):
“Sigma is another way of saying standard deviation. And the greater the number of standard deviations, the more unlikely the event.
A 6-sigma event is extremely rare. If you want to put a number to it, think 1 in 500 million. According to Business Insider quant reporter Andy Kiersz, it’s like flipping a coin 29 times in a row and getting heads each time. It’s like rolling a die 11 times in a row and getting 6 each time.”
Since the article is relying upon weekly fund flows data, a 1-in-500,000,000 chance of something happening means it should be expected about once every 9.6 million years. Let’s think about that for a moment, and ponder the simple reality that this article is relying on 22 measly years of data to make an estimate about something that might be expected to occur once every 9.6 million years. Does that seem reasonable?
Naturally, we were also reminded of Goldman Sachs CFO David Viniar’s s now infamous August 2007 claim that “[Goldman’s Global Alpha fund was] seeing things that were 25-standard deviation moves, several days in a row.” With respect, no it wasn’t. Even if Goldman’s hedge fund was engaging in high frequency trading that made nanosecond observations (1/1,000,000,000th of a second), a 25-sigma move would still only be expected once every vigintillion years (that’s a 1 followed by 120 zeros).
For perspective, the Earth is only 4.5 billion years old. That’s a paltry 9 zeros, folks.
Quantitatively, something appears to have gone horribly wrong. The most likely culprit is profoundly misguided assumptions about the true distribution of returns. This happens often in the world of finance, as both individuals and professionals alike underestimate the frequency and magnitude of “fat tail” events. In other words, extreme outcomes are a lot more common than they should be under common distribution assumptions, and when one attempts to determine the standard deviation of an extreme event against a distribution that suggests extreme events rarely happen, the result is a severe underestimate of the true frequency.
There are two ways to remedy this issue. First, reverse-engineer the implied volatility of the distribution assuming normality and incorporating the outlier case over the observation horizon to-date. Depending on how the outlier manifested (one day loss, drawdown, etc.), the math for this moves beyond the scope of this article. Or second, assume a different distribution, such as a Student’s t distribution well specified degrees of freedom, a uniform distribution, or a pareto distribution (beware the Dragon Kings!).
In any case, if you’re looking for a general rule of thumb, here’s what we go by: any claim relating to the financial markets that purports to be 5-sigma or greater is to be read only for amusement, or else completely ignored. This is because 4-sigma events (either positive or negative) ought to be expected about 1/31,560th of the time. In the case of daily observations (assuming 250 trading days/year), that means a 4-sigma event will hit about once every 126 years. By the same trading day standard, 5-sigma events should be expected once every 13,932 years!
Whether we’re talking about high-yield bonds, Goldman hedge funds, or any other financial or economic dataset, you simply cannot draw such grandiose conclusions on the basis of such small datasets. And due to the constant evolution of markets, you also cannot make predictions that go beyond the reasonable timeframes of political and institutional policy.
The bottom line is that like us, you should feel free to generally dismiss, or at least read with great skepticism, stories with headlines suffering from sigma sensationalism syndrome.