Negative skewness, or: bulls walk up stairs, bears jump out of windows

Recent market action is a good reminder of the asymmetry in markets. In general, stock market rises don't look like stock market declines. Stock indexes slowly eke out gains over a period of months, but lose all of those gains just a few days. There are plenty of famous meltdowns in stocks, including 1914, 1929, 1987, and 2008, but almost no famous "melt ups."

Just like the Inuit have multiple words for snow because they are surrounded by the stuff, equity commentators have many words for crashes (panics, selloff, etc). These events are not uncommon. In the same way that many indigenous African languages have no word for snow, we lack a good word to describe one or two day melt-ups in equity markets since these aren't part of our landscape.

There are a number of trader's adages that describe this pattern, including bulls walk up the stairs, bears jump out the window and variations on that theme. In the economic literature, this phenomenon is referred to as negative skewness. If you look at the distribution of daily percent returns for the S&P 500 Index over a long period of time, you'll notice that there are more extreme negative results than extreme positive results, with the majority of results being slightly positive. Whereas a normal distribution, or the bell shaped curve we've all seen in statistics class, is symmetrical with 95% of values dwelling within two standard deviations of the mean, a negatively skewed distribution has a fat left tail where declines extend far beyond what you would expect for a normally distributed data set.

The chart below illustrates this. Out of 22,013 trading days going back to 1928, just 47.8% of days resulted in negative outcomes while 52.2% resulted in positive outcomes. This makes sense given the generally upward trajectory of equity markets over that period. If we sort each day's return into buckets, we start to see asymmetries develop. For instance, there were 10,973 days on which markets moved higher or lower by by 0.5%, just 48.4% of which were lower. The majority of 0.5 to 1% and 1 to 2% changes were to the positive side as well. The distribution changes once we look at the 2% and over bucket. Out of 1485 days with "extreme" returns, the majority (51.9%) of changes were declines of 2% or greater rather than rises of +2% or greater.

Figure: Distribution of daily changes in the S&P 500 index going back to 1928

Financial economists have a number of hypothesis for negative skewness. One theory blames leverage, whereby a drop in a firm's equity price raises its leverage, or the amount of debt it uses to finance itself. This makes an investment in the company more risky and leads to higher volatility of its shares. Conversely, when a stock rises, its leverage decreases, making the shares less risky. For that reason, rises in equities are tame while falls are wild. While an attractive theory, data shows that as stock prices decline, all-equity financed companies experience jumps in volatility of the same magnitude as leveraged companies, indicating that leverage is not a good explanation for a pattern of negative skewness.

Another explanation is the existence of "volatility feedback." When important news arrives, this signals that market volatility has increased. If the news is good, investor jubilation will be partially offset by an increase in wariness over volatility, the final change in share price being smaller than it would otherwise have been. When the news is bad, disappointment will be reinforced by this wariness, amplifying the decline.

Other theories blame short sale constraints for the asymmetry. If bearish investors are restricted from expressing their pessimism, they will be forced to the sidelines and their information will not be fully incorporated into prices. When the bulls start to bail out of equities, the bearish group becomes the marginal buyer, at which point bearish information is finally "discovered" by the market, the result being large price declines.

Putting the reasons aside, behavioral finance types have some interesting things to say about how investors perceive skewness. According to prospect theory, investors are not perfectly rational decision makers. To begin with, returns are not appraised in a symmetrical manner; a 5% loss hurts investors more than a 5% gain feels good. Next, investors overweight unlikely events and underweight average ones. Given these two quirks, investors may prefer positively skewed assets (like government bonds), which have far fewer large declines than normally skewed assets, as this distribution reduces the potential for psychological damage. The possibility of large lottery-like returns, the odds of which investors overweight relative to the true odds of a positive payout, also drive preferences for positive skew assets. Negatively skewed assets like equity ETFs, which expose investors to tortuous drops while not offering much potential for large melt-ups, are to be avoided.

Put differently, positive skew is a feature that investors will pay to own. Negative skew is a "bad" and people need to be compensated for enduring it.

If you buy this theory, then in order to coax investors into holding negatively skewed assets like stocks, sellers need to offer buyers a higher expected return. The presence of this carrot could be one of the reasons why equities tend to outperform bonds over time. For equity owners who are suffering through the current downturn, here's the upshot: negative skew events like the current one, while stressful, may be the price you have to pay in order to harvest the superior returns provided by stocks over the long term.

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