Skewness in the distribution of economic outcomes is underappreciated. In this post, I summarize some basic insights regarding skewness in three themes. I also describe why I think now is a particularly interesting time to focus on skewness.

### Prelude

Skewness generally refers to an asymmetry in a statistical distribution. A conventional measure of skewness is the standardized third moment:

where $\mu$ and $\sigma$ are the mean and the standard deviation of random variable $X$.

There are broadly two ways to measure skewness. First is to obtain past values of the economic variable of interest and use sample moments of the above formula. The alternative is to use option prices. As is well-known, however, option prices deliver risk-neutral skewness, not the physical skewness. A recent paper addresses the limitations of both approaches and instead proposes using the cross-section to isolate the desired quantity of interest, which has been popular in the term structure literature. Furthermore, studies have suggested using information from volatility and trading volume to fine-tune the measurement.

### #1: Do investors care about skewness?

The discussion on skewness has been most active within asset pricing. In contemplating the role of skewness for equity risk premia, researchers have usually extended what they have learned about variance.

One intuitive argument is as follows.

• Holding mean and variance held constant, investors prefer positively skewed to negatively skewed portfolios. In such case, stocks with negative skewness — a larger chance of a large loss — should be associated with higher expected returns.
• Furthermore, CAPM tells us that only the stock's contribution to the total variance of a well-diversified portfolio should be priced. Extending this logic to skewness implies that only the stock's contribution to the portfolio's skewness should be priced.

This argument is indeed supported by Kraus and Litzenberger (1976) and Harvey and Siddique (2000) who find that coskewness has a significant impact on equity risk-premia. These two papers have been extended by papers that focuse on the role of systematic skewness (e.g. Simaan (1993), Dittmar (2002), Chabi-Yo, Leisen, and Renault (2014)).

However, we also know that idiosyncratic volatility seems to matter in addition to systemic risk, so it's also reasonable to examine a similar role of idiosyncratic skewness.

Indeed, theories have come up with various mechanisms to have idiosyncratic skewness matter — heterogeneous preferences (Mitton and Vorkink (2007)), prospect theory preferences (Barberis and Huang (2007)), and distorted beliefs (Brunnermeier and Parker (2005)). Empirically, it does seem to explain a lot of variation in expected returns as well.