# The possible preferences of investors for higher than first 2 moments of return distribution?

Can anyone explain in an intuitive manner a justification for possible preferences of investors for moments of return distribution beyond the first two moments (i.e. mean and variance). For example, why it is often argued that positive skewed distributions are preferred to negative ones? Is it related to known concept of risk aversion?

• A preference for positive skewness can be justified by Prospect Theory, specifically the idea that losses are more painful than gains are pleasant. Feb 13 '18 at 21:11
• Agree on the prospect theory comment. This newspaper article provides a nice summary: ft.com/content/329e1eb0-fd28-11e3-bc93-00144feab7de Feb 14 '18 at 16:44

Investor preferences for higher level moments are probably most easily explained by behavioral finance. Investors' tendency to overvalue out-sized positive and negative outcomes, such as gamblers' willingness to play negative expectancy casino games, is consistent with many of the intuitions underlying Prospect Theory. There are several possible behavioral biases which underlie these cogntitive errors. For example, the recency effect is thought to cause investors to overestimate the probability of outsized winnings based on forgetfulness of losses and or heightened focus on others’ recent winnings. Thus, it is thought that investors would be motivated moreso by the mean than the median ("typical") outcome.

Preferences for higher level moments are thought to work similarly. Since the skew is positive for a distribution where the mean exceeds the median, the recency bias predicts that securities with positive historical and/or implied skewness are more likely to have lower future returns since they are also more likely to be fully priced. This intuition is broadly supported by literature on cross-sectional asset returns.

Ang, Hodrick, Xing, and Zhang (2008) summarize this view:

Barberis and Huang (2005) develop a behavioral setting in which the individual skewness of stock returns may be priced.8 Under the cumulative prospect theory preferences of Tversky and Kahneman (1992), investors transform objective probabilities using a weighting function that overweights the tails of the probability distribution. This causes positively skewed securities to become overpriced and to earn negative average excess returns.

Additionally, the cross-section of equity returns themselves tend to be positively skewed. Two recent papers demonstrate that individual stocks' returns, in aggregate, do not outperform bonds. I.e., most of an index' performance is attributable to a few outliers. Motivated by these outliers, investors tend to in aggregate seek out future large winners thus resulting in their relative over-valuation.

Bessembinder (2017) finds:

Four out of every seven common stocks that have appeared in the CRSP database since 1926 have lifetime buy-and-hold returns less than one-month Treasuries. When stated in terms of lifetime dollar wealth creation, the best-performing four percent of listed companies explain the net gain for the entire U.S. stock market since 1926, as other stocks collectively matched Treasury bills. These results highlight the important role of positive skewness in the distribution of individual stock returns, attributable both to skewness in monthly returns and to the effects of compounding. The results help to explain why poorly-diversified active strategies most often underperform market averages.

Heaton, Polson, Witte (2017) also find:

...active equity managers tend to underperform a benchmark index. We motivate our model with the empirical observation that the best performing stocks in a broad market index often perform much better than the other stocks in the index. Randomly selecting a subset of securities from the index may dramatically increase the chance of underperforming the index. The relative likelihood of underperformance by investors choosing active management likely is much more important than the loss to those same investors from the higher fees for active management relative to passive index investing. Thus, active management may be even more challenging than previously believed, and the stakes for finding the best active managers may be larger than previously assumed.

Thus the desire for positive skew might make sense in the context that equity markets' out-performance is primarily due to cross-sectional skewness.

EDIT

On further research, it appears that investors’ preferences for positive skew can help explain why realized skew is negatively correlated to equity returns. If investors have a preference for stocks with positive skew, then they will bid up prices of stocks which in turn results in lower future returns.

In Amaya Vasquez (2015), the authors indicate:

Buying stocks in the lowest realized skewness decile and selling stocks in the highest realized skewness decile generates an average return of 19 basis points the following week with a t-statistic of 3.70. This result is robust across a wide variety of implementations and is not captured by the Fama-French and Carhart factors. The relation between realized kurtosis and next week׳s stock returns is positive but not always significant. We do not find a strong relation between realized volatility and next week׳s stock returns.

Furthermore, that stocks with higher realized skewness of return tend to be more expensive might also help explain their sensitivity to earnings shocks.

The finding of positive correlation between realized volatility and forward equity returns is disputed in the literature. A positive relationship can be seen as an artifact of investors’ aversion to perceived risk, for which realized variation is often a proxy. However, there are many other findings which contradict this one... namely a class of phenomena often called “betting against beta”. Where they consistently do agree is on the positive relationship between equity returns and the normalized differences between IV and RV.

I think your assumption of "why it is often argued that positive skewed distributions are preferred to negative ones? Is it related to known concept of risk aversion?" is incorrect or at least partially wrong.

For most of types of financial instruments, such as most stocks and indices, they show negative skewness on their return distributions. That's exactly the characteristics of these instruments, as the historical data show downside risk are less frequent but more severe, or people always call "fat tails". However, lots of commodities do show positive skewness, especially the ones with high elasticity on supply curve, such as grains, livestock, energy etc.

From forward-looking perspective, skewness also demonstrated in option world, that's why puts are usually much more expensive than calls with same level of deltas(here use stocks as examples again). However, the tail implied from option market is actually even fatter than historical real returns, that's caused by risk aversion as lots of market participants are willing to pay more on puts for protection, which you can also understand it as demand-driven.

For your question of moments, I would say two moments can't well address the real feature of returns distributions on financial instruments. Using stocks as examples one more time, they usually show way higher kurtosis and some other features. So if you fit return data to a normal distribution, the fitting will be very ugly. I would like to introduce you a distribution called meixner distribution, which is a 4-parameter distribution fits return data the best, as far as I know. https://www.eurandom.tue.nl/reports/2002/004-report.pdf

The argument I have seen for higher order moments follows from an expansion of log wealth: \begin{align} log(W) &= log(W_0 (1+r))\\ &= log(W_0) + log(1 + E[r] + r - E[r])\\ &= log(W_0) + log(1 + E[r]) + log\left(1 + \frac{r - E[r]}{1 + E[r]}\right), \end{align} where $W_0$ is initial wealth and $r$ are simple returns. Then apply a Taylor series expansion to the last log to get \begin{align} log(W) &= log(W_0) + log(1 + E[r]) + \frac{r - E[r]}{1 + E[r]} -\frac{1}{2}\left(\frac{r - E[r]}{1 + E[r]}\right)^2 +\frac{1}{3}\left(\frac{r - E[r]}{1 + E[r]}\right)^3 \ldots, \end{align} Now take expectations: \begin{align} E\left[log(W)\right] &= log(W_0) + log(1 + E[r]) -\frac{1}{2}\frac{E\left[\left(r - E[r]\right)^2\right]}{\left(1 + E[r]\right)^2} +\frac{1}{3}\frac{E\left[\left(r - E[r]\right)^3\right]}{\left(1 + E[r]\right)^3} \ldots, \end{align} Thus an investor maximizing expected log wealth should have a positive preference for larger odd order moments and smaller even order moments.

There is a different line of reasoning based on maximizing probability of a win which is based on Roy's criterion. Here the investor wishes to maximize $$\operatorname{Pr}\left(W > W_0(1+r_1)\right),$$ for some fixed return value $r_1$. The math is a bit more complicated here, but one result is that the preference for higher order moments can have mixed signs. For example, depending on the Sharpe ratio for the returns, an investor can have preference for either higher skew or lower skew in returns. An investor with a short-term view who needs to clear a certain threshold (like your investment manager) will sell lottery tickets to chase small returns, risking the possibility of a huge payout; the investor with long term view would buy lottery tickets and sacrifice their cost.